A Group cohomology

Z_3 symmetry-protected topological phases in the SU(3) AKLT model


We study symmetry-protected topological (SPT) phases in one-dimensional spin systems with symmetry. We construct ground-state wave functions of the matrix product form for nontrivial phases and their parent Hamiltonian from a cocycle of the group cohomology . The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU() case, the SU() AKLT Hamiltonian, is also presented which realizes nontrivial SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial SPT phase is found for a range of the parameter including the point of vanishing biquadratic term () as well as the SU(3) AKLT point []. A continuous phase transition to the SU(3) dimer phase takes place at , with a central charge . For SU(3) symmetric cases we define string order parameters for the SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.


I Introduction

The Haldane phaseHaldane (1983a, b) of antiferromagnetic spin chains is a representative topological phase of one-dimensional (1D) gapped quantum systems. In the Haldane phase, excitations are gapped in the bulk, while zero-energy states of effective spins are present at the boundaries. The essence of the Haldane phase is captured by the toy model proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT),Affleck et al. (1987, 1988) which is constructed from projection operators acting on two neighboring sites. Its ground state (the AKLT state) has the following structure. Each spin is decomposed into two virtual spins. On each site two spins are symmetrized to form an spin, while two spins from neighboring sites form a singlet on each bond. At each end of the spin chain, an effective spin is left without forming a singlet and realizes two-fold degenerate zero modes. The AKLT state shows no apparent symmetry breaking such as magnetic order and lattice symmetry breaking. However, it has a hidden order called the string order,den Nijs and Rommelse (1989) which corresponds to a ferromagnetic order in the system after a non-local unitary transformation.Kennedy and Tasaki (1992a, b) The string order signals a hidden symmetry breaking in the Haldane phase.

Recent advances in the understanding of 1D topological phases are brought by the notion of symmetry protected topological (SPT) phases.Pollmann et al. (2010); Turner et al. (2011); Fidkowski and Kitaev (2011); Chen et al. (2011, 2012) The Haldane phase is an SPT phase that is protected by any one of the following symmetries:Turner et al. (2011) (a) time-reversal symmetry, (b) link inversion symmetry, and (c) the dihedral group of rotations about the , , and axes. Here let us assume the symmetry of the dihedral group. The AKLT Hamiltonian is invariant under the rotation around the and axes, and these rotations commute with each other for the original spins. However, they do not commute (in fact anticommute) with each other for the virtual spins. This is an example of projective representations of symmetry groups, i.e., symmetry operations represented projectively on the effective (fractionalized) degrees of freedom which appear at the boundaries. This can be nicely formulated in the framework of matrix product states (MPSs) for 1D gapped systems. The AKLT wave function is written in the MPS form with matrices acting on the two states , of a virtual spin. Symmetry operations ( rotations) acting on the three states of each spin induce linear transformations of the matrices, which are then expressed as unitary transformations in the two-dimensional space spanned by and . The unitary matrices of this basis transformation give a projective representation of the symmetry group with a phase factor which is an element of the group cohomology . The Haldane phase is an example of SPT phases and corresponds to the nontrivial element of . In general 1D SPT phases protected by symmetry group are classified in terms of the second cohomology group of the group .Chen et al. (2011); Fidkowski and Kitaev (2011); Schuch et al. (2011); Chen et al. (2012)

In this paper we generalize the AKLT state of the Haldane phase to 1D SPT phases protected by symmetry. We focus on the case of and briefly discuss the general case . Our starting point is the observation that symmetry can be projectively represented by matrices, with a U(1) phase factor which is a nontrivial element of . This observation allows us to write down MPS wave functions with matrices as described below, as a natural generalization of the AKLT state. The MPS wave functions are ground states of an SU(3) generalization of the AKLT model and describe topological states in SPT phases.

