AAn index for two-dimensional SPT states
Nikita Sopenko
California Institute of Technology, Pasadena, CA 91125, USA
January 5, 2021
Abstract
We define an index for 2d G -invariant invertible states of bosonic lattice systemsin the thermodynamic limit for a finite symmetry group G . We show that this indexis an invariant of SPT phase. In the last few decades it became clear that many important physical properties ofmany-body system with a gapped local Hamiltonian are encoded in the entanglementpattern of its ground state, and do not depend on the details of the Hamiltonian[1]. In particular a gapped phase of such system, i.e. equivalence class under smalldeformations of the Hamiltonian which do not close the gap, can be determinedfrom its ground state. The latter is based on the fact that the ground states ofany two such systems in the same phase can be related by an evolution by a localHamiltonian [3], also known as quasi-adiabatic evolution.Importantly, the notion of a gapped phase makes sense only in the thermodynamiclimit. If one wants to define invariants of phases and eventually classify them,one is forced to work with quantum states in the infinite volume. Thereforecharacterization of such states is of fundamental importance.A particularly simple class of states was introduced by A. Kitaev [2] and calledinvertible. By definition a state is in invertible phase if it can be tensored withsome other state so that the resulting state is in the trivial phase, i.e. it can becompletely disentangled by a local Hamiltonian evolution. In the presence of asymmetry group G a G -equivariant version of such phases can be introduced, knownas symmetry protected (SPT) phases [4]. Invertible state is said to be in a non-trivialSPT phase, if it is G -invariant and can be inverted only by a Hamiltonian evolutionthat necessarily breaks G symmetry . More precisely one should consider stable equivalence [2] as we define in the main text. Many distinctive properties of a gapped phase can be well demonstrated on a Hilbert space of alarge but finite system. Though such approach is very insightful, from the theoretical standpoint it is notsatisfactory since it does not even allow to give a definition of a phase. See the main text for a precise definition. a r X i v : . [ m a t h - ph ] J a n hile one-dimensional SPT states are well understood by now , in higher di-mensions the theory is far from complete, despite a large number of examples ofSPT states. A powerful approach based on group cohomology was proposed in [10].However, soon after it was realized that there are states beyond this cohomologyclassification. It is believed that all such states are captured by a classification ofinvertible topological field theories [11, 12]. It is desirable to prove this fact.For 2d bosonic SPT states with a unitary action of a finite group G bothapproaches predict classification by H ( G, U (1)). In this paper we define an indexfor such systems taking values in H ( G, U (1)) using the formalism developed in[15] for G = U (1). Our approach is motivated by the work [16]. The absence ofnon-trivial invertible one-dimensional states needed for the definition of the indexwas proved recently in the work [9], to which this paper is complementary.This paper is organized as follows. In Section 2 we define the setup, introducenecessary concepts and give some of its properties. In Section 3 we define the indexand check that it gives expected results for states from [10]. The proof of the lemmaneeded for the definition is discussed in Appendix A. Acknowledgements:
I would like to thank Anton Kapustin for many inspiringdiscussions and comments on the draft. This research was supported in part by theU.S. Department of Energy, Office of Science, Office of High Energy Physics, underAward Number DE-SC0011632.
By a d -dimensional lattice Λ we mean a Delone set in R d . The elements of Λ arecalled sites. The distance between points j and k is denoted by | j − k | . An embedded d -dimensional submanifold of R d whose boundary has a finite number of connectedcomponents is called region. By a slight abuse of notation, we will identify a regionand its intersection with the lattice Λ. The complement of a region Γ is denoted by¯Γ. Let A j = End( V j ) be a matrix algebra on a finite dimensional Hilbert space V j , and let A Γ = N j ∈ Γ A j for a finite subset Γ ⊂ Λ. A bosonic d -dimensional spinsystem is defined by its algebra of observables A which is a C ∗ -algebra defined as anorm completion of the ∗ -algebra of the form A ‘ = lim −→ Γ A Γ (1)The numbers d j = dim V j are assumed to be uniformly bounded by d j < D . Theelements of A are called quasi-local observables, while the elements of A Γ for finite Based on earlier approaches using matrix product