key: cord-0615827-8bvqn8fe
authors: Quella, Thomas
title: Symmetry protected topological phases beyond groups: The q-deformed AKLT model
date: 2020-05-18
journal: nan
DOI: nan
sha: 97a01a2ab7503f21b488ee6710a82024f6b73252
doc_id: 615827
cord_uid: 8bvqn8fe

We argue that the $q$-deformed spin-1 AKLT Hamiltonian should be regarded as a representative of a symmetry protected topological phase. Even though it fails to exhibit any of the standard symmetries known to protect the Haldane phase it still displays all characteristics of this phase: Fractionalized spin-$frac{1}{2}$ boundary spins, non-trivial string order and -- when using an appropriate definition -- a two-fold degeneracy in the entanglement spectrum. We trace these properties back to the existence of an $SO_q(3)$ quantum group symmetry and speculate about potential links to discrete duality symmetries. We expect our findings and methods to be relevant for the identification, characterization and classification of other symmetry-protected topological phases with non-standard symmetries.

The Haldane phase of antiferromagnetic spin-1 SU (2) quantum spin chains is one of the prototypes of symmetry-protected topological (SPT) phases [1-3]. It exhibits a unique ground state, an excitation gap, a twofold degeneracy in the bipartite entanglement spectrum and, at least when SO(3) spin-rotation symmetry is preserved, fractional boundary spins as well as non-trivial string order. While these facts have only been verified numerically for the spin-1 Heisenberg model, the associated AKLT model [4, 5] provides an alternative representative of the same phase in which all these properties can be established with full analytical rigor and where the existence of non-trivial string order can be linked to the breaking of a hidden Z 2 × Z 2 symmetry [4] [5] [6] [7] [8] . This is due to the fact that the ground state of the AKLT Hamiltonian has a simple representation as a matrix product state (MPS) [4, 5, 9, 10] . (See also Refs. [11-13] for a general discussion of MPS and finitely correlated states.)

In view of the simple construction and intriguing properties of the AKLT model it is no surprise that variations of this construction have been applied to other symmetries, notably quantum group deformations of the SU (2) spin-rotation symmetry [14] [15] [16] [17] [18] [19] . We will call the resulting spin-1 model which is based on a U q su(2) quantum group symmetry the qAKLT model. From the construction of the qAKLT Hamiltonian as the parent Hamiltonian of an MPS it is evident that the model exhibits fractionalized boundary spins. Moreover, it has been known for a very long time that the model has non-trivial string order which is linked to the breaking of a generalized duality-type Z 2 × Z 2 -symmetry [18] . Its entanglement properties [20] and correlation functions [16] have also been studied in great detail.

However, while these properties are all well known, at least among experts, no one so far has established a link between the qAKLT model and SPT phases [21] . As we will discuss, this is (presumably) related to the extensive breaking of standard group symmetries and, in addition, to the puzzling lack of degeneracies in its entanglement spectrum [20] that would be expected in the presence of boundary spins and quantum group symmetry. With this Letter we aim to close this gap. We explain why and in which sense the quantum group symmetry U q su(2) is capable of protecting the non-trivial topology of the qAKLT model and we revisit the entanglement calculations, showing that the existing literature has been more concerned about implementing periodic boundary conditions than about actually enforcing the quantum group symmetry. We expect that our findings will trigger a systematic search for other SPT phases with non-standard symmetries and that our methods will be helpful in identifying, characterizing and classifying them.

The qAKLT model is defined on the Hilbert space of spin-1 quantum spins. It may be expressed in terms of standard SU (2)-spins S i and in that formulation the qAKLT Hamiltonian reads [14, 16] 

with c = 1 + q 2 + q −2 and b = [c (c − 1)] −1 . Except stated otherwise we will consider open boundary conditions or an infinite chain. The Hamiltonian is hermitian for real values of q. To keep the exposition simple, q > 0 will be assumed throughout this Letter. We note that both parameters b and c are invariant under the substitution q → q −1 . The standard AKLT model (obtained by setting q = 1) has a certain number of symmetries that are known to protect its topological properties [1]. These symmetries are i) SO(3) spin-rotation symmetry [22] , ii) its Z 2 × Z 2 subgroup of π-rotations around the principal axes, iii) inversion of the chain and iv) time-reversal symmetry, an antilinear symmetry implementing the transformation S → − S.