We construct the SU(3) AKLT states on a 1D lattice where the local Hilbert space on each site is spanned by eight states of the adjoint representation of su(3), which we call meson states. The eight meson states are represented by traceless bilinear forms of two sets of three virtual degrees of freedom, i.e., three quarks () in the fundamental representation and three antiquarks () in the conjugate representation . The SU(3) AKLT states are valence bond solids in which a quark and an antiquark on neighboring sites form a singlet state on the bond connecting the two sites, whereas a quark and an antiquark on the same site form a meson state. When the 1D chain has ends, three-fold degenerate boundary zeromodes appear at each end, which are either unpaired quark or antiquark states. The possibility of having two types (quark or antiquark) of zeromodes indicates that there are two distinct types of SU(3) AKLT states, each of which represents a distinct SPT phase. Both SU(3) AKLT states are ground states of the SU(3) AKLT Hamiltonian which consists of bilinear and biquadratic terms of su(3) generators in the representation with a particular ratio of the two terms. The SU(3) Hamiltonian and its ground-state wave functions were in fact presented earlier in Refs. Greiter et al., 2007; Greiter and Rachel, 2007; Katsura et al., 2008. In this paper we characterize the SU(3) AKLT states as SPT states in the classification in terms of group cohomology and report results of detailed study on their correlation functions and a quantum phase transition to a dimerized phase. We note that Refs. Duivenvoorden and Quella, 2012, 2013a studied PSU(3) symmetric spin chains which realize SPT phases corresponding to nontrivial elements of . The SU(3) AKLT Hamiltonian can also be considered as a PSU(3) symmetric model realizing SPT phases protected by PSU(3) symmetry in that the adjoint representation of SU(3) is also a representation of PSU(3).

We can further generalize the SU(3) AKLT Hamiltonian to the SU() AKLT Hamiltonian () consisting of bilinear and biquadratic terms of the su() generators in the adjoint representation . Its two-fold degenerate ground state (under periodic boundary conditions) is given by SU() AKLT states which are MPSs with matrices. The SU() AKLT states are valence bond solids in which states in the representation are decomposed into products of states from and representations, which form and singlet states on each site and bond, respectively. The SU() AKLT model has an energy gap as its two-point correlation functions of SU() operators are short-ranged with a correlation length being equal to . Realizations of SPT phases with SU() symmetry in other representations are proposed in the context of cold atoms.Nonne et al. (2013)

As in the SU(2) AKLT state, the SU() AKLT states have a hidden long-range order. To see this for the SU(3) AKLT model, we define string order parameters that characterize the SPT phase by making use of the system’s full SU(3) symmetry. Similar to the conventional string order parameter for the SU(2) AKLT state which indicates the antiferromagnetic order upon neglecting states, the string order parameters for the SU(3) AKLT states have string operators from SU(3) operators (analogous to the operator) which count the number of constituent quarks or antiquarks. We show the long-range order of string correlations by explicitly calculating string order parameters in the SU(3) AKLT states. Incidentally, the string orders that we define are different from those studied in Refs. Duivenvoorden and Quella, 2013a, 2012, b where only symmetry is assumed.

As the ratio of the two coupling constants in the SU(3) AKLT Hamiltonian is varied, a quantum phase transition occurs from a SPT phase to a topologically trivial dimer phase which breaks translation symmetry. We study this topological phase transition using the infinite-size variant of the density matrix renormalization group (iDMRG) method.White (1992, 1993); McCulloch () We obtain the phase diagram of the SU(3) bilinear-biquadratic model and determine the location of the critical point numerically. We find that the SPT phase occupies a finite region in the parameter space and survives even when the biquadratic term is absent. From scaling of entanglement entropy we obtain numerical evidence that the critical point is described by the level-2 SU(3) Wess-Zumino-Witten theory.

Finally, we demonstrate that the SU(3) AKLT Hamiltonian is realized by an spin chain with staggered quadrupole couplings in the strong-coupling limit. Using the fact that spin dipole and quadrupole operators of spins together form eight generators of su(3) in the fundamental representation , we construct Hamiltonians with staggered nearest-neighbor couplings of quadrupole operators whose ground states are smoothly connected to the SU(3) AKLT states in the limit where positive quadrupole couplings are very strong. In a similar manner, we propose that the SU(4) AKLT Hamiltonian is effectively realized in the strong-coupling limit of an spin-orbital model which is a variant of the Kugel-Khomskii model.Kugel and Khomskii (1982)

The paper is organized as follows. In Sec. II we review the MPS representation of gapped 1D quantum systems and the classification of 1D SPT phases in terms of group cohomology. In Sec. III we construct the SU(3) AKLT model from a nontrivial cocycle of and discuss its generalization to SU(). In Sec. IV we define string order parameters that characterize nontrivial SPT phases for the SU(3) symmetric case. In Sec. V we study the SU(3) bilinear-biquadratic model with the iDMRG method and show its ground-state phase diagram. In Sec. VI, we present realizations of the SU(3) and SU(4) AKLT Hamiltonians in an spin chain and an spin-orbital model. In Sec. VII we give a brief summary.

