Mosaic spin models with topological order
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Mosaic spin models with topological order
S. Yang , D. L. Zhou , and C. P. Sun Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
We study a class of two-dimensional spin models with the Kitaev-type couplings in mosaic struc-ture lattices to implement topological orders. We show that they are exactly solvable by reducingthem to some free Majorana fermion models with gauge symmetries. The typical case with a 4-8-8close packing is investigated in detail to display the quantum phases with Abelian and non-Abeliananyons. Its topological properties characterized by Chern numbers are revealed through the edgemodes of its spectrum.
PACS numbers: 75.10.Jm, 05.30.Pr, 71.10.Pm
Introduction-
The phenomenon of emergence (such as aphase transition) in a condensed matter system is usuallyunderstood according to the Landau symmetry-breakingtheory (LSBT) . There also exists a new kind of or-der called “topological order” which cannot be de-scribed in the frame of the LSBT (e.g., fractional quan-tum Hall effect). The study of topological order in theo-retical and experimental aspects has been an active areaof research . Since local pertur-bations hardly destroy the topological properties, suchtopologically ordered states show exciting potential to en-code and process quantum information robustly . There-fore it is significant and challenging to find more exactlysolvable models showing various topological orders.In this Rapid Communication, the Kitaev’s honeycombmodel is generalized to the general mosaic spin modelswith different two-dimensional Bravais lattices of com-plex unit cells. Then we study the 4-8-8 case in detail toreveal the general and special properties of mosaic spinmodels.Our mosaic spin models are constructed with the basicblock shown in Fig. 1(a), which is a vertex with three dif-ferent types of spin couplings along x - (black solid link), y - (blue dotted link), and z - (red double link) directions,respectively. In spite of the lattice symmetry, numer-ous spin models can be built based on this basic block.However, taking translational symmetry and rotationalsymmetry as much as possible into account, we regardeach basic block as the common vertex of three isogonswith n , n and n edges, so there are only four kinds ofmosaic spin models illustrated in Fig. 1(b)-1(e), called n - n - n mosaic models.Obviously, the 6-6-6 mosaic model is just Kitaev’s hon-eycomb model . Here, we remark that for given n , n ,and n , there exist some unequivalent kinds of plane ar-rangement of x links, y links, and z links, but we onlyillustrate one of them in Fig. 1. The general Hamiltonianof all mosaic spin models reads as H = − X u = x,y,z J u X ( j,k ) ∈ S ( u ) σ uj σ uk , (1)where S ( u ) is the set of links with u -direction couplings. Perturbation theory study and abelian anyons -
First, (b) 6-6-6 (c) 3-12-12(e) 4-8-8 (d) 4-6-12 (a) basic (Kitaev) (f) block eeeem mm m mm mmm y xz (g) (h) FIG. 1: (Color online) (a) Basic block for mosaic spin models,which consists of three branches with x- (black solid link), y-(blue dotted link), and z- (red double link) type couplings.(b) 6-6-6 mosaic model, i.e., Kitaev’s honeycomb model. (c)3-12-12 mosaic model. (d) 4-6-12 mosaic model. (e) 4-8-8mosaic model, e vortices lie on squares while m vortices lie onoctagons. (f) and (g) The possible nonconstant terms of theeffective Hamiltonian are obtained by flipping four spin pairsaround a octagon (f) and a quatrefoil (g). (h) Kitaev’s toriccode model is the effective model of the 4-8-8 mosaic modelwhen | J z | ≫ | J x | , | J y | . we study the 4-8-8 mosaic model as a typical illustra-tion in detail. To see its topological properties, we firstanalyze its low energy excitations when the system is ini-tially spontaneously polarized with the strong Ising in-teraction H = − J z P zlinks σ zj σ zk . The ground energy of H is E = − N J z , where N is the number of z links.For larger J z in comparison with J x and J y , we regardthe transverse part V = H − H as a perturbation andthen prove that the obtained effective Hamiltonian H eff just describes Kitaev’s toric code model , which supportsmany topological issues of the original mosaic spin model.The ground eigenstates of H are highly degenerate,where each two spins connected by