state approximation [5], the invariants of phaseswere defined in a series of works [6, 7, 8]. A completeness for bosonic systems with respect to localHamiltonian evolution was shown recently in [9]. For an earlier development see [13, 14] ⊂ Λ are called local observables localized on Γ. Following [15] we say that A is analmost local observable f -localized at a point p , if for any B ∈ A k we have k [ A , B ] k ≤ kAkkBk f ( | p − k | ) (2)for some monotonically decreasing positive (MDP) function f ( r ) = O ( r −∞ ). Suchobservables form a dense ∗ -sub-algebra A a‘ of A .We denote the partial (normalized) trace over a region ¯ B by A| B := Z Y k ∈ ¯ B dU k Ad Q k ∈ ¯ B U k ( A ) . (3)where the integration is over all on-site unitaries U k ∈ A k with Haar measure.Any A ∈ A a‘ almost local at site j can be decomposed as an infinite sum of localobservables A = P ∞ n =1 A ( n ) on disks Γ n ( j ) of radius n with the center at j , where A ( n ) are defined by m X n =1 A ( n ) = A| Γ m ( j ) (4)Note that kA ( n ) k ≤ g ( n ) for some MDP function g ( r ) = O ( r −∞ ) that depends on f ( r ) only (see Lemma A.1 of [15]).We say that a symmetry group G acts on A if for any site j we have homomor-phisms R j : G → U ( V j ) to a unitary group of V j . We denote the image of g ∈ G by R ( g ) j . For any (possibly infinite) region A we define automorphisms w ( g ) A ∈ Aut( A )as a conjugation Ad R ( g ) A with a (possibly formal) tensor product R ( g ) A := N j ∈ A R ( g ) j .When A = R d we omit A and simply write w ( g ) . In this paper we always assumethat G is a finite group.For a state ψ on A we denote its evaluation on observables by hAi ψ := ψ ( A ).We define the action β ( ψ ) of automorphisms β ∈ Aut( A ) on states via hAi β ( ψ ) = h β ( A ) i ψ . For a region A we denote the restriction of a state ψ on A by ψ | A . We calla pure state ψ factorized, if it satisfies hABi ψ = hAi ψ hBi ψ for any observables A and B localized on two different sites. Sometimes, if we want to specify the algebra A on which a given state ψ is defined, we write ( A , ψ ). In the following all stateson the whole algebra A will be pure.We say that an automorphism β ∈ Aut( A ) is almost local, if it is also anautomorphism of A a‘ . We denote the group of such automorphisms by Aut a‘ ( A ). The Hamiltonians of infinite systems are not elements of the algebra of observables.We can only define them by a formal sum F = X j ∈ Λ F j (5)of observables F j somehow localized at site j . We formalize this vague statementby requiring that F j is a self-adjoint almost local observable f -localized at j with niformly bounded k F j k (see Remark 2.2). Following the terminology of [15] we callsuch objects 0-chains. We denote the group of such 0-chains under on-site additionby C . On A a‘ it defines a derivation ad F ( A ) = P j ∈ Λ [ F j , A ] ∈ A a‘ .For a region A , by F A we denote a 0-chain ( F A ) j := ( δ j ∈ A ) F j , where δ j ∈ A is 1when j ∈ A and 0 otherwise. By F | A we denote a 0-chain ( F | A ) j := ( F j ) | A .We say that F is approximately localized on a submanifold A , if k F j k ≤ g (dist( j, A )) for some MDP function g ( r ) = O ( r −∞ ). We denote the group ofsuch chains under on-site addition by C A . Similarly, a unitary operator that implements an evolution by a Hamiltonian doesnot exist as an element of the algebra of observable for infinite systems. Instead,we can only define automorphisms of the algebra of observables generated by someHamiltonian.Let F ( s ) be a 0-chain, that depends on s ∈ [0 , α F ( s ) ∈ Aut( A ) by − i dds α F ( s )( A ) = α F ( s )(ad F ( A )) , α F (0) = Id (6)for A ∈ A al and by extending it to the whole algebra A . Note that α F ( s ) ∈ Aut a‘ ( A ).We call a pair ( α F ( s ) , F ( s )) locally generated automorphism (LGA), and if anautomorphism β ∈ Aut a‘ ( A ) coincides with α F (1) for some 0-chain F ( s ), we saythat it can be locally generated. In the following by a slight abuse of notation we use α F both for the pair ( α F ( s ) , F ( s )) and for the automorphism α F (1). We also omit s in F ( s ) and simply write F . We want to stress, that F is a part of the informationfor LGA. Remark 2.1.
Our definition of LGA is a generalization of the well-known notionof finite-depth quantum circuits (FDQC). While the latter is extremely useful inquantum information theory and for construction of examples of non-trivial states,it is not very suitable for a definition of phases and their invariants, since they onlygive some approximation to actual evolutions by Hamiltonians, which are not strictlylocal. Another reason why we introduce the notion of LGA is because it carriesthe information about the generating Hamiltonian, that will be important for ourdefinition of the index.
Remark 2.2.
