A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph
AA Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph
Jintae Kim, Hyun-Yong Lee,
2, 3 and Jung Hoon Han ∗ Department of Physics, Sungkyunkwan University, Suwon 16419, Korea Department of Applied Physics, Graduate School, Korea University, Sejong 30019, Korea Division of Display and Semiconductor Physics, Korea University, Sejong 30019, Korea (Dated: February 23, 2021)We analyze a model demonstrating the co-existence of subsystem symmetry breaking (SSB) andsymmetry-protected topological (SPT) order, or subsystem LSPT order for short. Its mathematicalorigin is the existence of both a subsystem and a local operator, both of which commute with theHamiltonian but anti-commute between themselves. The reason for the exponential growth of theground state degeneracy is attributed to the existence of subsystem symmetries, which allows oneto define both the Landau order parameter and the SPT-like order for each independent loop.
I. INTRODUCTION
Multiply degenerate ground states in a many-body sys-tem are usually generated as a consequence of globalsymmetry breaking (GSB). A prime example of GSB isthe twofold degeneracy of the ground states in the Isingmodel, where all spins are either up or all down. Oneof the remarkable advances in condensed matter theoryover the past decades is the identification of a new mech-anism by which multiple ground state degeneracy (GSD)is generated. One of these routes is the topological order,as most dramatically realized in fractional quantum Hallsystems . Another recently discovered path to havingmultiple GSD is the so-called SPT order . In modelswith SPT order, the ground state is unique if defined ona closed manifold but becomes multiply degenerate onan open geometry such as the open chain. Among thetell-tale signs of SPT order is the symmetry fractional-ization of the global symmetry at the edges which isalso responsible for the multiple GSD. A lot of differentmodels exhibiting SPT order has been examined in thepast, in particular in one dimension . Some modelsdemonstrate the GSB even for the closed geometry, to-gether with the SPT order . A recent twist to theoriginal idea of SPT order protected by the global sym-metry is the subsystem SPT (SSPT), being protected bythe symmetries in a sub-manifold of the overall system .The SPT models treated in the past considered analternating site and link variables arranged in a one-dimensional fashion. The case of m degrees of freedomresiding on the vertices and n degrees of freedom on thelinks was considered in Refs. 16–19. The model exhib-ited a multiple GSD even for a closed chain. When thesame model was placed on a graph, the GSD increasedexponentially with the number of holes, called the firstBetti number, in the graph. In this paper, we give a clearinterpretation of the results found in Ref. 19 in terms ofthe notion of subsystem Landau and SPT orders. Fea-tures that are characteristic of symmetry breaking as wellas SPT can be found in this model, and for each closedloop in the graph. We label such order as the subsystemLandau-SPT order, or SLSPT order for short. The ob-servation of the co-existence of Landau and SPT orders in a given chain had been made in the past , but notbeen extended to the case of a general graph. In Sec. IIwe review the V /L model introduced in Ref. 19, whichis a generalization of the V /L model (to be definedprecisely) studied by several authors in the past .The ground state degeneracy this model is understood interms of a properly defined Landau order parameter inSec. III. The V /L on a graph is analyzed in terms ofthe co-existing Landau and SPT orders in Sec. IV. Theorigin of the exponential growth of GSD is understood.We make a summary and conclude in Sec. VI. II. V2/L4 MODEL
For self-consistent reading of this paper, we make abrief review of the V /L model introduced in Ref. 17and analyzed in Ref. 19. In a family of Hamiltonianswe will call the V m /L n model, there reside m degrees offreedom, or levels, at the vertices ( v ) and n levels at thelinks ( l ). In the simplest case of a one-dimensional chain,the vertices and links appear alternatively, as illustratedin Fig. 1. Quantum states at the vertices and links arewritten | α (cid:105) v and | β (cid:105) l , with 0 ≤ α ≤ m − ≤ β ≤ n −
