Entanglement phases as holographic duals of anyon condensates

Entanglement phases as holographic duals of anyon condensates

Kasper Duivenvoorden JARA Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany    Mohsin Iqbal JARA Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany    Jutho Haegeman Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, 9000 Gent, Belgium    Frank Verstraete Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, 9000 Gent, Belgium Vienna Center for Quantum Science, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria    Norbert Schuch Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany

Anyon condensation forms a mechanism which allows to relate different topological phases. We study anyon condensation in the framework of Projected Entangled Pair States (PEPS) where topological order is characterized through local symmetries of the entanglement. We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for abelian double models through the structure of their entanglement spectrum. We illustrate our findings with the double model, which can give rise to both Toric Code and Doubled Semion order through condensation, distinguished by the SPT structure of their entanglement. Using the ability of our framework to directly measure order parameters for condensation and deconfinement, we numerically study the phase diagram of the model, including direct phase transitions between the Doubled Semion and the Toric Code phase which are not described by anyon condensation.

I Introduction

The study of topologically ordered phases, their relation, and the transitions between them has received steadily growing attention in the last decade. Their lack of local order parameters, the dependence of the ground space structure on their topology, and the exotic nature of their anyonic excitations puts them outside the Landau framework of symmetry breaking and local order parameters, and thus asks for novel ways of characterizing and relating different phases, for instance the structure of their ground space or the nature of their non-trivial excitations (anyons), and the way in which those are related throughout different phases.

Anyon condensation has been proposed as a mechanism for relating topological phases bais:anyon-condensation (). The main idea is that some mechanism drives a species of bosonic anyons to condense into the vacuum. This, in turn, forces any anyon which has non-trivial statistics with to become confined, as a deconfined anyon would have non-trivial statistics with the new vacuum, and moreover leads to the identification of anyons which differ by fusion with . At the same time, the relation between anyon types and ground space of a theory suggests that this condensation is accompanied by a change in the ground space structure. The formalism of anyon condensation allows to construct “simpler” anyon models from more rich ones, and suggests to think of the “condensate fraction” of the condensed anyon as an order parameter for a Landau-like description of the phase transition. Yet, it is a priori not clear how such an order parameter should be measured, and existing approaches describe anyon condensation as a breaking of the global symmetry of the quantum group or tensor category underlying the model bais:quantum-sym-breaking (); bais:hopf-symmetry-breaking-jhep (); kitaev:gapped-boundaries (); kong:anyon-condensation-tensor-categories ().

Projected Entangled Pair States (PEPS) verstraete:mbc-peps () form a natural framework for the local modelling of topologically ordered phases buerschaper:stringnet-peps (); gu:stringnet-peps (). They associate to any lattice site a tensor which describes both the physical system at that site, and the way in which it is correlated to the adjacent sites through entanglement degrees of freedom. It has been shown that in PEPS, topological order emerges from a local symmetry constraint on the entanglement degrees of freedom, characterized by a group action (for so-called double models of groups) schuch:peps-sym () or more generally by Matrix Product Operators for twisted doubles buerschaper:twisted-injectivity () and string-net models sahinoglu:mpo-injectivity (); bultinck:mpo-anyons (). In all cases, both ground states and excitations can be modelled from the very same symmetries which characterize the local tensors: Group actions and irreducible representations (irreps) in the former and Matrix Product Operators with suitable endpoints in the latter case schuch:peps-sym (); buerschaper:twisted-injectivity (); sahinoglu:mpo-injectivity (); bultinck:mpo-anyons (). Yet, it has been observed that the entanglement symmetry of the tensors is not in one-to-one correspondence with the topological order in the system: By adding a suitable deformation to the fixed point wavefunction, the system can be driven into a phase transition which is consistent with a description in terms of anyon condensation schuch:topo-top (); haegeman:shadows (); marien:fibonacci-condensation (); fernandez:symmetrized-tcode (). This raises the question: What is the exact relation between topological phase transitions in tensor networks and anyon condensation, and can we explain this transition “miscroscopically” using the local symmetries in the tensor network description?