In contrast, the Hamiltonian (1) is anisotropic for q = 1 and hence breaks SU (2) spin-rotation symmetry. The only continuous symmetry that is left is a U (1)-symmetry by rotations around the z-axis. The anisotropy also breaks the Z 2 × Z 2 symmetry group of π-rotations. Finally, inversion symmetry and time-reversal symmetry are broken by the term (S z j ) 2 S z j+1 − S z j (S z j+1 ) 2 . To summarize, all the discrete symmetries known to protect the Haldane phase are broken explicitly. According to the general classification [1-3], the Hamiltonian (1) should thus not be regarded as residing in an SPT phase.

As will be discussed in the following section, the Hamiltonian (1) is the parent Hamiltonian of an MPS that is constructed using the representation theory of U q su(2) . As such, it is naturally invariant under the action of this quantum group that defines a q-deformation of the su(2) spin algebra [23] . This deformation is defined in terms of q-spin generators S satisfying the relations

For the spin-1 representation these commutation relations are satisfied with the identification

Even though these expressions are just rescaled versions of the standard spin operators, there is another important signature of the q-deformation: The action of the qspins S on tensor products is not simply given by adding up the q-spins for individual factors but rather by applying a so-called coproduct. For two tensor factors this coproduct reads

and the action on multiple tensor factors (such as the whole spin chain) is obtained by iterating this action appropriately. We note that the standard action of su(2) is only recovered in the limit q → 1. The main result of this Letter is that the quantum group symmetry just sketched (or rather an incarnation adapted to the fact that we are dealing with a spin-1 chain) is capable of protecting the topological properties of the qAKLT state. It needs to be emphasized though that this symmetry is only present when the chain is (semi)infinite or considered with open boundary conditions. If periodic boundary conditions are used, as is the case in most of the literature on this subject, the Hamiltonian and also its ground state are not invariant under U q su(2) . This symmetry is only restored if specific twisted boundary conditions are used. These facts are well known in the community working on quantum group invariant integrable models, see e.g. Refs. [24, 25] , and will also play an important role for establishing our main result. However, to keep the present discussion focused we will restrict our attention to open boundary conditions and infinite systems.

We would also like to stress that the breaking of some of the original discrete symmetries is rather mild. The π-rotations around the x-and y-axis as well as inversion and time-reversal remain symmetries if they are accompanied by the transformation q → q −1 . However, these are not symmetries, strictly speaking, but rather duality transformations, mapping one model to another one that is physically equivalent. Such duality symmetries are not covered by the standard classification of SPT phases [1-3] and hence require a new approach [26] . We defer this interesting question to future research and content ourselves with a detailed discussion of the continuous U q su(2) symmetry. Judging from the special case q = 1 we expect that continuous and discrete duality-type symmetries lead to the same Z 2 -classification of SPT phases in the present situation.

The starting point of the qAKLT construction is a state (or rather a set of four states) that can be represented in the form of an MPS. More precisely we define

for a finite chain of length L with open boundary conditions. In this expression, α, β = ±1/2 denote the degrees of freedom associated with a left and right spin-1 2 boundary spin and the rest of the product is a mixed matrix/tensor product of matrices B i that all have the form [16] (see Appendix)

where Ω = (q +q −1 )/(1+q 2 +q −2 ). The index i indicates on which site the physical states |0 and |± live. The qAKLT state (5) arises from a valence bond construction involving the spin-1 representation as the physical spin and two spin-1 2 spins as auxiliary spins [16] (see Appendix). The associated transfer matrix has a nondegenerate (in modulus) eigenvalue 1 and this translates into the existence of a mass gap.

For completeness we note that the qAKLT state should be defined by

when considering closed boundary conditions. The insertion of the twist q 2S z acting on the auxiliary space guarantees invariance under the quantum group U q su(2) in this setting, see Appendix. As will be discussed in the following section it is precisely this twist (and other closely related ones) that enable us to derive entanglement properties that support the interpretation of the qAKLT model as a representative of an SPT phase.

