diff --git "a/DtAzT4oBgHgl3EQfwv4v/content/tmp_files/load_file.txt" "b/DtAzT4oBgHgl3EQfwv4v/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DtAzT4oBgHgl3EQfwv4v/content/tmp_files/load_file.txt" @@ -0,0 +1,1441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf,len=1440 +page_content='Anisotropic Quantum Hall Droplets Blagoje Oblak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1 Bastien Lapierre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='2 Per Moosavi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='3 Jean-Marie Stéphan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 and Benoit Estienne5 1CPHT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' École Polytechnique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IP Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' F-91128 Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' France 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' University of Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Winterthurerstrasse 190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 8057 Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Switzerland 3Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Wolfgang-Pauli-Strasse 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 8093 Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Switzerland 4Univ Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Université Claude Bernard Lyon 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Institut Camille Jordan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' UMR5208,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' F-69622 Villeurbanne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' France 5Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Laboratoire de Physique Théorique et Hautes Energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' LPTHE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' F-75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' France (Dated: January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 2023) We study two-dimensional (2D) anisotropic droplets of non-interacting electrons in the lowest Landau level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' confined by trapping potentials whose level curves have an arbitrary shape at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Using semiclassical methods, we show that energy eigenstates are localized on equipoten- tials of the trap, with angle-dependent local widths and heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We exploit this one-particle insight to deduce explicit formulas for many-body observables in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For instance, the droplet’s density falls off at the boundary with an angle-dependent width inherited from that of the underlying wave functions, while the many-body current is localized on the edge, to which it is tangent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Correlations along the edge are long-ranged, in accordance with the system’s low-energy edge modes which are described by a free chiral conformal field theory in terms of the angle variable of the trapping potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' These results are likely to be observable in solid-state systems or quantum simulators of 2D electron gases with a high degree of control on the confining potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Introduction 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Setup and main results 2 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Anisotropic states from area-preserving maps 4 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Edge-deformed anisotropic traps 6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Many-body observables 10 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Conclusion and outlook 14 Acknowledgments 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Isotropic droplets 15 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Semiclassical expansion of P V P 15 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The transport equation 16 References 19 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' INTRODUCTION Quantum Hall (QH) droplets are mesoscopic two- dimensional (2D) electron gases placed in a strong per- pendicular magnetic field and confined by some electro- static potential: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' They lie at the heart of the QH effect [1–3] and provide a key benchmark for topolog- ical phases of matter as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In practice, however, the vast majority of detailed analytical studies of QH droplets are limited to highly symmetric cases, typically involving isotropic traps or harmonic potentials that are translation-invariant in one direction [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is espe- cially troubling as far as edge properties are concerned, since these are sensitive to the shape of the trap and determine the system’s low-energy excitations [6–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The goal of this paper is to address this lack of analyt- ical results by predicting universal aspects of many-body observables near the edge of essentially any anisotropic droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We achieve this by providing general, explicit, one-line formulas for the density, current, and correla- tions in the regime of strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We also study the corresponding low-energy edge modes, which are described by a free-fermion chiral conformal field the- ory (CFT) whose Fermi velocity is constant provided dis- tances along the boundary are measured by the canonical angle variable determined by the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' These pre- dictions are likely to be observable thanks to direct local imaging techniques in condensed matter systems [11–16] or quantum simulators [17–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is not the first time such questions appear in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, random potentials with no symme- tries are essential to model disorder, whose importance for the robustness of QH physics is hard to overstate [25– 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' An especially relevant series of works in that con- text is [28, 29], which study the density and current of QH droplets with arbitrary potentials, at finite temper- ature, generally including Landau level mixing, in the semiclassical limit of strong magnetic fields and weak traps [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' However, the coherent states used in these references do not allow for any resolution at the single- particle level, precluding the computation of low-energy dynamics and long-range correlations along the bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Our objective here is instead to find explicit wave functions that depend on the shape of the edge, and use that as a starting point for many-body objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Regarding electronic edge correlations, similar ques- tions have been addressed in the context of classical 2D arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='01726v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='mes-hall] 4 Jan 2023 2 Energy Fermi energy Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 2D electron droplet (shaded area) placed in a strong perpendicular magnetic field and confined by a typical anisotropic edge-deformed potential well (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' At leading or- der in the thermodynamic limit, the droplet’s boundary (thick black curve) coincides with the equipotential of the trap at the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Coulomb gases, where holomorphic methods play a key role [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' More broadly, the results put forward here may be seen as microscopic, first-principles derivations of quantities that are normally studied within less con- trolled approximation schemes in the geometry of the QH effect [33–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Our hope is thus to build a bridge between these theoretical works and concrete observations that may soon be accessible in tabletop experiments with a high degree of control on the confining potential [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Here is the plan of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To begin, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' II sum- marizes our methods and results, avoiding all techni- cal details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The next two sections are devoted to one- body physics in the lowest Landau level: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III first discusses generalities on semiclassical holomorphic wave functions, while Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV presents a detailed computation of the semiclassical energy spectrum in a class of ‘edge- deformed’ potentials of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This finally leads to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' V, where we investigate many-body densi- ties, currents, correlations, and low-energy edge modes in anisotropic traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' VI by discussing several follow-ups and open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To streamline the text, non-essential details are deferred to Apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A–C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' SETUP AND MAIN RESULTS This section summarizes our methods and key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To start, we describe the general setup: a semiclassi- cal limit (strong magnetic field, small magnetic length) in the lowest Landau level (LLL) [26, 27, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We then introduce edge-deformed potentials and present their ap- proximate energy eigenstates, before finally giving simple formulas for the corresponding local many-body observ- ables and dynamics in the vicinity of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Some of these findings are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Semi-classical limit in the LLL This work concerns spin-polarized non-interacting elec- trons of mass M and charge q in a 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Each electron is governed by a Landau Hamiltonian with an anisotropic potential V (x), H1-body = 1 2M (p − qA)2 + V (x), (1) where A is the vector potential of the magnetic field B = dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (We view A as a one-form, which simplifies some notation but is otherwise inconsequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=') We shall assume that V (x) is ‘monotonous’, by which we mean that it has a unique global minimum away from which it grows monotonously, but it is otherwise general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Conse- quently, the level curves or ‘equipotentials’ of V (x) are nested and take the form shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We assume throughout that the potential is weak relative to the mag- netic field [26–29, 41, 42], and that it is nearly constant on length scales comparable to the magnetic length ℓ2 ≡ ℏ qB > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (2) In that regime, the potential is a small perturbation of the pure Landau Hamiltonian ∝ (p−qA)2 and the eigen- states of (1) are expected to be well-approximated by wave functions in the LLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For instance, if the potential V (x) = V0(r2/2) is isotropic, the eigenstates of (1) have some definite angular momentum and reduce at strong B to standard LLL wave functions in symmetric gauge: φm(x) = 1 √ 2πℓ2 zm √ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' e−|z|2/2, (3) where m ≥ 0 is an integer angular momentum and we have introduced the dimensionless complex coordinate z ≡ x + iy √ 2ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (4) Each wave function (3) reaches its maximum on the cir- cle |z| = √m, away from which it decays in a Gaussian manner within a magnetic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Our goal will be to ob- tain similar approximate eigenstates in anisotropic traps, using the squared magnetic length (2) as a small expan- sion parameter [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Equivalently, we shall carry out a semiclassical (small ℏ), high field expansion (large B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In practice, the projection to the LLL is implemented thanks to the (one-body) operator P ≡ �∞ m=0 |φm⟩⟨φm|, whose kernel can be read off from the wave functions (3): ⟨z, ¯z|P|w, ¯w⟩ = 1 2πℓ2 e−(|z|2+|w|2)/2 ez ¯ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (5) This is manifestly Gaussian and reduces to a delta func- tion in the formal semiclassical limit ℓ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' At small but finite ℓ, the projection (5) makes space non-commutative in the sense that the LLL-projected position operators (x, y) satisfy the Heisenberg algebra [PxP, PyP] = iℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (6) 3 ��� ��� ��� x x x y y y √ N − √ N − √ N √ N (a) (b) (c) x x x y y y ℓ √ 2N −ℓ √ 2N −ℓ √ 2N ℓ √ 2N Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (a) A planar plot of the many-body density (14) along with several equipotentials (dashed lines), for a droplet of N = 100 electrons confined by the edge-deformed