1, respectively.The Hamiltonian of the V m /L n model is a sum ofmutually commuting projectors. One type of projectorcalled A v is defined with respect to a vertex, while theother type C l is defined with respect to a link. Theirspecific expressions depend on the choice of ( m, n ) repre-senting the respective dimension of the Hilbert space atthe vertices and links. It turns out the m = n = 2 casehad been studied extensively in the past . Our focushere is on the extension to n >
2. Striking differencesfrom the V /L situation already show up for the V /L case, where we will be devoting most of our discussion.For a one-dimensional chain the vertices and sites canbe labeled as ( v i , l i ), with i running from 1 through L for a chain of length L (see Fig. 1). Then we write theHamiltonian H = − L (cid:88) i =1 ( A i + C i ) , (1) a r X i v : . [ c ond - m a t . s t r- e l ] F e b where i = ( v i , l i ) stands for a combined vertex+link unit.The vertex operator A i for the V /L model acts on agiven vertex v i and its two adjacent links ( l i − , l i ) as A i = 14 (cid:88) n =0 ( X l i − x v i X l i ) n = 14 (1 + X l i − x v i X l i + X l i − X l i + X l i − x v i X l i ) . (2)The link operator C i acts on the given link l i and its twoneighboring vertices ( v i , v i +1 ), C i = 12 (cid:88) n =0 ( z v i Z l i z v i +1 ) n = 12 (1 + z v i Z l i z v i +1 ) . (3)Such choice of A and C operators is specific to the V /L model.The operator form of A i depends on the direction of“arrows” on the links. The rule is to assign X to a linkwith an “incoming” arrow to a vertex and X to a linkwith an “outgoing” arrow. The A -operator for arbitraryarrow directions on the links is therefore given by A i = 14 (cid:88) n =0 ( (cid:89) l in X l in (cid:89) l out X l out x v i ) n . (4)The products (cid:81) l in and (cid:81) l out mean, for example, that ifboth arrows are “in”, then we must write ( X l i − x v i X l i ) n .For simplicity, we adhere to the arrow scheme shown inFig. 1 and the vertex operator definition in Eq. (2). FIG. 1. Schematic figure of alternating vertices and orientedlinks. For a closed chain the link l L attaches to the vertex v . The lower-case x v and z v operators act on the vertices.Meanwhile, upper-case X l and Z l operate on the linkstates. In general they satisfy X l | g (cid:105) l = | g + 1 (cid:105) l (mod 4)and Z l | g (cid:105) = ω g | g (cid:105) l with ω = i , g = 0 , , ,
3. Similarrelations hold for the vertex operators, with ω = −
1. Anidentity Z pl X ql = ω pq X ql Z pl (5)will be used repeatedly for many of the derivations thatwill follow. It is easily verified that all the projectors inthe Hamiltonian are mutually commuting and square toitself.This model we wrote down in Eq. (1) possesses someglobal symmetries. For the V /L Hamiltonian, there aretwo global symmetry operators φ = (cid:89) i x v i , θ = (cid:89) i Z l i , (6) which generate the Z × Z symmetry of the model. Onecan check that both symmetry generators commute withthe Hamiltonian. A crucial aspect of the V /L model isthe existence of a local Z operator X l i , which commuteswith the Hamiltonian [ H, X l i ] = 0 for any link l i . Suchlocal operator is absent in the V /L model. As a result,the eigenstates can be classified according to the set ofquantum numbers or “ p -sectors” p ≡ { p , · · · , p L } (7)where p i = ± X l i at thelink l i .Within each p -sector one can reduce the V /L modelto an effective V /L model. To accomplish this, firstone organizes the link states in terms of the eigenstatesof X l : | (cid:105) l = ( | (cid:105) + | (cid:105) l + | (cid:105) l + | (cid:105) l ) / , | (cid:105) l = ( | (cid:105) l + ω | (cid:105) l + ω | (cid:105) l + ω | (cid:105) l ) / , | (cid:105) l = ( | (cid:105) l − | (cid:105) l + | (cid:105) l − | (cid:105) l ) / , | (cid:105) l = ( | (cid:105) l + ω | (cid:105) l + ω | (cid:105) l + ω | (cid:105) l ) / . (8)It is easily shown that X l | n (cid:105) l = ω n | n (cid:105) l ,Z l | n (cid:105) l = | n − (cid:105) l . (9)Both | (cid:105) l i and | (cid:105) l i share the same eigenvalue X l i = p i =+1. For | (cid:105) l i and | (cid:105) l i the eigenvalue is p i = −
