diff --git "a/DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/load_file.txt" "b/DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DtE4T4oBgHgl3EQf6Q5X/content/tmp_files/load_file.txt" @@ -0,0 +1,751 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf,len=750 +page_content='Strain-induced Landau levels of Majorana fermions in an anisotropically interacting Kitaev model on a honeycomb lattice Takuto Yamada1 and Sei-ichiro Suga1 1Graduate School of Engineering, University of Hyogo, Himeji 671-2280, Japan (Dated: January 16, 2023) The low-energy states of an anisotropically interacting Kitaev model on a honeycomb lattice under triaxial strain are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' A numerical calculation shows that quantized states appear in the low-energy region and their energy is proportional to the square root of the quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Furthermore, the quantized state at zero energy appears only on one sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The obtained results are characteristic of the Landau levels of Dirac fermions with time-reversal symmetry, indicating the emergence of the strain-induced Landau levels of Majorana fermions, which is also determined in the anisotropic Kitaev model by an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' INTRODUCTION The Kitaev model is an S = 1/2 quantum spin model that has bond-dependent Ising-type interactions on a honeycomb lattice1, called Kitaev interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' A Ma- jorana representation of the spin operators has shown that this model is described by noninteracting itiner- ant Majorana fermions coupled with Z2 gauge fluxes1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the ground state, the model is equivalent to the noninteracting itinerant Majorana fermions on a hon- eycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefore, the low-lying dispersion is described by the type of Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Fascinating properties related with Majorana fermions have been revealed by intensive theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Since Majo- rana fermions are charge-neutral particles acting as their own antiparticles, they are difficult to interact directly to electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Materials exhibiting Kitaev in- teractions, called Kitaev candidate materials have been found, including A2IrO3 (A = Na, Ir)2–10, α-RuCl310–18, and H3LiIr2O619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The behavior caused by Majorana fermions in these materials has been studied using various methods20–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In their results, half-integer thermal quan- tum Hall effect can be a conclusive evidence for the emer- gent Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This phenomenon has been first pointed out theoretically1 and then observed experimen- tally in α-RuCl325,26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Strain fields can induce an artificial vector potential, which has opposite signs at two Dirac points due to time- reversal symmetry27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Experiments on artificial graphene have revealed a strong pseudomagnetic field in the range of 10 T–100 T and the presence of Landau levels28–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The strain-induced pseudomagnetic field can interact di- rectly with itinerant Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Indeed, numer- ical calculations have shown the emergence of Landau levels of itinerant Majorana fermions in the isotropically interacting Kitaev model under triaxial strain33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, the phenomena related to the strain-induced Landau lev- els in the Kitaev candidate materials can be a hallmark of itinerant Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' According to the ab-initio calculations for the Kitaev candidate materials, the Kitaev interactions include a spatial anisotropy6,7,15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' So far, the Landau levels of itinerant Majorana fermions and the related phenom- ena have been investigated for the strained Kitaev model with isotropic interactions33,34, while whether these Lan- dau levels could emerge in the anisotropically interact- ing strained Kitaev model is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Here, in the present study, we explore the low-energy properties of the anisotropically interacting Kitaev model on a hon- eycomb lattice under triaxial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We focus on the parameter region where the itinerant Majorana fermions exhibit a gapless dispersion relation in the absence of a strain field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Through a numerical calculation, we demon- strate that the strain-induced Landau levels of Majorana fermions emerge also in the anisotropically interacting Kitaev system, which is confirmed also by an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Sec- tion II outlines the deformation of the Kitaev model for the numerical calculation using a singular-value decom- position method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We then determine the Z2 gauge-flux sector of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Section III presents the nu- merical results for the local density of states (LDOS) of the itinerant Majorana fermions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' we show the presence of the Landau levels typical of massless Dirac fermions with time-reversal symmetry in the low-energy region of the considered model, indicating the emergence of the strain-induced Landau levels of Majorana fermions in the anisotropically interacting Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Section IV dis- cusses the low-energy states of the system based on the analytical calculation, illustrating results consistent with the numerical outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Finally, the study is summa- rized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' MODEL