trap of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The constancy of the bulk density and its fall at the boundary are manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (b) The current’s norm (15) for the same droplet, together with the edge (black line) on which it is localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (c) The correlation function (16) for the same droplet, seen as a function of x2 = (x, y) with x1 = (ℓ � Nλ(0), 0) denoted by a cross and fixed at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Long-range correlations along the boundary are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One can thus think of the plane R2 as a ‘phase space’ whose canonical variables are x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This interpretation pervades much of the QH literature [40, 44–53] and will similarly affect our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, projecting the Hamiltonian (1) to the LLL and looking for its spectrum leads to the eigenvalue equation P V P|ψ⟩ = E|ψ⟩, (7) where the unknowns are the energy E and the quantum state |ψ⟩ ∈ LLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that the kinetic term of (1) has disappeared in (7): the potential itself plays the role of an effective Hamiltonian in the non-commutative phase space (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Exact solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (7) are generally out of reach, so one has to resort to approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The semiclassical one that we shall use is well known in the QH context [26–29, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Accordingly, we will look for solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (7) labeled by a large quantum number m ∈ N, seen as a generalization of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This large m limit is accompanied by a small ℓ limit, such that the area 2πℓ2m remains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In that regime, the mth eigenstate is approximately Gaussian and localized on an equipotential γm of V (x), enclosing a quantized area such that the Bohr-Sommerfeld condition holds: � γm x dy = 2πℓ2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (8) Equivalently, the flux of the magnetic field through the area enclosed by γm is m times the flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The energy of the mth state is then Em = E0 m + ℓ2E1 m + O(ℓ4), (9) where E0 m = V (γm) is the leading classical approxima- tion and the quantum correction E1 m involves the Lapla- cian of the potential and the curvature of the equipo- tential γm [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that the more familiar Wentzel- Kramers-Brillouin (WKB) approximation of 1D quantum mechanics [56] includes (topological) Maslov corrections on the right-hand side of (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' we will find similar correc- tions below, although their interpretation as topological invariants is prevented by a subtle distinction between real and Kählerian polarizations in geometric quantiza- tion [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One-body results The semiclassical limit just outlined applies to any (monotonous) weak potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In practice, our main con- cern is the physics of QH droplets near the edge, where the details of the bulk potential are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Most of our explicit results will therefore be given for ‘edge- deformed’ potentials, obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Consider any monotonously increasing function V0(t) for t ≥ 0, and let λ(ϕ) be any strictly positive 2π-periodic function of the angle ϕ ∈ [0, 2π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' normalize λ so that � dϕ λ(ϕ) = 4π, where we write � dϕ as a shorthand for � 2π 0 dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then adopt polar coordinates in the plane such that x + iy = r eiϕ and define the potential V (r, ϕ) = V0 � r2 λ(ϕ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (10) We refer to this as an edge-deformed trap because it re- sults from a deformation r2 �→ r2/λ(ϕ) that changes the shape of the boundary of isotropic droplets in a finite and smooth way, even in the thermodynamic limit where the droplet’s area goes to infinity [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In fact, the cor- responding infinitesimal transformations are expected to 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 be conformal transformations of the edge CFT [58–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The class of potentials (10) is thus exhaustive, at least as far as edge effects are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The traps (10) turn out to allow for explicit calcula- tions of the semiclassical energy spectrum, generalizing the known isotropic results reviewed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, we show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV that the corresponding eigenfunc- tions, solving (7) in the LLL, are Gaussians localized on equipotentials r = ℓ � mλ(ϕ) at large quantum numbers m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' They can be written in polar coordinates as ψm(x) ∼ eiΘm(x) � 2πℓ2σ(ϕ) e−a2/σ(ϕ)2 (2πm)1/4 , (11) where Θm(x) is a position-dependent phase, a ≡ � r − ℓ � mλ(ϕ) �� ℓ � λ(ϕ) is a dimensionless radial coordinate that measures the distance from the equipotential, and σ(ϕ) ≡ � 2 λ(ϕ) � 1 + � λ′(ϕ) 2λ(ϕ) �2 (12) is an angle-dependent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We stress that this exhibits the expected ‘quantum smearing’ of wave functions in a strong but finite magnetic field [28, 29], which would be missed by the leading classical approximations ℓ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As for the energy of the state (11), its expansion (9) up to neglected O(ℓ4) contributions is Em ∼ V (γm) + ℓ2 2 Ωm � 1 + � 1 + Γm Ωm � � dϕ 4π λ(ϕ)σ(ϕ)2 � , (13) where V (γm) = V0 � ℓ2m � is the leading term, while the first quantum correction involves derivatives Ωm ≡ V ′ 0(ℓ2m) > 0 and Γm ≡ ℓ2m V ′′ 0 (ℓ2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Each mth energy is thus determined by the potential and its derivatives at an equipotential that satisfies the quantization condition (8), in accordance with general theorems for holomorphic WKB theory [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Many-body results Now consider the ground state of a large number N ≫ 1 of free spin-polarized electrons, each governed by the single-particle Hamiltonian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This ground state is a Slater determinant of wave functions whose large m behavior is the Gaussian (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As we show in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' V, the corresponding many-body density, current, correla- tions, and low-energy effective action can all be written in closed form in terms of λ(ϕ) and the number N of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The density ρ(x) = �N−1 m=0 |ψm(x)|2 thus sat- isfies the bulk behavior ρ ∼ 1 2πℓ2 , while its form near the edge is given by a complementary error function: ρ(r, ϕ) ∼ 1 4πℓ2 erfc �√ 2 a σ(ϕ) � , (14) where a ≡ � r − ℓ � Nλ(ϕ) �� ℓ � λ(ϕ) is again a dimen- sionless radial coordinate, now measuring the distance from the edge at redge = ℓ � Nλ(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As a result, the ground state forms a star-shaped droplet whose bound- ary has an angle-dependent width (12) inherited from that of one-body wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Turning to the current J = �N−1 m=0 1 2i(ψ∗ mdψm − ψmdψ∗ m − 2iqA|ψm|2), we write it as a one-form in polar coordinates to find J(r, ϕ) ∼ − exp � − 2a2 σ(ϕ)2 � (2πℓ2)3/2σ(ϕ) � ℓ √ N dϕ + λ′(ϕ) 2λ(ϕ)3/2 dr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (15) This is localized on the edge and tangent to it, miss- ing the bulk behavior Ji ∝ εij∂jV as expected in the LLL [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Finally, the two-point correlation function C(x1, x2) = �N−1 m=0 ψ∗ m(x1)ψm(x2) behaves near the edge as C(x1, x2) ∼ eiΘN(x1,x2) 4πℓ2� σ(ϕ1)σ(ϕ2) i exp � − a2 σ(ϕ1)2 − b2 σ(ϕ2)2 � √ 2πN sin �� ϕ1 ϕ2 dθ 4 λ(θ) � (16) with a ≡ � |x1| − ℓ � Nλ(ϕ1) �� ℓ � λ(ϕ1) and similarly for b in polar coordinates (|x1|, ϕ1) and (|x2|, ϕ2), respec- tively, while ΘN(x1, x2) is a complicated overall phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note again the Gaussian localization at the edge, as well as the long-range correlator ∝ sin(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=')−1 typical of gapless fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, we eventually confirm that the under- lying low-energy edge modes are described by a chiral CFT of free fermions: see the action functional (68) be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The corresponding Fermi velocity is constant along the boundary when measured in terms of the angle vari- able of the potential (10), namely θ(ϕ) ≡ � ϕ 0 dϕ′ λ(ϕ′)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' ANISOTROPIC STATES FROM AREA-PRESERVING MAPS This section presents the WKB ansatz [see (20)] that forms the basis of all our later considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The struc- ture is ultimately quite simple: given a monotonous po- tential V (x), we pick one of its equipotentials, γm, with quantized area (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We then build a wave function with winding m, perfectly localized on γm, and finally project it to the LLL using the operator (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' General theorems [54, 55] ensure that LLL-projected eigenstates satisfying (7) can indeed be built in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The detailed applica- tion of this method to edge-deformed traps (10) is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A note: what follows relies on the mathematics of area- preserving diffeomorphisms, which is not reviewed in de- tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We refer instead to [57] for an introduction whose language is similar to that adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For more gen- eral discussions in the symplectic context, see [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Potentials in action-angle variables Let us be more precise about the geometry of the setup, remaining at the classical level for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We pick 5 a smooth potential V (x) and assume as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' II that it is monotonous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Its unique global minimum is thus sur- rounded by nested level curves, and one can always find an area-preserving deformation of the plane that sends each equipotential on a circle [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In other words, one can find an invertible smooth map F : R2 → R2 with unit Jacobian such that V � F(x) � = V0(r2/2), (17) where the trap on the right-hand side is isotropic (it only depends on r = |x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' If F is the identity (or a rotation around the origin), then V was isotropic to begin with and its eigenstates satisfying (7) are the standard wave functions (3) with definite angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In the more general case of arbitrary V , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (17) suggests using F to map the eigenstates (3) on those corresponding to our general V (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The existence of F in (17) is guaranteed by the mono- tonicity of V , and is equivalent to the existence of glob- ally well-defined action-angle variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In fact, we can use this to write F in a more explicit form that will be useful below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let therefore (ℓ2K, θ) be action-angle co- ordinates for the potential V (x) [66], where K ≥ 0 is dimensionless and θ ∈ [0, 2π) is a genuine angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' They are normalized so that ℓ2dK ∧ dθ = dx ∧ dy, which is to say that their Poisson bracket reads {ℓ2K, θ} = ℓ2 in terms of the phase space (x, y) with bracket (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then the map (x, y) �→ (ℓ2K, θ) is an area-preserving diffeo- morphism in terms of which V (x) = V0(ℓ2K(x)) is in- variant under rotations of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To be specific, write these coordinates as functions K(x, y) and θ(x, y) and let the inverse be x = F(K, θ) and y = G(K, θ) for some func- tions (F, G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' this inverse is nothing but the deformation F in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In other words, knowing the action-angle vari- ables of a potential V allows us to map it on its (unique) isotropic cousin V0, which in turn can be used to relate the corresponding anisotropic eigenstates to those in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It should be clear that these considerations apply to any monotonous anisotropic trap, in which case one generally encounters intricate area-preserving maps with complicated action-angle variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV, we will argue that most of these difficulties wash away when focusing on edge physics, whereupon the only relevant maps are