Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Quantum antiferromagnetic Heisenberg half-odd integer spin model as theentanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
Wen-Jia Rao and Guang-Ming Zhang , ∗ and Kun Yang State Key Laboratory of Low-Dimensional Quantum Physics andDepartment of Physics, Tsinghua University, Beijing 100084, China. Collaborative Innovation Center of Quantum Matter, Beijing 100084, China. National High Magnetic Field Laboratory and Physics Department,Florida State University, Tallahassee, Florida 32310, USA. (Dated: August 11, 2018)Applying a symmetric bulk bipartition to the one-dimensional Affleck-Kennedy-Lieb-Tasaki va-lence bond solid (VBS) states for the integer spin-S Haldane gapped phase, we can create an arrayof fractionalized spin-S/2 edge states with the super unit cell l in the reduced bulk system, andthe topological properties encoded in the VBS wave functions can be revealed. The entanglementHamiltonian (EH) with l = even corresponds to the quantum antiferromagnetic Heisenberg spin-S/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For theodd integer spins, however, the EH just corresponds to the quantum antiferromagnetic Heisenberghalf-odd integer spin model with spinon excitations, characterizing the critical point separating thetopological Haldane phase from the trivial gapped phase. Our results thus demonstrate that thetopological bulk property not only determines its fractionalized edge states, but also the quantumcriticality associated with the topological phase, where the elementary excitations are precisely thosefractionalized edge degrees of freedom confined in the bulk of the topological phase. PACS numbers: 05.30.Rt, 05.30.-d, 03.65.Vf
INTRODUCTION
Topological phases of matter including those requiresymmetry protection have been the subject of intenseinterest in quantum information science, condensed mat-ter physics and quantum field theory. Much effort hasbeen devoted to classification of these topological phases,and tremendous success is achieved in our understand-ing of quantum Hall states[1], topological insulators[2–4], and symmetry protected topological (SPT) phases[5–7]. The SPT phases possess bulk energy gaps and donot break any symmetry, but have robust gapless edgeexcitations. These SPT states can not be continuouslyconnected to a trivial gapped state without closing theenergy gap. So there exists a topological phase transi-tion between a SPT phase and its adjacent trivial phase,and the corresponding critical theory does not belong tothe conventional Landau-Ginzburg-Wilson paradigm[8–11]. Such a critical point is a prototype of “deconfinedquantum critical point (QCP)” with fractionalized ele-mentary excitations[12]. A crucial question is how to ex-tract the critical properties from the ground state wavefunction of the SPT phases.In one dimension, Haldane[13] predicted that quantumantiferromagnetic Heisenberg spin chains are classifiedinto two universality classes: half-odd integer spins withgapless excitations and integer spins with gapped excita-tions. Recent studies[14, 15] indicated that the Haldanegapped phase for odd integer spin chains is a typical SPTphase, while the even integer spin chains correspond tothe topologically trivial phase, because their edge statesare not protected by the projective representation of the bulk SO(3) symmetry. According to the classificationtheory[7], there exists only one nontrivial SPT phase forthe SO(3) symmetric quantum Heisenberg spin model,whose fixed point wave function is given by the Affleck-Kennedy-Lieb-Tasaki (AKLT) valence bond solid state(VBS)[16]. Since the symmetry protection of the SPTphase in the bulk can be analyzed in terms of symmetryprotection of the fractionalized edge spins, it motivates usto question if there exists a general connection betweenthe SPT phase and the quantum critical phases of thequantum antiferromagnetic Heisenberg half-odd-integerspin chains.In this paper, we first review the entanglement prop-erty of a single block in the one-dimensional integer spin-S AKLT VBS states, and