AND METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Formulation for numerical calculations The Hamiltonian is described by H = − � ⟨jk⟩x Jx jkσx j σx k − � ⟨jk⟩y Jy jkσy j σy k − � ⟨jk⟩z Jz jkσz j σz k, (1) where σα j (α = x, y, z) is an α component of the Pauli matrix at the j site and Jα jk is the coupling constant be- tween the nearest-neighbor atoms on the α bond in the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='05330v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='str-el] 12 Jan 2023 2 C C C rz ry rx Jz Jy Jx A B R = 3 R = 2 R = 1 Jz Jy Jx (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1: (Color online) (a) Unstrained honeycomb flakes ex- pressed by R: R = 1 is a central hexagon (a cross denotes its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' ), R = 2 consists of a central hexagon and six surround- ing hexagons, R = 3 consists of the R = 2 and twelve sur- rounding hexagons, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, R honeycomb flake in- cludes 2N = 6R2 spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The A and B sublattices are shown in black and white, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b) Central hexagon of the un- strained honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants, Jx, Jy, and Jz, on the X, Y , and Z bonds are represented in blue, red, and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The vectors connect correspondingly the nearest-neighbor sites along these bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Triaxial strain C is represented schematically using three brown arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We use a zigzag-terminated honey- comb lattice with an open boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The size of the honeycomb flakes is expressed by R [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(a)]33, and it includes 2N = 6R2 spins, where N is the number of the unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The triaxial strain originates at the center of the central hexagon marked by an cross in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the unstrained honeycomb lattice, the coupling con- stants are independent of the site: Jα jk = Jα(> 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When a weak triaxial strain is applied as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(b), the coupling constant Jα jk becomes33,35–37 Jα jk ≈ Jα [1 − β (1 − |rj − rk|/a0)], where β is the mag- netoelastic coupling and a0 is the unstrained bond length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The position vector of an atom is given by rj = r0 j + uj, where r0 j = (x0 j, y0 j ) is the position vector in the un- strained lattice and uj is the displacement vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' they are expressed respectively as r0 j = |r0 j |(cos θ0 j, sin θ0 j) and uj = (C/a0) |r0 j |2(cos 3θ0 j, sin 3θ0 j) using the polar coordi- nate, where C is the triaxial strain strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jα jk must be positive on the whole nearest-neighbor bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' According to our numerical calculation, this condition is satisfied for CR ⪅ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We thus set CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2 in the following nu- merical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the honeycomb flakes possessing the same constant CR, a scaling holds concerning the honeycomb flake shapes for different R values38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To diagonalize the Hamiltonian, four Majorana fermions, cj and bα j , are set at each site1, satisfying {cj, ck} = 2δjk, {cj, bα k} = 0, and {bα j , bβ k} = 2δαβδjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To project the enlarged Hilbert space into the physi- cal Hilbert space, the constraint cjbx j by j bz j = 1 is im- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In this procedure, the spin operator is repre- sented as σα j = icjbα j and the Hamiltonian reads as Hu = i � α∈{x,y,z} � ⟨jk⟩α Jα jkuα jkcjck, where uα jk = ibα j bα k is a bond operator with an eigenvalue of ±1 and satisfies [Hu, uα jk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, uα jk is identified with a static Z2 gauge field between the nearest-neighbor j and k sites on the α bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We then introduce a relevant gauge-flux op- erator defined as a product of the six Z2 gauge fields sur- rounding a hexagon1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The gauge-flux operator commutes with Hu and its eigenvalue becomes ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefor, the system can be mapped to itinerant Majorana fermions coupled with the Z2 gauge fluxes on the hexagonal pla- quettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' For every configurations of the Z2 gauge fluxes, the Hamiltonian Hu can be expressed as33 Hu = i 2 � ¯cT A ¯cT B � � 0 M −M T 0 � � ¯cA ¯cB � , (2) where Mjk = Jα jkuα jk and ¯cA(B) is an N-component vec- tor representing the itinerant Majorana fermions on the A(B) sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We call the Z2 gauge-flux having −1 ‘flux’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When at least two of the three coupling constants are equal in the unstrained system, the Lieb’s theorem39 states that the exact ground state is in the sector where all the Z2 gauge fluxes take unity (the flux-free sector)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The sector where the n gauge fluxes become −1 is called the n-flux sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' By using a singular-value decomposition method, we calculate the eigenvalues ϵm,n (m = 1, 2, · · · , N) and the eigenvectors for a given n-flux configuration n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' then we obtain the LDOS, ρj,A(B)(E), of the itinerant Majorana fermions on the A(B) sublattice in the j-th unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The magnetoelastic coupling is set as β = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants