Topological magnons in ferromagnetic Kitaev-Heisenberg model on CaVO lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Multiple topological phases of Kitaev-Heisenberg model on square-octagon lattice
Moumita Deb ∗ and Asim Kumar Ghosh † Department of Physics, Jadavpur University, 188 Raja Subodh Chandra Mallik Road, Kolkata 700032, India
A number of first order topological phases are found to emerge in the ferromagnetic Kitaev-Heisenberg model on square-octagon lattice in the presence of Dzyaloshinskii-Moriya interaction.Heisenberg and Kitaev terms have been considered on nearest and next-nearest neighbor bonds in avariety of ways. Both isotropic and anisotropic couplings are taken into account. Topological phasesare characterized by Chern numbers for the distinct magnon bands as well as the number of modesfor topologically protected gapless magnon edge states. Band structure, dispersion relation alongthe high-symmetric points of first Brillouin zone, density of states and thermal Hall conductancehave been evaluated for every phase. Phase diagrams have been constructed. Topological phasetransition is also noted in the parameter space.
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I. INTRODUCTION
Study of topological properties on condensed mattersystems hails a new era in theoretical as well as exper-imental investigations. Chern number ( C ) is recognizedas the topological invariant for the characterization oftopological phase of that particular class of topologicalinsulators (TI), where time-reversal symmetry (TRS) isbroken . The relation between C and number of topolog-ically protected modes of edge state is governed by ‘bulk-edge-correspondence’ rule . C of a definite band is deter-mined by integrating the Berry curvature over the firstBrillouin zone (1BZ). Nontrivial topological phase corre-sponds to that state where the system exhibits at leasta pair of nonzero values of C . A variety of many particleinteractions are found responsible behind the emergenceof non-trivial topological phases. Several kinds of inter-actions deform the Berry curvatures in so many differentways that they eventually lead to numerous topologicalphases. Performance of few potential two-spin interac-tions will be discussed here, those are found crucial toinduce non-triviality in magnetic systems.The first magnetic material who demonstrates non-triviality is Lu V O . The experimental results areexplained in terms of a spin-1/2 ferromagnetic (FM)Heisenberg model on a kagom´e lattice in the pres-ence of antisymmetric Dzyaloshinskii-Moriya interaction(DMI) . Two distinct topological phases character-ized by C =(10¯1) and (¯101) appear around the zero DMIstrength, where ¯ x means − x . Emergence of those phasescan be regarded as the handiwork of DMI term . Thosestates are observed again in another kagom´e ferromagnet,Cu[1,3-benzenedicarboxylate (bdc)] . This kind of stateis now termed as topological magnon insulating (TMI)phase. In both cases, TMI phase is observed in the pres-ence of an external magnetic field which acts as a TRSbreaking component in the system.In another development, FM Heisenberg models con-sisting of both nearest-neighbor (NN) Kitaev and sym-metric spin-anisotropic interactions (SAI) are found toexhibit TMI phases on the honeycomb lattice in the pres-ence of magnetic field. This two-band system hosts the TMI phase with C =(1¯1), when both the two-spin inter-acting terms are present . This model is extended be-yond NN interactions by including third-neighbor Kitaevand SAI terms which is found to host multiple novel TMIphases with higher Chern numbers . DMI term failsto induce non-triviality in this honeycomb model. Thussearch for new combinations of two-spin terms continuesthose are capable to induce non-triviality in other latticegeometries.In this article, emergence of multiple TMI phases willbe reported in a four-band FM Heisenberg model formu-lated on a square-octagon lattice which includes both NNand next-nearest-neighbor (NNN) Heisenberg and Kitaevterms along with NNN DMI term. NN DMI term has noeffect on the topological phases, while presence of NNNDMI turnes out to be indispensable for the emergence oftopological phases. Again, DMI only on the NN bondsalone fails to drive the system into the topologically non-trivial regime. Remarkably, SAI term has no role in thissystem while magnetic field is assumed to break the TRS.Emergence of photo-induced multiple topological phasesis reported before in a tight-binding model on square-octagon lattice . Topological phases in the presence ofspin-orbit coupling and magnetic field are also investi-gated on this lattice .Square-octagon lattice has been brought to light beforein the context of antiferromagnetic (AFM) compound,CaV O . The spin-1/2 V ions in this spin-liquid con-stitute square-octagon lattice structure . Coordinationnumber for both honeycomb and square-octagon latticesis three while their symmetries are different. For ex-ample, honeycomb (square-octagon) lattice has the six(four) -fold rotational symmetry, C ( C ). Primitive cellcontains two and four lattice points for honeycomb andsquare-octagon lattices, respectively. FM Kitaev mod-els with anisotropic NN bond strengths based on thosenon-Bravais lattices have been solved exactly in termsof Majorana fermions . The gapless phases for boththe lattices become topologically nontrivial as soon asthe magnetic field is switched on. The topological phaseof both honeycomb and square-octagon Kitaev modelsis unique in a sense that the resulting two-band systemcarries C =(1¯1) in the Majorana fermion representation.In contrast, no topological phase based on the bosonicmagnon