the ‘edge deformations’ mentioned below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For now, we remain general and turn to quantum considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Anisotropic eigenstates Using the action-angle variables (ℓ2K, θ) for V (x), one can concretize the statements around Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (8)–(9) into formulas and eventually obtain anisotropic eigen- functions that satisfy (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, the Bohr-Sommerfeld quantization condition (8) implies that the equipotential γm is the set of points in R2 where K = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Now consider the following quantum state, perfectly localized on γm: |Ψm⟩ ≡ 2πℓ2 � dθ n(θ) eimθ��F(m, θ), G(m, θ) � , (18) where the normalization 2πℓ2 is included for later conve- nience, the ‘wave function’ ⟨x|F(m, θ), G(m, θ)⟩ = δ2� x− F(m, θ) � is a delta function, and n(θ) is some complex periodic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The latter does not wind upon com- pleting one turn in the plane along the equipotential, meaning that all the winding of (18) is encoded in the phase eimθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We stress that (18) is analogous to the standard WKB ansatz ψ(x) ∼ eiS0(x)/ℏeiS1(x) in 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, the phase eimθ is the leading classical contribution eiS0/ℏ for m ≫ 1, corresponding to the ‘geometrical optics’ approximation of the wave function, while n(θ) is the ‘physical optics’ quantum correction eiS1 that eventually needs to satisfy a transport equation in order for the Schrödinger equation to hold [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The only difference lies in the interpretation of areas in the plane as values of an ‘action’, which ul- timately stems from the non-commutative geometry (6) of LLL physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that n(θ) is the only unknown in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, most of the WKB method below will con- cern the derivation of a transport equation for n(θ) from the requirement that (7) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (18), it is straightforward to build a state in the LLL thanks to the projector (5): denoting ψm(z, ¯z) ≡ ⟨z, ¯z|P|Ψm⟩, (19) one finds the wave function ψm(z, ¯z) = e−|z|2/2 � dθ n(θ) eimθ × e−[F (m,θ)2+G(m,θ)2]/4ℓ2 ez[F (m,θ)−iG(m,θ)]/ √ 2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (20) This is manifestly of the form e−|z|2/2 times a holomor- phic function that depends on the action variable ℓ2m and the uniformizing map F of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It will be our starting point for the semiclassical solution of the eigen- value equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As a consistency check, note that (20) simplifies for isotropic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In that case, action-angle variables are just polar coordinates ℓ2K = r2/2 and θ = ϕ, and the map in (17) is F(x) = x, merely implementing a change from polar to Cartesian coordinates: F(m, θ) = ℓ √ 2m cos(θ) and G(m, θ) = ℓ √ 2m sin(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One can then verify that (20) with n(θ) = const coincides (up to nor- malization) with the standard LLL wave function (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Similarly to that case, any projected wave function (20) reaches its maximum on the equipotential γm and is ap- proximately Gaussian close to it, as ensured by the kernel (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This will be confirmed explicitly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV for edge deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Expanding the eigenvalue equation None of what we wrote so far involves a manifest semi- classical expansion: it is hidden in the eigenvalue equa- tion (7) and the function n(θ) in (20), since n(θ) should be expanded as a power series n(θ) = n0(θ) + ℓ2n1(θ) + O(ℓ4) (as before, there are no odd powers of ℓ since ℓ2 ∝ ℏ is really the semiclassical parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It is therefore worth anticipating the first few terms of the semiclas- sical approximation of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We stress that the expansion below will eventually be limited to the leading order of the transport equation, so that only n0(θ) will be calcu- lated in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In principle, one could of course push the expansion to higher orders for more detailed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The semiclassical expansion of the right-hand side of (7) is clear: it is given by the large m, small ℓ2 expan- sion of the projected wave function (20), including an expansion of n(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As for the energy, its expansion was written in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The left-hand side of (7) is more subtle, as its semiclassical expansion involves that of the operator P V P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The latter is a ‘Berezin-Toeplitz operator’ [54, 55] that will play an important role for edge-deformed po- tentials, so we now explain its expansion in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' First, given Cartesian coordinates (x, y), express the po- tential in complex coordinates (4) as V (x, y) ≡ V(z, ¯z) for some function V(z, ¯w) which is holomorphic in its first argument and anti-holomorphic in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then re- calling that P is the LLL projector with kernel (5), one finds ⟨z, ¯z|P V P|w, ¯w⟩ = 1 2πℓ2 e− 1 2 (|z|2+|w|2) × � R2 du dv V (u, v) e−|X|2+z ¯ X+ ¯ wX (21) with X ≡ (u + iv)/ √ 2ℓ similarly to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Our task is to expand the integral on the right-hand side in the semi- classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The key is to assume that the potential varies slowly on the scale of the magnetic length [26–29], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' we choose once and for all a smooth potential V (x), independent of ℓ, and let ℓ be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In that regime, the integrals in (21) are approximately Gaussian, which gives (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A 2) ⟨z, ¯z|P V P|w, ¯w⟩ ℓ≪1 ∼ 1 2πℓ2 e− |z−w|2 2 e z ¯ w−¯zw 2 × � V(z, ¯w) + ℓ2 2 (∇2V )(z, ¯w) � (22) where (∇2V )(z, ¯w) is the bicomplex function that corresponds to the Laplacian of the potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (∇2V )(z, ¯w) = 4 2ℓ2 ∂z∂ ¯ wV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is the standard semiclas- sical expansion of a Berezin-Toeplitz operator [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note the general structure: the entire P V P operator boils down to P itself, with kernel (5), multiplied by a function that coincides with V at leading order but also includes quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In the ‘zoomed-out’ limit where the kernel of P is a delta function, the first term of (22) becomes V(z, ¯z)δ2(z − w, ¯z − ¯w) as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' More- over, for harmonic potentials, the truncated expression (22) is actually exact since the next term ∇4V and all subsequent ones vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This agrees with the common lore that ‘WKB is exact for quadratic Hamiltonians’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' EDGE-DEFORMED ANISOTROPIC TRAPS Here we apply the WKB ansatz of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III to poten- tials (10) with scale-invariant level curves, obtained by acting with edge deformations [57] on an isotropic trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As we explain below, these are the most general traps one expects to find close to the edge of star-shaped QH droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The plan is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' First, we introduce edge deformations and give a few examples for later ref- erence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Second, we apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (7) to edge-deformed traps and expand it in the classical limit (large m, small ℓ2 with ℓ2m = O(1) kept fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We keep track of all terms up to order O(ℓ2), so as to capture the leading part of the transport equation for the function n(θ) in (18)–(20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This eventually yields an explicit energy spectrum [see (37)] along with approximately Gaussian eigenfunctions [see (43)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Lastly, we conclude with a consistency check by showing that our wave functions reproduce the asymp- totic (large m) form of the known LLL-projected spec- trum for anisotropic harmonic traps [67–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Edge deformations We have seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III that area-preserving defor- mations play a key role for the semiclassical solution of the eigenvalue equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The group of all such de- formations is obviously huge, so it is essential to identify the subset of transformations that are likely to be im- portant for low-energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In fact, part of this work has already been carried out, at least implicitly, in the seminal series of papers [58–63], which we now use as a basis for the definition of edge deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (A similar motivation was put forward in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=') Label points on the plane by their polar coordinates (r, ϕ), defined as usual by x + iy = r eiϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then, the boundary of any isotropic QH droplet is located at some fixed radius redge = O(ℓ √ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' What is the most general area-preserving deformation that preserves this order of magnitude?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The answer is readily found by realizing that the constraint of preserving redge = O(ℓ √ N) is equiva- lent, at leading order in 1/N, to the condition that the de- formation commutes with overall dilations r �→ const×r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The most general deformation satisfying this criterion is an edge deformation �r2 2 , ϕ � �→ � r2 2f ′(ϕ), f(ϕ) � , (23) where f(ϕ) is an (orientation-preserving) deformation of the circle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' any smooth map satisfying f(ϕ + 2π) = 7 f(ϕ) + 2π and f ′(ϕ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The angle-dependent rescaling of r on the right-hand side ensures that the map preserves area, and reproduces the argument of the potential (10) with λ = 2f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that the set of maps (23) is isomor- phic to the group of diffeomorphisms of the circle, whose central extension famously leads to the Virasoro algebra encountered in CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, this motivates the state- ment in [60, 61] that generators of maps (23) in the QH effect produce conformal transformations of edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We stress that the subset of transformations (23) orig- inates from an asymptotic analysis of the relevant or- ders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One can undoubtedly consider other families of deformations, motivated by different consid- erations, but those are irrelevant for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For instance, the transformations r2 �→ r2 + α(ϕ) are crucial for the effective low-energy description of QH droplets [6, 10, 61], but they are subleading compared to (23) since they deform the radius redge = O(ℓ √ N) by terms of order O(1/N) instead of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Conversely, one might consider ‘higher-spin transformations’ [58, 60, 61] that change the radius in a dramatic way such as r2 �→ β(ϕ)r4[1+O(1/r)], but these stretch QH droplets to an infinite extent in the thermodynamic limit, which is why we discard them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let us provide a few examples of edge deformations for future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' First, (23) includes rigid rotations around the origin given by f(ϕ) = ϕ + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A richer class is obtained by fixing some positive integer k and considering all maps of the form eikf(ϕ) = α eikϕ + β ¯β eikϕ + ¯α (24) where α, β are complex and satisfy |α|2 − |β|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For fixed k, such maps span a group locally isomorphic to SL(2, R), always containing a subgroup of rigid rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We will return to these deformations below, since they can be seen as Fourier modes for circle diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In particular, setting α = cosh λ and β = sinh λ for some real parameter λ turns the map (24) into an analogue of a Lorentz boost with rapidity λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In terms of the bulk action (23), any deformation (24) turns a circle into a ‘flower’ with k petals: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 5 for k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For k = 2, this