in the unstrained lattice satisfy Jx + Jy + Jz = 1, forming the triangle in the parameter space expressed by Jx, Jy, and Jz [left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a)]1, while the central downward triangle enlarged in the right panel represents the gapless phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' One-flux gap and ground-state sector In the strained honeycomb lattice, the translational invariance is broken, and hence the Lieb’s theorem can- not be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, we must confirm whether the ground state is in the flux-free sector for the given R with CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The ground-state energy for a n-flux configuration, n, is given by EGS,n = − � m ϵm,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the open boundary system, the one-flux state is possible and can be a candidate competing with the flux-free state38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We calculate the one-flux gap ∆1 = EGS,1 − EGS,0 for all the one-flux configurations at various R up to 90 for the given Jx, Jy, and Jz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 2 depicts the typical behavior of the minimum one-flux gap ∆min 1 for a given R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' For R ≥ 35, ∆min 1 is well described by the follow- ing polynomial: ∆min 1 = aR−4 + bR−2 + c, where a, b, and c are the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The extrapolated c values for R → ∞ are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='61 × 10−3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='44 × 10−4, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='73 × 10−4 in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 2(a)-2(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We perform the same cal- culations for the given coupling constants used in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(b)-3(g), finding that all the extrapolated c values for R → ∞ positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, we can deduce that the ground state of the anisotropically interacting Kitaev model for CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2 is in the flux-free sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the following nu- 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='010 ∆min 1 (a) Jx = 1/3, Jy = 1/3, Jz = 1/3 ∆min 1 |R→∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='61 × 10−3 R ≥ 35 R < 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0020 ∆min 1 (b) Jx = 7/24, Jy = 7/24, Jz = 5/12 ∆min 1 |R→∞ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='44 × 10−4 R ≥ 35 R < 35 0 1 202 1 252 1 302 1 352 1 502 1 802 1/R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00100 ∆min 1 (c) Jx = 11/48, Jy = 17/48, Jz = 5/12 ∆min 1 |R→∞ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='73 × 10−4 R ≥ 35 R < 35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 2: (Color online) (a) Minimum one-flux gap (∆min 1 ) for a given R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The dashed lines reprsent the polynomial, ∆min 1 = aR−4 + bR−2 + c, that well describes ∆min 1 for R ≥ 35 for the constants a, b, and c having the following values: (a) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='95 × 103, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='04 × 101, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='61 × 10−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='56 × 101, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='81 × 10−1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='44 × 10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (c) −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='77 × 101, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='42 × 10−1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='73 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' merical calculations, we set R = 60 (2N = 21600) and C = 1/300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' NUMERICAL RESULTS Figures 3(b)-3(g) illustrate the LDOS, ρj,A(E) and ρj,B(E), at the site in the central hexagon of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The left and right panels show the results for the A and B sublattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We also evaluate the integral value: Ij,A(B)(E) = � E 0 ρj,A(B)(E′)dE′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 3(b) displays the results for the isotropic interactions that are plotted using the small open circle in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants of the top, middle, and bottom panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(c)-3(g) correspond to the black dots from close to the center toward the edge along the lines A-E [right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We find that Ij,A(B)(E) forms plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We mea- sure them in units of the lowest-energy plateau in the A sublattice, finding that the pronounced plateaus are described as 2n + 1 (n = 0, 1, 2, · · · ) on the A sublattice and 2n (n = 0, 1, 2, · · · ) on the B sublattice from the low energy in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' At or in the vicinity of the bound- ary between pronounced neighboring plateaus, ρj,A/B(E) reaches a peak, as indicated by the vertical dashed lines in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(b)-3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The peak at E = 0 generally ap- pears only on the A sublattice, and it is called the sub- lattice polarization40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We then plot the peak energies, En (n = 0, 1, 2, · · · ), on the A sublattice (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 4), whose coupling constants correspond to the black dots along the lines A-E [right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 4, En satisfies the relation En ∝ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' These results are char- acteristic of the Landau levels of massless Dirac fermions with time-reversal