a z link can be ei-ther |↑↑i or |↓↓i . The fusion projection Υ † l can mapthe l th aligned spin pair | m, m i l to an effective spin | m i l ( m = ↑ or ↓ ), i.e., Υ † l | m, m i l = | m i l . We use thefusion projection and the Green function formalism tocalculate the effective Hamiltonian H eff = P ∞ l =0 H ( l ) eff = E + Υ † V [1 + G ( E ) + G ( E ) V G ( E )] V Υ + · · · where G ( E ) = ( E − H ) − . We first obtain the con-stant zeroth order one, the vanishing first order and thirdorder ones. Here, each terms σ xj σ xk or σ yj σ yk in V flips twospins, increasing the energy by 4 J z . Up to the second or-der perturbation, one V flips two spins and the other V flips them back, giving H (2) eff = − N ( J x + J y ) / (4 J z ) asa constant. As shown in Figs. 1(f) and 1(g), we taketwo σ xj σ xk and two σ yj σ yk from four V around one octagonor one quatrefoil in a particular order. Taking all the2 ×
4! = 48 possible cases into account, we obtain thefourth order effective Hamiltonian H (4) eff = − J x J y J z (5 X O σ yl σ yr σ yu σ yd + X Q σ zl σ zr σ zu σ zd ) , (2)where the constant term was dropped, O and Q repre-sent the octagon and quatrefoil in the two-dimensional(2D) lattice. Up to a unitary transformation for spinrotation σ y → σ z , σ z → σ x , σ x → σ y , the above Hamil-tonian represents the Kitaev’s toric code model . Thusthe above fusion projection constructs a new Bravais lat-tice illustrated in Fig. 1(h) with the effective spins layingon its links. Considering Kitaev model (2) possesses richtopological features characterized by m and e anyons,we conclude that m particles live on octagons while e particles live on squares in our model with original spinrepresentation. Majorana fermion mapping with Z -gauge symmetry- The 4-8-8 mosaic model consists of four equivalentsimple sublattices, and a unit cell [see the green rhombustablet in Fig. 2(a)] contains each of four kinds of vorticesreferred to as 1, 2, 3, and 4. According to Kitaev , weuse the Majorana fermion operators to represent Paulioperators as σ x = ib x c , σ y = ib y c , and σ z = ib z c , whereMajorana operators b x , b y , b z , and c satisfy α = 1, αβ = − βα for α, β ∈ { b x , b y , b z , c } and α = β . Then, theHamiltonian (1) can be rewritten as H = P j,k G jk c j c k ,where the operator-valued coupling G jk ≡ iJ u Z jk ( u = x, y, z ) if ( j, k ) ∈ S ( u ); G jk = 0 when ( j, k ) / ∈ S ( u ). Here,a link ( j, k ) determines a type of coupling u = u ( j, k ).Due to the vanishing anticommutator of b uj and b uk , wehave Z jk = − Z kj for j = k .For each site, the above-mentioned Majorana operatorsact on a 4 D space, but the physical subspace is only 2 D .Thus we need to invoke a gauge transformation of Z group to project the extended space into the physicalsubspace through the projection operator D = b x b y b z c : | ψ i belongs to the physical subspace if and only if D | ψ i = | ψ i . With this physical projection, some eigenstates of H can be found exactly because G jk lays on the center of anAbelian algebra generated by Z jk with [ Z jk , H ] = 0 and[ Z jk , Z ml ] = 0. Since ( Z jk ) = 1 , Z jk = ib uj b uk generatesa Z group and its eigenvalues are z jk = ± . Therefore, { Z jk , I | ( j, k ) ∈ S ( u ) , u = x, y, z } generate the symmetrygroup Z ⊗ Z ⊗ · · · ⊗ Z of the model; the whole Hilbertspace is then decomposed according to the direct sum of some irreducible representations, and each irreduciblesector is characterized by { z jk | ( j, k ) ∈ S ( u ) , u = x, y, z } ,i.e., the directions shown in Figs. 2(a)-2(c).Obviously, in each irreducible representation space,we can reduce the Hamiltonian (1) into a quadraticform, which represents an effective Hamiltonian of freefermions for a given vortex arrangement. To character-ize the vortex configuration, we introduce square and oc-tagon plaquette operators W (4) p = σ z σ z σ z σ z and W (8) p = σ y σ y σ x σ x σ y σ y σ x σ x or W (4) p = − Y ( j,k ) ∈ ∂p (4) Z jk , W (8) p = − Y ( j,k ) ∈ ∂p (8) Z jk , (3)where ∂p (4) and ∂p (8) represent the sets of boundarylinks of square and octagon plaquettes with label p ; the( j, k ) links are ordered clockwise around the plaquette.The operators W ( j ) p ( j = 4 ,
8) commute with each other, h W ( j ) p , H i = 0, W (4)2 p = W (8)2 p = I , and thus each pla-quette operator has two eigenvalues w p = ±