maps the circle on an ellipse [57], which will be useful for anisotropic harmonic traps in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Edge-deformed potentials Given an isotropic potential V0(r2/2), how is it affected by an edge deformation (23)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The answer is provided by the anisotropic trap (10) with λ(ϕ) = 2f ′(ϕ): V (r, ϕ) ≡ V0 � r2 2f ′(ϕ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (25) In what follows, we exclusively consider this class of po- tentials and refer to them as ‘edge-deformed traps’, for the reasons stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Having fixed once and for all some circle deformation f(ϕ), our goal is to solve the cor- responding eigenvalue equation (7) in the classical limit of high quantum numbers and small magnetic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We begin by listing the key classical data of the prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The action-angle variables corresponding to (25) are (ℓ2K, θ) = � r2� (2f ′(ϕ)), f(ϕ) � with an inverse given by (r2/2, ϕ) = � ℓ2K/(f −1)′(θ), f −1(θ) � , where f −1 denotes the 1D inverse of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Points at constant K are equipoten- tials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' level curves of (25), each of which is a set of points such that r2 2f ′(ϕ) = ℓ2K (26) with constant K ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In Cartesian coordinates, this is the set of points x = � 2ℓ2Kf ′(ϕ) cos(ϕ), y = � 2ℓ2Kf ′(ϕ) sin(ϕ) for ϕ ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Equivalently, in terms of the angle variable θ = f(ϕ) ∈ [0, 2π], the equipo- tential is x = � 2ℓ2K (f −1)′(θ) cos(f −1(θ)) ≡ F(K, θ), y = � 2ℓ2K (f −1)′(θ) sin(f −1(θ)) ≡ G(K, θ), (27) where the notation (F, G) was introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that we will eventually focus on the regime where K ≫ 1 is very large in such a way that the dimensionful area 2πℓ2K be an O(1) quantity as ℓ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Moving just slightly away from the classical regime, we have seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III that the expansion of the operator P V P involves a bicomplex potential function V(z, ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In the case of edge-deformed potentials (25), with the con- ventions used there and above for complex coordinates, one finds V(z, ¯w) = V0 � ℓ2 z ¯w f ′� 1 2i log[z/ ¯w] � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (28) Note that this only makes sense for z and w close to each other, otherwise taking z → e2πiz affects the argument of f ′ on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' By contrast, when z and w remain close, taking z → e2πiz also requires w → e2πiw, and this time the angle 1 2i log[z/ ¯w] is indeed invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Finally, the expansion (22) also involves the complexi- fied Laplacian of the potential, but only its real value will be relevant at the order studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let us therefore express the Laplacian of (25) in polar coordinates: ∇2V = 1 f ′ � 2 − 1 2 f ′′′ f ′ + f ′′2 f ′2 � V ′ 0 � r2/2f ′� + r2 f ′2 � 1 + f ′′2 4f ′2 � V ′′ 0 � r2/2f ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (29) Here the prime means differentiation with respect to the argument, namely ϕ for f(ϕ) and r2/2 for V0(r2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We shall rely on (28) and (29) below, since they directly affect the eigenvalue equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Eigenvalue equation and energy Having studied the potential (25), let us turn to the quantum state meant to solve the eigenvalue equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III B, we begin by building a state (18) that is perfectly localized on the equipotential, project to the LLL using the operator (5), and obtain the wave function (20) that now reads ψm(z, ¯z) = e−|z|2/2 � dϕ f ′(ϕ) n(f(ϕ)) × exp � imf(ϕ) − 1 2mf ′(ϕ) + z � mf ′(ϕ) e−iϕ� , (30) where we changed variables using θ = f(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It remains to show that this solves the eigenvalue equation (7) for edge- deformed traps (25) in the semiclassical regime, provided the function n(θ) satisfies a suitable transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The latter is derived by expanding the energy (9) and the potential (22) to get 0 = � dϕ f ′(ϕ) n(f(ϕ)) × exp � imf(ϕ) − 1 2mf ′(ϕ) + z � mf ′(ϕ) e−iϕ� × � V � z, � mf ′(ϕ) e−iϕ� + ℓ2 2 ∇2V − E0 m − ℓ2E1 m � (31) where V(z, ¯w) is the bicomplex function (28) and the equation holds up to neglected O(ℓ4) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' At leading order in the classical limit, the potential expan- sion (22) boils down to ⟨z|P V P|w⟩ ∼ V(z, ¯z)δ2(z − w), so (31) merely states that E0 m = V0(ℓ2m) = V (γm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The issue is to find the two remaining unknowns: the function n(f(ϕ)) and the first-order energy correction E1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To determine these, the crucial step is to evaluate (31) along the equipotential (26) labeled by K = m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' for z = � mf ′(α) eiα with α ∈ [0, 2π), where as usual we assume m ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, if (31) holds on a level curve, then it holds for all z by holomorphicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is writ- ten in more detail in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' C, where we show that the integrand of (31) has a saddle point at ϕ = α, eventu- ally resulting in a transport equation for the unknown function n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Here we skip the computation and analyse separately the real and imaginary parts of the transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We start with the real part, which will allow us to deduce the LLL-projected energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The imaginary part is postponed to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV D, where we also display the resulting nearly Gaussian wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let Φ(ϕ) denote the phase of n(f(ϕ)) ≡ N(ϕ) eiΦ(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then the real part of the transport equation [see (C17)] yields Φ′(ϕ) = E1 m Ωm f ′(ϕ) − 1 2 � Γm Ωm + 1 �� 1 + f ′′(ϕ)2 4f ′(ϕ)2 � − 1 2 + ∂ϕ � f ′′(ϕ) 8f ′(ϕ) � + 1 2 ∂ϕ[f ′′(ϕ)/2f ′(ϕ)] 1 + f ′′(ϕ)2/4f ′(ϕ)2 , (32) where E1 m is the first order correction to the energy (9) and we introduced the parameters Ωm ≡ V ′ 0(ℓ2m) > 0, Γm ≡ ℓ2m V ′′ 0 (ℓ2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (33) In many-body droplets with N electrons, these will re- spectively measure the Fermi velocity and the curvature of the potential at the Fermi surface when m = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that all terms in (32) are total derivatives save for the factor 1 + [f ′′/2f ′]2, so the solution is Φ(ϕ) = E1 m Ωm f(ϕ) − 1 2 � Γm Ωm + 1 � � ϕ 0 dθ � 1 + f ′′(θ)2 4f ′(θ)2 � − ϕ 2 + f ′′(ϕ) 8f ′(ϕ) + 1 2 arctan � f ′′(ϕ) 2f ′(ϕ) � + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (34) This turns out to imply a quantization condition for en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, when we initially introduced the function n(θ) in (18), we mentioned that it must have a vanishing winding number along the equipotential, so that all the winding of the wave function is contained in the expo- nential factor eimθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The phase Φ(ϕ) must therefore be strictly 2π-periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Φ(2π) = Φ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Using (34), this fixes the first quantum correction of the energy (9): E1 m Ωm = 1 2 + � Γm Ωm + 1 � � dϕ 4π � 1 + f ′′(ϕ)2 4f ′(ϕ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (35) The latter generally generally depends on m through Γm and Ωm in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A simplification only occurs in ‘har- monic’ setups where Γm = 0 and the right-hand side of (35) is an f-dependent constant, for all m [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In any case, the full mth energy (9) in the semiclassical limit can be written as Em ∼ V0 � ℓ2m � + ℓ2 2 � Ωm + � Γm + Ωm � � dϕ 2π � 1 + f ′′(ϕ)2 4f ′(ϕ)2 �� , (36) reproducing the result announced in (12)–(13) with λ = 2f ′ and generalizing the isotropic value obtained for f ′ = 1 [see (A3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The leading-order Bohr-Sommerfeld quantization condition (8) is manifestly satisfied, while the first quantum correction can be expressed in terms of a Maslov-like shift and an integral of the Laplacian, confirming the general result in [55]: Em = V0 � ℓ2� m + 1 2 �� + ℓ2 4 � dϕ 2π f ′(ϕ)∇2V �� r2=2ℓ2(m+1/2)f ′(ϕ) + O(ℓ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (37) (In the language of [55], our ‘Maslov-like’ term actually stems from an integral of the curvature of γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=') It now remains to write the corresponding wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Gaussian wave functions As above, write n(f(ϕ)) = N(ϕ) eiΦ(ϕ) for the un- known function of the WKB ansatz, with a norm N(ϕ) = 9 |n(f(ϕ))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then the imaginary part of the transport equation [see (C18)] can be recast into N ′(ϕ) N(ϕ) = 1 4∂ϕ log � 1 f ′(ϕ) � 1 + f ′′(ϕ)2 4f ′(ϕ)2 �� , (38) which remarkably has the form of an overall logarithmic derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The general solution is thus ��n � f(ϕ) ��� = N0 � 1 f ′(ϕ) � 1 + f ′′(ϕ)2 4f ′(ϕ)2 ��1/4 , (39) where the normalization N0 will soon be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note the exponent 1/4, typical of WKB approximations [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We can now use (39) to evaluate approximate eigen- functions (30) near their maximum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' close to the equipotential (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To see this, zoom in on the equipo- tential by writing z ≡ �√m + a ���� f ′(α) eiα (40) for m ≫ 1 and some finite a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The integral (30) then has a unique saddle point at ϕ = α−iδ1/√m+O(1/m), with δ1 = a � 1 − i f ′′(α) 2f ′(α) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The saddle-point approximation of the wave function (30) thus yields ψm(z, ¯z) ∼ 1 √ 2πℓ2 1 (2πm)1/4 eimf(α)+iΦ(α) × 1 � σ(α) exp � �− f ′(α) a2 1 − i f ′′(α) 2f ′(α) � � , (41) where we used Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (34)–(39) for the phase and norm of n(f(α)), fixed the integration constant in (39) to N0 = 1 2πℓ( m 2π)1/4, and introduced the ubiquitous width σ(ϕ)2 ≡ 1 f ′(ϕ) � 1 + f ′′(ϕ)2 4f ′(ϕ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (42) This is the angle-dependent variance of the probability density of (41), written in (12) in terms of λ = 2f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In- deed, one has |ψm(z, ¯z)|2 ∼ 1 2πℓ2 e−2a2/σ2(α) √ 2πm σ(α), (43) which satisfies the desired normalization condition � d2x |ψ|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (41) reproduces the wave function stated in (11), once more with λ = 2f ′ and now involving the phase Θm(x) = mf(ϕ) + Φ(ϕ) − a2f ′′(ϕ) 2f ′(ϕ)σ(ϕ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (44) The Gaussian behavior of LLL-projected is thus mani- fest, as anticipated at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III B for the gen- eral WKB ansatz (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 3 for two choices of the confining potential (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Finally, (41) gen- eralizes the behavior of isotropic states (3) [see (A1)], in- cluding the O(1/√m) contribution that we did not state here but that can be computed by incorporating the next order term δ2/m for the saddle point and repeating the analysis [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Out[� ]= 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='6 0 1 |ψm|2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The density of a wave function (41) at m = 30 in an edge-deformed trap (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The Gaussian behavior is manifest, as is the angle-dependent ‘roller coaster’ predicted by (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Left: Elliptic potential given by (25) with f of the form (24) and k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Right: Same edge-deformed trap as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Comparison with elliptic wave functions To conclude this section, we now focus on the ‘flower’ deformations (24) and show that the resulting transport equation is integrable: both the phase (34) and the norm (39) can be expressed in terms of elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' These results are valuable in themselves since flower maps are the simplest edge deformations—they are analogues of Fourier modes for circle diffeomorphisms—, but also because their special case k = 2 reproduces known wave functions in elliptic harmonic traps [68, 74], providing an important benchmark for our WKB approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Consider first the deformation (24) with α = cosh λ and β = sinh λ for an arbitrary integer k and a real pa- rameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then