The entanglement properties of the qAKLT state were discussed in Ref. [20] . These considerations were based on an MPS with periodic boundary conditions which, as we have pointed out, is not invariant under the action of U q su(2) . It is thus no surprise that the results do not reflect the degeneracies appropriately that would be expected as a result of U q su(2) -invariance and the presence of fractionalized spin-1 2 boundary spins. To avoid finite-size effects we work in an infinite chain where the MPS can be interpreted as a translation invariant iMPS [13, 27, 28] . The MPS tensor (6) is in right canonical form and normalized, i.e. it satisfies BB † = I where the product is taken both over the physical and the auxiliary index [13] . It may be checked that the al-

The importance of these tensors lies in the fact that the two sets of semi-infinite states

with α = ±1/2 are orthonormal on the left and right semi-infinite Hilbert space, respectively. For this reason the expression

is a Schmidt decomposition and permits to read off the (non-degenerate) entanglement spectrum ǫ α = − log Λ 2 α and entanglement entropy S EE = − α Λ 2 α log Λ 2 α from the diagonal matrix Λ. We note in passing that this result precisely corresponds to the entanglement present in the normalized U q su(2) -singlet

This could of course be expected since the latter is precisely what is used to describe the singlet bonds in the valence bond construction of the MPS. It is clear that this state has lower entanglement than a Bell state, at least for generic values of q. However, the two spins are still fully entangled in the sense that they form a singlet with respect to the quantum group symmetry, so the states are always entangled, regardless of the value of q, and all coefficients are completely fixed up to normalization. This type of entanglement can be captured by a qdeformed definition of the reduced density matrix. Following Ref.

[29] we define

where ρ is the density matrix associated with |qAKLT ∞ and S z L corresponds to the action of S z on the left part of the chain which is traced out. The MPS tensor satisfies the equivariance property

where the symbol ⊲ on the left hand side denotes an action on the physical space and the conjugation on the right hand side acts on the virtual spins by means of S = S where S are the standard spin operators in the spin-1 2 representation. Using the (trivial) coproduct for S z it can then be shown that

i.e. the action of q −2S z can be pushed to the auxiliary level. The q-deformed entanglement spectrum thus reads

and shows a 2-fold degeneracy which arises from the presence of virtual fractionalized spin-1 2 boundary spins. Given the q-deformed reduced density matrix it is then straightforward, using again the equivariance property (13), to calculate the associated q-deformed entanglement entropy [29] which is given by

We note that the result is just the logarithm of the socalled qantum dimension of the spin-1 2 representation describing the virtual fractionalized boundary spin. The full degeneracy of the q-deformed entanglement spectrum and the reduction of the q-deformed entanglement entropy to the logarithm of the quantum dimension are easily verified to generalize to singlet bonds between arbitrary spin-S representations of U q su(2) (see Appendix).

Let us finally adopt a slightly more general perspective. It is known that any gapped ground state |ψ can be well approximated by means of an MPS. With U q su(2) symmetry and integer physical spins there are two distinct classes of MPS, just as for the standard su(2) case. This is due to the fact that the representation theory of U q su(2) at real values q = 0 precisely mimics the representation theory of su(2), including labeling and dimensions of irreducible representations and tensor product decompositions [30] .

In particular, integer physical spins can only arise from either two integer or two half-integer auxiliary spins. For su(2), all of these representations lift to SU (2) while only integer spin representations lift to SO(3) = SU (2)/Z 2 . In contrast, half-integer spins are only projective representations of SO(3) since they have a non-trivial action of the central subgroup Z 2 ⊂ SU (2). This representation of the center Z 2 can be interpreted as a topological invariant [31] . As discussed in more detail in Ref. [32] , similar statements hold true for the quantum group U q su(2) which allows to define two associated mathematical structures SU q (2) and SO q (3) which should be interpreted as distinct exponentiated versions of U q su(2) .