excitation emerges in the NN Kitav models onhoneycomb and square-octagon lattices in the presenceof magnetic field.The Hamiltonian based on this model is formulated inSec II. The linear spin-wave theory (LSWT) is developedin Sec III. Methods adopted for the numerical evaluationof ( C ), number of edge state and thermal Hall conduc-tance (THC) are described in Sec IV. THC experiencessudden jump in the vicinity of phase transition points.The system hosts one short of one and a half dozen ofdistinct topological phases in total. They are presentedin Sec V. Half dozen of them appears when isotropic Ki-taev interaction is assumed while a dozen appears foranisotropic case. Only one phase is found common inboth the cases. Topological phase diagrams have beenproduced in various forms. A discussion based on thoseresults is presented in Sec VI. II. KITAEV-HEISENBERG MODEL WITHDZYALOSHINSKII-MORIYA INTERACTIONSON THE SQUARE-OCTAGON LATTICE
In order to develop the spin wave theory, a general formof spin Hamiltonian is considered which contains threespecific two-spin interacting terms, say, Kitaev, Heisen-berg and DMI in the presence of an external magneticfield. Both Kitaev and Heisenberg terms are present onthe NN and NNN bonds of the square-octagon latticewhile DMI is present on a pair of opposite NNN bondswithin the octagon plaquette. The Hamiltonian (1) iswritten as H = J X h ij i S i · S j + 2 X h ij i γ K γ S γi S γj + J ′ X hh ij ii S i · S j + 2 X hh ij ii γ K ′ γ S γi S γj + D m · X hh ij ii S i × S j − h · X i S i . (1)Here, J ( J ′ ) and K γ ( K ′ γ ) are the Heisenberg and Ki-taev interaction strengths respectively for the NN (NNN)bonds. DMI strength is denoted by D m . Every site ofthe square-octagon lattice is connected with the othersby three NN bonds as well as three NNN bonds. In orderto assign the Kitaev interactions, three different com-ponents are represented by γ = x, y, z , both for NN andNNN bonds. h = gµ B H , where H is the strength of mag-netic field which is acting along the +ˆ z direction. S αi isthe α -th component of spin operator, S i , at the i -th site,where α = x, y, z . Summations over NN and NNN bondsare shown by the indices h·i and hh·ii , respectively. Pe-riodic boundary condition (PBC) is assumed along both x and y directions. Values of J and J ′ are always nega-tive when they are nonzero. Both positive and negativevalues of K and K ′ are considered, but magnitudes of J and J ′ are always greater than those of K and K ′ in theisotropic case to ensure the FM ground state. J and J ′ are assumed zero in the anisotropic case without any lossof generality.Schematic view of this spin model is given in Fig 1 (a),where NN (NNN) bonds are indicated by solid (dashed)lines. δ and δ are the two primitive vectors to consti-tute the primitive cell (a square of arm length √ a ) ofthe square-octagon lattice. The primitive cell representedby the square encloses four nonequivalent sites A , B , C and D , shown by red, blue, yellow and green spheres,respectively. Obviously, the square-octagon lattice canbe decomposed in terms of four interpenetrating squarelattices made of each for four sites A , B , C and D , sep-arately. Alternately, the resulting lattice can be thoughtof as a specific structure of 1/5-depleted square latticewhich preserves the four-fold rotational symmetry, C , ofthe square lattice itself.No topological phase will appear in this model if DMIis absent. Additionally, the direction of D m as well asthe combination of bonds on which DMI is acting arevery crucial for the emergence of topological phases. Forexample, DMI on every NN and NNN bond within thesquare plaquette does not lead to non-trivial topologicalphase by any means. It is found that only two specificcombinations comprising of two pairs of opposite NNNbonds within the octagon plaquette where the directionsof D m are also opposite to each other could lead to thenontriviality. One of such combination is shown in Fig 1(a), in which DMI is acting over AD and BC bonds butopposite in directions. Opposite arrowhead over AD and BC bonds imply the directions of D m , which are ± ˆ z ,for the respective bonds. DMI over AB and CD bondsmay form another potential combination if the directionsof that are chosen opposite to each other. However, thelater choice is not assumed here, since it fails to host newtopological phases anymore. III. SPIN WAVE ANALYSIS
The magnon dispersion relations based on the exactFM ground state are obtained by converting the spin op-erators in terms of bosonic creation ( b † ) and annihilation( b ) operators via the Holstein-Primakoff transformation, S zj = S − b † j b j , S + j ≃ √ S b j , S − j ≃ √ S b † j . (2)Hamiltonian (Eq.1) has been expressed in the momen-tum space by Fourier-transforming the operators in thatspace, b j = √ N P k b k e i k · R j , where N is the total num-ber of primitive cells. Thus, the Hamiltonian with respectto the ground state energy, E G = 2 N S [3( J + J ′ )+2( K z + K ′ z )] − N S , is H = S X k Ψ † k H k Ψ k , (3)where, Ψ † k = [ b † A, k , b † B, k b † C, k , b † D, k , b A, − k , b B, − k , b C, − k , b D, − k ].Now, b † A , b † B , b † C and b † D are the bosonic creation oper-ators on the sublattices A , B , C and D , respectively. PSfrag replacements (a) (c) (b)
A BCD δ δ a ˆ x ˆ y K x K y K z K ′ x K ′ y K ′ z D m N un i t s k x k y Γ XMFIG. 1: (a) Geometrical view of the model with NN and NNN interactions on the square-octagon lattice, (b) geometry ofthe lattice used for edge state calculation, upper and lower edges are drawn in blue and red colors, respectively, (c) the firstBrillouin zone showing the high-symmetry points, Γ, M and X.