In this paper, we derive a comprehensive framework for the explanation, classification, and study of anyon condensation in PEPS. Our framework explains and classifies anyon condensation in terms of the different “entanglement phases” emerging at the boundary under the action of the local entanglement symmetry of the tensor, and provides us with the tools to explicitly study the behavior of order parameters measuring condensation and confimement of anyons. More specifically, we show that the symmetry constraint in the entanglement degrees of freedom of the tensor gives rise to a corresponding “doubled” symmetry in the fixed point of the transfer operator, this is, in the entanglement spectrum at the boundary. Anyon condensation can then be understood in terms of the different phases at the boundary, this is, the symmetry breaking pattern together with a possibly symmetry-protected phase of the residual unbroken symmetry. We give necessary and sufficient conditions for the condensation of anyons in abelian double models in terms of the symmetry at the boundary, and show that this completely classifies all condensation patterns in double models of cyclic groups, giving rise to all twisted double models. We also show that these conditions allow to independently derive the anyon condensation rules described above, providing a tensor network derivation of these conditions. The central idea is to relate anyon condensation and confinement to the behavior of string order parameters, which in turn can be related to symmetry breaking and symmetry-protected order, and combine this with the constraints arising from the positivity of the boundary state.

We illustrate our framework by discussing all possible phases which can be obtained by condensation from a double model, which can give rise to Toric Code, Doubled Semion, and trivial phases. Specifically, we show that the Toric Code and Double Semion can exhibit the same symmetry breaking pattern at the boundary, yet are distinguished by different SPT orders, corresponding to the condensation of a charge or a dyon (a combined charge-flux particle), respectively, and thus a different string order parameter. Finally, we apply our framework to numerically study topological phases and the transitions between them along a range of different interpolations. Specifically, the interpretation of condensation and confinement in terms of string order parameters allows us to directly measure order parameters for the different topological phases, namely condensate fractions and order parameters for deconfinement, which allow us to study the nature and order of the phase transitions. Our framework also allows us to set up interpolations between the Toric Code and Double Semion phase, which are a priori not related by anyon condensation, and we find that depending on the nature of the interpolation, we can either find a second-order simultaneous confinement-deconfinement transition, or a first-order transition not characterized by anyon condensation.

The paper is structured as follows: In Sec. II, we introduce PEPS, explain how topological order and topological excitations are modelled within this framework, and define condensation and confinement in PEPS models. Sec. III contains the classification of anyon condensation and confinement through the behavior of the boundary: We start by giving the intuition and the main technical assumption, then derive the conditions imposed by the symmetry structure and positivity of the boundary state, and finally show that this classification gives rise to the well-known anyon condensation rules. In Sec. IV, we apply this classification to the case of quantum doubles and show that it precisely gives rise to all twisted double models. Finally, in Sec. V, we illustrate our framework with a detailed discussion of the condensation from a double, and study the corresponding family of models and the transitions between them numerically.

Ii Symmetries in PEPS and anyons

In this section, we will first introduce the general PEPS framework. We will then explain how certain symmetries in PEPS naturally lead to objects defined on the entanglement degrees of freedom which behave like anyonic excitations. The natural question is then to understand the conditions under which these objects describe observable anyons, or whether they fail to do so by either leaving the state invariant (condensation) or by evaluating to zero (confinement) in the thermodynamic limit.

We will focus our discussion to the case of abelian groups; however, several of our arguments in fact apply to general groups, and even beyond that for so-called MPO-injective PEPS; we will discuss these aspects in Sec. VI.

ii.1 PEPS, parent Hamiltonians, and excitations

Let us start by introducing Projected Entangled Pair States (PEPS). We focus on a translational invariant system on a square lattice with periodic boundary conditions, where we take the system size to infinity. PEPS are constructed from a local tensor , where is the physical index and are the virtual indices, and is called the bond dimension. Graphically, they are depicted as a sphere with five legs, one for each index, cf. Fig. LABEL:fig:peps-defa; equivalently, we can consider as a linear map from virtual to physical system. The tensor is then arranged on a square lattice, Fig. LABEL:fig:peps-defb, and adjacent virtual indices are contracted (i.e., identified and summed over), which is graphically depicted by connecting the corresponding legs. We thus finally obtain a tensor which only has physical indices, and thus describes a quantum many-body state . A useful property of PEPS is the possibility to block sites – we can take the tensors on some patch and define them as a new tensor with correspondingly larger . This allows us to restrict statements about properties of localized regions to fixed-size (e.g., single-site or overlapping ) patches.