Just as for the undeformed case, the entanglement in MPS with integer auxiliary spins can be removed while this is not the case for half-integer auxiliary spins if we insist on the preservation of SO q (3) symmetry. We thus expect a Z 2 -classification of U q su(2) -invariant quantum spin chains based on integer spins, the qAKLT model being a representative of the non-trivial phase [33] .

We note that singlet bonds between two spin-S representations lead to a (2S + 1)-dimensional degeneracy in the q-deformed entanglement spectrum (see Appendix). For half-integer S this degeneracy is even while it is odd for integer spins. Just as for the ordinary Haldane phase [8] there is thus a characteristic entanglement signature of the topologically non-trivial phase which is protected by SO q (3) symmetry.

We have presented overwhelming evidence that the qAKLT model (with q > 0) should be regarded as a representative of a novel type of SPT phase, protected by the q-deformed symmetry SO q (3). Even though the qAKLT state on an infinite chain shows no degeneracy in the standard mid-cut entanglement spectrum, a twofold degeneracy is recovered when defining the reduced density matrix with an appropriate quantum trace. This statement remains true even if the entanglement in the chain is very low (i.e. for q ≪ 1 or q ≫ 1). In addition, the state exhibits fractionalized spin-1 2 boundary spins and non-trivial string order, as already found in earlier studies [18] . Our findings open many directions of further investigations. Some of the models of recent physical interest involve quantum group symmetries with q a root of unity. This is for instance the case for the anyonic chains that have been introduced in Ref. [34] and for the abstract classification of topological field theories that arise from intertwiner dynamics [35] . In both cases, AKLT-like states are known to exist. It would thus be interesting to investigate whether our results and methods carry over to the case |q| = 1 and, in particular, roots of unity where the representation theory of U q su(2) becomes considerably more intricate [30] . Obviously, it would also be natural to revisit higher spin instances of the qAKLT model [36] [37] [38] [39] .

However, the most important and far-reaching question is whether there are other kinds of generalized symmetries that are capable of protecting topological order in 1D or even higher-dimensional systems. In a companion paper [32] we show that the ideas presented in this Letter readily generalize to spin chains with arbitrary quantum group symmetry U q [g] (with real q = 0) where g is a finite dimensional simple Lie algebra such as su(N ), so(N ) or sp(2N ). These results of Ref. [32] extend the classification of Ref.

[31] which has been established for simple Lie groups.

A similar and hence natural class of symmetries deserving further investigation are elliptic quantum groups, i.e. two-parameter deformations of Lie algebras. On general grounds one would expect arbitrary Hopf- * algebras to be good candidates for generalized symmetries, potentially with additional restrictions on their structure.

In this context one should also ask about the role of the discrete duality symmetries that were discussed in the main text and that involve the transformation q → q −1 . Are these capable of protecting topological phases by themselves and, if yes, how does the resulting classification relate to the one obtained with respect to the continuous quantum group? For the undeformed case these questions were answered in Refs. [40, 41] .

Since even the examples of generalized symmetries that have been discussed here and in Ref. [32] are not amenable to the standard definition of projective representations a new mathematical framework will need to be developed to describe the topological invariants in full generality. It is likely that this framework will employ tools from non-commutative geometry, the natural generalization of geometric objects to an abstract algebraic setting.

Finally, thinking of implications of our findings beyond one dimension it should be noted that matrix product operators played a significant role in the treatment of two-dimensional topological phases [42, 43] . Our findings may help to generalize some of these considerations. There exist different conventions for the quantum group U q su(2) and hence it is useful to be explicit about the convention we use. Our definition follows Ref.

[30] even though it should be noted that what we call U q su(2) is calledȖ q (sl 2 ) in that book.