prove that the entanglementHamiltonian can be expressed in terms of the Heisenbergexchange of two edge spin-S/2’s. By using a symmet-ric bulk bipartition[9, 17, 18], we can create an array offractionalized spin-S/2 edge states with super unit cell l in the reduced bulk system. Then the reduced densitymatrix and entanglement Hamiltonian (EH) can be de-rived in terms of the fractionalized edge spins, leadingto the quantum antiferromagnetic Heisenberg spin-S/2model when the super unit cell l includes even numberof lattice sites. For S = 4 n + 2 with integer n , the EHstill describes the nontrivial Haldane gapped phase withodd integer spins, and for S = 4 n the EH correspondsto the even integer Haldane gapped phase. For the oddinteger spin- S , however, the quantum antiferromagneticHeisenberg half-odd integer spin model emerges, charac-terizing the quantum critical point separating the non-trivial Haldane phase from the trivial phase. So our re-sults demonstrate that the topological bulk property notonly determines its fractionalized edge states, but alsothe critical point at the continuous phase transition toits nearby trivial phase. SINGLE BLOCK ENTANGLEMENT
The spin- S AKLT VBS state as the fixed point stateof the Haldane gapped phase is defined by | VBS i = N Y i =0 (cid:16) a † i b † i +1 − b † i a † i +1 (cid:17) S | vac i , (1)where a † i and b † i are the Schwinger boson creation opera-tors with a local constraint a † i a i + b † i b i = 2 S , and the spinoperators are expressed as S + i = a † i b i , S − i = b † i a i , and S zi = (cid:16) a † i a i − b † i b i (cid:17) /
2. In this construction, each phys-ical spin is composed of two spin- S/ S state, while each neighboring sites are linkedby spin- S/ l sites denoted by the part A. With the help ofthe spin coherent state representation, the reduced den-sity matrix ρ A can be obtained by tracing out the degreesof freedom without the part A, and its nonzero eigenval-ues λ j with degeneracy 2 j + 1 have been derived[19, 20] λ j = 1( S + 1) S X k =0 (2 k + 1) [ f ( k )] l − × I k (cid:20) j ( j + 1) − S ( S + 2) (cid:21) , (2) f ( k ) = ( − k S ! ( S + 1)!( S − k )! ( S + k + 1)! , (3)where j = 0 , , ...S and the recursion function I k [ x ] isdefined by I k +1 [ x ] = 2 k + 1( S + k + 2) (cid:18) k + 4 xk + 1 (cid:19) I k [ x ] − kk + 1 (cid:18) S − k + 1 S + k + 2 (cid:19) I k − [ x ] , (4)with I [ x ] = 1 and I [ x ] = x ( S +2) . Since the function | f ( k ) | decreases with k very quickly, only the first twoterms ( k = 0 ,
1) dominate in the summation for a longblock length l . Thus the eigenvalues are approximatedas λ j ≈ S + 1) + 3 (cid:18) − SS + 2 (cid:19) l − [2 j ( j + 1) − S ( S + 2)]( S + 2) , and up to the first order of δ = (cid:16) − SS +2 (cid:17) l the entanglementspectrum is thus derived as ξ j ≈ J ( l ) (cid:20) j ( j + 1) − S (cid:18) S (cid:19)(cid:21) , (5) FIG. 1: (a) The picture of AKLT VBS state. Each blue dotrepresents a spin- S/
2, yellow circle stands for the physicalspin- S , and solid lines denote the singlet bonds. A block with l sites is chosen as the subsystem A. (b) The entanglementspectra of the single block are given for l =even and l =odd,respectively. with J ( l ) = S ( S +2) (cid:16) − SS +2 (cid:17) l . Then the correspondingEH can be recognized as: H E = J ( l ) s · s where s and s are the fractionalized edge spins. Therefore, for a longblock length l , the entanglement properties of the singleblock are just described by the quantum Heisenberg spinmodel, and the corresponding entanglement spectra aredisplayed in Fig. 1(b). SYMMETRIC BULK BIPARTITION