To any PEPS, one can naturally associate a family of parent Hamiltonians which have this PEPS as their exact zero-energy ground state perez-garcia:parent-ham-2d (); schuch:peps-sym (). Such a Hamiltonian is a sum of local terms , each of which ensures that the state “looks locally correct” on a small patch, i.e., as if it had been built from the tensor on that patch. This is accomplished by choosing such that is zero on the physical subspace spanned by the tensors on that patch (for arbitrary virtual boundary conditions) and positive otherwise; note that by choosing a sufficiently large patch, it is always possible to find a non-trivial such Hamiltonian (the dimension of the allowed physical subspace scales with the boundary, while the available degrees of freedom scale with the volume).

Clearly, the global PEPS wavefunction is a zero-energy state and thus a ground state of the parent Hamiltonian . At the same time, conditions on are known under which this ground state is unique (in a finite volume) perez-garcia:parent-ham-2d (): Specifically, it is sufficient if the map from the virtual to the physical system described by (possibly after blocking) is injective; equivalently, this means that the full auxiliary space can be accessed by acting on the physical space only, i.e., that one can apply a linear map which “cuts out” a tensor and gives direct access to the auxiliary indices.

Parent Hamiltonians naturally give rise to the notion of localized excitations, this is, states whose energy differs from the ground state only in some local regions. To this end, one replaces some tensors by “excitation tensors” , while keeping the original tensor everywhere else, cf. Fig. 2a. For injective PEPS, these are in fact the only possible localized excitations, since due to the one-to-one correspondence between virtual and physical system any tensor will yield an increased energy w.r.t. the parent Hamiltonian perez-garcia:parent-ham-2d ().

Figure 2: (a) Replacing one tensor by a different tensor results in a localized excitation, i.e., which is not detected by the parent Hamiltonian anywhere else. (b) An excitation is topologically trivial if can be obtained from by acting with a map on the physical system.

A key question in the context of this work is when an excitation is topologically non-trivial. We will use the following definition: An excitation is topologically trivial exactly if it can be created (with some non-zero probability) by acting locally on the system, i.e., if there exists a linear (not necessarily unitary) map on the physical system which will create that excitation on top of the ground state, this is, which transforms to . It is now straightforward to see that for an injective PEPS, all localized excitations (Fig. 2a) are topologically trivial: Injectivity implies that (as a map from virtual to physical system) has a left-inverse , and thus will act as , i.e., create the desired excitation locally, as shown in Fig. 2b.

ii.2 G-injective PEPS and anyonic excitations

Let us now turn towards PEPS which can support topologically non-trivial excitations. To this end, we consider PEPS which are no longer injective, but enjoy a virtual symmetry under some group action,


with a unitary representation of some finite group ; we will denote such tensors as -invariant. Graphically, this is expressed as


where we use the convention that matrices act on the indices from left to right and down to up, such that in Eq. (1) turns into . An important property of -invariance is its stability under concatenation: When grouping together several -invariant tensors, the resulting block is still -invariant, as the and on the contracted indices exactly cancel out. In the following, we will focus on abelian groups (though various parts of the discussion generalize to the non-abelian case), and denote the neutral element by .

If -invariance is the only symmetry of the tensor , i.e., if is injective on the subspace left invariant by the symmetry, we call -injective. The parent Hamiltonians of -injective PEPS have a topological ground space degeneracy and can support anyonic excitations schuch:peps-sym (), as we will also discuss in the following. We will generally assume that the tensors are -injective, since otherwise we might be missing a symmetry, likely rendering the discussion incomplete.

ii.2.1 Electric excitations

In order to understand how these excitations look like, let us consider again the possible localized excitations w.r.t. the parent Hamiltonian. As we have seen earlier, any state where one tensor has been replaced by a different tensor is by construction a localized excitation. In the injective case, any such could be obtained by acting locally on the physical degrees of freedom, rendering the excitation topologically trivial. However, it is easy to see that this is no longer the case for -invariant tensors: Local operations (Fig. 2b) can only produce tensors which are again -invariant, i.e., transform trivially under the action of the symmetry group, since it is exactly the invariant virtual subspace which is accessible by acting on the physical indices. In contrast, ’s which transform non-trivially can no longer be created locally, and thus are topologically non-trivial excitations. It is natural to label these excitations by irreducible representations of the abelian symmetry group , this is, we can write



This is, any such excitation can be understood as a superposition of excitations with fixed , and we will focus on excitations with a fixed in the following. These excitations will be denoted as electric excitations with charge . (For non-abelian groups, we would require instead that each is supported on the irrep of the group action.)

It is straightforward to see that for -injective PEPS, the topological part of the excitation is fully characterized by : In case itself is injective on the irrep , this is immediate since it can be transformed into any other by locally acting on the physical index; in case is not injective, the same can be done by acting on a block centered around (due to -injectivity, this allows to access all degrees of freedom at the boundary in the irrep ).