One may wonder why we define LGA with this particular decay forthe generating Hamiltonian. The reason is twofold. First, quasi-adiabatic evolutionbetween ground states of gapped local Hamiltonians can be implemented only if werequire the decay to be not faster that superpolynomial [17]. Second, we want thedecay to be independent of the dimension of the system, so that the classification ofdefect states (defined below) depends only on the dimension of the defect, not of theunderlying space.LGAs form a group. Indeed, the associative composition α F ◦ α G of two LGAs We use symbol ◦ both for composition of LGAs as defined in the main text and for composition ofautomorphisms of A . F and α G is an LGA generated by G ( s )+( α G ( s )) − ( F ( s )), while the inverse ( α F ) − of α F is generated by − α F ( s )( F ( s )). The unit element is given by α which is acanonical LGA for the identity automorphism Id. We denote this group by G ( A ) orsimply G if the choice of the algebra is clear from the context.Any automorphism β ∈ Aut a‘ ( A ) gives an automorphism of the group G definedby β ( α F ) := α β ( F ) (7)where β ( F ) j := β ( F j ).We say that α F is approximately localized on a submanifold A , if F ∈ C A .If α F and α G are approximately localized on A , then their composition is alsoapproximately localized on A . The same is true for the inverse ( α F ) − . That allowsus to define a group G A of LGAs approximately localized on A , which is a subgroupof G . Moreover, any automorphism β ∈ Aut a‘ ( A ) gives an element of Aut( G A ) since β ( F ) ∈ C A for any F ∈ C A .If α F ∈ G p for a point p , then P ( s ) = P j F j ( s ) is an almost local observable. Inthis case there is a canonical unitary observable V defined by V = T s { e i R dsP ( s ) } ,where T s { ... } is an ordered exponential with respect to s , such that α F (1) = Ad V .We say that V is the corresponding unitary observable for α F . Note that if V and V are the corresponding unitary observables for α F , α F ∈ G p , then ( V V )is the corresponding unitary observable for α F ◦ α F . Also note that if V is thecorresponding observable for α F ∈ G p , then α G ( V ) is the corresponding observablefor α G ◦ α F ◦ ( α G ) − ∈ G p (see Lemma 2.2 below).A few elementary lemmas below will be useful for us later. Lemma 2.1. If G ∈ C A , then α F + G ◦ ( α F ) − ∈ G A . Proof. α F + G = α G ◦ α F for G ( s ) = α F ( s )( G ( s )) ∈ C A . Lemma 2.2. If G ∈ C A , then α F ◦ α G ◦ ( α F ) − ∈ G A . Proof.
Note that α F ◦ α G = α F + G with G = G + (( α G ( s )) − ( F ( s )) − F ( s )) ∈ C A .By Lemma 2.1 we have α F + G ◦ ( α F ) − ∈ G A . Lemma 2.3.
For any given region A , any α F ∈ G can be decomposed as α F = α F (0) ◦ α F | A ◦ α F | ¯ A (8) for some α F (0) ∈ G ∂A . Proof.
We choose G = − ( F − F | A − F | ¯ A ) ∈ C ∂A and apply Lemma 2.1. Lemma 2.4. If α F ∈ G A − ∪ A + for A + and A − lying at non-intersecting cones withthe same origin p , then we can split α F = α F − ◦ α F + , such that α F ± ∈ G A ± . Proof.
That is a direct consequence of the Lemma 2.3
Lemma 2.5. If α F − ◦ α F + = α for α F ± ∈ G A ± with A + and A − lying at non-intersecting cones with the same origin p , then α F ± ∈ G p . roof. If F − + F + = 0 for F ± ∈ C A ± , then F ± ∈ C p . Since α F − ◦ α F + is generatedby F + ( s ) + ( α F + ( s )) − ( F − ( s )) satisfying this condition, we get the statement of thelemma. Lemma 2.6.
Any LGA α F on a one-dimensional lattice Λ with sites j ∈ Z ⊂ R can be equivalently defined as an ordered composition α F = −−−→ ∞ Y n = −∞ α F n := ... ◦ α F n − ◦ α F n ◦ α F n +1 ◦ ... (9) of LGAs α F n generated by almost local self-adjoint observables F n . Proof.