1. Withina given p -sector, we have Z l i | (cid:105) l i = | (cid:105) l i , ( p i = +1) Z l i | (cid:105) l i = | (cid:105) l i , ( p i = − . (10)In other words, the Z l operator acts effectively as a spin-1/2 Pauli operator, Z l i → x l i , in the two-dimensionalsubspace of fixed p i . One can also show that X l i acts asthe Pauli- z l i if p i = +1, and as X l i ≡ ωz l i if p i = − p -sector: X l → ω (1 − p l ) / z l ,X l → p l ,X l → p l ω (1 − p l ) / z l . (11)With these considerations one can reduce the A i and C i operators in Eqs. (2) and (3) as A i → A ( p ) i = 14 (cid:104) p i − p i + ( p i − + p i ) ω − ( p i − + p i ) / z l i − x v i z l i (cid:105) C i → C ( p ) i = 12 (1 + z v i x l i z v i +1 ) . (12)Each p -sector then gives rise to an effective V /L Hamil-tonian H ( p ) = − (cid:88) i ( A ( p ) i + C ( p ) i ) (13)with the A ( p ) i and C ( p ) i terms given in Eq. (12). One canview the original V /L model as the direct sum H = ⊕ p H ( p ) , (14)where each H ( p ) is defined in the V /L subspace. FromEq. (12) we obtain identical operators A ( p ) i when all the p i ’s are reversed, p i → − p i , implying that each V /L sector ought to be doubly degenerate . The two degeneratesubspaces are connected by the global operation (cid:81) i Z l i or (cid:81) i Z l i , both of which implement p i → − p i . In otherwords, H ( − p ) = ( (cid:89) i Z l i ) H ( p ) ( (cid:89) i Z l i )= ( (cid:89) i Z l i ) H ( p ) ( (cid:89) i Z l i ) . (15)In particular, the ground states of the V /L modelcomes from the sector p = { , · · · , } and p = {− , · · · , − } where one can write A i and C i as A i = 12 (1 + z l i − x v i z l i ) ,C i = 12 (1 + z v i x l i z v i +1 ) . (16)In fact, this V /L model has been studied extensivelyas a model for one-dimensional SPT . The V /L model is a natural extension of the V /L model. Thepure V /L model, with all p l = 1 or −
1, has the global Z × Z symmetry generated by (cid:89) i x v i and (cid:89) i x l i , (17)but no extra local symmetry. III. GROUND STATES OF V2/L4 MODEL
There are two ways to go about writing down theground states of the V /L Hamiltonian, Eq. (1). Thefirst one is to identify the ground states of − (cid:80) i C i andthen act on them with the projector (cid:81) i A i . The otherway is to first identify the ground states of − (cid:80) i A i , andthen act on them with the projector (cid:81) i C i . It is not hardto see that both ways lead to states with the eigenvaluesof A i and C i all equal to +1, which by definition givesthe ground states of the projector Hamiltonian.Given our analysis in the previous section, it seemsmore enlightening the analyze the ground states in theeigenbasis of X l i , where the operator A i is diagonalizedeasily. We obtain the two degenerate ground states ofthe V /L model as | G (cid:105) = P C | S (cid:105) , | G (cid:105) = P C | S (cid:105)| S (cid:105) = ( ⊗ i | (cid:105) v i )( ⊗ j | (cid:105) l j ) | S (cid:105) = ( ⊗ i | (cid:105) v i )( ⊗ j | (cid:105) l j ) . (18) The vertex eigenstates | (cid:105) v i = ( | (cid:105) v i + | (cid:105) v i ) / √ | (cid:105) v i = ( | (cid:105) v i − | (cid:105) v i ) / √ x v operator.The ground states of the V /L model are obtained asthe projection by P C = (cid:81) i C i on the two “seed states” | S (cid:105) and | S (cid:105) . The (unique) ground state of the V /L model is obtained from | G (cid:105) above, by rewriting C i asin Eq. (16). The two ground states of the V /L modelshare the properties( (cid:89) i Z l i ) | G (cid:105) = | G (cid:105) ,X l i | G (cid:105) = + | G (cid:105) X l i | G (cid:105) = −| G (cid:105) . (19)We also show how to write down the ground statesin the Z -basis, where the C i ’s are diagonalized first and P A = (cid:81) i A i acts as a projector. The two ground statesare | G (cid:48) (cid:105) = P A (cid:104) ( ⊗ i | (cid:105) v i )( ⊗ j | (cid:105) l j ) (cid:105) , | G (cid:48) (cid:105) = P A (cid:104) ( ⊗ i | (cid:105) v i )( ⊗ j (cid:54) = j (cid:48) | (cid:105) l j ) ⊗ | (cid:105) l j (cid:48) (cid:105) (20)where j (cid:48) is arbitrary. These ground states share the prop-erties X l i | G (cid:48) (cid:105) = | G (cid:48) (cid:105) , ( (cid:89) i Z l i ) | G (cid:48) (cid:105) = + | G (cid:48) (cid:105) , ( (cid:89) i Z l i ) | G (cid:48) (cid:105) = −| G (cid:48) (cid:105) . (21)Comparing Eqs. (19) and (21), one concludes | G (cid:105) = ( | G (cid:48) (cid:105) + | G (cid:48) (cid:105) ) / √ | G (cid:105) = ( | G (cid:48) (cid:105) − | G (cid:48) (cid:105) ) / √ . (22)In what