In the following, we will focus our attention on electric excitations of the form


where (the yellow diamond) transforms as ; the general case will be discussed in Appendix B.

An important point to note about electric excitations is that for any system with periodic boundaries, they must come in pairs (or groups) which together transform trivially under the symmetry action, i.e., have total trivial charge, since otherwise the state would vanish.

ii.2.2 Magnetic excitations

For injective PEPS, locally changing tensors was the only way to obtain localized excitations, due to the one-to-one correspondence of physical and virtual system perez-garcia:parent-ham-2d (). For -injective PEPS, however, there exist ways to non-locally change the tensor network without creating an excitation, or only creating a localized excitation schuch:peps-sym (). To this end, note that Eq. (2) can be reformulated as


and rotated versions thereof. This has the natural interpretation of the and forming strings (symbolized by the dashed blue lines above), which can be freely moved through the lattice (“pulling though condition”). (Whether or has to be used depends on the orientation of the string relative to the lattice schuch:peps-sym ().) Thus, any string of ’s is naturally invisible to the parent Hamiltonian, as it can be moved away from any patch the parent Hamiltonian acts on. Indeed, if -injectivity holds, one can use the equivalence of physical and virtual system on the invariant subspace to prove that such strings are the only non-local objects which cannot be detected by the parent Hamiltonian schuch:peps-sym (). This yields a natural way to build localized excitations by placing a string of ’s with open ends on the lattice, as illustrated in Fig. LABEL:fig:excitation-string-pair: Any such string can only be detected at its endpoints, thereby forming a localized excitation. These excitations are topological by construction, since by acting on the endpoints alone, we are not able to create such a string. At the same time, using -injectivity one can prove that the endpoints can always be detected in a finite system. Thus, we arrive at a second type of topological non-trivial excitations, namely strings of ’s with an endpoint,

(We have followed the notation introduced in Fig. LABEL:fig:excitation-string-pair, where blue dots denote or , and the dashed blue line highlights the string formed.) Again, is an arbitrary -invariant tensor which can be used to dress the endpoint with an arbitrary topologically trivial excitation; under blocking, it can always be assumed to only sit on a single site as shown. Again, given periodic boundaries any such string must end in a second anyon (or more generally the strings emerging from several anyons can fuse as long as the corresponding group elements multiply to the identity).

We will denote these excitations as magnetic excitations with flux .

ii.2.3 Dyonic excitations

Beyond electric and magnetic excitations, it is also possible to combine the two into a so-called dyon which is of the form


Note that we have made the choice that the irrep sits on the same leg at which the -string ends. While this choice is arbitrary, it is related to any other endpoint, e.g. one where the string ends on the leg before , by a local -string, i.e., a pair of magnetic excitations, which can be created locally and can thus be accounted for by an appropriate choice of , or even incorporated in .

A general anyonic excitation is thus up to local modifications labeled by a tuple and ; we denote the anyon by , and an anyon string with the two conjugate anyons and at its endpoints by .

ii.2.4 Braiding statistics

Let us briefly comment on the braiding statistics of these excitations; we refer to Ref. schuch:peps-sym () for details. Any physical procedure for moving anyons will result in the -string being pulled along the path. Thus, a half-exchange of two identical anyons transforms

(where for simplicity we have set , as it transforms trivially). Straightening the string by pulling it through the right excitation requires to commute with , which gives rise to a phase ; since the resulting two crossing strings are identical to two non-crossing strings, we thus obtain a overall phase of due to the half exchange.

Similarly, full exchange of two different anyons and gives rise to two such exchanges, and thus to a mutual statistics for a full exchange.

We therefore see that the strings defined this way indeed exhibit the same statistics as , the quantum double model of  kitaev:toriccode (); schuch:peps-sym ().

ii.3 Virtual level vs. observable excitations: Condensation and confinement

ii.3.1 Anyon condensation and confinement

It is suggestive to assume that this is the complete picture, and -injective PEPS always exhibit an anyon theory given by the quantum double . However, it has by now been understood that this is not the case schuch:topo-top (): By adding a physical deformation to the tensor, , one can drive the system towards a product state, eventually crossing a phase transition. E.g., in the toric code this induces string tension (or more precisely loop fugacity), which eventually leads to the breakdown of topological order castelnovo:tc-tension-topoentropy ().