Indeed, we have α F n + F ( n, + ∞ ) = α α F ( n, + ∞ ) ( F n ) ◦ α F ( n, + ∞ ) . (10)On the one hand, if we have a 0-chain F we can define F n to be α F ( n, + ∞ ) ( F n ). Onthe other, if we have LGAs α F n , we can define 0-chain F n by F n = −−−→ ∞ Y k = n +1 α F k − ( F n ) . (11) There is a natural operation on states known as stacking. Given two state ( A , ψ )and ( A , ψ ) we can construct a new state ( A ⊗ A , ψ ⊗ ψ ). In the following wedenote it by ψ ⊗ ψ since it is clear that the resulting state is defined on the tensorproduct of the corresponding quasi-local algebras.We say that two states ψ and ψ on A are LGA-equivalent, if there is α F ∈ G ,such that ψ = α F ( ψ ). We call states which are LGA-equivalent to a factorizedstate short-range entangled (SRE). We say that two states ψ and ψ are stablyLGA-equivalent, if there are factorized states ( A , ψ ) and ( A , ψ ) such that ψ ⊗ ψ and ψ ⊗ ψ are LGA-equivalent. The latter defines an equivalence class on statescalled phase .Stacking induces a commutative monoid structure on phases with an identitybeing the trivial phase τ (the phase of a factorized state). We denote this monoidby (Φ , • , τ ). Invertible elements of Φ form an abelian group Φ ∗ and are called invertible phases . States ψ which are representatives of such phases [ ψ ] ∈ Φ ∗ arecalled invertible .In the presence of a symmetry group G we can consider a class of G -invariantstates and define the same notions using G -invariant LGA equivalence and G -invariantfactorized state. The latter is defined to be a factorized state ψ with G -invariantvectors | v j i ∈ V j , such that hA j i ψ = h v j |A j | v j i for any A j ∈ A j (see [9] for moredetails). Similarly, we have a monoid (Φ G , • , τ G ) of G -invariant phases with abeliangroup Φ ∗ G of G -invertible phases. We call a G -invariant phase symmetry protected (SPT) if is mapped to Φ ∗ under a forgetful map Φ G → Φ. Representatives of SPTphases are called
SPT states . A Figure 1: Definition of a cone-like region.
We call a state ψ SRE defect state on a submanifold A for a pure factorized state ψ , if it is LGA-equivalent to ψ via α F ∈ G A .We call a state ψ invertible defect state on a submanifold A , if there is anothersystem ( A , ψ ) with a pure factorized state ψ and a defect state ψ on A for ψ ,such that ψ ⊗ ψ is an SRE defect state for ψ ⊗ ψ . Remark 2.3.
If a lattice Λ in R n is a sublattice of some lattice ˜Λ in R m for n < m ,then SRE and invertible states on Λ naturally give SRE and invertible defect stateson R n ⊂ R m .A natural way to produce an invertible defect state on a ( d − ∂A of a region A for a factorized state ψ is to take some α F ∈ G , suchthat α F ( ψ ) = ψ , and consider ψ = α F A ( ψ ). Lemma 2.7.
All such states are invertible defect states.
Proof.
First, note that ( F A − F | A ) ∈ C ∂A , and therefore by Lemma 2.1 ψ = α F A ( ψ )and α F | A ( ψ ) are LGA equivalent by an element of G ∂A . Second, by Lemma 2.3the state α F | A ◦ α F | ¯ A ( ψ ) is LGA equivalent to ψ by an element of G ∂A . Let ustake a copy ( A , ψ ) of the system ( A , ψ ) with a defect state α F | ¯ A ( ψ ). Since therestrictions ψ | ¯ A and ψ | A are factorized pure states, the state α F | A ( ψ ) ⊗ α F | ¯ A ( ψ )is LGA equivalent to ψ ⊗ ψ by an element of G ∂A ( A ⊗ A ). Remark 2.4.
In fact in this way any invertible state for a lattice Λ ⊂ R n can berealized as an invertible defect state for some lattice in R n +1 on the boundary ofa half-plane. For that we can take a lattice that coincides with Λ at even valuesof x n +1 and with Λ (the lattice of the inverse system) at odd values of x n +1 . Theautomorphism α F can be constructed using Eilenberg swindle.From now on we consider two dimensional lattices Λ ⊂ R only. We define acone-like region A with the origin at the point p to be a region which asymptoticallybehaves as a cone with positive angle (see Fig. 1).Using the results of [9] we have the following lemma, which we prove in theappendix. Lemma 2.8.
All invertible defect states of 2d systems on the boundary ∂A of acone-like region A are SRE defect states. A ( ∂A ) − ( ∂A ) + Figure 2: A is a cone-like region at p with boundary components ( ∂A ) − and( ∂A ) + . Corollary 2.8.1.
All invertible defect states at a point are SRE defects states.
The last corollary is also a direct consequence of Lemma 4.1 from [9].
Corollary 2.8.2.
Let ψ be a 2d SRE state, and let α Q ∈ G , such that α Q ( ψ ) = ψ .Then for any region A there is K ∈ G ∂A , such that ( α Q A ◦ α K )( ψ ) = ψ . Proof.
Note that ( α F ◦ α Q A ◦ ( α F ) − )( ψ ) = α Q A ( ψ ) for Q = α F ( Q ) and any state ψ . If ψ = α F ( ψ ) for some factorized state ψ , then ( α F ◦ α Q ◦ ( α F ) − )( ψ ) = α Q A ( ψ ) = ψ . By Lemma 2.7 and Lemma 2.8 there is α K ∈ G ∂A such that( α Q A ◦ α K )( ψ ) = ψ . Therefore we can take α K = ( α F ) − ◦ α K ◦ α F . Corollary 2.8.3.