Ii Matrix product states and group cohomology

In this section, we give a brief review on the classification of the 1D SPT phases in terms of the group cohomologyChen et al. (2011); Fidkowski and Kitaev (2011); Schuch et al. (2011); Chen et al. (2012) and its application to the AKLT model for the Haldane phase. This will serve as a basis for the generalization of the AKLT model to the SU(3) case in the next section.

ii.1 Matrix product state

We consider a gapped ground state of an infinite spin chain described by a wave function , which we assume to be translation invariant. Let us consider bipartitioning of the chain between the site and the site . Then we decompose the wave function


where ’s are singular values, and and are wave functions on the left and the right semi-infinite chains that form orthonormal basis for the left and right Hilbert spaces. Alternatively we can decompose the wave function between the site and the site :


where the set of singular values are the same as in Eq. (1) because of the translation symmetry. Now we write in terms of and local states at the site as


where is a matrix defined for each local state and is independent of the site where we cut the spin chain, again due to the translation symmetry.

If we repeat this procedure, we can relate any two left singular vectors and with as


The reduced density matrix for the finite region and physical quantities derived from it can be obtained from the above equation relating singular vectors. If we extend this procedure to a periodic chain of length , then we obtain the MPS form of the ground-state wave function,


where the trace is over the product of matrices .

ii.2 Symmetry operation and MPS

Let us suppose that the system of our interest has a symmetry group and its ground-state wave function is invariant under global action of any element in . We assume that the symmetry action is local (e.g., on-site) and unitary. Local states are transformed by action of as


with a unitary matrix . The wave function is written in the form of an MPS of Eq. (5), whose transformation by is obtained by applying Eq. (6) to the local states at every site:


We see that the wave function is an MPS made from the matrices


We demand that the ground state be invariant up to a phase factor, i.e., . This is achieved if


where is a -dependent unitary matrix which is independent of the local states . It is known that is unique up to a phase when the transfer matrix has only one eigenvalue of the largest magnitudePérez-García et al. (2008); Fidkowski and Kitaev (2011) (the state is not a macroscopic superposition of orthogonal states).

Let us consider successive actions of on , which induce transformations


where we have used the fact that is a unitary symmetry (which does not include an anti-unitary operator such as time reversal), as we assume throughout this paper. Equation (10) should coincide with the transformation induced by an action of ,


We thus have


where the second equation has a U(1) phase. Equation (12b) shows that ’s give a projective representation of the symmetry group . The phase function encodes topological data of the ground-state wave function and has the following two properties [Eqs. (14) and (16)] that define group cohomology.

Cocycle: Let us calculate the product in two different ways (associativity):


The consistency between the two results requires the phase function to satisfy


This is the cocycle condition. (For more mathematical details, see Appendix A.)

Coboundary: The ambiguity of a U(1) phase in defining a unitary matrix in Eq. (9) implies that we are free to take another set of unitary matrices,


Accordingly, the phase function appearing in the projective representation in Eq. (12b) is changed from to ,


where the three terms in the square brackets [ ] are called 2-coboundary; see Appendix A. The two phase functions and are equivalent up to a 2-coboundary and describe the same topological phase.

The set of phase functions that satisfy the cocycle condition (14) is quotiented with the equivalence relation of Eq. (16). This equivalences class is an element of , the second cohomology group of the group cohomology of over U(1). Apparently, when phase functions of two states belong to different elements of , we cannot adiabatically deform one state to the other while preserving the symmetry. Thus the cohomology group classifies topological phases protected by symmetry group .Fidkowski and Kitaev (2011); Chen et al. (2011, 2012) The definition of group cohomology and a useful formula (Künneth formula) in the calculation of non-trivial cocycles are briefly summarized in Appendix A.