symmetry29,36,37,41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' thus, the itinerant Majorana fermions under triaxial strain are quantized to the Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 3(c)-3(g) indicate that as the system leaves the isotropically interacting point, the Landau levels of Ma- jorana fermions are smeared at the higher energies and their number is reduces at the lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Within the shaded areas on the lines A-E in the right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a), at least three Landau levels of Majorana fermions from n = 0 appear on the A sublattice, confirming the relation En ∝ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' From the permutation of Jx, Jy, and Jz, there are six equivalent regions in the phase diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We apply our results to the five ad- ditional regions and summarize the results in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the shaded area, the Landau lev- els of Majorana fermions emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the outer unshaded area, instead, one or two peaks appear at and next to E = 0 in ρj,A(E), and the sublattice polarization is sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We perform the same calculations for R = 40, 45, and 50 while keeping CR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' As R increases, the region where the Landau levels of Majorana fermions ap- pear expands toward the boundary between gapless and gapped phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We, therefore, expect the Landau levels of Majorana fermions to emerge in the whole unstrained gapless phase, when the system becomes large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' EFFECTIVE LOW-ENERGY THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Formulation, eigenenergy, and sublattice polarization We now discuss the low-energy states of the itiner- ant Majorana fermions on the triaxially strained honey- comb lattice through an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='40,42, we adopt the effects of weak triaxial strain as Jα(r) = Jα � 1 + τ � r · rα/3a02�� , where rz = a0(0, 1), rx = a0(− √ 3/2, −1/2), and ry = a0( √ 3/2, −1/2) are vectors that connect the unstrained nearest-neighbor sites, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1(b), and τ controls the strain strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Also in the effective low-energy theory, the cou- pling constants in the unstrained lattice satisfy Jx+Jy + Jz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The strain effect is considered around the Dirac points, K and K′, of the isotropically interacting system in the unstrained honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefore, this for- mulation is effective for a system with weakly anisotropic 4 A B C D E (a) Phase diagram (c) A (d) B (e) C (f) D (g) E (b) Isotropic point FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3: (Color online) (a) Phase diagram of the unstrained Kitaev model on the plane Jx +Jy +Jz = 1, where Jα (α = x, y, z) are the coupling constants of the unstrained system with Jα ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the left panel, the inner triangle represents the gapless phase and the three outer triangles are gapped phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' A gapless phase is enlarged in the right panel, where over three Landau levels of Majorana fermions appear in the inner shaded region and the sublattice polarization is satisfied for R = 60 and C = 1/300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In the outer unshaded area, one or two peaks appear at and next to E = 0 in ρj,A(E), and the sublattice polarization is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b)-(g) Local density of states, ρj,A/B(E), at the central hexagon of the system and the integral values, Ij,A/B(E) = � E 0 ρj,A/B(E′)dE′, for R = 60 and C = 1/300;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (b) ρj,A/B(E) and Ij,A/B(E) for the isotropic interactions plotted using the small open circle in the right panel of (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (c)-(g) the coupling constants of the top, middle, and bottom panels correspond to the black dots from close to the center toward the edge along the lines A-E [right panel of (a)], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kitaev interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The pseudovector potential, Aξ = (ξAξ x, ξAξ y), induced by triaxial strain is given as Aξ x = vξ x −1 �� Jz − 1 2Jx − 1 2Jy � + (Jx − Jy) xτ 4 √ 3a0 + � Jz + 1 4Jx + 1 4Jy � yτ 3a0 � , (3) 5 (a) A (b) B (c) C (d) D (e) E FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 4: (Color online) Peak energies, En (n = 0, 1, 2, · · · ), of ρj,A(E) as functions of √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants in (a)-(e) correspond to those in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(c)-3(g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Aξ y = vξ y −1 √ 3 2 � (Jx − Jy) − (Jx + Jy) xτ 2 √ 3a0 − (Jx − Jy) yτ 6a0 � , (4) where ξ = ±1 for K/K′ and vξ x = √ 3a0 4ℏ �√ 3 (Jx + Jy) + iξ (Jx − Jy) � ≡ |vx|eiξφx, vξ y = √ 3a0 4ℏ � 1 √ 3 (3Jz + 1) − iξ (Jx − Jy) � ≡ |vy|eiξφy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The pseudomagnetic field, Bξ = (0, 0, ξBξ z), is given as Bξ z = cos φx∂xAξ y−cos φy∂yAξ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The Hamiltonian around K and K′ reads Hξ = ξ � |vxvy| � 0 Πξ x ∗ − iΠξ y ∗ Πξ x + iΠξ y 0 � , (5) where Πξ x = � |vx/vy| � eiξφxpx + ξAξ x � and Πξ y = � |vy/vx| � eiξφypy + ξAξ y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' By defining the annihilation operators as aK = lB √ 2ℏ � ΠK x ∗ − iΠK y ∗� , aK′ = lB √ 2ℏ � ΠK′ x + iΠK′ y � (6) with lB 2 = ℏ/|Bξ z|, the eigenenergy is obtained as En = � 2 √ 2ℏ/lB � � |vxvy|√n, (n = 0, 1, 2, · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The n = 0 eigenenstates are |Ψ+ 0 ⟩ = (0, |ψ0⟩)T for K and |Ψ− 0 ⟩ = (|ψ0⟩, 0)T for K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Since the components of the two-dimensional spinor in the sublattice basis