The symmetric bulk bipartition is the most effectivetool to generate an extensive array of fractionalized edgespin-S/2’s in the bulk subsystem, i.e., the spin chain is di-vided into two subsystems both including the same num-ber of disjoint blocks[9, 17]. The fractionalized edge spinscan thus percolate in the reduced bulk system and emergeas coherent elementary excitations of the effective fieldtheory of the subsystem. It is convenient to write theAKLT VBS wave function in the form of matrix productstate (MPS) representation shown in Fig. 2(a) | V BS i = X { s i } Tr h A [ s ] A [ s ] ..A [ s N ] i | s , s , ..s N i , (6)where A [ s i ] are the ( S + 1) × ( S + 1) local matrices, whoseelements can be obtained from the Schwinger boson rep-resentation, and the periodic boundary condition are as-sumed. When we group each continuous l lattice sitesinto a block, all the even blocks are denoted by thepart A and the rest by the part B. Then by tracing outthe part B, the reduced density matrix ρ A and the EH( H E = − ln ρ A ) can be derived. The general procedureis described as the following four steps.Step 1. Conduct the coarse graining and distill relevantstates within each block[21]. We pick out a block with l sites, and perform the singular value decomposition (cid:16) A [ s ] A [ s ] ..A [ s l ] (cid:17) α,β = κ − X p =0 X ( { s i } ) ,p Λ p Y p, ( α,β ) , (7)where the number of nonzero singular values κ recordsthe number of relevant states in the block. For the spin- S AKLT VBS state, κ = ( S + 1) , and the relevant states | p i are effectively composed by two edge spin- S/ | p i = X m,n χ pm,n | m, n i , which are the combination of the degenerate edge states | m, n i with m, n ∈ [ − S/ , S/ | Ψ i = X { p i } Tr (cid:16) B [ p ] B [ p ] ..B [ p N/l ] (cid:17) | p , p , ..p N/l i , (8)where the block matrices are given by B [ p ] α,β =Λ p,p Y p, ( α,β ) .Step 2. Trace out the degrees of freedom in the part B.Such a procedure can be presented elegantly by a graph-ical notation described in Fig. 2(c). The contribution ofthe subsystem B is represented by the transfer matrix T = P p B [ p ] ⊗ ¯ B [ p ] . The expression ρ A can be writteninto a matrix product operator form, which is displayedin Fig. 2(c) ρ A = T r Y j R j , (9) R j = X p j ,q j | p j ih q j | (cid:16) B [ p j ] ⊗ B [ q j ] (cid:17) T. (10)Step 3. To derive the EH, we have to express the pro-jection operator | p j ih q j | in terms of product of spin oper-ators. Note that each | p j i is composed of two spin- S/ | m ih n | = P i Γ ( m,n ) ,i O i ,where O i ( i = 0,1 , ...S + 2 S ) are the spin- S/ O = I . With these considerations, the fullexpression R j is written as R j = X { p j ,q j } X { m,n,α } (cid:20)(cid:18) B [ p j ] ⊗ B [ q j ] (cid:19) T (cid:21) χ p j m ,m χ q j n ,n × Γ ( m ,n ) ,α j − Γ ( m ,n ) ,α j O α j − O α j . (11) It is emphasized that no approximation has been madeso far.Step 4. Since the form R j is complicated, a controlledapproximation can be introduced. For a long block length l , the only dominant coupling in ρ A is δ = (cid:16) − SS +2 (cid:17) l , whichimplies that the exchange coupling between two edgespins decays exponentially. Then R j can be separated FIG. 2: (a) The MPS representation of the AKLT VBS state.(b) The blocked AKLT VBS state. (c) The reduced den-sity matrix under symmetric bulk bipartition with a repeatingstructure. into two individual edge spins, and the final result for H E is given by H E ≈ S ( S + 2) (cid:18) − SS + 2 (cid:19) l X i s i · s i +1 , (12)where s i is the fractionalized edge spin- S/ S = 1 and S = 2 cases are included in the supplementary material.Therefore, the resulting entanglement properties canbe divided into three categories: (i) For l = odd , H E represents a ferromagnetic ordered phase with spin waveexcitations (the numerical result for S = 1 is includedin the supplementary material); (ii) For l = even and S = even , H E describes the Haldane gapped phase withinteger spins. In particular, for S = 4 n + 2 with inte-ger n , it represents the SPT phase of the odd integerspin Haldane phase even though the original VBS statecorresponds to the topologically trivial state. (iii) For l = even and S = odd , H E is just the quantum Heisen-berg antiferromagnetic half-odd integer spin model withquantum critical ground state[22]. The correspondingeffective field theory for S > S Wess-Zumino-Witten (WZW) theory, butthe stable fixed point of these critical phases is deter-mined by the SU(2) level-1 WZW theory[22, 23]. Theseare the important properties encoded in the AKLT VBSstates with integer spins.