Terms containing product of four bosonic operatorshave been neglected since they invoke inter-magnoninteractions. H k is a 8 × × X k and Y k as H k = (cid:20) X k Y k Y † k X T − k (cid:21) , (4)where, both X k and Y k are Hermitian. X k = a a , k a , k a , k a ∗ , k a a , k a , k a ∗ , k a ∗ , k a a , k a ∗ , k a ∗ , k a ∗ , k a ,Y k = b , k b , k b , k b , − k b , k b , k b , − k b , − k b , k b , − k b , − k b , − k . (5)Considering the spin polarization along the + z directioncomponents of X k and Y k are obtained below. a = − J − K z − J ′ − K ′ z + h/S,a , k = ( J + K y ) + ( J ′ + K ′ y ) e i k · δ ,a , k = ( J + K x ) e i k · δ + ( J ′ + K ′ x ) ,a , k = J + J ′ e − i k · δ − iD m e − i k · δ ,a , k = J + J ′ e − i k · δ + iD m e − i k · δ ,a , k = ( J + K x ) e − i k · δ + ( J ′ + K ′ x ) ,a , k = ( J + K y ) + ( J ′ + K ′ y ) e − i k · δ ,b , k = − K y − K ′ y e i k · δ ,b , k = K x e i k · δ + K ′ x ,b , k = b , k = 0 ,b , k = K x e − i k · δ + K ′ x ,b , k = − K y − K ′ y e − i k · δ , with , δ = a √ i, δ = a √ j. Here, a is the length of NN bond which is assumed to be1. Following the Bogoliubov diagonalization method ap-plicable for the bosonic operators, the non-Hermitian ma-trix, I B H k has been diagonalized, instead of H k , in orderto obtain the eigenenergies and eigenmodes where I B =diag[1 , , , , − , − , − , − I B H k are treated as the magnon excitation ener-gies of the system. Accuracy of the results increases withthe value of S . Topological phases have been obtainedin the regime where the real eigenenergies are available.Eigenenergies thus constitute the four-band magnon dis-persion relations. Band structure varies with the valueof magnetic field in such a fashion that no alteration ofthe topological phases is found. IV. CHERN NUMBER, EDGE STATES ANDTHERMAL HALL CONDUCTANCE
In order to characterize the first order topologicalphases of this four-bands system, the Chern number foreach distinct magnon band is calculated, when there re-mains a definite gap between adjacent bands. Chernnumber for a particular band i , C i , is obtained by in-tegrating the Berry curvature of that band, F i ( k ) overthe 1BZ. C i = 12 π x F i ( k ) d k , (6)where, F i ( k ) is expressed in terms of the correspond-ing Berry connection, A iµ ( k ) = h ψ i ( k ) | ∂ k µ | ψ i ( k ) i as F i ( k ) = ∂ k x A iy ( k ) − ∂ k y A ix ( k ), and | ψ i ( k ) i is the eigen-vector of the i -th magnon band. C i has been evaluatednumerically .According to the ‘bulk-edge-correspondence’ rule thepresence of nonzero value of C implies the existence of -1 0 1 -1 0 1 2 4 6 PSfrag replacements E kx ky (a) C C C C -1 0 1 -1 0 1 2 4 6 PSfrag replacements E kx ky (a) C C C C (b) C C C C -1 0 1 -1 0 1 2 4 PSfrag replacements E kx ky (a) C C C C E (c) C C C C -1 0 1 -1 0 1 3 6 9 PSfrag replacements E kx ky (a) C C C C E(c) C C C C (d) C C C C -1 0 1 -1 0 1 2 4 6 PSfrag replacements E kxky (a) C C C C E(c) C C C C E kx ky (e) C C C C -1 0 1 -1 0 1 2 4 6 PSfrag replacements E kxky (a) C C C C E(c) C C C C E kx ky (e) C C C C (f) C C C C FIG. 2: Three-dimensional magnon bands. a) J = − D m =0 . C = (11¯1¯1), (b) J = − K ′ = − . D m = 0 . C = (11¯20), (c) J = − K = 0 . D m = 0 . C = (1¯11¯1),(d) J = − J ′ = − . K ′ = − D m = 1 for C = (002¯2),(e) J ′ = − . K ′ = − . D m = 1 for C = (¯222¯2), and (f) J = − K = 0 . K ′ = − D m = 1 for C = (02¯20), with h = 1. Parameters of non-zero values are indicated here. edge states. Edge state corresponds to the surface prop-erty of the system. To calculate the bulk-edge energyspectrum a pair of edges parallel to the x -axis is createdhere by breaking the PBC along the y -axis. As a result,a strip of square-octagon lattice is constructed which has N primitive cells along the ˆ y and infinitely long towardsthe ˆ x . Fourier transform of the bosonic operators is takenonly along the x direction and 4 N × N Hamiltonian hasbeen obtained.THC, κ xy , of the system can be expressed in terms of F ( k ) for the system as , κ xy ( T ) = − k B T π ~ X i x c ( ρ i ( k )) F i ( k ) d k . (7)Here T is the temperature, k B is the Boltzmann con-stant and ~ is the reduced Planck’s constant. c ( x ) =(1 + x ) (cid:0) ln xx (cid:1) − (ln x ) − ( − x ), where Li ( z ) = DOS