are ex- changed in K and K′, the eigenstates at n = 0 are nonzero only on the A sublattice (sublattice polariza- tion), while they are nonzero on both the A and B sub- lattices at n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' These eigenenergy and eigenstate fea- tures agree with the numerical results, providing the evi- dence of the emergence of the Landau levels of Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Numerical results in terms of the effective low-energy theory FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 5: (Color online) Peak energy, E1, of ρj,A(E) as a func- tion of √ C at the isotropically interacting system obtained in the numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' R =40, 50, and 60 are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Corre- sponding C is obtained for the fixed CR =1/25, 2/25, 3/25, 4/25, 5/25, 6/25, and 7/25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Let us discuss the numerical results in terms of the effective low-energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' As the system leaves the isotropically interacting point in the numerical calcula- tion, the Landau levels of Majorana fermions are smeared at the higher energies and their number is reduced at the lower energies, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(c)-3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When the anisotropy of the interactions becomes strong, the Dirac points deviate considerably from those of the isotropi- cally interacting unstrained system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This situation is op- 6 TABLE I: Parameters a and b to fit the coefficient (E1) of En ∝ √n using a linear regression, E1 = a √ 8C +b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coefficients (E1) are obtained by the numerical calculation for R =40, 50, and 60 at the fixed CR =1/25, 2/25, 3/25, 4/25, 5/25, 6/25, and 7/25, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coupling constants are for the isotropically interacting system and for the system closest to the isotropically interacting point on each line A-E in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The third line (below the results for a) denotes the ratios of a to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 at the isotropic point are denoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (Jx, Jy, Jz) ( 1 3, 1 3, 1 3) A ( 31 96, 31 96, 17 48) B ( 121 384, 127 384, 17 48) C ( 59 192, 65 192, 17 48) D ( 115 384, 133 384, 17 48) E ( 7 24, 17 48, 17 48) a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99966 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00420 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00720 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00989 Ration of a to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00047 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00769 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='01039 b 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='03667 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='48732 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='51854 × 10−5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='13822 × 10−4 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='83096 × 10−4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='69701 × 10−4 posed to the condition for the effective low-energy the- ory to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Thus, as the system leaves the isotropically interacting point, the higher-order terms of Aξ become relevant, reducing the number of the Landau levels of Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We next investigate the relation between the control parameters of the strain strength, C and τ, in the numer- ical and analytical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To this end, we evaluate the coefficient of En ∝ √n obtained in the numerical cal- culation for R =40, 50, and 60 at the seven values of the fixed CR within CR < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Figure 5 illustrates the coef- ficient (E1) as a function of √ C in the isotropically inter- acting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The data follows the line E1 = a √ 8C + b with a ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='99951 and b ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='03667×10−5, indicating that the relation, En = √ 8Cn, is deduced within our numer- ical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coefficient E1 in the system closest to the isotropically interacting point on each line A-E is evaluated in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The evaluated a and b are summarized in Table I, indicating that a ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='0 and b ≈ 0, and thus En ≈ √ 8Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The coefficient of En ∝ √n in the effective low-energy theory, � 2 √ 2ℏ/lB � � |vxvy|, is given by using τ and one of the three coupling constants, leading to the expression for En as En = � (Jz − 1)2 + � Jz + 1 3 �2� 1 2 � 3τn 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (7) The eigenenergy in the isotropically interacting system is obtained as En = 2 � τn/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Comparing En in the numerical and analytical calculations for the isotropi- cally interacting system, the relation, τ = 6C, is de- rived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This relation is consistent with that derived by comparing the pseudomagnetic field in the numerical43 and analytical42 calculations for the isotropically inter- acting system, where |Bξ z| = 4Cℏ/a02 and 2ℏτ/(3a02), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Since Jz’s in the anisotropically interacting system in Table I are the same, their coefficients of √τ take the same value, � 1025/768, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The ratio of this coefficient to that in the isotropically interacting system is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='00049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This ratio is close to that denoted