the energy quantization condition (35) can be integrated exactly, yielding E1 m Ωm = 1 2 + 1 2 � 1 + Γm Ωm �� 1 + k2 4 � cosh(2λ) − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (45) As for the solution of the transport equation, consisting of the phase (34) and norm (39), it is found to be n(θ) = N0 e i 8 Γm Ωm k sin(kθ) sinh(2λ)e i θ 2 � ( Γm Ωm +1) � 1− k2 4 � +1− k 2 � × �cosh λ + eikθ sinh λ eikθ cosh λ + sinh λ � k2−4 8k (1+ Γm Ωm )− 1 2k + 1 4 × � −4eikθ + (e2ikθ − 1)k sinh(2λ) eikθ cosh λ + sinh λ (46) up to an overall constant phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that (46) depends in a non-trivial way on the po- tential derivatives (33), with some simplification in the ‘harmonic’ regime Γm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let us therefore apply Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (45)–(46) to the case of an elliptic potential k = 2 with constant stiffness Ωm = Ω > 0 (hence Γm = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The cor- responding edge deformation (23) maps the isotropic har- monic potential V0(x) = Ω r2/2 on its anisotropic cousin, V (x) = Ω e−2λx2 + e2λy2 2 cosh(2λ) , (47) 10 whose equipotentials are ellipses rather than circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The energy correction (45) then becomes E1 m = 1 2Ω[1 + cosh(2λ)] and the solution (46) of the transport equation can be written as n(θ) = N0 � cosh λ − e2iθ sinh λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (48) It is straightforward to use this data to obtain the elliptic version of the normalized Gaussian wave function (41): ψm(z, ¯z) ∼ �2π m �1/4 1 2πℓ eimθ √ cosh λ − e−2iθ sinh λ × exp � −e2iθ + tanh(λ) e2iθ − tanh(λ)a2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (49) Crucially, the latter coincides with the large m approxi- mation of the exact LLL-projected eigenstates of the har- monic potential (47), as can be verified thanks to known asymptotic formulas for Hermite polynomials [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is actually true even at subleading order in 1/√m, which we omit here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' MANY-BODY OBSERVABLES This section finally applies the results of Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' III–IV to fully-fledged QH droplets consisting of a large num- ber N ≫ 1 of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Specifically, we exploit our insights on near-Gaussian single-particle wave functions (41) to compute many-body observables and read off the universal shape-dependent effects implied by the width (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The density is treated first: it equals 1 2πℓ2 in the bulk, then drops to zero as an error function at the edge redge = ℓ � 2Nf ′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We then turn to the current and show that it is localized as a Gaussian on the edge, to which it is tangent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Third, correlations near the edge are found to display the usual power-law behavior of free fermions, dressed by radial Gaussian factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This re- duces to known expressions in isotropic traps [59] and in the case of flower deformations (24) with k = 2, where one recovers the elliptic results of [68, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Finally, the radial behavior of correlations is shown to be consistent with the effective low-energy field theory of edge modes: we derive it microscopically and obtain a chiral CFT in terms of the angle variable on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Density Consider a QH droplet of N ≫ 1 non-interacting 2D electrons subject to the Hamiltonian (1), with a very strong magnetic field B = dA and a weak edge-deformed potential (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The ground state of this many-body sys- tem is a Slater determinant of the wave functions ψm for occupied states m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' , N −1, where we recall that m is a quantized action variable generalizing angular mo- mentum (see the red dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Each ψm yields a m Energy ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦•••• N Fermi energy ••••••••◦◦◦◦◦◦· · · ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦· · · 2Λ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The one-body spectrum (37), where the states that are occupied in the many-body ground state are highlighted in red and those that contribute to the low-energy Hamiltonian (64) are filled (black for ‘particles’, red for ‘holes’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The cutoff Λ is large but much smaller than N in the sense that Λ = O(1) in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' single-particle probability density |ψm(x)|2, the sum of which gives the many-body density ρ(x) = N−1 � m=0 |ψm(x)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (50) While WKB theory does not give access to the form of ψm at low m, large values of m should be correctly captured by the analysis of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' IV, in which case the one-body density is approximately Gaussian and given by (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We now exploit this Gaussian form to evaluate the many- body density, both in the bulk and close to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (Some technical details are highlighted along the way, as the same method will later allow us to study the many- body current and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=') The key point is that each wave function (43) is local- ized on an equipotential of V (x) with area 2πℓ2m, so the density close to some equipotential |z| = const × � f ′(ϕ) only receives sizeable contributions from wave functions whose quantum number is close to |z|2/f ′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Accord- ingly, the bulk density for 1 ≪ |z| ≪ √ N is obtained by letting the upper summation bound of (50) go to infinity and writing the approximate density as ρ(x) ∼ 1 2πℓ2 ∞ � m=m0 e − 2 σ2(ϕ) � |z| √ f′(ϕ) −√m �2 √ 2πm σ(ϕ) , (51) where the lower summation bound m0 ≫ 1 is irrele- vant as long as it is much smaller than |z|2, and σ(ϕ) is the angle-dependent width (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' At large |z|, the Euler- Maclaurin formula allows us to approximate the sum over m by a (Gaussian) integral over √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This yields the ex- pected density ρ(x) ∼ 1 2πℓ2 (52) in the bulk of a QH droplet with filling fraction ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' An analogous argument can be carried out close to the droplet’s edge, with one key difference: the upper summation bound of (50) is now crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Thus, letting |z| = �√ N + a � f ′(ϕ) with finite a in the large N limit 11 and using once more the approximate Gaussian behavior (43), the density (50) near the edge behaves as ρ(x) ∼ 1 2πℓ2 ∞ � k=1 e − 2 σ2(ϕ) � a+ k 2 √ N �2 √ 2πN σ(ϕ) , (53) where we changed variables as m ≡ N − k with k = O( √ N) at large N and only kept track of leading-order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For N ≫ 1, the sum over k can once more be converted into an integral, now over k/2 √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This yields the asymptotic behavior ρ(r, ϕ) ∼ 1 4πℓ2 erfc � 1 σ(ϕ) r − ℓ � 2Nf ′(ϕ) ℓ � f ′(ϕ) � (54) where erfc denotes the complementary error function and the width (42) is inherited from that of LLL wave func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is a remarkably explicit result, announced in (14) with λ = 2f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It confirms that the density is roughly constant and equal to (52) in the bulk, then drops to zero within a distance of the order of the magnetic length (2) around the edge at r = ℓ � 2N f ′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' See Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 2(a)–5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We stress that, in contrast to wave functions, the den- sity (54) only depends on the potential near the edge of the droplet: bulk deformations of the potential do not af- fect the quantized bulk density (52) in the limit of strong magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In this sense, (54) is a universal formula for the density of any QH droplet of LLL states whose edge traces an equipotential of the form r2 = 2ℓ2N f ′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It would be instructive to probe this local density in mod- ern quantum simulators [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that the leading-order formula (54) receives a number of subleading corrections that can be systemati- cally computed in our formalism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' these are omitted here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A related comment is that the bulk value density (52) is only valid at extremely strong magnetic fields, which stems from the simplification provided by the LLL projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The actual density profile depends on the gradient of the potential, but this involves higher Landau levels that are beyond our scope [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Current The current of a droplet of N ≫ 1 electrons can simi- larly be evaluated as a sum over single-particle currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To this end, recall that the gauge-invariant one-body probability current of a charged wave function ψ with mass M is a one-form ℏ j/M given by j = 1 2i � ψ∗dψ − ψdψ∗ − 2iq ℏA|ψ|2� , (55) where the first term is only sensitive to the phase of ψ and A = 1 2Br2 dϕ = ℏ q |z|2 dϕ in symmetric gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Thus, the many-body current of a Slater determinant of the occupied states ψm with m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' , N − 1 is J = N−1 � m=0 jm, (56) where jm is the single-particle current (55) of each ψm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As before, the WKB approximation does not give ac- cess to wave functions for small m, but this is unim- portant close to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In that regime, we have al- ready gathered all ingredients needed to evaluate the currents (55) up to small quantum corrections: the one- body density is given by (43), while the derivative of the phase is contained in (41) and the phase transport equation (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In practice, the WKB phase Φ turns out to be negligible at leading order, and the only rel- evant parts of the phase are those already visible in (41): the (fast) phase eimf(ϕ) together with the contribution from A = � ℏ|z|2/q � dϕ eventually gives rise to the lead- ing ϕ component of the current, while the (slow) phase e−i[f ′′(ϕ)/2f ′(ϕ)]a2/σ2 yields its radial component that is non-zero whenever f ′′(ϕ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Starting from these facts, it is straightforward to adapt the method of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' V A to the many-body current (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Writing |z| = �√ N +a �� f ′(ϕ), the sum over m ≡ N −k becomes an integral over k/(2 √ N) = O(1), which yields the leading order result quoted in (15) with λ = 2f ′: J(r, ϕ) ∼ − e − 2a2 σ(ϕ)2 (2πℓ2)3/2σ(ϕ) ℓ � 2Nf ′(ϕ) dϕ + f ′′(ϕ) 2f ′(ϕ) dr � 2f ′(ϕ) (57) where a = � r−ℓ � 2Nf ′(ϕ) �� ℓ � 2f ′(ϕ) and σ(ϕ) is given by (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Both components in (57) receive subleading cor- rections that are omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In particular, there is an O(1) term in Jϕ that is non-zero on the edge, even in the isotropic case f ′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The computation of that term re- quires the O(1/√m) correction that was omitted in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Using the metric ds2 = dr2 + r2dϕ2, one verifies that the one-form ℓ√2N f ′ dϕ+(f ′′/2f ′) dr in (57) is the dual of a vector tangent to the equipotential at the droplet’s edge [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Moreover, the norm squared of (57) is ∥J(r, ϕ)∥2 ∼ 1 2(2πℓ2)3 exp � −2 � r − ℓ � 2Nf ′(ϕ) �2 ℓ2σ(ϕ)2f ′(ϕ) � , (58) showing that the current has a constant maximum along the edge but a varying width: see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Similarly to the density, it is important to remember that the LLL projection misses some important physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, the actual bulk current is the symplectic gra- dient of the confining potential multiplied by the Hall conductance [28, 29, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' No such effect occurs in (57) because it requires higher Landau levels, which are beyond our scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Correlations The methods that we have applied to density and cur- rent can also be used to compute electronic correlations near the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, consider as before an anisotropic droplet whose occupied one-body states have quantum 12 ��� ��� ��� x x x y y y √ N − √ N − √ N √ N (a) (b) (c) x x x y y y ℓ √ 2N −ℓ √ 2N −ℓ √ 2N ℓ √ 2N Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (a) The density (54) for N = 100 electrons and a ‘flower’ edge deformation (24) of order