Entanglement phases as holographic duals of anyon condensates
Kasper Duivenvoorden Mohsin Iqbal Jutho Haegeman Frank Verstraete Norbert Schuch








JARA Institute for Quantum Information, RWTH Aachen
University, 52056 Aachen, Germany

JARA Institute for Quantum Information, RWTH Aachen
University, 52056 Aachen, Germany
Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Str.1, 85748 Garching, Germany

Ghent University, Department of Physics and Astronomy,
Krijgslaan 281-S9, 9000 Gent, Belgium

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

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 ℤ_4
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.  

Entanglement phases as holographic duals of anyon condensates
    Norbert Schuch
    Accepted .  Received ; in original form 
=============================================================





§ 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 <cit.>. The main idea is that
some mechanism drives a species a of bosonic anyons to condense into the
vacuum. This, in turn, forces any anyon b which has non-trivial
statistics with a to become confined, as a deconfined b anyon would
have non-trivial statistics with the new vacuum, and moreover leads to the
identification of anyons which differ by fusion with a. 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 <cit.>.



Projected Entangled Pair States (PEPS) <cit.> form a
natural framework for the local modelling of topologically ordered
phases <cit.>. 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) <cit.> or
more generally by Matrix Product Operators for twisted
doubles <cit.> and string-net
models <cit.>. 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 <cit.>.
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 <cit.>.
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 ℤ_N 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 ℤ_4 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. <ref>, we introduce PEPS, explain how
topological order and topological excitations are modelled within this
framework, and define condensation and confinement in PEPS models.
Sec. <ref> 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. <ref>, we apply this classification to the case of
ℤ_N quantum doubles and show that it precisely gives rise to all
twisted ℤ_M double models. Finally, in
Sec. <ref>, we illustrate our framework with a detailed
discussion of the condensation from a ℤ_4 double, and study the
corresponding family of models and the transitions between them
numerically.












§ 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. <ref>.



 §.§ 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 A^i_αβγδ, where
i=1,…,d is the physical index and
α,β,γ,δ=1,…,D are the virtual indices, and
D is called the bond dimension.  Graphically, they are depicted as a
sphere with five legs, one for each index, cf. Fig. <ref>a;
equivalently, we can consider A=∑ A^i_αβγδ|i⟩⟨α,β,γ,δ|
as a linear map from virtual to physical system.  The tensor A is then
arranged on a square lattice, Fig. <ref>b, 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 c_i_1… i_N which only has physical indices, and thus
describes a quantum
many-body state |Ψ⟩= ∑ c_i_1… i_N|i_1,…,i_N⟩. 
A useful property of PEPS is the possibility to block sites – we can take
the tensors on some k_1× k_2 patch and define them as a new tensor
A' with correspondingly larger D. This allows us to restrict
statements about properties of localized regions to fixed-size (e.g.,
single-site or overlapping 2× 2) patches.

To any PEPS, one can naturally associate a family of parent Hamiltonians
which have this PEPS as their exact zero-energy ground
state <cit.>. Such a
Hamiltonian is a sum of local terms h, 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 A on that patch.  This is accomplished by choosing
h≥0 such that h 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 H=∑ h≥0.  At the same time,
conditions on A are known under which this ground state is unique (in a
finite volume) <cit.>: Specifically, it is
sufficient if the map from the virtual to the physical system described
by A (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” B, while keeping the original tensor A everywhere
else, cf. Fig. <ref>a.  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 B A will yield an increased energy w.r.t. the parent
Hamiltonian <cit.>. 




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 L on the physical
system which will create that excitation on top of the ground state, this
is, which transforms A to B.  It is now straightforward to see that
for an injective PEPS, all localized excitations
(Fig. <ref>a) are topologically trivial: Injectivity
implies that A (as a map from virtual to physical system) has a
left-inverse A^-1, and thus L:=BA^-1 will act as LA=B, i.e.,
create the desired excitation locally, as shown in
Fig. <ref>b.