in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Therefore, the relation τ = 6C is approximately satisfied in the anisotropically interacting systems denoted in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' SUMMARY We have investigated the low-energy states of an anisotropically interacting Kitaev model under triaxial strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Based on the numerical results, we argue that the Landau levels of Majorana fermions emerge in the wide gapless region around the isotropically interacting point in the phase diagram of the unstraind system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The emer- gence of the strain-induced Landau levels of Majorana fermions in the anisotropically interacting Kitaev system is confirmed also by an analytical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' When uni- axial strain is applied, the same features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=', En ∝ √n and sublattice polarization) are expected to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' To observe the features of the strain-induced Landau lev- els of Majorana fermions, scanning tunneling microscopy (STM) is promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' According to the STM theory for the Kitaev model, the differential conductance through a single site provides direct information on the LDOS of Majorana fermions at low temperatures44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' In fact, the two features of the strain-induced Landau levels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=', En ∝ √n and sublattice polarization) have been observed in artificial graphene under simulated triaxial strain eval- uated pseudomagnetic fields up to 60 T29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' The fabrica- tion of the α-RuCl3 thin films has been studied actively in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This situation favors experiments for in- vestigating whether the strain-induced Landau levels of Majorana fermions emerge in Kitaev candidate materials such as α-RuCl3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' We hope that our results contribute to such studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Acknowledgments We would like to thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Suzuki and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Taniguchi for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' This work was supported by JSPS KAKENHI (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 19K03721) from MEXT, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kitaev, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 321, 2 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Chaloupka, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Chaloupka, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jackeli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Khaliullin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kim, Na- ture Physics 11, 462 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Dinnebier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jackeli, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Takagi, Nature 554, 341 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Hermanns, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Kimchi, and J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Knolle and and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Moessner, Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Mat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 89, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 24 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Trebst and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Hickey, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Katsnelson, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Guinea, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 496, 109 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 28 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Levy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Szameit, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Photonics 7, 153 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 31 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Power, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jauho, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 035456 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 38 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 73, 2158 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 40 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Poli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Arkinstall, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Schomerus, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 90, 155418 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 41 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Venderbos and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Fu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 195126 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 42 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Salerno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Ozawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Price, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Carusotto, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 95, 245418 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 43 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Settnes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Power, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Jauho, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' B 93, 035456 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 44 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Udagawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Takayoshi, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Oka, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'} +page_content=' 126, 127201 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE4T4oBgHgl3EQf6Q5X/content/2301.05330v1.pdf'}