follows, we provide a geometrical interpre-tation of the ground state | G (cid:105) . Due to z v = 1,the product of neighboring z v i Z l i z v i +1 is simply Q = z v (cid:0)(cid:81) i S Z l i (cid:1) z v where i S stands for the links betweenthe left-most vertex v and the right-most one v . Its ac-tion on the seed state | S (cid:105) permutes the link states along S , i.e., | (cid:105) l → | (cid:105) l and flips the vertex states at edges | (cid:105) v / → | (cid:105) v / , or graphically illustrated as , where blue dot stands for each edge vertex while the redstring for all links and vertices along S . Consequently,the expansion of the projector P C leads to the super po-sition of all possible string configurations on the circle: . (23)Due to | G (cid:105) = (cid:81) i Z l i | G (cid:105) , the geometrical interpretationof | G (cid:105) is given in the same manner. Only difference isthat all link states are raised by one, i.e., | n (cid:105) l → | n + 1 (cid:105) l .In an open chain of the same model, some edge statesappear as a consequence of SPT, as thoroughly analyzedin Ref. 19. IV. V2/L4 MODEL ON A GRAPHA. GSD on a Graph
The discussion of the ground states of the V /L modelboth in the closed and the open chain thus far mightsuggest that we are merely dealing with what seems to betwo copies of the well-known V /L model. Interestingly,the real point of departure between the two families ofmodels occurs when these models are put on a graph .The closed circle is a simplest example of a graph with thefirst Betti number B = 1. Now, one can imagine puttingthe model on a more intricate graph such as shown inFig. 2(b), which has B = 2. Intuitively, the first Bettinumber measures the number of cycles or independentloops in a given graph. FIG. 2. Examples of closed and connected planar graphs withvarious Betti numbers (a) B = 1, (b) B = 2, and (c), (d) B = 3. Dots are the vertices and the arrows represent thelinks of the graph. Dashed lines are the independent closedloops where we can perform a subsystem Z symmetry trans-formation. Because of the orientation, the loop operators cor-respond to the right semi-circular loop are different for (c) and(d). Considering a B = 2 graph such as shown in Fig. 2(b),one can write the same A i and C i projectors as in the B = 1 graph, namely a circle, except at the two vertices v i and v j where two lines become connected at the vertex.At these vertices, there are three (not two) links which are connected to a single vertex. The definition of thevertex operators which are projectors and commute withother vertex operators must generalize accordingly, A i = 14 (1+ X l i − X l i X l i (cid:48) x v i + X l i − X l i X l i (cid:48) + X l i − X l i X l i (cid:48) x v i ) A j = 14 (1+ X l j − X l j X l i (cid:48)(cid:48) x v j + X l j − X l j X l i (cid:48)(cid:48) + X l j − X l j X l i (cid:48)(cid:48) x v j ) (24)where v i ( v j ) has one (two) arrowhead towards it and two(one) arrowheads away from it. In fact, a completely gen-eral definition of the vertex operator for arbitrary graphis possible as A i = 14 (cid:88) j =0 ( (cid:89) q ∈ in X l q,i (cid:89) q (cid:48) ∈ out X l q (cid:48) ,i ) x v i j . (25)There is a factor X ( X ) for the links whose arrows comeinto (out of) the vertex. The link operator C i is the sameas in Eq. (3) regardless of the graph type. Despite themuch complex forms, A i and C j always remain mutuallycommuting projectors. The V /L Hamiltonian on anarbitrary graph generalizes accordingly, H = − N v (cid:88) i =1 A i − N l (cid:88) j =1 C j , (26)spanning all the vertices and the links in the graph andusing Eq. (25) for the vertex operator. The number ofvertices ( N v ) and of links ( N l ) are no longer equal fora general graph but are rather related by the first Bettinumber, N l − N v + 1 = B . (27) B. Subsystem Symmetries