 §.§ 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, 

    A=A(U̅_g⊗U̅_g  ⊗ U_g⊗ U_g)

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

    -2em
        < g r a p h i c s >
      ,

where we use the convention that matrices act on the indices from left to
right and down to up, such that U̅_g in Eq. (<ref>)
turns into U_g^†.  An important property of G-invariance is its
stability under concatenation: When grouping together several G-invariant
tensors, the resulting block is still G-invariant, as the U_g and
U_g^† 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 e∈ G.

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




  §.§.§ 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 B is by construction a localized excitation.  In the
injective case, any such B 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
G-invariant tensors:  Local operations
(Fig. <ref>b) can only produce tensors B
which are again G-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, B'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 α(g)∈ℂ of
the abelian symmetry group G, this is, we can write 

    B=∑_α B_α ,

where 

    < g r a p h i c s >

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

It is straightforward to see that for G-injective PEPS, the topological
part of the excitation is fully characterized by α: In case
B_α itself is injective on the irrep α, this is immediate
since it can be transformed into any other B_α' by locally acting
on the physical index; in case B_α is not injective, the same can
be done by acting on a 3× 3 block centered around α (due to
G-injectivity, this allows to access all degrees of freedom at the
boundary in the irrep α).

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

    < g r a p h i c s >

where R_α (the yellow diamond) transforms as 
R_α U_g = α(g) U_g R_α;
the general case will be discussed in Appendix <ref>.

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.





  §.§.§ 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 <cit.>. For G-injective
PEPS, however, there exist ways to non-locally change the tensor network
without creating an excitation, or only creating a localized
excitation <cit.>. To this end, note that
Eq. (<ref>) can be reformulated as

    -1.5em
        < g r a p h i c s >
    -1.5em
        < g r a p h i c s >

and rotated versions thereof.
This has the natural interpretation of the U_g and U_g^† forming
strings (symbolized by the dashed blue lines above), which can be freely
moved through the lattice (“pulling though condition”).  (Whether U_g
or U_g^† has to be used depends on the orientation of the string
relative to the lattice <cit.>.) Thus, any string of
U_g's is naturally invisible to the parent Hamiltonian, as it can be
moved away from any patch the parent Hamiltonian acts on. Indeed, if
G-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 <cit.>. This yields a natural way to build
localized excitations by placing a string of U_g's with
open ends on the lattice, as illustrated in
Fig. <ref>: 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 G-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 U_g's
with an endpoint,

    < g r a p h i c s >
       .
 
(We have followed the notation introduced in
Fig. <ref>, where blue dots denote U_g or
U_g^†, and the dashed blue line highlights the string formed.)
Again, C is an arbitrary G-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 g.






  §.§.§ 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

    -3em
        < g r a p h i c s >

Note that we have made the choice that the irrep R_α sits on the
same leg at which the U_g-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 R_α, by a local U_g-string, i.e., a pair of
magnetic excitations, which can be created locally and can thus be
accounted for by an appropriate choice of C, or even incorporated in
R_α.

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




  §.§.§ Braiding statistics


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

    < g r a p h i c s >

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

Similarly, full exchange of two different anyons gα and
hβ gives rise to two such exchanges, and thus to a mutual
statistics α(h)β(g) for a full exchange.

We therefore see that the strings defined this way indeed exhibit the same
statistics as D(G), the quantum double model of
G <cit.>.





 §.§ Virtual level vs. observable excitations: Condensation and confinement




  §.§.§ Anyon condensation and confinement


It is suggestive to assume that this is the complete picture, and
G-injective PEPS always exhibit an anyon theory given by the quantum
double D(G). However, it has by now been understood that this is not the
case <cit.>: By adding a physical deformation Λ to
the tensor, A→Λ A, 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 <cit.>.

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 <cit.> that the endpoints of a
virtual string gα correspond to observable excitations,
this only applies in a finite volume.  However, it is perfectly possible
that—depending on the choice of A—new behavior emerges in the
thermodynamic limit, which is reflected in a non-trivial environment
imposed on a virtual anyon string gαℓ (with
ℓ the separation between the endpoints) which can prevent it from
describing an observable anyonic excitation as ℓ→∞.
This can happen in at least two distinct ways: Either the environment
transforms trivially under gαℓ, in which case the
PEPS with gαℓ still describes the ground
state, or the environment is orthogonal to gαℓ,
in which case the state has norm zero and is thus unphysical.