This is directly related to the question as to whether the objects which we have just identified as anyonic excitations on the virtual level actually describe observable anyons in the thermodynamic limit, and in the limit of large separation between the individual anyons. While, as we have argued, one can prove schuch:peps-sym () that the endpoints of a virtual string correspond to observable excitations, this only applies in a finite volume. However, it is perfectly possible that—depending on the choice of —new behavior emerges in the thermodynamic limit, which is reflected in a non-trivial environment imposed on a virtual anyon string (with the separation between the endpoints) which can prevent it from describing an observable anyonic excitation as . This can happen in at least two distinct ways: Either the environment transforms trivially under , in which case the PEPS with still describes the ground state, or the environment is orthogonal to , in which case the state has norm zero and is thus unphysical.

Figure 4: Tensor networks for detecing (a) confinement (the network for the norm of the state evaluates to zero) and (b) condensation (the network for the overlap with the ground state evaluates to non-zero, both as ) for a general anyon pair of the form Eq. (6).

We will thus distinguish two different ways in which non-trivial virtual excitations might fail to describe observable anyonic excitations:
1. Confinement: The state of the system with an anyon string does not describe a properly normalizable quantum state, i.e.,


where first the system size and then the separation is taken to infinity. The expectation value in Eq. (7) corresponds to the tensor network in Fig. 4a, this is, the expectation value of the string operator in the double layer ket+bra tensor network.
2. Condensation: is not orthogonal to the ground state in the thermodynamic limit,


i.e., the individual endpoints are not distinguished any more from the ground state by a topological symmetry, and thus differ from it at most in local properties. The corresponding tensor network is shown in Fig. 4b and corresponds to the expectation value of the string operator .

In the remainder of this paper, we will explore the conditions under which condensation and confinement occurs in PEPS models, and provide a classification of the possibly ways in which this can happen.

ii.3.2 Condensation, confinement, and string order parameters

In order to understand condensation and confinement of anyons in PEPS models, we need to assess the behavior of overlaps , corresponding to string operators on the virtual level, cf. Fig. 4, in the thermodynamic limit and as . In what follows, we will assume for simplicity; we discuss how to adapt the arguments to the general case in Appendix B.

It is instrumental to introduce the transfer operator

which is a completely positive map (from left to right) acting on a one-dimensional chain of -level systems; if we disregard complete positivity, we can equally think of as a map on a 1D chain of systems. In the following, we will restrict to the case of hermitian (corresponding e.g. to a system with combined reflection and time-reversal symmetry), which in particular implies that the left and right fixed points of are equal.

Let us now see how the symmetry of the tensor is reflected in the transfer operator. -invariance of the is inherited by , which thus enjoys the symmetries (with the system size); this is, carries an on-site symmetry with representation , with . The irreps of are given by , where and are irreps of ; there is thus a correpondence between irreps of and pairs of irreps of , and we will write . The trivial irrep will be denoted by . Finally, we define . Generally, we will stick to the convention that we use boldface letters for objects living on ket+bra.

In terms of the transfer operator, we can now re-express our quantities of interest for condensation and confinement as expectation values of in some left and right fixed points and of

where we assume . [We use round brackets to denote vectors on the joint ket+bra virtual level.] The can also be understood as operators acting between ket and bra level, in which case we will denote them by . Specifically, has been shown to exactly reproduce the entanglement spectrum of a bipartition of the system cirac:peps-boundaries (), and thus any statement about the translates into a property of the entanglement spectrum. Note that is formed exactly by a string of symmetry operations and terminated by irreps of the doubled symmetry group , i.e., a string order parameter, and it is thus suggestive to understand the condensation and confinement of anyons by studying the possible behavior of string order parameters for the group .

Iii Classification of string order parameters and condensation

The following section presents the core result of the paper: We classify all different behaviors which the string operators in a -invariant PEPS can exhibit by relating them to the classification of symmetry-protected (SPT) phases in one dimension, as given by the fixed point of the transfer operator. We start in Sec. III.1 by explaining the intuition why the classification of anyon behaviors should be related to the classification of 1D phases. In Sec. III.2 we explicitly state the technical assumptions made (specifically, the form of the fixed point space). Secs. III.3III.6 contain the classification: In Sec. III.3, we study the structure of symmetry breaking of the fixed point space and show that the endpoints decouple as , allowing us to restrict to semi-infinite strings in the following; in Sec. III.4, we derive the constraints imposed by the symmetry breaking on the anyons and show how it allows to decouple anyon pairs; in Sec. III.5, we make the connection between the behavior of anyons and the SPT structure of the fixed points, and in Sec. III.6, we show that there exists an additional non-trivial restriction on the SPTs which can appear as fixed points of , and thus to the possible anyon behavior, arising from the (complete) positivity of . Finally, in Sec. III.7, we show that the conditions derived in the preceding sections precisely give rise to the known anyon condensation rules.