PSfrag replacements E X ΓΓ k M (a) C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a) (b) C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a) (c) E C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a)(c) E (d) C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a)(c) E (e) E C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a)(c) E(e)E (f) C C C C C C FIG. 3: Dispersion relation along the high-symmetry pointsof Brillouin zone. The side panel shows the DOS. Values ofthe parameters are the same as Fig 2 for the respective plots. − R z du ln (1 − u ) u and ρ i ( k ) is the Bose-Einstein distribu-tion, i.e. , ρ i ( k ) = 1 / ( e E i k /k B T − κ xy ( T ) also gets saturated at hightemperatures. As κ xy ( T ) directly depends on the Berrycurvature so it behaves differently in different topologi-cal phases. As a result, κ xy suffers sudden change in itsvalue at the phase transition points. V. TOPOLOGICAL PHASES
In this study, isotropic Heisenberg interaction on theNN and NNN bonds is assumed, while both isotropicand anisotropic Kitaev couplings are taken into account.Anisotropic XXZ Heisenberg interaction on the NN andNNN bonds is not considered here because of the fol-lowing reason. Total Hamiltonian containing isotropicHeisenberg and DMI terms can be mapped on to theanisotropic XXZ Heisenberg Hamiltonian via a canonicaltransformation of the spin operators, which is valid forany values of S . It is true even for arbitrary directionsof D m over different bonds as long as they are parallelor antiparallel to each other . Which means that effectof DMI term on the Heisenberg model can be studied interms of a suitable XXZ Heisenberg Hamiltonian wherethe value of anisotropic parameter depends on the val-ues of exchange and DMI strengths. So, in other words,inclusion of anisotropic Heisenberg interaction could notlead to the emergence of new topological phases anymore.For example, the same set of two distinct topologi-cal phases with C =(10¯1) and (¯101), appear in two pre-viously studied models where isotropic and anisotropicXXZ Heisenberg Hamiltonians are formulated on kagom´elattice in the presence of DMI . A closer scrutiny onthose two models reveals that only the diagonal termsof those Hamiltonian matrices ( H ij ) are different, wherethe values of C s are insensitive to them. Off-diagonal ma-trix elements satisfy the relation, H ij ( − D m ) = H ∗ ij ( D m ),which on the other hand corresponds to the band inver-sion about D m = 0, in this particular case . And as aresult, C s of the two topological phases exhibit mirrorsymmetry around the middle band.Here, the system hosts seventeen distinct topologicalphases in total. Six are found for the isotropic Kitaevcoupling but twelve for the anisotropic case. All of themare described in the following two subsections. One phaseis found common in both the cases. Every topologicalphase is described by band structure, dispersion relationalong the high-symmetric points of first Brillouin zone,density of states (DOS) and thermal Hall conductance.Value of C for each distinct band has been evaluated inassociation with the bulk-edge energy spectrum for theHamiltonian formulated on the strip of square-octagonlattice of finite length along y -axis. π / √ π / √ DOS0.00.20.40.6
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a)(b)(c)(d)(e) site upper edgelower edge π / √ π / √ DOS0.00.20.40.6
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a) (b)(c)(d)(e) site upper edgelower edge π / √ π / √ DOS0.00.20.40.6
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a)(b) (c)(d)(e) site upper edgelower edge E | ψ | π / √ π / √ DOS0.00.20.40.6
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a)(b)(c) (d)(e) site upper edgelower edge E | ψ | π / √ π / √ DOS0.00.20.4
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a)(b)(c)(d) (e) site upper edgelower edge E | ψ | | ψ | π / √ π / √ DOS0.00.20.4