NUMERICAL CALCULATIONSSpin-1 AKLT state
In order to put the above analytical results on a solidground, we perform the exact numerical diagonalizationfor the reduced density matrix ρ A for the spin-1 AKLTstate without any approximations. The full entanglementspectrum (ES) for the block length l = 4 is displayed inFig. 3(a). We use the effective length L A to denote thereduced system length, independent of the block lengthin the original scale. The degeneracies of each levels cor-respond to 1 , , S A /L A saturates to 0 . . ξ − ξ is found to scale linearly withthe inverse subsystem length ξ − ξ = k L − A shown inFig. 3(b), suggesting the bulk ES is gapless in the ther-modynamic limit. Moreover, the second excited entan-glement level is also fitted as ξ − ξ = k L − A displayedin Fig. 3(b), and the ratio of these two excited levels isdetermined as k /k = 1 . ∼
2, implying the differenceof scaling dimensions for these two excited levels is 2.To determine the universality class of this spec-trum, we focus on the wave function of the low-est level | ψ i . By further cutting the reduced sys-tem into two halves with lengths l a and ( L A − l a ),respectively, we calculate the entanglement entropy: s ( l a , L A ) =Tr l a +1 ,l a +2 ..L A ( | ψ ih ψ | ). Fitting to theCalabrese-Cardy formula[24], s ( l a , L A ) = c (cid:20) L A π sin (cid:18) πl a L A (cid:19)(cid:21) + s , (13)we obtain the central charge c = 1 . ± .
02 in Fig. 3(c).This result confirms that the obtained ES belongs to theuniversality class of the 1+1 (space-time) dimensional SU (2) WZW conformal field theory, which is the sameas the quantum antiferromagnetic Heisenberg spin-1 / WZW theory with c = 3 / l = 3, the bulk ES is calculated and displayedin Fig. 4(a). The entanglement entropy density is foundto saturate to 0 . .
3% to the value ofln 2. The lowest entanglement level ξ is linear with sys-tem size. However, the lowest entanglement level has thedegeneracy L A + 1 in each system size. We computed themagnetization distribution m ztot = P i m zi for these statesand found they are well located in [ − L A / , L A / L A / ξ − ξ ∼ L − A , which isa direct sign of spin-wave excitations. To further con-firm the this spectrum, we fit the second excitation level n - L A (a) L -1A (b) g(l a ,L A ) L A =6 L A =8 L A =10 L A =12 L A =14 s ( l a , L A ) (c) FIG. 3: (a) Bulk ES with S = 1 and the block length l = 4. (b) Two lowest entanglement levels are linear with L − A . (c) The entanglement entropy s ( l a , l A ) as a function of g ( l a , L A ) = ln h L A π sin (cid:16) πl a L A (cid:17)i . ξ − ξ ∼ L − A , as dictated in Fig. 4(b). If we take theHeisenberg interaction as the EH, the coupling constantis fitted to be J ∼ − . J = ( − / = − . Spin-2 AKLT state
Another important calculation is performed for thespin-2 AKLT VBS state. The ES with l = 6 under open boundary condition is presented in Fig. 5(a). The low-est level is singlet and the first excited level is triplet.However, the level spacing between these two states isfitted as an exponential decay with the subsystem size:( ξ − ξ ) /J ( l ) ∼ e − L A / ∆ with ∆ = 4 . J ( l ) is fitted to be 0 . . ξ − ξ ) /J ( l ) shown in Fig. 5(c) approaches to the finitevalue 0 . .
41 from n L A (a) J(l)=-0.037 L -2A (b) FIG. 4: (a) The bulk ES with S = 1 and the block length l = 3, the lowest level is ( L A + 1)-fold degenerate. (b) Theentanglement spectral gap is linear with square of the inversesubsystem size, indicating a spin-wave excitation for the fer-romagnetic Heisenberg spin chain. The second excited levelis also plotted, but the data from small sizes slightly deviatefrom the line. the density matrix renormalization group calculation[26].The difference can be improved when the longer lengthof the effective spin chain is calculated. DISCUSSION AND CONCLUSION