The algebra structure of the quantum group is encoded in the commutation relations

which define a q-deformation of the Lie algebra su(2) which is recovered as q → 1. Alternatively, the first two relations can be written as

The quantum group U q su(2) carries the structure of a Hopf algebra. To complete the definition we need to specify a coproduct ∆ : U q su(2) → U q su(2) ⊗ U q su(2) which, from a practical perspective, permits to define the notion of tensor product representations. There are different choices available but for reasons to be described below we work with the definition

In addition to the coproduct we also need to define the unit η : C → U q su(2) , the counit ǫ : U q su(2) → C and the antipode S : U q su(2) → U q su(2) . The latter is given by

The other functions have the form ǫ ≡ 0 (on the generators S z and S ± ) and η(1) = I. Since quantum spin chains are defined on a Hilbert space we also need to introduce a suitable notion of hermitian conjugation. This is captured by a Hopf- * structure. With our choice of coproduct and for real values of q the Hopf- * structure is given by

We note that a different choice of coproduct (as is sometimes preferred in the mathematics literature) would lead to different and physically rather unnatural expressions for the hermitean conjugation.

The representation theory of U q su(2) for real values of q = 0 very much mimics the well-known representation theory of su(2) [30]. All finite dimensional representations are fully reducible. The quantum group has irreducible representations V j that are labelled by a spin j = 0, 1 2 , 1, . . . and have dimension 2j + 1. Moreover, the decomposition of a tensor product j 1 ⊗ j 2 precisely corresponds to the well-known decomposition for su(2). In view of the non-trivial (and q-dependent) coproduct, concrete expressions for Clebsch-Gordan coefficients are different though.

The representation V j is spanned by (orthonormal) vectors |m with m = −j, . . . , j on which the generators of U q su(2) act by

These expressions make use of so-called q-numbers [x] q that are defined via

An important invariant of the representation V j is the quantum dimension dim q (j) that is defined by

and approaches the standard dimension dim(j) = 2j + 1 in the limit q → 1.

In what follows we will mainly be concerned with the representations V 0 , V 1 2 and V 1 , so let us consider these more explicitly. The representation V 0 is the one-dimensional trivial representation in which all generators act by zero. The representation V 1 2 is the two-dimensional fundamental representation with basis {|↑ , |↓ } on which the generators act by

We note that these expressions precisely agree with the expressions for su(2)-spins in the fundamental representation, so S = S here. Finally, the representation V 1 is the three-dimensional representation with basis {|+ , |0 , |− } where S z = diag(1, 0, −1) and, defining ξ = q + q −1 ,

In this case we have an identification S z = S z and S ± = ξ/2 S ± with the standard su(2) spins. We note that all these expressions are consistent with our choice of Hopf- * structure.

In the main text we used an expression for the MPS that is different from others that can be found in the literature. We therefore include the simple derivation here. For the construction of the MPS tensor we look at one physical site which is comprised of two auxiliary spins plus the left auxiliary spin of its neighbor to the right. We thus consider the tensor product V 1 2 ⊗ V 1 2 ⊗ V 1 2 and start by inducing a singlet in the right two factors. This is done by means of the map I 0 :

Up to normalization this yields the two states

where α ∈ {↑, ↓}. On these states we act with the projector onto the S = 1 component in the first two factors which can easily be confirmed to be given by

Writing the resulting states in matrix form in the standard basis of V 1 2 we find

This expression turns out to be right canonical but not normalized. Including the correct normalization leads to the MPS tensor (6) used in the main text.

The MPS tensor defined in Eq. (6) satisfies the equivariance relations

as can easily be checked case by case. We note that the action on the right hand side of these equations just corresponds to the use of (I ⊗ S)∆(• * ) as expected from the general Hopf algebra structure of U q (sl 2 ). The first two lines reflect what one would have for the action of an ordinary Lie algebra and a group, respectively. (See Ref. [44] for a discussion of equivariance properties of general MPS tensors in the group case.)

The identities above are valid for a single site. In an MPS one has mixed tensor/matrix products of the form

where the index is referring to the site of the physical spin. This product has the equivariance properties

which are an immediate consequence of the relations (30) for the individual tensors. Indeed, the terms created between two Bs simply drop out. When verifying these relations it is important to work with the correct coproduct for a multiple tensor product.

An important consequence of the relations (32) is that an MPS with periodic boundary conditions is not invariant under the action of U q su(2) . Instead one will need to work with the quantum trace |MPS = tr q 2S z B 1 · · · B L .