1. A plaque-tte with w p = 1 is a vortex-free plaquette while w p = − Let us denote the site index j in detail by( s, λ ), where s refers to a unit cell, and λ to a posi-tion type inside the cell. The Hamiltonian then reads H = P s,λ,t,µ G sλ,tµ c sλ c tµ /
2. Due to the translationalinvariance of the lattice along the unit direction vec-tors n = (1 ,
0) , n = (0 , G sλ,tµ actually dependson λ, µ and t − s , and thus exp [ i q · ( r t − r s )] G sλ,tµ =exp ( i q · r t ) G λ,tµ . To study the spectral structure ofthe system, we invoke the generic fermion operator a q ,µ = P t e i q · r t c tµ / √ N where N is the total number of theunit cells and a p ,µ a † q ,λ + a † q ,λ a p ,µ = δ pq δ µλ . e G λµ ( q )is the Fourier transformation of G λ,tµ . In the momen-tum space, the fermion representation of the Hamiltonianreads H = 12 X q A † q e G ( q ) A q . (4)Case I: In the vortex-free (VF) sector, we choose aparticular direction ( z jk = +1 or −
1) for each link[see Fig. 2(a)], so that translational symmetry holdsand w (4) p = w (8) p = 1 for all plaquettes. Since A † q =( a † q , , a † q , , a † q , , a † q , ), we have the 4 × e G ( q ) = e G V F or e G V F = (cid:18) J x σ y − iJ y σ x + iJ z αiJ y σ x − iJ z α † J x σ y (cid:19) , (5)where α = diag. [exp( − iq ) , − exp( iq )], q = q · n , q = q · n .The single particle spectrum ε ( q ) = − ε ( − q ) is givenby the eigenvalues of the spectral matrix e G ( q ). An im- x J y J z J B x A y A z A ( ) e ( ) d x J y J z J B xy A z A y J z J x J z A y A x A B ( ) f ( ) a n n x y zz ( ) c
12 34 ( ) b
12 34 56 78
FIG. 2: (Color online) (a)-(c) 4-8-8 mosaic spin models in(a) vortex-free (VF) sector, (b) vortex-half occupied (VHO)sector, and (c) vortex-full occupied (VFO) sector. (d)-(f) Thecorresponding phase graphs of the above lattices with gaplessphase B and gapped phases A : (d) VF, (e) VHO, and (f)VFO. portant property of the spectrum is whether it is gap-less, i.e., whether ε ( q ) vanishes for some q . Obviously,the vanishing of determinant Det ( e G V F ) enjoys the zeroeigenvalues of e G V F . Then the gapless condition is J x + J y = J z . (6)As shown in Fig. 2(d), the phase diagram of our modelconsists of three phases, the gapless phase B , which is ac-tually a conical surface, distinguishing from two gappedphases A z and A xy . Since the possible zero energy degen-erate points are (0 , ± π ) and ( ± π,
0) in the first Brillouinzone, we choose J x = J y = 1, J z = √
2, and q = π toplot the profile graph of the energy spectrum with respectto q ∈ [ − π, π ] in Fig. 3(b) by solid lines. The eigenval-ues of e G V F are chosen in the concourse {±√ q / ±√ q / } . Thus in the vicinity of the energy de-generate points, the low-energy excited spectrum is ap-proximately linear. This property maybe helpful to studyquantum state transfer problems .Case II: We choose another particular direction foreach link as shown in Fig. 2(b), and the plaquettes with w (4) p = − w (8) p = − A †′ q = ( a † q , , a † q , , a † q , , a † q , , a † q , , a † q , , a † q , , a † q , ), the corresponding 8 × e G V HO = J x σ y − iJ y σ x iJ z e − iq ′ α † iJ y σ x J x σ y − iJ z β † iJ z β J x σ y − iJ y σ x − iJ z e iq ′ α iJ y σ x − J x σ y , (7)where α = diag (cid:16) , − e − iq ′ (cid:17) , β = diag (cid:16) e − iq ′ , − (cid:17) , q ′ = q · n ′ , q ′ = q · n ′ , n ′ = (1 , n ′ = ( − , J x < J y + J z , J y < J x + J z , J z < J x + J y (8) and the corresponding phase graph is plotted in Fig. 2(e).We notice that the same phase graph has been obtainedby Pachos for the Kitaev model.Case III: We choose the directions of links as shown inFig. 2(c) so that the translational symmetry still holdsand w (4) p = w (8) p = − α , q , and q . Therefore e G V F O = (cid:18) J x σ y − iJ y σ x + iJ z αiJ y σ x − iJ z α † − J x σ y (cid:19) . (9)The gapless condition is found as( J x − J z ) ≤ J y ≤ ( J x + J z ) . (10)If J x , J y , J z ≥