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a)(b)(c)(d)(e) site upper edgelower edge E | ψ | | ψ | (f) FIG. 4: Magnon dispersions of bulk-edge states in the one-dimensional BZ. Upper and lower edge modes are drawn inblue and red lines, respectively, while bulk modes are ingolden points. The side panel shows the DOS. The lowerpanel indicates variation of probability density of both edgemodes with respect to site number for a fixed k x . Values ofthe parameters are: a) J = − D m = 0 . C = (11¯1¯1), (b) J = − K ′ = − . D m = 0 . C = (11¯20), (c) J = − K = 0 . D m = 0 . C = (1¯11¯1), (d) J = − J ′ = − . K ′ = − D m = 1 for C = (002¯2), (e) J ′ = − . K ′ = − . D m = 1 for C = (¯222¯2), and (f) J = − K = 0 . K ′ = − D m = 1 for C = (02¯20), with h = 1. Parameters with onlynonzero values are mentioned here. A. Isotropic Kitaev coupling
In this case, the same value of Kitaev interaction alongthree different links of the lattice is considered, whichmeans, K γ = K and K ′ γ = K ′ . Similar model on thetwo-band honeycomb lattice exhibits multiple TMI phasewith higher values of C s, when third neighbor interactionsare invoked . In this study, isotropic Kitaev model onthe square-octagon lattice is found to exhibit six topolog-ical phases. Bulk energy dispersion with specific valuesof C have been plotted in Fig 2 for different TMI phases.Dispersion relations along the high-symmetry points of1BZ, in addition to the DOS are shown in Fig 3. DOSconfirms the existence of band gap in every case. Disper-sions of bulk-edge states in the one-dimensional BZ havebeen shown in Fig 4. Edge state dispersion branches forupper (blue) and lower (red) edges are indicated in dif-ferent colors. -2-1 0 1 2 -1.0-0.8-0.6-0.4-0.20.0 -1.0-0.8-0.6-0.4-0.2 PSfrag replacements (a) (b) K ′ K ′ C E E E E FIG. 5: Variation of Chern numbers with K ′ for (a) K = − . J ′ = −
1, and (b) K = 1, J ′ = − .
5. For each case J = − D m = 1 and h = 1. PSfrag replacements κ x y ~ / k B K ′ K ′ (a) (b) J = − J = − K = − . J ′ = − D m = 1 D m = 1 h = 1 h = 1 kBT = 20 kBT = 20 K = − J ′ = − . FIG. 6: Variation of κ xy ~ /k B in the parameter space when T is fixed. Different regions are identified with distinct colors. TMI phases as well as band gaps appear as soon asNNN DMI is switched on. Let us now describe the TMIphases shown in Fig 2. Two distinct topological phasesappear when all other NNN bond strengths are zero.These are C = (11¯1¯1) and C = (1¯11¯1), as shown in Fig2 (a) and (c), respectively, when J = − D m = 0 . J = − K = 0 . D m = 0 .
5, in two respective cases. C s are expressed following the ascending order of energyvalue in every case. The remaining four phases appearin the presence of other NNN bond strengths. Among −0.20.20.61.0 0 5 10 15 PSfrag replacements κ x y ~ / k B kBT (a)(b)(c)(d)(e)(f) FIG. 7: Variation of κ xy ( T ) with T for a) J = − D m = 0 . C = (11¯1¯1), (b) J = − K ′ = − . D m = 0 . C = (11¯20), (c) J = − K = 0 . D m = 0 . C = (1¯11¯1),(d) J = − J ′ = − . K ′ = − D m = 1 for C = (002¯2),(e) J ′ = − . K ′ = − . D m = 1 for C = (¯222¯2), and (f) J = − K = 0 . K ′ = − D m = 1 for C = (02¯20), with h = 1. No value is assigned to those parameters when theyare zero. them C = (¯222¯2) appears when all the NN interactionsare absent. Therefore, TMIs with C = (11¯20), C = (002¯2)and C = (02¯20) emerges when both NN and NNN termsare present. However, appearance of those phases is byno means fixed for those particular values of the param-eters. Those phases may appear for other combinationsof parameter with different values also. But, no addi-tional phase other that those six is by any means foundto appear.Gapless edge states are shown in Fig 4, where the num-ber of edge states are found to satisfy the ‘bulk-edge cor-respondence’ rule which states that sum of the Chernnumber upto the i -th band, ν i = P j i C j , is equal to thenumber of pair of edge states in the gap . Which meansthat the values of the Chern numbers can be derived,otherwise, from the edge state pattern itself.Topological phase transition (TPT) may be noted inthe parameter space upon changing the values of the pa-rameters. One such transition occurs when K ′ becomesnon-zero but J = − D m = 0 .