In an SPT phase characterized by a projective representation of symmetry group , the ground-state wave function possesses non-trivial boundary modes of which symmetry transformations become anomalous. To see this, we consider an MPS wave function on a finite chain of length ,


where and are boundary vectors specifying boundary conditions at the end sites and . From Eq. (9), the action of an element of symmetry group transforms the MPS wave function as


Thus the boundary states determined by and are transformed according to . This indicates that the symmetry operations for effective boundary states are not given by the original action of but by its projective representation . In this sense the symmetry actions become anomalous at the boundaries.

ii.3 Haldane phase and SU(2) AKLT model

The Haldane phase of antiferromagnetic spin chains is known as an example of an SPT phase with symmetry group .Pollmann et al. (2010); Turner et al. (2011) Let us consider the AKLT model,Affleck et al. (1987, 1988) of which Hamiltonian reads


where is a spin operator of :


The AKLT Hamiltonian has SU(2) symmetry generated by the above three spin operators. In particular, they are invariant under its subgroup generated by a -rotation around the -axis,


and a -rotation around the -axis,


that commute with each other,


The ground state of the AKLT Hamiltonian is best described in terms of the MPS in the following way. We first decompose every spin into two spins. Then the ground state is given as a valence-bond solid state of virtual spins. Namely, the ground-state wave function is obtained by (i) projecting two spins from two neighboring sites into a singlet state () and (ii) projecting the spins on each site into a triplet state (). This is expressed in the MPS with 2 by 2 matrices acting on the two-dimensional Hilbert space of a virtual spin spanned by . Two spins forming an spin on one site are coupled through three types of matrices which are the projection operators onto triplet states and labeled by the values of the total :


Two spins from neighboring sites are coupled by the matrix which is a projection operator to a singlet state,


The ground-state wave function is then written in the MPS form,


where the matrices are given by


in terms of the Pauli matrices . This construction from projection operators is natural because the AKLT Hamiltonian in Eq. (19) consists of a product of Casimir operators of neighboring spins that project states onto states such that the ground state is made of either or states of neighboring spins, i.e., two out of four spins on two neighboring sites form a singlet.

Now let us discuss transformation of the MPS by operators from the symmetry group, i.e., , and . We can easily check that


Comparing these equations with Eq. (9), we find a projective representation of the symmetry group


and the associated phase function,


which is a 2-cocycle corresponding to a nontrivial element of given in Appendix B [Eq. (159)]. We note that commuting operations and are represented projectively, and their projective representations and anticommute with each other.

Iii SPT phase and SU(3) AKLT model

In this section we study 1D SPT phases which are protected by global symmetry and characterized by a topological number. They are natural generalizations of the Haldane phase with symmetry discussed in the previous section.Duivenvoorden and Quella (2013a, 2012, b) We show that SPT phases are realized in an SU(3) extension of the AKLT model.Greiter et al. (2007); Greiter and Rachel (2007)

iii.1 Group cohomology of

Here we present a projective representation for the symmetry group , summarizing the results from Appendix B.

The group elements of are given by Eq. (158), and its second cohomology group is , generated by a 2-cocycle shown in Eq. (159). A projective representation of with the phase function is generated by matrices


which satisfy the algebra


In the case of our main interest, , the projective representation is given by matrices,


where . We thus expect a ground-state wave function of a SPT phase to have the MPS form of matrices which are subject to symmetry transformations generated by and in Eq. (33). We will demonstrate this below.

iii.2 SU(3) AKLT model

In this section we show that an SPT phase protected by global symmetry is realized in an SU(3) extension of the AKLT model. We begin with a brief review on representations of the Lie algebra su(3).Georgi (1999) In this paper, the three basis states of the fundamental representation of su(3) are denoted by three quarks . Similarly, its conjugate representation is spanned by antiquarks . We write the eight generators of su(3) in each representation as (). For the fundamental representation, the su(3) generators are given by


where ’s are the Gell-Mann matrices:


For the conjugate representation , the su(3) generators are given by


Cartan subalgebra of su(3) consists of and that allow us to define weight vectors. The weight diagrams of the fundamental representation and its conjugate representation are shown in the - plane in Fig. 1. The raising and lowering operators defined by


are also indicated for the fundamental representation in Fig. 1.