0, we have J x ≤ J y + J z , J y ≤ J x + J z , J z ≤ J x + J y . Thus in this case the phase diagram ofour model is the same as that of Kitaev’s honeycombmodel. As shown in Fig. 2(f), the region within the redlines labeled by B is gapless. The other three gappedphases A x , A y , and A z are algebraically distinct. How-ever, the energy spectrum of the 4-8-8 mosaic model ismore complex than that of the Kitaev model. When J x = J y = J z = 1 and q = − q = q , the eigenval-ues of the single fermion are chosen in the concourse {− / ± cos( q/ π/ / ± cos( q/ − π/ } . Similarly,the different energy spectrums of mosaic spin models im-ply their different dynamic properties.In order to see the stability of the ground state inthe VF sector, we compare the ground energy E = − P q ε q / J x = J y = J z = 1, we find the ground energy per site is E ,V F = − . E ,V HO = − . E ,V F O = − . E ,V F < E ,V HO < E ,V F O . The othercases can be studied similarly. Actually, as it was pointedout by Kitaev , Lieb’s theorem ensures that the VF lat-tice has the lowest energy to form a ground state. In thefollowing, we will focus on the stable VF lattice and in-vestigate the nontrivial topological properties in the B phase. Topological properties of B phase in the pres-ence of magnetic field -
The perturbation V = − P j (cid:0) h x σ xj + h y σ yj + h z σ zj (cid:1) introduced by Kitaev canbreak the time-reversal symmetry. Then the nontrivialthird-order term becomes H (3) eff = κ P j,k,l ( iZ jl Z kl ) c j c k ,where κ ∼ h x h y h z /J . As illustrated in Fig. 3(a),the thin dashed arrows represent the effective secondnearest-neighbor interactions between fermions inducedby H (3) eff , and their directions denote the chosen gauge Z jl Z kl . When κ = 0 .
025 , the changed profiled spec-trum is figured by dashed lines in Fig. 3(b). Thereforethe system in the B phase acquires an energy gap in thepresence of a magnetic field, which is helpful for protect-ing non-Abelian anyons.According to Kitaev , the topological properties of atwo-dimensional noninteracting fermion system with an e -1 - p p q e -1 p q p ( ) a ( ) b ( ) c FIG. 3: (Color online) (a) Thin dashed arrows describe the ef-fective second nearest-neighbor interactions between fermionsand the corresponding gauge induced by a magnetic field.(b) Profile graph of an energy spectrum with J x = J y = 1, J z = √ q = π axis in the absence (solid lines)and presence (dashed lines) of a magnetic field. (c) Energyspectrum of the above system with finite size along the n di-rection in the magnetic field. Two chiral edge modes crossingat E = 0 correspond to Chern number ± energy gap are usually characterized by Chern number,which can be determined by observing the edge modes ofthe spectrum . If the system illustrated in Fig. 2(a)is finite along the n direction while still periodic in the n direction, its energy spectrum is shown in Fig. 3(c)with J x = J y = 1, J z = √
2, and κ = 0 . E = 0. Since the Fermi energy lies inthe central gap, only these two edge states around zeroenergy are relevant to Chern number . We also noticethat the two edge modes have a universal chiral feature ,i.e., even if the edges are changed, the energy curves ofthe edge modes do not change their tendencies, respec- tively. Therefore we conclude that the Chern number is ± B phase. We also get zero Chernnumber in the Abelian phases A xy and A z with similarstudies. Compared with Kitaev’s honeycomb model andthe 4-8-8 mosaic model with even cycles in the lattice,the 3-12-12 mosaic model with odd cycles spontaneouslybreaks time reversal symmetry to obtain Chern number ± . Conclusion -
We generalize Kitaev’s honeycomb modelto various mosaic spin models with translation and rota-tion symmetries and study the 4-8-8 case in detail. It isfound that when | J z | ≫ | J x | , | J y | , our model is equiva-lent to Kitaev’s toric code model with Abelian anyons.Different vortex excitations result in different phase dia-grams with a gapless and gapped spectral structure. Inthe stable vortex-free case, the zero-energy Dirac pointsappear and the external magnetic field can induce an en-ergy gap. The nontrivial Chern number in B phase isobtained by studying the edge modes of the spectrum.We thank X. G. Wen, T. Xiang, Y. S. Wu, Y. Yu,H. Q. Lin, G. M. Zhang, and J. Vidal for helpful dis-cussions. One of the authors (D.L.Z.) acknowledgesthe hospitality of the Program on “Quantum Phases ofMatter” by KITPC. The project was supported by theNSFC (Grants No. 90203018, No. 10474104, and No.60433050) and the NFRPC (Grants No. 2006CB921206,No. 2005CB724508, and No. 2006AA06Z104).When this work is nearly finished, we notice that H.Yao and S. A. Kivelson have just studied the 3-12-12mosaic model in detail . X. G. Wen,
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