iii.1 Intuition

Let us first present the intuition behind this classification. To this end, we use that we are interested in gapped phases and thus the system is short-range correlated: This suggests that the fixed point of the transfer operator is short range correlated as well, and thus has the same structure as the ground state of a local Hamiltonian with the identical symmetry .

Let us now consider the different phases of such a Hamiltonian. We first restrict to the the regime of Landau theory, where phases are classified by order parameters, i.e., irreps of the symmetry group. Depending on the phase, different irreps will have zero or non-zero expectation values, which implies condensation [for a non-zero expectation value of an irrep with ] and confinement [for a vanishing expectation value of an irrep ] of charges, corresponding to broken diagonal or unbroken non-diagonal symmetries, respectively. On the other hand, assuming a mean-field ansatz (which is exact in a long-wavelength limit), we find that strings of group actions either create a domain wall (for a broken symmetry) or act trivially (for an unbroken symmetry), relating the symmetry breaking patterns also to the condensation and confinement of magnons. We thus see that the condensation and confinement of electric and magnetic excitations corresponds to Landau-type symmetry breaking in the fixed point of the transfer operator, as observed in Ref. haegeman:shadows (). As we will see in the following, this picture becomes more rich when we go beyond Landau theory and allow for SPT phases: These phases are not captured by mean-field theory and are rather characterized by the behavior or string order parameters, i.e., strings of group actions terminated by order parameters, which give rise to condensation and confinement of dyonic excitations.

iii.2 The assumption: Matrix Product fixed points

We start by stating our main technical assumption: The fixed point space of (possibly after blocking) is spanned by a set of injective Matrix Product States (MPS), which are related by the action of the symmetry group.

Let us be more specific. Let denote a joint ket+bra index of the blocked transfer operator. Then, we assume there exists a set of matrices which describe distinct MPS

on a finite chain with periodic boundary conditions. We require that these MPS fulfill the following conditions:

  1. The span the full fixed point space of . (This is, evaluating any quantity of interest either in the fixed point space of or in yields the same result in the thermodynamic limit.)

  2. The are injective, i.e., has a unique eigenvalue with maximal magnitude, where is the mixed transfer operator of the MPS. W.l.o.g., we choose to normalize such that .

  3. For each and , there is a such that , and for each pair , , there is a corresponding . (Here and in the following, we use as a shorthand for the global symmetry action whenever the meaning is clear from the context.)

Note that we make no assumption that the are positive, and in fact in many cases the fixed point space cannot be spanned by positive and injective MPS.

Assumption 1 is the main technical assumption here. Note that to some extent a similar assumption underlies the classification of phases of 1D Hamiltonians chen:1d-phases-rg (); schuch:mps-phases (), where the ground space is approximated by MPS as well: While this is motivated by the known result that MPS can approximate ground states of finite systems efficiently hastings:arealaw (); verstraete:faithfully (); schuch:mps-entropies (), also in that scenario it is yet unproven whether this rigorously implies that MPS are sufficiently general to classify phases in the thermodynamic limit.

Assumptions 2 and 3 can be replaced by the weaker assumption that the fixed point space is spanned by some MPS, together with the assumption that we are not missing any symmetries. Specifically, given an MPS with periodic boundary conditions, it can be brought into a standard form (possibly involving blocking of sites) where it can be understood as a superposition of distinct injective MPS (possibly with size-dependent amplitudes) perez-garcia:mps-reps (); cirac:mpdo-rgfp (). While the are not necessarily fixed points of the transfer operator themselves, such as in the case of an antiferromagnet where the transfer operator acts by permuting the , they will be fixed points of the transfer operator obtained after suitable blocking. Since, as we will see in a moment, cross-terms between different vanish when computing physical quantities of interest, we can instead work with a fixed point space spanned by the  111Note that this does not imply that the fixed point space is actually spanned by the . In fact, it is easy to see that this would require extra conditions such as rotational invariance, since e.g. a transfer operator projecting onto a GHZ-type state would have a unique fixed point (the GHZ state) which is not an injective MPS., corresponding to Assumption 2.