It is evident that this state is invariant under the action of S z and q αS z . When acting with S ± we find

where we have introduced the abbreviation X = B 1 · · · B L . The necessity for the use of quantum traces in a quantum group context has been known for a long time in the context of quantum integrable systems, see e.g. Refs. [25, 29] .

In the main text we focused on the qAKLT state for the spin-1 representation of U q su(2) . This state is ob-tained from a valence-bond construction involving two spin-1 2 auxiliary spins. It is straightforward to generalize this construction to spin-S auxiliary spins, resulting in a spin-2S analogue of the qAKLT state [36] . Just as for the ordinary qAKLT state, the correlation functions and entanglement properties of these states have been discussed in great detail [37] [38] [39] . As these entanglement considerations were based on periodic boundary conditions which are not compatible with invariance under U q su(2) we now revisit this issue from an iMPS perspective.

To be precise, let us consider the singlet bond between two spin-S representations which serves as a higher spin model for the Schmidt decomposition (10) in the main text. Using the explicit action (23) of the quantum group generators it may easily be verified that the normalized singlet state can be written as

where the quantum dimension dim q (S) has been defined in Eq. (25) . If ρ denotes the associated density matrix then the q-deformed reduced density matrix in this state is given by

We see that the resulting q-deformed entanglement spectrum exhibits a full degeneracy of its dim(S) q-deformed entanglement energies

where m = −S, . . . , S. It is then also straightforward to determine the associated q-deformed entanglement entropy which is given by

We recognize that the q-deformed entanglement entropy precisely captures the quantum dimension of the two spins forming the singlet bond.

q-deformations of the O(3) symmetric spin-1 Heisenberg chain

Equivalence and solution of anisotropic spin-1 models and generalized t-J fermion models in one dimension

Groundstate properties of a generalized VBS-model

q-deformations of quantum spin chains with exact valence-bond ground states

Hidden symmetry breaking in a generalized valence-bond solid model

Quantum spin chains with quantum group symmetry

Entanglement spectra of the q-deformed Affleck-Kennedy-Lieb-Tasaki model and matrix product states

A connection between quantum group symmetries and SPT phases was already anticipated in [31]. Speculations about the qAKLT model (and its generalizations to higher spin) realizing an SPT phase have also appeared in

However, the latter suggestion seems to boil down to a pure analogy

We could say SU (2) here since SO(3) = SU (2)/Z2 and the Z2 subgroup acts trivially on integer spins. However, from a more general perspective it is more appropriate to think of SO(3) as the protecting symmetry

Our conventions are based on what is calledȖq(sl2) in Ref. [30]. We note that Ref. [30] has reserved the symbol Uq(sl2) to denote the same quantum group with a different coproduct

Common structures between finite systems and conformal field theories through quantum groups

Bethe ansatz solution of a closed spin 1 XXZ Heisenberg chain with quantum algebra symmetry

We note that the combination of two of these tansformations is still a symmetry of the Hamiltonian (1). However, inversion symmetry for instance can easily be broken by staggering the couplings without changing the ground state or its topological properties

Classical simulation of infinite-size quantum lattice systems in one spatial dimension

Infinite time-evolving block

Symmetry protected topological phases beyond groups: q-deformed symmetries

There should be no symmetry-protection if half-integer physical spins are involved

Anyonic quantum spin chains: Spin-1 generalizations and topological stability

Topological lattice field theories from intertwiner dynamics, arXiv e-prints

The matrix product representation for the q-VBS state of one-dimensional higher integer spin model

Spin-spin correlation functions of the q-valence-bond-solid state of an integer spin model

Entanglement spectra of q-deformed higher spin VBS states

Entanglement Properties of a Higher-Integer-Spin AKLT Model with Quantum Group Symmetry

Hidden symmetry-breaking picture of symmetry-protected topological order

From symmetryprotected topological order to Landau order

Twisted injectivity in projected entangled pair states and the classification of quantum phases

Anyons and matrix product operator algebras

Matrix product states: Symmetries and twobody Hamiltonians

APPENDIX: SUPPLEMENTAL MATERIAL The quantum group Uq su