Let ψ be a 2d SRE state, and let α Q ∈ G ∂A for a cone-like region A , such that α Q ( ψ ) = ψ . Then for a splitting Q = Q − + Q + , such that Q ± ∈ C ∂A ) ± ,there is N ∈ C p , such that ( α Q − ◦ α N )( ψ ) = ψ . Proof.
In the same way as in the proof of Corollary 2.8.2 we can choose Q = α F ( Q )and argue that there is α N ∈ G p such that ( α Q ◦ α N )( ψ ) = ψ . Then α N =( α F ) − ◦ α N ◦ α F . Let ψ be a 2d SPT state on A with a symmetry group G . By definition we canstack it with a state ( A , ψ ) with a trivial G -action, so that the resulting state isSRE. We redefine ψ by ψ ⊗ ψ for the rest of this subsection. Let ψ be a factorizedpure state such that ψ = α F ( ψ ) for some α F ∈ G . Let w ( g ) be an automorphism,that corresponds to on-site symmetry action.The automorphism w ( g ) can be locally generated, but there is no canonical LGAfor it. Instead, we consider it as an automorphism of the group G . For any region A the automorphisms w ( g ) A ( · ) of the group G satisfy the evident relations w ( g ) A ◦ w ( h ) A ◦ (cid:16) w ( gh ) A (cid:17) − = Id , (12) ( g ) A ◦ w ( h ) A ◦ w ( k ) A ◦ (cid:16) w ( ghk ) A (cid:17) − = Id , (13) w ( g ) A ◦ w ( h ) A ◦ w ( k ) A ◦ w ( l ) A ◦ (cid:16) w ( ghkl ) A (cid:17) − = Id . (14)Let A be a cone-like region as shown on fig. 2. Let us split w ( g ) into w ( g ) − ◦ w ( g )+ with w ( g )+ = w ( g ) A and w ( g ) − = w ( g )¯ A . Since w ( g ) ( ψ ) = ψ , and since w ( g ) can be locallygenerated, by Corollary 2.8.2 there is α − K ( g ) ∈ G ∂A , such that (cid:16) w ( g ) − ◦ α − K ( g ) (cid:17) ( ψ ) = ψ = (cid:16) α K ( g ) ◦ w ( g )+ (cid:17) ( ψ ) . (15)We define an automorphism ˜ w ( g )+ ( · ) of G by˜ w ( g )+ ( α F ) := α K ( g ) ◦ w ( g )+ ( α F ) ◦ ( α K ( g ) ) − (16)which by Lemma 2.2 is also an automorphism of G B for any submanifold B . Wedefine an element of G ∂A by υ ( g,h ) := α K ( g ) ◦ w ( g )+ ( α K ( h ) ) ◦ ( α K ( gh ) ) − (17)which depends on the choice of K only. Since w ( g )+ ◦ w ( h )+ ◦ ( w ( gh )+ ) − = Id we have υ ( g,h ) ( ψ ) = ψ . By Lemma 2.4 it can be split as υ ( g,h ) = υ ( g,h ) − ◦ υ ( g,h )+ (18)for υ ( g,h ) ± ∈ G ( ∂A ) ± .Since υ ( g,h ) ( ψ ) = ψ , by Corollary 2.8.3 there is α − N ( g,h ) ∈ G p , such that (cid:16) υ ( g,h ) − ◦ α − N ( g,h ) (cid:17) = ψ = (cid:16) α N ( g,h ) ◦ υ ( g,h )+ (cid:17) ( ψ ) . (19)We define LGAs ˜ υ ( g,h )+ := α N ( g,h ) ◦ υ ( g,h )+ , (20)˜ υ ( g,h ) − := υ ( g,h ) − ◦ ( α N ( g,h ) ) − (21)that gives another decomposition υ ( g,h ) = ˜ υ ( g,h ) − ◦ ˜ υ ( g,h )+ . (22)Using an identity on LGAs υ ( g,h ) ◦ υ ( gh,k ) ◦ ( υ ( g,hk ) ) − ◦ (cid:16) ˜ w ( g )+ ( υ ( h,k ) ) (cid:17) − = α (23)and Lemma 2.5 we can define an LGA ι ( g,h,k ) = ˜ υ ( g,h )+ ◦ ˜ υ ( gh,k )+ ◦ (cid:16) ˜ υ ( g,hk )+ (cid:17) − ◦ (cid:16) ˜ w ( g )+ (˜ υ ( h,k )+ ) (cid:17) − (24)which is an element of G p and depends on the choice of K and N only. Note that ι ( g,h,k ) ( ψ ) = ψ . We denote the corresponding unitary