5. In this case, thesystem undergoes a transition from the state (11¯1¯1) toanother state (11¯20). Distribution of C s of those twophases around the transition point can be understood inthe following way. Gap between the upper two bandsvanishes at the transition point in the parameter spacedue to the presence of a Dirac cone at the band touchingpoint. When the gap reopens C s of the respective uppertwo bands change by ±
1, resulting in the redistributionof them. Occurrence of other TPTs may be explained insimilar fashion. For example, transition from (11¯1¯1) to(1¯11¯1) takes place by switching on the K . In this case,two intermediate bands touch in such a way that a Diraccone is formed at the band touching point.Another pair of TPT is shown in Fig 5, where the ap-pearance of topological phases is noted with the variationof K ′ . The energies, E , E , E and E are denoted ac-cording to the ascending order of their values. Fig 5 (a)shows that two nontrivial topological phases, (1¯11¯1) and -1 0 1 -1 0 1 4 6 8 PSfrag replacements E kx ky (a) C C C C -1 0 1 -1 0 1 6 8 PSfrag replacements
E kx ky (a) C C C C (b) C C C C -1 0 1 -1 0 1 4 6 8 PSfrag replacements
E kx ky (a) C C C C E (c) C C C C -1 0 1 -1 0 1 6 8 PSfrag replacements
E kx ky (a) C C C C (d) C C C C -1 0 1 -1 0 1 6 8 PSfrag replacements
Ekxky (a) C C C C E kxky (e) C C C C -1 0 1 -1 0 1 4 6 8 PSfrag replacements
E kx ky (a) C C C C (f) C C C C FIG. 8: Three-dimensional magnon bands. Values of the pa-rameters are: (a) K x = − . K y = − . K ′ x = − K ′ y = 0 . C = (01¯10), (b) K x = − . K y = − . K ′ x = − . K ′ y = 0 . C = (100¯1), (c) K x = − . K ′ x = − K ′ y = − . C = (2¯33¯2), (d) K x = − K y = − . K ′ x = − . K ′ y = − . C = (1¯22¯1), (e) K x = − . K y = − . K ′ x = − . K ′ y = − . C = (1¯11¯1), and(f) K x = − . K y = − . K ′ x = − . K ′ y = − . C = (2¯22¯2), with K z = − K ′ z = − D m = 1, h = 1. Novalue is assigned to those parameters when they are zero. (002¯2) appear around K ′ = − .
5. TMI with (11¯1¯1) isfound when K ′ < − .
16, as shown in Fig 5 (b). Uponincrease of K ′ , Chern numbers of the lower two bandsremain unchanged while those of upper two bands areexchanged by ± κ xy with respect to K ′ for four different TMI phases isshown in Fig 6 with different colors. Those are plottedwhen k B T = 20. Sudden jump in κ xy corresponds to thepoint where TPT occurs. Similarly, variation of κ xy withrespect to T is shown in Fig 7. DOS
PSfrag replacements E X ΓΓ k M (a) C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a) (b) C C C C DOS
PSfrag replacements E X ΓΓ k M (a) E (c) C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a) (d) C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a) E (e) C C C C C C DOS
PSfrag replacements E X ΓΓ k M (a) (f) C C C C C C FIG. 9: Dispersion relation along the high-symmetry pointsof Brillouin zone. The side panel shows the DOS. Values ofthe parameters are the same as Fig 8 for the respective plots.
B. Anisotropic Kitaev coupling
Anisotropic Kitaev coupling corresponds to K x = K y = K z as well as K ′ x = K ′ y = K ′ z . Kitaev modelwith anisotropic NN coupling, ( K x = K y = K z ), onboth the honeycomb and square-octagon lattices havebeen solved exactly . Both the systems host gaplessand gapped phases in the ground state phase diagram.Identically, both the systems exhibit a unique topologi-cal phase, C =(1¯1), in the presence of magnetic field, whenthey are studied in terms of Majorana fermion.However, in this study, a dozen of TMI phases is foundin the presence of anisotropic Kitaev couplings on the NNand NNN bonds together with NNN DMI and externalmagnetic field. Band structures for six different TMIphases, are shown in Fig 8. Dispersion relations alongthe paths in 1BZ in addition to the DOS are shown in Fig9, for six different cases. Dispersions of bulk-edge statesin the one-dimensional BZ have been shown in Fig 10. Acomprehensive topological phase diagram of the systemfor the anisotropic case is shown in Fig 11. Variation of κ xy with respect to K y and K ′ y for different TMI phasesis shown in Fig 12. Those are plotted for k B T = 20.Similarly, variation of κ xy with respect to T is shown inFig 13. All the figures are drawn for fixed values of theparameters, K z = − K ′ z = − D m = 1 and h = 1.The remaining six TMI phases are defined by simulta-neously reversing the sign of C for all four bands, whichare henceforth termed as the conjugate phases. Thoseconjugate phases are obtained by reversing the sign of ei-ther K ′ y alone or that of both K y and K ′ y simultaneouslyin the parameter space, but, without changing the signs π / √ π / √ DOS0.00.20.40.6