The symmetric bulk bipartition allows us to estab-lish a general description of QCP separating the SPTphase from its trivial gapped phase directly from thefixed point wave function of the topological phase. Forthe one-dimensional SPT phase with the protecting sym-metry of G = SO (3) Lie group, its fundamental groupis Π ( G ) = Z . So there are only two different phases:the odd integer spin Haldane gapped phase and its trivialgapped phase adiabatically connected to the even integerspin Haldane gapped phase. A QCP exists to separatethese two phases, and the effective model Hamiltonianfor this QCP is just given by the quantum antiferromag-netic Heisenberg half-odd integer spin chain. The cor-responding critical theory is characterized by the 1+1 n L A ...... (a) Log [ ( - ) / J ] L A (b) (- ) / J L -1A (c) FIG. 5: (a) The bulk ES with S = 2 and the block length l = 6 for an open chain. The lowest level is singlet, andthe first excited level is triplet. (b) The first excited leveldecays exponentially with the subsystem size. (c) The bulkexcitation energy ( ξ − ξ ) /J ( l ) saturates to a finite value inthermodynamic limit. (space-time) dimensional SU (2) WZW conformal fieldtheory with the Lie group ˜ G = SU (2), where ˜ G is justthe universal covering group of G and has a trivial funda-mental group, Π ( ˜ G ) = 1. Our results may thus gener-alize the widely discussed bulk-edge correspondence: thebulk topological property of the topological phase notonly determines its symmetry-protected edge degrees offreedom, but also the critical properties of the second or-der phase transition to the trivial phase. Furthermore,the fundamental degrees of freedom of the critical theoryare precisely these edge degrees of freedom confined inthe bulk of the topological phase. As a result this QCPis a typical deconfined critical point. These results canbe generalized for other SPT phases with the protectingsymmetry of continuous Lie group.To summarize, we have applied a symmetric bulk bi-partition to the one-dimensional AKLT VBS states forthe integer spin-S Haldane gapped phase, and an arrayof fractionalized spin-S/2 edge spins can be created inthe reduced bulk system. Via the calculations of thebulk entanglement spectra for the reduced system, thetopological properties encoded in the original VBS wavefunctions are revealed. Acknowledgements.-
G. M. Zhang would like to thankD. H. Lee and X. Wan for the helpful discussions andacknowledge the support of NSF-China through GrantNo.20121302227. K. Yang is supported by NSF grantsDMP-1442366 and DMP-1157490. ∗ Electronic address: [email protected][1] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M.den Nijs, Phys. Rev. Lett. 49, 405 (1982).[2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[3] X. L. Qi and S. C. Zhang, Rev. Mod. Phys. , 1057(2011).[4] J. E. Moore, Nature (London) , 194 (2010).[5] X. Chen, Z. C. Gu, and X. G. Wen, Phys. Rev. B ,035107 (2011).[6] N. Schuch, D. Perez-Garcia, and I. Cirac, Phys. Rev. B , 165139 (2011).[7] X. Chen, Z. C. Gu, Z. X. Liu, and X. G. Wen, Phys. Rev.B , 155114 (2013).[8] X. Chen, F. Wang, Y. M. Lu, and D. H. Lee, Nucl. Phys.B (FS) 248 (2013).[9] W. J. Rao, X. Wan, and G. M. Zhang,Phys. Rev. B ,075151 (2014).[10] T. H. Hsieh, L. Fu, X. L. Qi, Phys. Rev. B , 085137(2014); T. H. Hsieh and L. Fu, Phys. Rev. Lett. ,106801 (2014). [11] L. Tsui, Y. M. Lu, H. C. Jiang, and D. H. Lee, Nucl.Phys. B (FS) 330 (2015); L. Tsui, F. Wang and D.H. Lee, arXiv:1511.07460.[12] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, andM. P. A. Fisher, Science , 1490 (2004).[13] F. D. M. Haldane, Phys. Lett. , 464 (1983); Phys.Rev. Lett. , 1153 (1983).[14] Z. C. Gu and X. G. Wen, Phys. Rev. B , 155131 (2009).[15] F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa,Phys. Rev. B , 075125 (2012).[16] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys.Rev. Lett. , 799 (1987); Commun. Math. Phys. ,477 (1988).[17] W. J. Rao, K. Cai, X. Wan, and G. M. Zhang, Phys.Rev. B , 214430 (2015).[18] R. A. Santos, C. M. Jian, and R. Lundgren,arXiv:1511.01489.[19] H. Katsura, T. Hirano, and Y. Hatsugai, Phys. Rev. B , 012401 (2007).[20] V. E. Korepin and Y. Xu, Int. J. Modern Phys. B ,1361 (2010).[21] F. Verstraete, J. I. Cirac, J. I. Latorre, E. Rico, and M.M. Wolf,Phys. Rev. Lett. , 140601(2005).[22] I. Affleck and F. D. M. Haldane, Phys. Rev. B , 5291(1987).[23] S. C. Furuya and M. Oshikawa, arXiv:1503.07292.[24] P. Calabrese and J. Cardy, J. Phys. A , 504005 (2009).[25] A. Kitazawa and K. Nomura, Phys. Rev. B , 11358(1999).[26] S. R. White, Phys. Rev. Lett.69