observable for ι ( g,h,k ) by I ( g,h,k ) . e define ω ( g, h, k ) := hI ( g,h,k ) i ψ (25)which due to Ad I ( g,h,k ) ( ψ ) = ψ is a pure phase, and therefore takes values in U (1).We have the following relations between LGAs ι ( g,h,k ) ◦ ( ˜ w ( g )+ (˜ υ ( h,k )+ )) ◦ ι ( g,hk,l ) ◦ ( ˜ w ( g )+ (˜ υ ( hk,l )+ )) ◦ ˜ υ ( g,hkl )+ == ι ( g,h,k ) ◦ ( ˜ w ( g )+ (˜ υ ( h,k )+ )) ◦ ˜ υ ( g,hk )+ ◦ ˜ υ ( ghk,l )+ =˜ υ ( g,h )+ ◦ ˜ υ ( gh,k )+ ◦ ˜ υ ( ghk,l )+ == ˜ υ ( g,h )+ ◦ ι ( gh,k,l ) ◦ ( ˜ w ( gh )+ (˜ υ ( k,l )+ )) ◦ ˜ υ ( gh,kl )+ == ˜ υ ( g,h )+ ◦ ι ( gh,k,l ) ◦ ( ˜ w ( gh )+ (˜ υ ( k,l )+ )) ◦ (˜ υ ( g,h )+ ) − ◦ ι ( g,h,kl ) ◦ ( ˜ w ( g )+ (˜ υ ( h,kl )+ )) ◦ ˜ υ ( g,hkl )+ . (26)Multiplying the last line by the inverse of the first and using˜ υ ( hk,l )+ ◦ (cid:16) ˜ υ ( h,kl )+ (cid:17) − = (˜ υ ( h,k )+ ) − ◦ ( ι ( h,k,l ) ) − ◦ (cid:16) ˜ w ( h )+ (˜ υ ( k,l )+ ) (cid:17) − (27)we get the pentagon relation on LGAs˜ υ ( g,h )+ ◦ ι ( gh,k,l ) ◦ ( ˜ w ( gh )+ (˜ υ ( k,l )+ )) ◦ (˜ υ ( g,h )+ ) − ◦◦ ι ( g,h,kl ) ◦ ˜ w ( g )+ (cid:18) (˜ υ ( h,k )+ ) − ◦ ( ι ( h,k,l ) ) − ◦ (cid:16) ˜ w ( h )+ (˜ υ ( k,l )+ ) (cid:17) − (cid:19) ◦ ( ι ( g,hk,l ) ) − ◦ (cid:16) ˜ w ( g )+ (˜ υ ( h,k )+ ) (cid:17) − ◦ ( ι ( g,h,k ) ) − = α . (28)Since for unitary observables U a , such that Ad U a ( ψ ) = ψ , we have hU a i ψ = h ˜ υ ( g,h )+ ( U a ) i ψ = h α K ( g ) ◦ w ( g )+ ( U a ) i ψ (29) hU U ... U n i ψ = hU i ψ hU i ψ ... hU n i ψ (30)from pentagon relation we get ω ( g, h, k ) ω ( g, hk, l ) ω ( h, k, l ) = ω ( gh, k, l ) ω ( g, h, kl ) (31)Therefore ω ( g, h, k ) defines a cohomology class[ ω ] ∈ H ( G, U (1)) . (32)Let us show that this class does not depend on the choice of K ( g ) and N ( g,h ) andthe cone-like region A .Any change of N ( g,h ) corresponds to a redefinition ˜ υ ( g,h )+ → α N ( g,h ) ◦ ˜ υ ( g,h )+ forsome α N ( g,h ) ∈ G p , such that α N ( g,h ) ( ψ ) = ψ . Let V ( g,h ) be the correspondingunitary observable for α N ( g,h ) . Using eq. (29) and eq. (30) we get ω ( g, h, k ) → ω ( g, h, k ) µ ( h, k ) µ ( g, hk ) µ ( gh, k ) µ ( g, h ) (33) or µ ( g, h ) = hV ( g,h ) i ψ , so that [ ω ] is not affected.Since [ ω ] in independent of N ( g,h ) , we can change the position of p and thesplitting υ ( g,h ) = υ ( g,h ) − ◦ υ ( g,h )+ to any point on ∂A without changing [ ω ].Let C + be a cone containing ( ∂A ) + . Suppose we change K ( g ) or the boundary( ∂A ) + itself (so that we still have a cone-like region A ) inside C + . We can take p to be at distance r from C + , and such change can only affect ω ( g, h, k ) by O ( r −∞ )terms. Since r can be arbitrary large, this term is actually vanishing. Similarly, anychange of K ( g ) or ( ∂A ) − inside the complement of C + can’t change [ ω ]. Remark 3.1.
Note that our index is defined only using the algebra of observablesand structural properties of invertible states of lower dimensions. We don’t have tointroduce any representation of this algebra.
Remark 3.2.