Figure 5: Possible symmetries in the single-site tensor of the transfer operator. (a) Joint ket+bra single-site tensor. (b) Single-layer symmetry, related to topological order, Eq. (1). (c) Double-layer symmetry. (d) Local encoding of a global physical symmetry related to the symmetry (c). Degeneracy of the transfer operator under such a joint symmetry implies breaking of the physical symmetry rispler:peps-symmetrybreaking ().

Assumption 3 can be justified by requiring that any degeneracy is due to some symmetry of the transfer operator—otherwise, it would be an accidental degeneracy and thus not stable against perturbations. Since the transfer operator has itself a Matrix Product structure, any symmetry of the transfer operator must be encoded locally, i.e., it will show up as a symmetry of the single-site ket+bra object shown in Fig. 5perez-garcia:inj-peps-syms (). There can be two distinct types of such symmetries: Those which act identically on ket and bra layer, shown in Fig. 5b for on-site symmetries, and those which only act on one layer, shown in Fig. 5c. (Symmetries which act on the two layers in distinct ways can be split into a product of the former two symmetries, cf. the argument at the beginning of Sec. III.3.) Symmetries which only act on one layer correspond to topological symmetries of the PEPS tensor, such as those of Eq. (1), and thus need to be incorporated into the description from the very beginning. Symmetries acting identically on ket and bra layer, on the other hand, give rise to a non-trivial physical symmetry action through the identity in Fig. 5d and thus correspond to a global physical symmetry of the system; since their corresponding symmetry sectors are degenerate in the transfer operator, they are susceptible to physical perturbations which lead to symmetry breaking rispler:peps-symmetrybreaking (), and we can therefore assume that the system is in one of the symmetry-broken sectors, in which all fixed points are related by the action of the topological symmetry. This in particular includes breaking of translational symmetry, which warrants that we can obtain injective tensors by blocking sites. Note that it is conceivable that different symmetry-broken sectors are described by a different condensation scheme (a simple example can be obtained by coupling different deformations of the system to an Ising model).

iii.3 Symmetry breaking structure

In this section, we clarify the symmetry breaking structure of the fixed point space, and show that the relevant expecation values do not depend on which vector in the fixed point space we choose.

To this end, consider the set of satisfying the three assumptions just laid out. For each , let . It is clear that is a subgroup of ; furthermore, for abelian is independent of , since for any and s.th. ,

and we write .

What is the structure of ? To this end, consider arbitrary s.th. . For any , we have that , and thus

and thus . Since , this implies that as well, or


and similarly . Now choose s.th. , the fixed point of obtained when starting from , and pick some . Then, for sufficiently small , and , and thus and are both positive fixed points and therefore satisfy Eq. (9), which implies that also satisfies . We thus find that whenever , we must also have that and .

Now consider a general element . Then, , and thus . It follows that and form groups, and since , .

Condition 1

The conserved symmetry is isomorphic to a direct product with , where labels the the diagonal and the off-diagonal symmetry, i.e., with and .

To distinguish it from the ket/bra product, we will denote the diagonal/off-diagonal product by .

Let us now consider the evaluation of a anyonic string order parameter inside general left and right boundary conditions . This results in a sum over terms of the form


where we supress the dependency of on and . In case , the largest eigenvalue of the mixed transfer operator is strictly smaller than one (a straightforward application of Cauchy-Schwarz, see e.g. Lemma 8 of Ref. fernandez-gonzalez:uncle-long ()), and thus, exponentially as , i.e., only terms with survive in the thermodynamic limit. In case , we use that for some fiducial with , and thus


and since and commute, and the phases from commuting and cancel out, we find that . We thus find that the expectation value for any string is the same regardless of the boundary conditions, and we will therefore omit the subscript from now on and write and (in fact, we will most of the time also omit the label of the tensor).

After these considerations, we are left with the following question: Given a symmetry , , and an invariant fixed point given by an injective MPS with tensor , what are the the different possible ways in which strings describing the behavior of anyons can behave regarding condensation and confinement.

iii.4 Behavior of string order parameters I: Symmetry breaking and decoupling

Let us now consider what happens when we separate the two ends of a string . Evaluated in the fixed point MPS , this corresponds to


We now distinguish two cases: If , then with , and since different representations of an injective MPS are related by a local gauge transformation perez-garcia:mps-reps (); cirac:mpdo-rgfp (), it holds that

and thus

and since , as . We thus obtain

Condition 2

unless . In particular, all anyons with are confined.