We can discuss how to explicitly construct the multi-tude of degenerate ground states on a graph. For exam-ple, the V /L model defined on a graph shown in Fig.2(b) allows two loop operators that commute with theHamiltonian. We call them θ and θ , and they con-sist of the product of Z l ’s along the left semi-circularloop and the large circular loop, respectively. These twoloops are drawn as dashed lines in Fig. 2(b). Note thatthe definition of these loop operators are obtained bysimply “following the arrows” drawn on the graph. On a B = 2 graph we have two such independent loops. Ac-cordingly one can write down four independent groundstates, which are | G (cid:105) = P C (cid:2) ( ⊗ i | (cid:105) v i )( ⊗ j | (cid:105) l j ) (cid:3) , and | G (cid:105) = θ | G (cid:105) , | G (cid:105) = θ | G (cid:105) , | G (cid:105) = θ θ | G (cid:105) . (28)At first sight the construction of degenerate ground stateson a graph bears resemblance to the way that topologi-cally distinct states are generated on a finite-genus space.The GSD formula GSD = 2 B has resemblance to theformula for the topological GSD = 2 g , which appliesto two-dimensional topological models such as the toriccode, where g (also known as the second Betti number)is the genus of two-dimensional surface.Unlike the topologically ordered states which do nothave local order parameters, the four ground states de-rived above can be distinguished by their “order pa-rameter” X l . With | G (cid:105) , the expectation value is (cid:104) G | X l | G (cid:105) = +1 on every link of the graph. For | G (cid:105) ,the links along the loop where θ acts have (cid:104) X l (cid:105) = − | G (cid:105) , it is the links along the outer perimeter wherethe order parameters are reversed. Finally in the fourthground state | G (cid:105) it is the other inner loop where thelinks have (cid:104) X l (cid:105) = −
1. The four order parameter pat-terns are depicted in Fig. 3. There is a clear parallelto the usual classification of states by order parameters,but one must carefully note that its nature is not en-tirely global. Perhaps a better termnology is the subsys-tem symmetry breaking (SSB, not to be confused with the spontaneous symmetry breaking ) and distinguish it fromthe global symmetry breaking (GSB) of most many-bodymodels.Along a similar line of reasoning, the loop operatorconsisting of the product of Z l ’s and Z l ’s along the inde-pendent loops as dictated by the flow of arrows com-mute with the Hamiltonian and performs the subsys-tem Z symmetry transformation . In the case of Fig.2(c), all such loop operators consist of the product of Z l ’s only. On the other hand, the loop operator corre-sponding to the right semi-circuplar loop in Fig. 2(d)is (cid:81) i Z l i (cid:81) i (cid:48) Z l i (cid:48) where i ’s are the links on the arc and i (cid:48) ’s are the links on the inner segment. The number ofindependent loops equals the Betti number B , and thedegenerate ground states are generated by applying thesubsystem loop operators and their products on one par-ticular ground state | G (cid:105) . There are exactly 2 B − B ground states on a graph with the Bettinumber B .The V /L Hamiltonian on a graph still has the exactsymmetry [
H, X l ] = 0 for all the links l . As a result, X l operators in the graph model can be replaced by theirrespective quantum numbers p l . In particular, the vertexoperators at the junction given in Eq. (24) become, in agiven p -sector, A ( p ) i = 14 (cid:2) p i − p i p i (cid:48) + ( p i p i (cid:48) + p i − ) ω (3 − p i − − p i − p i (cid:48) ) / z l i − z l i z l i (cid:48) x v i (cid:3) A ( p ) j = 14 (cid:2) p j − p j p i (cid:48) + ( p j − p i (cid:48) + p j ) ω (3 − p j − − p j − p j (cid:48) ) / z l j − z l j z l i (cid:48) x v i (cid:3) . (29) FIG. 3. Four kinds of configurations of p ’s that reduce the V /L model to the V /L model when B = 2. The ± signson the links represent the eigenvalues of X l , or equivalently,the p l ’s. One can show, by explicit calculation, that A ( p ) i be-comes (cid:0) z l i − z l i z l i (cid:48) x v i (cid:1) for { p i − , p i , p i (cid:48) } = { , , } , {− , , − } , {− , − , } . In a similar manner, we have A ( p ) j = (cid:0) z l j − z l j z l i (cid:48) x v j (cid:1) for { p j − , p j , p i (cid:48) } = { , , } , {− , − , } , { , − , − } . The V /L model on a graphcomes from having the following choice of vertex opera-tors: A ( p ) i = 12 (cid:0) z l i − z l i z l i (cid:48) x v i (cid:1) A ( p ) j = 12 (cid:0) z l j − z l j z l i (cid:48) x v j (cid:1) (30)at the two junctions i and j shown in Fig. 2(b). For allother vertices and all the links one has the usual defini-tion of the vertex and link operators given in Eq. (16).After some enumeration, one finds the four configura-tions of p ’s shown in Fig. 3 can reduce the V /L