The structural properties of d -dimensional invertible states for d > d -dimensional systems with d >
2, since it is believed that in d > E state [13]).However, this is not surprising since it is known that there are states which arenot captured by cohomology classification in higher dimensions even for unitarysymmetries (see [18]). Multiplicativity.
For a stack ( A , ψ ) = ( A , ψ ) ⊗ ( A , ψ ) of two differentstates ( A , ψ ) and ( A , ψ ) we have hI ( g,h,k )12 i ψ = h ( I ( g,h,k )1 ⊗ I ( g,h,k )2 ) i ψ = hI ( g,h,k )1 i ψ hI ( g,h,k )2 i ψ , (34)Therefore, the index is multiplicative. Invariant of SPT phase.
First, note that the index does not depend on the state ψ with a trivial G -action chosen at the beginning to produce SRE state ψ ⊗ ψ .Indeed, if there is another such state ψ , we can realize both computations of theindex using ψ ⊗ ψ and ψ ⊗ ψ on a system ψ ⊗ ψ ⊗ ψ ⊗ ψ , where ψ is a copyof ψ with a trivial G -action.Second, the index is not affected if we stack the state ψ with a G -invariant purefactorized state, since the index is multiplicative and the latter has the trivial index.It is left to show that the index is not affected by G -invariant LGA.Let α F be a G -invariant LGA, such that ψ = α F ( ψ ) for G -invariant SPT states ψ and ψ . Note that α F ◦ w ( g )+ ( α F ) − ∈ G ∂A . Suppose we have a choice of K ( g ) and N ( g,h ) for the state ψ . For the state ψ we can choose α K ( g ) = α F ◦ α K ( g ) ◦ w ( g )+ ( α F ) − ∈ G ∂A , (35) υ ( g,h )+ = α F ◦ υ ( g,h )+ ◦ ( α F ) − ∈ G ( ∂A ) + , (36) α N ( g,h ) = α F ◦ α N ( g,h ) ◦ ( α F ) − ∈ G p . (37) ( ∂A ) + ( ∂A ) − w ( g ) act non-trivially on the blue shaded region.The automorphisms α K ( g ) , υ ( g,h ) act non-trivially on the green shaded region.The automorphisms ˜ υ ( g,h )+ act non-trivially on the yellow shaded region. Then for the state ψ we have ι ( g,h,k ) = α F ◦ ι ( g,h,k ) ◦ ( α F ) − with the correspondingobservable I ( g,h,k ) = α F ( I ( g,h,k ) ). Therefore, the index of the state ψ hI ( g,h,k ) i ψ = h α F ( I ( g,h,k ) ) i ψ = hI ( g,h,k ) i ψ . (38) As was shown in [10], for any representative of a class [ ω ] ∈ H ( G, U (1)) one canconstruct a tensor network state that is SPT. Let us show that our index for suchstate is [ ω ].Let Λ be a square lattice with sites ( x, y ) ∈ Z × Z . Let V be a regular representationof G with a basis h l | for l ∈ G . The on-site Hilbert space is V ( x,y ) = N a =1 V ( a )( x,y ) for V ( a )( x,y ) ∼ = V . The induced basis for V ( x,y ) is h l , l , l , l | := h l | (1) ⊗ h l | (2) ⊗ h l | (3) ⊗ h l | (4) . (39)The on-site action of G is defined by h l , l , l , l |R ( g ) = h l g, l g, l g, l g | ω ( l l − , l , g ) ω ( l l − , l , g ) ω ( l l − , l , g ) ω ( l l − , l , g ) (40)for some a representative ω ( g, h, k ) of [ ω ]. The state ψ is chosen to be a tensorproduct of vector states with h ψ ( x − / ,y − / | ∼ X l ∈ G h l | (1)( x,y ) ⊗ h l | (2)( x − ,y ) ⊗ h l | (3)( x − ,y − ⊗ h l | (4)( x,y − (41) n V (1)( x,y ) ⊗V (2)( x − ,y ) ⊗V (3)( x − ,y − ⊗V (4)( x,y − . One can check that such vector is invariantunder the on-site action of G on a region that contains four involved sites.Let A be an upper half-plane y >
0, so that w ( g )+ acts on sites ( x, y ) ∈ Z × Z > .To define a 0-chain K ( g ) on Z × { } we use the representation from Lemma 2.6. Let V linkx be a diagonal subspace of V (1)( x − , ⊗ V (2)( x, . Let e iK ( g ) x be a unitary observablelocalized on V linkx − ⊗ V linkx which acts on basis vectors as h l x − | ⊗ h l x | e iK ( g ) x = h l x − g | ⊗ h l x | ω ( l x − l − x , l x , g ) (42)and as identity otherwise. Then α K ( g ) = −−−→ ∞ Y x = −∞ α K ( g ) x (43)gives an LGA υ ( g,h ) = α K ( g ) ◦ w ( g ) ( α K ( h ) ) ◦ ( α K ( gh ) ) − = −−−→ ∞ Y x = −∞ α Q ( g,h ) x (44)with Q ( g,h ) x such that for basis vectors h l x − | ⊗ h l x | e iQ ( g,h ) x = h l x − | ⊗ h l x | ω ( l x − , g, h ) ω ( l x , g, h ) (45)where we have used ω ( l x − l − x , l x g, h ) ω ( l x − l − x , l x , g ) ω ( l x − l − x , l x , gh ) = ω ( l x − , g, h ) ω ( l x , g, h ) . (46)We can take υ ( g,h )+ = −→ ∞ Y x =1 α Q ( g,h ) x (47)and define α N ( g,h ) local on V link by h l | e iN ( g,h ) = h l | ω ( l , g, h ) − . (48)Finally, using ω ( l g, h, k ) ω ( l , g, hk ) ω ( l , g, h ) ω ( l , gh, k ) = ω ( g, h, k ) (49)we get ι ( g,h,k ) with the corresponding unitary I ( g,h,k ) = ω ( g, h, k ). Thus, hI ( g,h,k ) i ψ = ω ( g, h, k ) . (50) All invertible defect states are SRE
Since the proof of Lemma 2.8 is essentially the same as the proof that all invertiblestates on a one-dimensional lattice are SRE from Section 4 of [9], we just sketch thedifferences, assuming the reader is familiar with this proof and the terminology. Alllemmas mentioned below are from Section 4 of [9].We also assume that A is a half-plane y > A is a cone-like region.We say that a set of ordered eigenvalues { λ j } has g ( r ) -decay if ε ( k ) ≤ g (log( k ))for some MDP function g ( r ) = O ( r −∞ ), where ε ( k ) = P ∞ j = k +1 λ j .Let ψ be an invertible defect state on ∂A with an inverse state ( A , ψ ) and let α F be an LGA in G ∂A ( A ⊗ A ), such that α F (Ψ) = Ψ for Ψ = ψ ⊗ ψ and somepure factorized state Ψ on A ⊗ A .Let b n, m c be a region ( x, y ) ∈ ( n, m ] × ( −∞ , + ∞ ), and let us define B n = b n − , n c and C n = b−∞ , n c . Note that F B n is a well-defined almost local observable.Note that since F is approximately localized on { y = 0 } by Fannes inequalitythe entropy of each site j = ( x, y ) in B n is bounded by f ( y ) for some MDP function f ( r ) = O ( r −∞ ), and therefore the entropy of a rectangle Ξ R = ( n − , n ] × [ − R, R ]is bounded for R → ∞ . By the result of [19] the state Ψ is quasi-equivalentto (Ψ | B n ) ⊗ (Ψ | ¯ B n ), that allows to decompose the GNS Hilbert space H for Ψ as H = H B n ⊗ H ¯ B n . Furthermore, by the fact that for any region I n,m = b n, m c wehave a decomposition α F = α F | Cn ◦ Ad U n ◦ α F | In,m ◦ Ad U m ◦ α F | ¯ Cm (A.1)for some f -local unitaries U n and U m at ( n,
0) and ( m,
0) and the proof of Lemma4.2, the entropy of I n,m is also uniformly bounded in n and m , and therefore Ψ isquasi-equivalent to (Ψ | C n ) ⊗ (Ψ | ¯ C n ), that allows to decompose the GNS Hilbert space H for Ψ as H = H C n ⊗ H ¯ C n . These two facts allow us in the following to replaceintervals ( n, m ] of a one-dimensional chain in the proofs of lemmas from Section 4 of[9] by regions b n, m c in our situation.The Proposition 2 from section 4 of [9] implies that Ψ | C n can be described bya density matrix with g ( r )-decay of Schmidt coefficients for some MDP function g ( r ) = O ( r −∞ ). Since Ψ | C n = ( ψ | C n ) ⊗ ( ψ | C n ), the state ψ | C n can be also describedby a density matrix with g ( r )-decay of Schmidt coefficients. Therefore, by the sameProposition 2 the state ψ is SRE defect state. References [1] Bei Zeng, Xie Chen, Duan-Lu Zhou, and Xiao-Gang Wen.
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Quantum Science and Technology.Springer, New York, 2019. The proof of Theorem 1.5 from [19] was given for a splitting of a one-dimensional chain into twohalves. However, the same proof applies word for word to the splitting of the plane into B n and ¯ B n for anSRE defect state Ψ on ∂A . url : http://scgp.stonybrook.edu/archives/7874 .[3] Matthew B Hastings and Xiao-Gang Wen. “Quasiadiabatic continuationof quantum states: The stability of topological ground-state degener-acy and emergent gauge invariance”. In: Physical review b
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