k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The constancy of density in the bulk and its sharp decay at the boundary are manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (b) The current’s norm (58) for the same flower-shaped droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The localization on the edge equipotential (black line) is clearly visible, as is the position-dependent width of the Gaussian jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (c) The correlation function C(x1, x2) (61) for the same flower-shaped droplet, seen as a function of x2 when x1 = (ℓ � Nλ(0), 0) is fixed close to the edge of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' numbers m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then the correlation func- tion between the points x1 and x2 is C(x1, x2) = N−1 � m=0 ψ∗ m(x1) ψm(x2), (59) which reduces to the density (50) when x1 = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As before, we rename m ≡ N − k and let the complex coor- dinates z, w corresponding to x1, x2 be such that z = �√ N + a �� f ′(ϕ1) eiϕ1, w = �√ N + b �� f ′(ϕ2) eiϕ2, (60) where a, b are finite at large N and ϕ1, ϕ2 are the po- lar angles of x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One can then plug the Gaussian wave functions (41) into (59), this time assuming k fi- nite, and performing the sum over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The gradient ex- pansion of the potential implies that the ratio Γm/Ωm ∼ ΓN/ΩN + O(ℓ2) is nearly constant in this regime, so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (59) becomes a geometric sum over k that reproduces the result stated in (16) with λ = 2f ′: C(x1, x2) ∼ eiΘN(x1,x2) (2π)3/2ℓ2√ N 1 � σ(ϕ1)σ(ϕ2) × i e − a2 σ(ϕ1)2 − b2 σ(ϕ2)2 2 sin � [f(ϕ1) − f(ϕ2)]/2 �, (61) where σ(ϕ) is the angle-dependent width (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The over- all phase ΘN(x1, x2) = ΘN(x2) − ΘN(x1) is given by (44), and involves in particular the WKB phase (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Some features of (61) are worth emphasizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' First, note the Gaussian jump of power-law correlations near the edge, involving a free fermion correlation ∝ sin([f(ϕ1)−f(ϕ2)]/2)−1 expressed in terms of f(ϕ1) and f(ϕ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' this is the standard static diagnostic of the pres- ence of edge modes [10, 59, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' A second striking aspect is the apparent lack of translation-invariance along the edge, caused not only by the argument f(ϕ1) − f(ϕ2) = � ϕ1 ϕ2 dθ f ′(θ) but also by the widths σ(ϕ1) and σ(ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In particular, the product σ(ϕ1)−1/2σ(ϕ2)−1/2 is reminis- cent of prefactors picked up by primary fields in CFT under conformal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Since the boundary correlator (61) holds in any edge- deformed trap (25), it also applies to special cases of in- terest such as the anisotropic harmonic potential (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The corresponding correlations were actually computed long ago in [68], and they coincide with our result (61) upon using the map (24) with k = 2, α = cosh λ, β = sinh λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In fact, this specific setup is also well known in the context of the Coulomb gas, since edge correla- tions can then be related by a conformal map to the standard Euclidean correlation function (z1 − z2)−1 of a free fermion CFT [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Finally, it is a simple matter to include time- dependence in the correlator (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, the occupied one-particle states in (59) have definite energies Em given by (37) at large m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This spectrum is approximately linear close to the Fermi energy: changing variables according to m = N + k with k finite at large N, one has EN+k − EN ∼ ℏω k (62) where ω ≡ ℓ2ΩN/ℏ is the angular Fermi velocity given by the derivative (33) at m = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that ω re- ceives a number of subleading quantum corrections in- volving e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' the curvature ΓN in (33) [78–80];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' we ne- glect those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In the linear regime (62), one can repeat the asymptotic computation of the correlator and find 1 1 1 1 4 r 1 1 1 1 1 1 1 1 1 1 1 1 113 once more a result of the form (61), now with a time- dependent overall phase and a time-dependent denom- inator 2 sin � [f(ϕ1) − f(ϕ2) − ω(t1 − t2)]/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This ex- hibits the standard ballistic propagation of correlations in a CFT, which we confirm below by deriving the effec- tive low-energy dynamics of our droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Edge modes The effective low-energy description of anisotropic QH droplets can be derived similarly to the isotropic case [59] inspired by Luttinger-liquid theory [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This has the advantage of circumventing topological field theory, at the cost of failing to apply in fractional QH states [6– 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We now provide such a first-principles calculation, eventually concluding that edge modes span a free chiral CFT expressed in terms of the angle coordinate f(ϕ) along the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Aside from its intrinsic interest, this computation provides an independent check of the validity of the correlation (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Our starting point is the second-quantized Hamilto- nian in a Fock space of free fermions, H = � m,n (Em,n − µ)a† m,nam,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (63) Here the sum runs over eigenstates of the one-body Hamiltonian (1), labeled by the Landau level n ∈ N and the ‘action variable’ quantum number m ∈ N within each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Their energies are denoted Em,n, and µ is some chemical potential to be fixed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For each pair (m, n), the operator a† m,n creates the corresponding eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The exact energy spectrum is unknown, but this is not an issue since low-energy excitations all be- long to the LLL, with an approximately linear dispersion (62) within a window [−Λ, Λ] around the Fermi momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then choosing the chemical potential of (63) as µ = const − 1 2ℏω, the low-energy approximation of the many-body Hamiltonian (63) becomes H ∼ � k∈[−Λ,Λ] ℏω (k + 1/2) a† N+kaN+k, (64) where the sum is over all integers k in the specified win- dow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Since aN+k are fermionic Fock space operators, the Λ → ∞ limit of (64) yields a gapless Hamiltonian for free fermions written in Fourier modes, and the chemical po- tential enforces antiperiodic (Neveu-Schwarz) boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It now only remains to relate the edge CFT to bulk wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To this end, note that the Fock space operators in (64) create states in the LLL that are given by the Gaussian wave function (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One can therefore express them in terms of 2D creation operators c†(x): a† N+k ∼ � f ′(ϕ) dϕ √ 2π ei(k+1/2)f(ϕ) Ψ†(f(ϕ)), (65) where the 1D fermionic field Ψ is defined as a radial in- tegral involving the wave function (41), Ψ†(f(ϕ)) ≡ ei(N−1/2)f(ϕ) eiΦ(ϕ) f ′(ϕ) � σ(ϕ) × � ∞ 0 r dr ℓ(2πN)1/4 c†(x) e − 1 σ2(ϕ) � r √ 2ℓ2f′(ϕ) − √ N �2 (66) with σ the width (42) and Φ the WKB phase (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that in writing (66) we assumed that the momentum in- dex k and the cutoff Λ are of order O(1) in the thermo- dynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It is then clear that the operator Ψ†(ϕ) creates an electron at the position ϕ on the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' From this point onward, the derivation of the low- energy effective field theory is essentially done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, the Hamiltonian (64) expressed in terms of the edge field (66) reads H = ℏω � dθ Ψ†(θ) (−i∂θ)Ψ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (67) Furthermore, the normalization of the field Ψ defined by (66) is canonical in angle variables: using the standard anticommutator {c(x1), c†(x2)} = δ(2)(x1 −x2), one sim- ilarly finds {Ψ(f(ϕ1)), Ψ†(f(ϕ2))} = δ(f(ϕ1) − f(ϕ2)) in terms of the Dirac delta function on a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This de- termines the kinetic term of the action functional for Ψ, which is thus a canonical fermionic expression ∼ Ψ†∂tΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It can be combined with the Hamiltonian (64) to write down the fermionic action functional of edge modes, S[Ψ, Ψ†] = ℏ � dt dθ iΨ†(θ) � ∂t + ω∂θ � Ψ(θ), (68) where we recall that the angular Fermi velocity is ω = ℓ2ΩN/ℏ, given by (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is manifestly a (1+1)D free chiral CFT in terms of the angle variable θ = f(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The low-energy effective theory (68) is universal: for any trapping potential, edge modes are described by a chiral fermionic CFT expressed in terms of the angle co- ordinate of the trap at the boundary, as could have been guessed from the dynamics of electronic guiding centers induced by the potential V in the non-commutative plane (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In the present case, the angle coordinate was just θ = f(ϕ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' more general cases involve more complicated action-angle variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We stress that the angle variable generally has nothing to do with other obvious position coordinates, such as the polar angle ϕ or the arc length s(ϕ) = ℓ √ 2N � ϕ 0 dα � f ′(α) + f ′′(α)2 4f ′(α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (69) Any such ‘wrong’ coordinate makes the apparent Fermi velocity of edge modes position-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is rem- iniscent of inhomogeneous CFTs whose light-cones are curved owing to the presence of a non-zero space-time curvature [82–89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' However, one should keep in mind that our edge modes sense a flat metric ω2dt2 − dθ2 = 14 ω2dt2 − f ′(ϕ)2dϕ2 whose light-cones are straight lines in terms of the angle variable θ = f(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let us conclude this section by showing that the action (68) is consistent with the seemingly complicated corre- lator (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We start from the definition (66) to write the 1D correlation function ⟨Ψ†(θ1)Ψ(θ2)⟩ as a double radial integral of the 2D quantity ⟨c†(x1)c(x2)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Now using the asymptotic relation (61), one finds that all normaliza- tions simplify and the correlator of the edge field boils down to ⟨Ψ†(θ1)Ψ(θ2)⟩ = 1 2π i 2 sin � [θ1 − θ2]/2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (70) The same result would have been obtained directly from the low-energy action (68): it is a correlation function of free gapless fermions written in terms of the angle coordinates θ1 = f(ϕ1) and θ2 = f(ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As a bonus, time-dependent correlations automatically satisfy the be- haviour ∝ sin � [θ1 − θ2 − ω(t1 − t2)]/2 �−1 stated at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' V C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK This work was devoted to a detailed study of meso- scopic droplets of non-interacting planar electrons placed in a perpendicular magnetic field and confined by star- shaped anisotropic traps with scale-invariant level curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In particular, we provided explicit formulas for the corre- sponding wave functions and energy spectrum, allowing us to compute the many-body density, current, and cor- relations of an entire droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The resulting low-energy edge modes were eventually shown to behave as a chiral CFT in terms of the angle variable along the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This was all achieved in a regime of high magnetic fields, ultimately equivalent to a semiclassical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' These results pave the way for a number of applications and follow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, recent advances in quantum sim- ulation suggest the tantalizing possibility of probing lo- cal properties of QH-like droplets in the lab [21–24], both for static ground states and their dynamical edge excita- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The density (54) or the current (57) then predict observable shape-dependent widths, while the low-energy theory (68) predicts the ballistic propagation of local boundary disturbances with a position-dependent veloc- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' More generally, the geometry of the QH effect [33– 39] could soon become relevant for experiments involv- ing ultracold atoms or photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Our framework pro- vides a bridge between this field of mathematical physics and concrete observables in mesoscopic quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Verifying the predictions put forward here, through lin- ear response experiments or direct imaging, would be a fascinating example of many-body quantum mechanics at work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Turning now to theory, the link between our formal- ism and QH symmetries deserves further study: following the series of works [58–63], one can think of edge defor- mations as approximately unitary operators acting on many-body QH states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It is then natural to wonder how these operators get composed together;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' do they span a Virasoro group as in CFT?