We will thus distinguish two different ways in which non-trivial virtual
excitations gα might fail to describe observable anyonic
excitations:


1. Confinement: The state |ψ[gαℓ]⟩
of the system with an anyon string does not describe a properly
normalizable quantum state, i.e., 

    ⟨ψ[gαℓ]
    |ψ[gαℓ]⟩→ 0
    N,ℓ→∞

where first the system size N and then the separation ℓ is taken to
infinity.  The expectation value in Eq. (<ref>)
corresponds to the tensor network in Fig. <ref>a, this is,
the expectation value of the string operator
gαℓ⊗gαℓ in the
double layer ket+bra tensor network.



2. Condensation: |ψ[gαℓ]⟩ is not 
orthogonal to the ground state |ψ⟩ in the thermodynamic limit,

    ⟨ψ|ψ[gαℓ]⟩0
         N,ℓ→∞ ,

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. <ref>b and corresponds to the expectation value of
the string operator ⊗gαℓ.  

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.






  §.§.§ 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 
⟨ψ[g'α'α̅'ℓ]
|ψ[gαℓ]⟩,
corresponding to string operators gαℓ⊗g'α'α̅'ℓ on the virtual level,
cf. Fig. <ref>, in the thermodynamic limit and as
ℓ→∞. In what follows, we will assume C=C'=A for
simplicity; we discuss how to adapt the arguments to the general case in
Appendix <ref>.  

It is instrumental to introduce the transfer operator 

    𝕋:=
    -5em
        < g r a p h i c s >

which is a completely positive map (from left to right) acting on a
one-dimensional chain of D-level systems; if we disregard complete
positivity, we can equally think of 𝕋 as a map on a 1D chain of
ℂ^D⊗ℂ^D systems.  In the following, we will
restrict to the case of hermitian 𝕋 (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 𝕋 are
equal.  

Let us now see how the symmetry of the tensor A is reflected in the
transfer operator.  G-invariance of the A is inherited by 𝕋,
which thus enjoys the symmetries [𝕋,U^⊗ N⊗]=[𝕋,⊗U̅^⊗ N]=0 (with
N→∞ the system size); this is, 𝕋 carries an
on-site G :=G× G symmetry with representation U_ g =
U_g⊗ U_g', with g≡ (g,g') ∈ G.  The irreps of G are given by α((h,h'))=
α((h,e)) α((e,h'))≡α(h)α̅'(h'),
where α(·):=α((·,e)) and
α'(·):=α̅((e,·)) are irreps of G; there is
thus a correpondence between irreps of G and pairs of irreps of G,
and we will write α = (α,α').  The trivial irrep
will be denoted by 1. Finally, we define
gαℓ:= gαℓ⊗g'α'α̅'ℓ. 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
gαℓ in some left and right fixed
points (ρ_L| and
|ρ_R) of 𝕋

    ⟨ψ[g'α'α̅'ℓ]
    |ψ[gαℓ]⟩
    =
    (ρ_L|
    gαℓ
    |ρ_R) ,

where we assume (ρ_L|ρ_R)=1.
[We use round brackets |·) to denote vectors on the joint ket+bra
virtual level.] The |ρ_∙) can also be understood as 
operators acting between ket and bra level, in which case we will denote
them by ρ_∙.  Specifically, ρ_Lρ_R has been shown to
exactly reproduce the entanglement spectrum of a bipartition of the
system <cit.>, and thus any statement about the
ρ_∙ translates into a property of the entanglement spectrum.
Note that gαℓ is formed exactly by a string of
symmetry operations and terminated by irreps of the doubled symmetry group
G≡ G× G, 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
G.








§ 
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
gαℓ in a G-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. <ref> by explaining the intuition why
the classification of anyon behaviors should be related to the
classification of 1D phases.  In Sec. <ref> we explicitly
state the technical assumptions made (specifically, the form of the fixed
point space). 
Secs. <ref>–<ref>
contain the classification: In Sec. <ref>, we 
study the structure of symmetry breaking of the fixed point space and
show that the endpoints 
gαℓ decouple as ℓ→∞,
allowing us to restrict to semi-infinite strings in the following; 
in Sec. <ref>, we derive the constraints
imposed by the symmetry breaking on the anyons and show how it allows to 
decouple anyon pairs;
in
Sec. <ref>, we make the connection between the behavior of
anyons and the SPT structure of the fixed points, and in
Sec. <ref>, we show that there exists an additional
non-trivial restriction on the SPTs which can appear as fixed points of
𝕋, and thus to the possible anyon behavior, arising from the
(complete) positivity of 𝕋. Finally, in
Sec. <ref>, we show that the
conditions derived in the preceding sections precisely give rise to the
known anyon condensation rules.