On the other hand, if , and thus there exist such that


where forms a projective representation of which can be chosen unitary by a suitable gauge of the MPS sanz:mps-syms (). Injectivity of the MPS further implies that its transfer operator has a unique fixed point

(w.l.o.g., we choose , and normalization implies ), and using Eq. (13), this implies that

Also, since , uniqueness of the fixed point of implies that , where , and the ordering of the indices of is chosen accordingly. With this, we can rewrite Eq. (12) as




and correspondingly . This implies that the expectation value of any string order parameter decouples into a product of two expectation values corresponding to semi-infinite strings, and in order to study condensation and confinement, it is thus sufficient to to consider the behavior of , Eq. (15). In order to highlight the role played by the two layers, we will sometimes also write , with , .

iii.5 Behavior of string order parameters II: Symmetry protected phases and group cohomology

We will now study the behavior of string order parameters , Eq. (15), with more closely and show that they are directly related to the classification of symmetry-protected phases through group cohomology. The crucial point here is that, following Eq. (13), a physical symmetry action can be replaced by a virtual symmetry action , where the form a projective representation of the symmetry group, i.e., , where is a so-called -cocycle – i.e., it satisfies due to associativity – which, up to gauge choices is classified by the second cohomology group ; this discrete classification of the is what is underlying the classification of symmetry-protected phases in one dimensions pollmann:symprot-1d (); chen:spt-order-and-cohomology (); schuch:mps-phases ().

The -cocycle also encodes what happens when we commute and :

Here, is called the slant product propitius:phd-thesis () of with ; for abelian groups, it forms a one-dimensional representation of ,  222This can be seen using the cocycle conditions and the fact that the group is abelian as follows: . Note that we can always construct (non-unique) representations and of such that for : To this end, let for , extend to a representation of (formally, this corresponds to an induced representation), and define ; finally, both and can be extended independently to representations of .

We will now derive conditions on and under which must be zero and demonstrate how in the remaining cases, it can be made non-zero by an appropriate choice of , and we find that this is in one-to-one correspondence to the inequivalent -cocycles, i.e., elements of ; the no-go part part of this discussion has been first given in Ref. pollmann:spt-detection-1d () in the context of string order parameters for SPT phases. To this end, let us consider an MPS with a specific projective representation with corresponding , and consider a string order parameter evaluated in that MPS,


We now insert a resolution of the identity before and use , which gives

We thus find that for to be non-zero, it must hold that , the irreducible representation obtained as the slant product of the -cocycle . Conversely, by choosing such that


– which is always possible due to the injectivity of – we have that

i.e., transforms as on as required, and

It remains to see how transforms under the action of the full symmetry group , and more specifically that the construction can be generalized to any irrep of with restriction ; this, together with how to separate into independent ket and bra actions, is discussed in Appendix A.

We thus see that the behavior of string order parameters is in one-to-one correspondence with the different SPT phases appearing in the fixed point of the transfer matrix: For a given SPT phase, a string order parameter can only be non-zero if , and at the same time, it is always possible to set up the endpoint of the string order parameter such that actually is non-zero.

Condition 3

A string operator with exists if and only if for all , with , , and , where is the -cocycle classifying the fixed point of the transfer operator.

Clearly, the same derivation for the other endpoint of the string, , yields exactly the same condition.

Note that Conditions 2 and 3 together show that the “amount of topological order” – this is, the number of anyons – is related to “symmetry breaking gap” between ket and bra, , where : Deconfined anyons satisfy , i.e., , and , which fixes on and thus leaves possibilities to extend it to , yielding a total of deconfined anyons. Out of those, pairs and are indistinguishable if , i.e., , and , which fixes on , leaving possible extensions; the size of each set of indistinguishable anyons is thus . The total number of anyons – the ratio of these numbers – is thus , and the total quantum dimension is , the “symmetry breaking gap” between ket and bra.

iii.6 Constraints from positivity

The condition that iff (Condition 3) has been derived for a general fixed point of MPO form. However, as we have seen in Sec. III.3, we can w.l.o.g. take the fixed point to be positive semidefinite, which gave rise to the structure of the unbroken symmetry subgroup (Condition 1). As we will see now, positivity induces yet another constraint, namely on the -cocycles realizable in the fixed point.

To this end, consider a positive fixed point with an SPT characterized by some -cocycle , and consider some ,