Figure 1: Weight diagrams of the Lie algebra su(3). Fundamental representation consists of three basis states (quarks). Its conjugate representation consists of three basis states (antiquarks). Adjoint representation is spanned by eight bilinear forms of quarks and antiquarks, i.e., mesons.

The SU(3) extension of the AKLT state is obtained as follows. We assume that both representation (, , ) and representation (, , ) are placed on each site. From their tensor product,


we keep the octet representation on each site. This is analogous to keeping an on-site triplet in the SU(2) case. For each pair of neighboring sites, we combine from one site and from the other and project them onto singlet , again similarly to the SU(2) case. At each end of a finite open chain, we have unpaired or states, which form a triplet zero-energy boundary mode. Figure 2 shows schematic pictures of the SU(3) AKLT states. We note that there are two ways of constructing such states; see Fig. 2(a) and (b).Greiter and Rachel (2007) Here we first discuss the state shown in Fig. 2(a) in detail. The other state will be discussed in Sec. III.4.

Figure 2: Schematic pictures of the two types of MPS wave functions for the SU(3) AKLT model. On every site there are states (quark ) and states (antiquark ) which are projected onto states (mesons represented by ovals) through the eight traceless matrices . (a) is coupled through the matrix to on the left neighboring site to form a singlet ( meson). (b) is coupled through the matrix to on the right neighboring site to form a singlet ( meson).

On each site we have states in the bilinear form of and states coupled by traceless matrices ,


where the su(3) operators are the ones defined in Eq. (34) and the raising and lowering operators are defined in Eq. (37). The eight states form the adjoint representation of su(3), corresponding to the octet of mesons:


whose weight diagram is shown in Fig. 1. The su(3) generators for the representation are given by matrices, which are written in this basis as


With a basis transformation [replacing with in Eqs. (40) and (41)], we can rewrite in the standard form


where is the structure constant of su(3) defined from the commutation relation

The singlet state on each bond is given by the bilinear form


which is composed of and states from neighboring sites coupled through the matrix


The MPS wave function of the SU(3) AKLT state shown in Fig. 2(a) is constructed as follows. First, and states from neighboring sites are projected onto the singlet using the matrix,


where and label states of representation and representation, respectively, on site in a 1D periodic chain of length . Second, and states on the same site are projected onto states using the eight traceless matrices in Eq. (39),


where labels physical states in the representation of Eq. (40). Finally, the SU(3) AKLT wave function is obtained as

The normalization constant is

The wave function has the same MPS form as Eq. (5) up to the normalization factor. Alternatively, we can write the MPS wave function asGreiter and Rachel (2007)

with matrices taking values in the local Hilbert space,

Let us construct a Hamiltonian having the above MPS wave function as a ground state, using projection operators acting on the representations (mesons) on every pair of neighboring states, in the same way as in the SU(2) AKLT model. The product of two sets of states from neighboring sites is decomposed as


However, the formation of a singlet on every bond, which was imposed in Eq. (45), implies that the maximum multiplets that can be formed by states from two neighboring sites are actually limited to


Therefore, if a Hamiltonian is a projection operator annihilating both and representations for every pair of states of neighboring sites in Eq. (49), then the MPS wave function in Eq. (47) becomes a zero-energy eigenstate. We can write down such a Hamiltonian using Casimir operators as


where ’s are su(3) operators in the representation given in Eq. (41), and is the eigenvalue of the quadratic Casimir operator, , for -dimensional representations. The MPS wave function in Eq. (47) is a zero-energy eigenstate of , whose eigenvalues are non-negative by construction. Hence the MPS state is an exact ground state. The other SU(3) AKLT state shown in Fig. 2(b) is another zero-energy ground state of , and there is a finite energy gap to excited states.Greiter and Rachel (2007) Using


we can reduce the Hamiltonian to the simpler form


which we shall call the SU(3) AKLT Hamiltonian in the rest of this paper. We note once again that the su(3) generators are in the representation.

iii.3 Symmetry operations of

Here we derive symmetry actions of the symmetry on the eight physical states by using the projective representation [Eq. (33)] and Eq. (9). The operations of and on the octet of matrices yield


These relations determine actions of the symmetry operators on the eight matrices , which we write in the form of Eq. (9) as