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a) site upper edgelower edge π / √ π / √ DOS0.00.20.40.60.8
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a) site upper edgelower edge (b) π / √ π / √ DOS0.00.20.40.6
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a) site upper edgelower edge (b) E | ψ | (c) π / √ π / √ DOS0.00.20.40.60.8
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a) site upper edgelower edge (b) (d) π / √ π / √ DOS0.00.20.40.6
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a) site upper edgelower edge (b) E | ψ | (e) π / √ π / √ DOS0.00.20.4
0 40 80 120 160 200 240 280 320 360 400
PSfrag replacements E k x | ψ | (a) site upper edgelower edge (b) (f) FIG. 10: Magnon dispersions of bulk-edge states in the one-dimensional BZ. Upper and lower edge modes are drawn inblue and red lines, respectively, while bulk modes are ingolden points. The side panel shows the density of edge states.The lower panel indicates variation of probability density ofboth edge modes with respect to site number for a fixed k x .Values of the parameters are: (a) K x = − . K y = − . K ′ x = − K ′ y = 0 . C = (01¯10), (b) K x = − . K y = − . K ′ x = − . K ′ y = 0 . C = (100¯1), (c) K x = − . K ′ x = − K ′ y = − . C = (2¯33¯2), (d) K x = − K y = − . K ′ x = − . K ′ y = − . C = (1¯22¯1),(e) K x = − . K y = − . K ′ x = − . K ′ y = − . C = (1¯11¯1), and (f) K x = − . K y = − . K ′ x = − . K ′ y = − . C = (2¯22¯2), with K z = − K ′ z = − D m = 1, h = 1. No value is assigned to those parameters when theyare zero. and values of remaining other parameters. The specificvalues of K y and K ′ y , in addition, will determine whichcriterion will be obeyed for the emergence of a particu-lar conjugate phase. However, no figure corresponding tothose conjugate phases is shown in this article.The band structure for TMI phase having C = (01¯10)is shown in Fig 10 (a), which is obtained for K x = − . K y = − . K ′ x = − K ′ y = 0 .
4. The conjugateTMI phase with C = (0¯110) appears if the value of K ′ y is changed to − .
4. System exhibits another TMI phasewith C = (2¯33¯2) when NN Kitaev interactions along the y bond is made zero, ( K y = 0), keeping other parametersunchanged. The corresponding band structure is shownin Fig 10 (c). The conjugate TMI with C = (¯23¯32) isfound just reversing the sign of K ′ y . TMI phase with C = (100¯1) emerges when K x = − . K y = − . K ′ x = − . K ′ y = 0 .
1, which is shown in Fig 10(b). But the conjugate TMI phase with C = (¯1001) ap-pears in this case if both K y and K ′ y reverse their sign.The band structure of the system obtained for K x = − K y = − . K ′ x = − . K ′ y = − .
3, is shown inFig 10 (d). This corresponds to the topological phasehaving C = (1¯22¯1). Conjugate of this TMI phase ap-pears if K ′ y picks up the reverse sign. At K x = − . K y = − . K ′ x = − .
1, TMI phase with C = (1¯11¯1)and its conjugate C = (¯11¯11) appear for K ′ y = − .
5, and K ′ y = 0 .
5, respectively. The former is shown in Fig 10(e). Finally, Fig 10 (f) corresponds to the band struc-ture of another TMI phase with C = (2¯22¯2). The cor-responding conjugate TMI phase, C = (¯22¯22), appearswhen signs of both K y and K ′ y are reversed. The TMIphase, C = (1¯11¯1), is found to appear in both the casesof isotropic and anisotropic Kitaev couplings. However,no conjugate phase is found in the isotropic case. -1.0-0.50.00.51.0-1.0 -0.5 0.0 0.5 1.0-1.0-0.50.00.51.0 PSfrag replacements (a) K y K ′ y K ′ y (01¯10) (0¯110)(2¯33¯2)(¯23¯32)(2¯22¯2) (¯22¯22)(b)(100¯1) (¯1001)(1¯22¯1)(¯12¯21)(1¯11¯1) (¯11¯11) (0000)(0000) FIG. 11: Regions of TMI phases of the system in K y - K ′ y parameter space for (a) K ′ x = − .
8, and (b) K ′ x = − .
6, with K x = − . K z = − K ′ z = − D m = 1, h = 1. Trivialregion is indicated by (0000). A comprehensive topological phase diagram of thesystem for the anisotropic case is shown in Fig 11 for K x = − .