 §.§ 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 𝕋 is short range correlated as well, and thus has the
same structure as the ground state of a local Hamiltonian with the
identical symmetry [𝕋, U_ g^⊗ N]=0.  

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 (α,e) with α e] and confinement [for a vanishing
expectation value of an irrep (α,α)] 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. <cit.>. 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.





 §.§ The assumption: Matrix Product fixed points






We start by stating our main technical assumption:
The fixed point space of 𝕋 (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 i=(i,i') denote a joint ket+bra index
of the blocked transfer operator.  Then, we assume there exists a set of
matrices M^ i, c which describe distinct MPS

    ρ_ c=
    -0.75em
        < g r a p h i c s >

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


  * 
The ρ_ c span the full fixed point space of 𝕋. (This
is, evaluating any quantity of interest either in the fixed point space of
𝕋 or in span{ρ_ c} yields the same result
in the thermodynamic limit.)


  * The ρ_ c are injective, i.e., 𝔼_ c^ c has a
unique eigenvalue with maximal magnitude, where 𝔼_ c^ c':=
∑_ i M^ i, cM̅^ i, c' is the mixed transfer operator
of the MPS. W.l.o.g., we choose to normalize  M^ i, c
such that λ_max(𝔼_ c^ c)=1.



  * For each c and g, there is a c' such that
U_g|ρ_ c⟩=|ρ_ c'⟩, and for each pair c, c', there is a corresponding g. (Here and in the following,
we use U_g as a shorthand for the global symmetry action U_ g^⊗ N whenever the meaning is clear from the context.)



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

Assumption <ref> is the main technical assumption
here. Note that to some extent a similar assumption underlies the
classification of phases of 1D
Hamiltonians <cit.>, 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 <cit.>,
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 ρ_ c (possibly with
size-dependent amplitudes) <cit.>.
While the ρ_ c 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 ρ_
c, 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
ρ_ c vanish when computing physical quantities of interest, we
can instead work with a fixed point space spanned by the
ρ_ c [

Note that this does not imply that the fixed point space is actually
spanned by the ρ_ c. 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.




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. <ref>a <cit.>.  There
can be two distinct types of such symmetries: Those which act identically
on ket and bra layer, shown in Fig. <ref>b for on-site
symmetries, and those which only act on one layer, shown in
Fig. <ref>c.  (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. <ref>.)
Symmetries which only act on one layer correspond to topological
symmetries of the PEPS tensor, such as those of
Eq. (<ref>), 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. <ref>d 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 <cit.>, 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).




 §.§ 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 ρ_ c satisfying the
three assumptions just laid out.  For each c, let H_
c:={ h∈ G :  U_ hρ_ c=ρ_
c}. It is clear that H_ c⊂ G is a subgroup of
G; furthermore, for G abelian  H_ c is independent of c, since for any h∈ H_
c and g∈ G s.th. 
U_ g|ρ_ c⟩=|ρ_ c'⟩,

    |ρ_ c'⟩ =  U_ g|ρ_ c⟩ =
      U_ g U_ h|ρ_ c⟩ =
      U_ h U_ g|ρ_ c⟩ = 
      U_ h|ρ_ c'⟩ ,

and we write H≡ H_ c.