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' If yes, how to derive the cen- tral charge c = 1 in terms of microscopic wave functions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' More broadly, what are the operators implementing area- preserving deformations in the sense of the WKB ansatz (20)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One expects these operators to provide a finite (ex- ponentiated) form of the operators studied in [58, 60, 61], with non-commutative composition laws consistent with the geometry (6) of LLL physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that the discussion above was mostly focused on leading-order properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For instance, one may wonder what are irrelevant corrections to the edge field theory (68), especially following the recent numerical observa- tion [79] that density waves on the edge satisfy a non- linear Korteweg-de Vries equation at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This regime is presumably described by small droplet defor- mations of the form r2 �→ r2 + α(ϕ), spanning a U(1) Kac-Moody algebra whose level is sensitive to the fill- ing fraction [58, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The resulting non-linear edge waves would then be described by an evolution equa- tion in an infinite-dimensional group manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This per- spective is standard in hydrodynamics [90–92], but it has only recently come to be appreciated in condensed matter physics [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The geometric study initiated here provides a basis for considerations of this kind in the QH effect, including the possibility of inhomogeneous (position-dependent) corrections in anisotropic traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Another obvious extension of this work is the fractional QH regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In that context, no single-particle descrip- tion is available, but many-body predictions such as the edge density (54) or the current (57) conceivably display universal geometric features that would remain true in interacting many-body ground states [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It would be thrilling to derive such predictions from the family of edge transformations studied here, either from a micro- scopic analysis of the Laughlin wave function, or thanks to the reformulation of fractional QH states as CFT cor- relation functions [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to Laurent Charles for illuminating discussions on semiclassical methods in Kählerian geo- metric quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' also thanks Mathieu Beauvil- lain, Nathan Goldman, and Marios Petropoulos for col- laboration on closely related subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Finally, we ac- knowledge useful and motivating interactions with Jean Dalibard, Benoit Douçot, Jean-Noël Fuchs, Marc Geiller, Gian Michele Graf, Semyon Klevtsov, Titus Neupert and Nicolas Regnault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The work of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' is supported by the European Union’s Horizon 2020 research and in- novation programme under the Marie Skłodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 846244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' acknowledges fund- ing from the European Research Council (ERC) under the European Union’s Horizon 2020 research and inno- vation program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' ERC-StG-Neupert-757867- 15 PARATOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' gratefully acknowledges financial sup- port from the Wenner-Gren Foundations (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' WGF2019- 0061).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' is supported by the ANR grant TopO No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' ANR-17-CE30-0013-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Appendix A: Isotropic droplets Most of this work is concerned with anisotropic prop- erties, so isotropic results provide a useful benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' They are simpler than their anisotropic counterparts and mostly well-known in the literature, so their properties are concisely summarized here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We begin by recalling el- ementary aspects of the one-body energy spectrum based on the exact wave functions (3), then turn to many-body observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' One-body spectrum Consider a spin-polarized 2D electron governed by the Landau Hamiltonian (1) with an isotropic confining po- tential V (x) = V0(r2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' At very strong magnetic fields, the resulting one-body spectrum is well approximated by the solution of the LLL-projected eigenvalue equa- tion (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As the potential is isotropic, it commutes with angular momentum, so the eigenstates of P V P are wave functions (3) with definite angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that these confirm the general near-Gaussian behavior found in (43): letting |z| = √m+a with finite a, one finds that (3) behaves at large m as φm(x) = eimϕ √ 2πℓ2 1 (2πm)1/4 e−a2 � 1 + O(1/√m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (A1) The energy Em of each state (3) is readily found by com- puting the wave function ⟨z, ¯z|PV0(r2/2)P|φm⟩, which yields the exact eigenvalue Em = ⟨φm|V |φm⟩ = 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' � ∞ 0 dt tm e−t V0(ℓ2t) (A2) in terms of the integration variable t ≡ |z|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note in pass- ing that this is the value one would find from first-order perturbation theory of the full Landau Hamiltonian (1): by construction, LLL-projected physics is only sensitive to first-order effects of the potential, while higher-orders ultimately involve higher Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Now fix an index m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' What is the corresponding equipotential in the sense of (9)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To answer this in the classical limit, we let m → ∞ while fixing the value of ℓ2m = O(1), and evaluate the integral (A2) thanks to a saddle-point approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The result is Em = V0(ℓ2m) + ℓ2Ωm + ℓ2 2 Γm + O(ℓ4), (A3) where Ωm and Γm were defined in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is consistent with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (13) and (36) with λ = 2f ′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Many-body aspects The sequence followed here is the same as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' V: we start with the density, then consider the current and cor- relations close to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In all cases, the edge asymp- totics reproduce the formulas of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' V for the simplest case where f ′(ϕ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let N ≫ 1 non-interacting planar electrons be subjected to the Hamiltonian (1), with a very strong magnetic field B = dA and a weak isotropic poten- tial V (x) = V0(r2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The ground state wave func- tion of this many-body system is a Slater determinant of the occupied single-particle eigenstates φ0, φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' , φN−1 given by (3), each of which has a one-body density |φm(x)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The resulting many-body density is thus (50), which can be computed in closed form in the very spe- cial case of states (3) with definite angular momentum: it is a normalized incomplete gamma function ρ(x) = (2πℓ2)−1Γ(N, |z|2)/Γ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Constancy of density in the bulk is then manifest, as is its drop to zero close to the edge |z| = √ N, with an error function behavior that can be deduced from known asymptotic formulas for gamma functions [76] and reproduces Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (14)–(54) with λ = 2f ′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For the LLL states (3) with definite angular momentum, each one-body current (55) is purely angu- lar, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' it reads jm = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' )dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The sum (56) can then be evaluated in closed form owing to an exact cancella- tion between the contribution of the states m and m+ 1, eventually leading to a current only due to the (N − 1)th wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It is then trivial to show that the current is localized as a Gaussian close to the edge, since this is also true of the underlying single-particle wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This reproduces Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (15)–(57) with λ = 2f ′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The computation of electronic correla- tions close to the edge is similar to that of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Indeed, since the many-body ground state wave function is a Slater determinant, the two-point correlation func- tion in the ground state can be expressed as in (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The exact wave functions (3) can then be used to write the correlation (59) as an incomplete gamma function (this time with a complex argument): C(z, ¯z, w, ¯w) = 1 2πℓ2 Γ(N, ¯zw) Γ(N) e−(|z|2+|w|2)/2 ez ¯ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (A4) It is then manifest that bulk correlations coincide with the kernel (5) at leading order in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As for the edge behavior, it can be extracted e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' from a steepest descent argument [76] and reproduces Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (16)–(61) with λ = 2f ′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Appendix B: Semiclassical expansion of P V P In this appendix, we derive (22) starting from (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To this end, think of V (x, y) as some smooth function of 16 (x, y) whose arguments can be complexified, and change the integration variables (x, y) of (21) to s ≡ x − ℓ √ 2(z + ¯w), t ≡ y + iℓ √ 2(z − ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (B1) In terms of (s, t), the integrals in (21) are two line in- tegrals in the complex plane, each along a path from −∞ + ic to +∞ + ic, where c is some irrelevant real con- stant (a different one for s and t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The advantage of the change of variables (B1) is to make the exponential factor in (21) purely Gaussian: ⟨z, ¯z|P V (x)P|w, ¯w⟩ = 1 (2πℓ2)2 e− |z−w|2 2 e z ¯ w−¯zw 2 × � ds dt V � s+ ℓ √ 2(z + ¯w), t− iℓ √ 2(z − ¯w) � e− s2+t2 2ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (B2) We then complexify V , thus replacing V (x, y) by V(z, ¯z), where V(z, ¯w) is a function of two complex variables, holomorphic in the first one and anti-holomorphic in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We can then deform independently both inte- gration contours for s and t back to the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For small ℓ, the Gaussian factor of (B2) localizes everything to s = t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We now use our assumption of slow varia- tion of V (x) to Taylor-expand it as V � s + ℓ √ 2(z + ¯w), t − iℓ √ 2(z − ¯w) � ∼ � V + s2 2 ∂2 xV + t2 2 ∂2 yV ����� ℓ √ 2 (z+ ¯ w),− iℓ √ 2 (z− ¯ w) �, (B3) where we only kept terms that give non-zero contribu- tions to the O(ℓ2) approximation of the integral (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that everything is evaluated at (x, y) = ( ℓ √ 2(z + ¯w), − iℓ √ 2(z − ¯w));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' in complex coordinates, this is just the point (z, ¯w), so it is simpler to write the potential as V(z, ¯w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Plugging the expansion (B3) into (B2) then yields the result (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Appendix C: The transport equation The purpose of this appendix is to derive the real and imaginary parts of the transport equation in (32) and (38), respectively, by imposing the eigenvalue equation (7) based on our WKB ansatz (30) in the case of edge- deformed droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The derivation relies on expanding the energy and the potential as in (9) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' It is divided in two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' First, we use the eigenvalue equa- tion to derive the constraint (31), and let z belong to an equipotential so that the whole equation boils down to a 1D integral identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Second, we show that the inte- gral has a sharp saddle point in the large m limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' this allows us to rephrase the integral constraint as a first- order transport equation for the unknown function n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Evaluation along an equipotential Using the wave functions (18)–(19) and the expansion (22) of the potential along with the projector property P 2 = P, the eigenvalue problem (7) reads 0 = � R2 d2w 2πℓ2 e− |z−w|2 2 + z ¯ w−¯zw 2 �� V+ ℓ2 2 ∇2V ���� (z, ¯ w)−Em � × � dθ n(θ) eimθ δ2� w − � F(m, θ), G(m, θ) �� (C1) up to O(ℓ4) corrections [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In the case of edge-deformed traps, V(z, ¯w) is the bicomplex potential given in (28) and the delta function localizes the whole integral over w to a level curve (27) with K = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Integrating over w and changing the integration variable from θ = f(ϕ) to ϕ yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note that the structure of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C1) and (31) is 0 = e−|z|2/2 F(z) for a holomorphic function F(z), so setting F(z) = 0 on a closed curve implies F(z) = 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Accordingly, we will solve (C1)–(31) along the equipo- tential (26) by fixing K = m and parametrizing z = � mf ′(α) eiα, α ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C2) This ensures that all three terms in the exponent of (31) are of the same order O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Then (31) with the choice (C2) and ϕ ≡ α + ε becomes 0 = � π −π dε f ′(α + ε) n(f(α + ε)) exp � imf(α + ε) − 1 2mf ′(α + ε) + m � f ′(α)f ′(α + ε) e−iε� × � V �� mf ′(α) eiα, � mf ′(α + ε) e−i(α+ε)� + ℓ2 2 ∇2V − E0 m − ℓ2E1 m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C3) This rewriting will allow us to carry out the integral thanks to the saddle-point approximation, obtained by expanding all terms in powers of ε and leading to a differential equation for n(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Saddle point and transport equation The saddle-point expansion of the integral (C3) is cumbersome but straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The strategy is to expand all factors in the integrand up to a suitable power of ε, then perform the resulting integrals of the form � dε ε# e−Cε2, where C is some f-dependent coefficient [see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C5)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The powers of ε involved are typically small, as higher powers are suppressed in the classical limit [large m and ℓ2m = O(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The fact that the argument of n(θ) also involves a factor ε eventually converts the integral into a transport equation of the form n′(θ) ∝ n(θ) [see (C16)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We start with (C3) and first expand the exponential, then the potential with its Laplacian, and finally the simplest f ′(ϕ)n(f(ϕ)) prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For convenience, we introduce the notation A ≡ f ′′ f ′ , B ≡ f ′′′ f ′ (C4) for combinations of derivatives of f that often appear below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' from now on, expressions of the form f or f ′, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=', are all implicitly evaluated at α unless specified otherwise (so f ≡ f(α), f ′ ≡ f ′(α), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Note for future reference the useful relation A′ = B − A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Using the notation (C4), one has exp � imf(α + ε) − 1 2mf ′(α + ε) + m � f ′(α)f ′(α + ε) e−iε� ∼ eimf+ 1 2 mf ′ exp � − 1 2mf ′ � 1 + A2 4 � ε2� � 1 + mf ′ε3 � i 6 − A 4 − iB 12 + iA2 8 − AB 8 + A3 16 �� (C5) where the factor exp � imf + mf ′/2 � is ultimately irrelevant for the eigenvalue equation (C3), so we will not include it in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The main point of (C5) is to exhibit the leading Gaussian behavior exp � −(mf ′/2)(1 + A2/4)ε2� of the integrand, which will eventually allow us to convert (C3) into a differential equation for the unknown function n(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In fact, the same exponential term appears in the approximately Gaussian wave function (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' We now turn to the expansions of the potential and of its Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As a first step,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' our task is to expand the potential V �� mf ′ eiα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' � mf ′(α + ε) e−i(α+ε)� = V0 � � �ℓ2m √f ′ � f ′(α + ε) e−iε f ′ � 1 2i log � √f ′ e2iα+iε √ f ′(α+ε) �� � � � ∼ V0 � ℓ2m � 1 − iε � 1 + A2 4 � + ε2 � − 1 2 + B 8 − 3A2 8 − A3 4i − A4 16 + AB 4i + A2B 32 � �� ∼ V0 � ℓ2m � − iℓ2m ε � 1 + A2 4 � V ′ 0 � ℓ2m � − 1 2ℓ4m2ε2 � 1 + A2 4 �2 V ′′ 0 � ℓ2m � + ℓ2m ε2 � − 1 2 + B 8 − 3A2 8 − A3 4i − A4 16 + AB 4i + A2B 32 � V ′ 0 � ℓ2m � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C6) where we used (28) and then the notation (C4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Aside from the contribution of the Laplacian, these are all the terms of the potential needed in the eigenvalue equation (C3) along an equipotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' As expected, they all ul- timately involve the potential and its derivatives at the equipotential (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For ε = 0, the whole expression boils down to V0(ℓ2m) alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let us now turn to the Laplacian term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' The eigenvalue equation (C3) requires the Laplacian evaluated at the complexified point (z, ¯w) = �� mf ′(α) eiα, � mf ′(α + ε) e−i(α+ε)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' In practice, the Laplacian term is multiplied by ℓ2 in (C3), so we may safely set ε = 0 when computing it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' this removes the complexification and allows us to write the Laplacian contribution in (C3) as ℓ2 2 ∇2V ∼ ℓ2 f ′ � 1 − B 4 + A2 2 � V ′ 0(ℓ2m) + ℓ4m f ′ � 1 + A2 4 � V ′′ 0 (ℓ2m), (C7) which follows from the general expression (29) evaluated on the equipotential (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' All together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Let us finally consider the very first factor on the right-hand side of (C3), namely f ′(α + ε) n(f(α + ε)) ∼ f ′n(f) + ε � f ′′n(f) + f ′2n′(f) � (C8) where higher powers of ε are negligible at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' To see why they may be neglected, it is helpful to investigate 18 the general structure of the small ℓ expansion of (C3): the exponential term in (C5) has the form exp[imf(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=')] ∼ const × e−mΛε2(1 + mLε3) (C9) with m ≫ 1 and Λ, L some O(1) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Sim- ilarly, the potential expansion (C6) together with the Laplacian correction (C7) can schematically be written as V0+ ℓ2 2 ∇2V0 ∼ V0+ℓ2W0+Gε+Hε2, where V0 ≡ V0(ℓ2m) while W0, G, H are again some O(1) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Finally, the expansion (C8) of the prefactor roughly has the form f ′n e(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=') ∼ const × (f ′n + εIn′ + εJn), (C10) where I, J are O(1) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Putting together the schematic expressions (C9)–(C10) and using the fact that constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' ε-independent) contributions are irrelevant, the eigenvalue equation (C3) becomes 0 = � dε (f ′n + εIn′ + εJn)e−mΛε2(1 + mLε3) × � V0 + ℓ2W0 + Gε + Hε2 − E0 m − ℓ2E1 m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C11) Here the right-hand side is a sum of integrals whose in- tegrand has the form εn e−mΛε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' For odd n, each such integral vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' for even n, it is non-zero and scales as m−n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is why only the first order in ε is needed in the expansion (C8): higher powers of ε would yield sub- leading corrections to (C11), which can be consistently taken into account only by expanding the exponential, potential and Laplacian terms up to orders in ε higher than what we did above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Here we content ourselves with the zeroth and first order terms in ℓ2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' in 1/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' At that level of approximation, (C11) yields the zeroth order statement V0 − E0 m = 0 (C12) and the first-order result f ′n � Λℓ2m[W0 −E1 m]+ H 2 + 3LG 4Λ � + G 2 � In′ +Jn � = 0, (C13) where ℓ2m = O(1) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C12) confirms that the eigenvalue equation holds if E0 m = V0(ℓ2m), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' if the en- ergy of the eigenstate |ψm⟩ is that of its equipotential at leading order [recall (9)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' More important, (C13) yields a transport equation for n, whose schematic form is GI 2 n′ n +f ′� Λℓ2m(W0−E1 m)+ H 2 + 3LG 4Λ � + GJ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='We now rely on the expansions (C5)–(C8) to write this ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='transport equation explicitly: using the notation (33) and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='plugging (C5)–(C8) into (C3) yields the condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='0 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='dε e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='− Kf ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1+ A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='ε2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1 + ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='A + f ′ n′(f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='n(f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='�� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1 + Kf ′ε3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='6 − A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 − iB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='12 + iA2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='8 − AB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='8 + A3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='−iℓ2Kε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1 + A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='Ωm − ℓ2Kε2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1 + A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='Γm + ℓ2Kε2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='2 + B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='8 − 3A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='− A3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4i − A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='16 + AB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4i + A2B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='Ωm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='+ ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='f ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1 − B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 + A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='Ωm + ℓ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='f ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='1 + A2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='Γm − ℓ2E1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C15) whose structure is that announced in (C11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' as had to be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' What remains is to multiply all the factors in the integrand, keep track of powers of ε and integrate over ε, which leads to iR′/R = � 1 + A2 4 � Γm 2Ωm + 1 − B 4 + A2 2 − f ′ E1 m Ωm − 1 1 + A2 4 �� B 8 + A4 16 − A2B 32 � + i � A 4 + 3A3 16 − AB 8 �� , (C16) where we introduced R ≡ R(α) ≡ n(f(α)) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' This is the transport equation for the O(1) multiplica- tive factor of the WKB ansatz (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Its real and imag- inary parts, respectively, govern the phase and norm of n(f(ϕ)) ≡ N(ϕ) eiΦ(ϕ): −Φ′ = � 1 + A2 4 � Γm 2Ωm − f ′ E1 m Ωm + 1 1+ A2 4 � 1 + 3A2 4 + A4 16 − 3B 8 − A2B 32 � , (C17) N ′/N = − 1 1 + A2 4 � A 4 + 3A3 16 − AB 8 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (C18) The identity B = A′+A2 then reduces these two relations to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' (32) and (38) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' 19 [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' von Klitzing, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} +page_content=' Dorda, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfwv4v/content/2301.01726v1.pdf'} 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