7. Two diagrams (a) and (b) are drawn byvarying K y and K ′ y , respectively, where the value of K ′ x remains fixed at − . − . C s are zero. -0.29-0.28-0.27-0.260.23 0.26 0.29 0.32 0.260.270.280.29 -0.32-0.29-0.26-0.230.080.120.16-0.3 -0.2 -0.1 0.00.220.240.260.280.30-0.2 -0.1 0.0 0.1 -0.30-0.28-0.26-0.24-0.22-0.1 0.0 0.1 0.2-0.010.01-0.1 0.0 0.1 PSfrag replacements κ x y ~ / k B κ x y ~ / k B K y K y K ′ y K ′ y K ′ y K ′ y (a) (b) (c)(d) (e) (f) K ′ x = − . K ′ x = − . K ′ x = − . K ′ y = 0 . Ky = − . K ′ y = − . K ′ x = − . K ′ x = − . K ′ x = − . Ky = − . Ky = 0 Ky = 0 . FIG. 12: Variation of κ xy ~ /k B in the parameter space when K x = − . K z = − K ′ z = − D m = 1, h = 1, k B T = 20.Different regions are identified with distinct colors, those areused before in Fig 11. -0.10.30.7-0.7-0.30.1 0 5 10 15 PSfrag replacements (I)(II) κ x y ~ / k B κ x y ~ / k B kBT (a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l) FIG. 13: (I) Variation of κ xy ( T ) with T for (a) K x = − . K y = − . K ′ x = − K ′ y = 0 . C = (01¯10), (b) K x = − . K y = − . K ′ x = − . K ′ y = 0 . C = (100¯1),(c) K x = − . K ′ x = − K ′ y = − . C = (2¯33¯2), (d) K x = − K y = − . K ′ x = − . K ′ y = − . C = (1¯22¯1),(e) K x = − . K y = − . K ′ x = − . K ′ y = − . C = (1¯11¯1), and (f) K x = − . K y = − . K ′ x = − . K ′ y = − . C = (2¯22¯2), with K z = − K ′ z = − D m = 1, h = 1. (II) Variation of κ xy ( T ) for the conjugate phases. Novalue is assigned to those parameters when they are zero. Variation of κ xy ~ /k B in the parameter space is shownin Fig 12 when K x = − . K z = − K ′ z = − D m = 1, h = 1, and k B T = 20. Here, different regions are iden-tified with distinct colors, those are used before in Fig11. Sudden jump in κ xy is noted where TPT takes place.Variation of κ xy with respect to T for six different TMIphases along with their conjugate phases is shown in Fig13, with different colors. Signs of κ xy for a particularTMI phase and its conjugate are of opposite to eachother. This corresponds to the fact that signs of the C s ofa definite phase are opposite to those of the correspond-ing conjugate phase. Thus, κ xy for all the twelve distincttopological phases have been shown in this figure. VI. DISCUSSION
Topological properties based on the bosonic magnonexcitation of the FM Kitaev-Heisenberg model on square-octagon lattice in the presence of DMI have been inves-tigated extensively in this study. The model comprisesof Heisenberg and Kitaev terms both on NN and NNNbonds. Both isotropic as well as anisotropic couplingsare considered. A sizable number of TMI phase appearsupon variation of parameter values. Topological phaseshave been characterized in terms of Chern numbers whichare evaluated numerically. DMI on a particular combi-nation of NNN bonds is found crucial for the emergenceof the TMIs. On the other hand, NN DMI has no rolefor the same.It has been indicated in the previous studies thatFM Kitaev model with NN anisotropic couplings ex-hibits a unique topological phase based on the Majo-rana fermion representation both for the honeycomband square-octagon lattices in the presence of magneticfield . Interestingly, the situation is different in case ofFM Kitaev-Heisenberg system, however, when solved interms of bosonic magnon excitations. The present inves-tigation reveals that multiple topological phases arise inKitaev-Heisenberg model on square-octagon lattice whenboth NN and NNN terms are there. No topologicalphase is there for NN term alone. In addition, topo-logical properties of the Kitaev-Heisenberg model on the honeycomb lattice are drastically different from those ofsquare-octagon lattice perhaps because of their differentsymmetries.In case of honeycomb lattice, nontriviality in the two-band Kitaev-Heisenberg system is induced by the pres-ence of SAI term, Γ ( S αi S βj + S βi S αj ) . Conjugate topolog-ical phases appear upon sign reversal of Γ, the strengthof SAI. But no role of DMI is found there. On the otherhand, no effect of SAI term is found on the topologi-cal properties of four-band Kitaev-Heisenberg system forsquare-octagon lattice, where, NNN DMI term is foundindispensable. Conjugate phases appear when the signsof Kitaev terms, K y and K ′ y are reversed depending onthe situation.NNN Kitaev and Heisenberg terms could not lead tonew topological phase for the honeycomb lattice . Buta number of new topological phases emerge as soon asthe third neighbor Kitaev and Heisenberg terms are in-troduced. On the other hand, the system hosts multipletopological phases in the presence of NNN Kitaev andHeisenberg terms, in case of square-octagon lattice. Itis expected that numerous novel topological phases withhigher values of Chern numbers will come up if thirdneighbor Kitaev and Heisenberg terms are taken intoaccount. Therefore, the topological properties of FMKitaev-Heisenberg models on both square-octagon andhoneycomb lattices are different when studied in termsof bosonic magnon excitations in comparison to the ex-actly solvable FM Kitaev models on the same latticesbased on the Majorana representation. VII. ACKNOWLEDGMENTS
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