modelto the V /L model on the graph with B = 2. Thisargument again shows why there is fourfold degeneracyof the ground states on the B = 2 graph.It is worth examining the general character of the pure V /L model on a graph. It can be shown that GSD ofthe V /L model remains at GSD=1 regardless of theBetti number of the graph. The global Z × Z sym-metry of the V /L model on a simply connected graphwith B = 1 is partially lost due to the vertex terms atthe junction, Eq. (30). One can prove quite easily thatalthough (cid:81) i x v i remains a symmetry, (cid:81) i x l i no longercommutes with the vertex operators at the junction andhence fails to be a symmetry operator. The global sym-metry of the V /L model is lowered from Z × Z to Z on a multiply connected graph. On further observa-tion, however, we realize that a partial product of linkoperators (cid:81) (cid:48) i x l i for the links forming a closed loop doescommute with the junction terms in Eq. (30) and restorethe Z × Z symmetry for that loop. Contrasted with theusual SPT, this is a realization of the subsystem SPT, orSSPT . By implication, when we cut open any segmentof the graph, the emerging edge behavior and symmetryfractionalization will be exactly those of the open chaincase already analyzed. Since the ground states of thepure V /L model are ground states of the V /L model,even in V /L model there is SSPT. V. EXCITATIONS
The concept of excitation arises naturally infrustration-free models to which our V /L model be-longs. The ground state(s) has all of the eigenvalues of A i and C i equal to +1, and excited states should have oneof these equal to zero instead . Depending on whetherthe link or the vertex operator eigenvalues change from1 to 0, one can make a distinction between link exci-tations and vertex excitations. Another useful way toclassify excitations is in terms of the changes in the p -eigenvalues, p = { p , · · · , p L } , of the chain. Changes inany of the eigenvalues in the p -set leads to different sec-tors of the block, H = ⊕ p H ( p ) , where H is the original V /L Hamiltonian and each H ( p ) represents some real-ization of the V /L Hamiltonian. We will examine thenature of excitations from both perspectives. Eigenval-ues of A i and C i will be denoted a i and c i from nowon.The link excitation is attained by rewriting one of theoperators C i in the projector P C by its orthogonal com-plement C ⊥ i = 12 (1 − z v i Z l i z v i +1 ) , C ⊥ i C i = 0 . (31)We can define the projector P C ( j ) = ( (cid:89) i
In this paper we have analyzed the properties of the V /L model on a general graph . The Hilbert space ofthe model is block-diagonalized by a set of local quantumnumbers { p l i = ± } , and we have shown that the V /L model maps to a general V /L model in each p -sector.The GSD of our model grows exponentially with theBetti number B characterizing the number of cycles ina graph. In fact one can easily show that the exponentialgrowth of GSD with the Betti number is not unique tothe V /L model. Even a simple Potts model on a graphcan be defined in a way that exhibits the same GSD be-havior. For that, one considers the same kind of graph asbefore and place n degrees of freedom at each link labeledby the variable z l = 0 , · · · , n −
1, and none on the ver-tices. We then have the Potts interaction − δ z l ,z l (cid:48) for theneighboring links ( l, l (cid:48) ), except when more than two linesmeet at a vertex. In that case we have the three-linkinteraction − δ z l ,z l ,z l among the three links ( l , l , l )joined at a vertex. For more than three links, one simplytakes the delta function of all the links − δ z l ,z l , ··· withan overall minus sign. It is easily verified that the groundstates have the z l ’s distributed in exactly the same pat-tern as those of X l ’s (or p l ’s) shown in Fig. 3 when n = 2.For general graphs and n degrees of freedom, GSD equals n B , but no feature of SPT or symmetry fractionalizationwould be present in such models.To sum up, the subsystem symmetry and its breakingobserved in the V /L model is not related to the SPTnature of the phase but rather co-exist with it. The ideaof subsystem symmetries is applicable for both the Lan-dau order parameter characterized by X l , and the SPTorder. ACKNOWLEDGMENTS
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