What is the structure of H?  To this end, consider arbitrary
γ_ c s.th. ρ:=∑γ_ cρ_ c≥0.  For
any h=(h,h') ∈ H, we have that ρ = U_hρ
U_h'^†, and thus

    ρ^2 = ρρ^† = (U_hρ U_h'^†)(U_h'ρ
    U_h^†) = U_hρ^2 U_h^† ,

and thus [ρ^2,U_h]=0.  Since ρ≥0, this implies that
[ρ,U_h]=0 as well, or

    ρ = U_hρ U_h^† ,

and similarly ρ = U_h'ρ U_h'^†. Now choose γ_
c s.th. ρ=∑γ_ cρ_ c=𝕋^∞(),
the fixed point of 𝕋 obtained when starting from , and
pick some c_0.  Then, for sufficiently small ϵ≥0,
σ':=+ϵ(ρ_ c_0+ρ_ c_0^†)≥0 and 
σ”:=+iϵ(ρ_ c_0-ρ_ c_0^†)≥0,
and thus ρ':=𝕋^∞(σ') and ρ”:=𝕋^∞(σ”) are both positive fixed points and therefore satisfy 
Eq. (<ref>), which implies that also 
ρ_ c_0=12ϵ[(ρ'-ρ)-i(ρ”-ρ)] 
satisfies
ρ_ c_0 = U_hρ_ c_0 U_h^†. We thus find that
whenever (h,h')∈ H, we must also have that (h,h)∈ H and
(h',h')∈ H.  

Now consider a general element (kℓ,k)∈ H. Then, (k,k)∈
H, and thus (ℓ,e)=(kℓ,k)·(k,k)^-1∈ H. It follows that
K∋ k and L∋ℓ form groups, and since (ℓ,e)∈
H ⇒ (ℓ,ℓ)∈ H, L⊂ K. 
 

The conserved symmetry H is
isomorphic to a direct product K×
L with L⊂ K, where K labels the the diagonal and L the
off-diagonal symmetry, i.e., H∋ h = (kℓ,k) with k∈ K and
ℓ∈ L. 

To distinguish it from the ket/bra product, we will denote the
diagonal/off-diagonal product by H = K ⊠ L.

Let us now consider the evaluation of a anyonic string order parameter
gαℓ inside general left and right boundary
conditions ∑λ_ c^l/r|ρ_ c).
This results in a sum over terms of the form

    O_ c^ c':=
    -2.8em
        < g r a p h i c s >
     ,

where we supress the dependency of O_ c^ c' on α and
g.
In case c c', the largest eigenvalue of the mixed transfer operator
𝔼_ c^ c' is strictly smaller than one (a
straightforward application of Cauchy-Schwarz, see e.g. Lemma 8 of
Ref. <cit.>), and thus, 
O_ c^ c'→ 0 exponentially as
N→∞, i.e., only terms with c= c' survive in
the thermodynamic limit.  In case c= c', we use that
ρ_ c =  U_ hρ_ c_0 for some
fiducial c_0 with h∈ G,
and thus 

    O_ c^ c=
    -3.6em
        < g r a p h i c s >

and since U_ g and U_ h commute, and the phases from
commuting U_ h R_α=α( h) 
 R_α U_ h and 
U_ h R_α̅=
α̅( h)  R_α̅ U_ h cancel out, we
find that O_ c^ c=O_ c_0^ c_0. 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 c_0 from now
on and write ρ≡ρ_ c_0 and M≡ M^c_0 (in fact,
we will most of the time also omit the label M of the tensor).

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






 §.§ 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 
gαℓ. 
Evaluated in the fixed point MPS ρ≡ρ_ c_0,
this corresponds to 

    [                            (ρ|gαℓρ=; -1.2em
        < g r a p h i c s >
       . ]

We now distinguish two cases: If g∉ H, then
U_g^⊗ N|ρ_ c_0⟩ = |ρ_ c'⟩ with c' c_0, and since different representations of an injective MPS are
related by a local gauge
transformation <cit.>, 
it holds that

    -0.8em
        < g r a p h i c s >
       .

and thus

    -3.2em
        < g r a p h i c s >
      ,

and since λ_max(𝔼^ c'_ c_0)<1,
(ρ|gαℓρ→0 as
ℓ→∞. We thus obtain


(ρ|gαℓρ→0
 unless g∈ H.  In particular, all anyons gα
with g∉K are confined.


On the other hand, if g∈ H, 
U_g^⊗ N|ρ_ c_0⟩ = |ρ_ c_0⟩ and thus
there exist V_ g such that

    -1em
        < g r a p h i c s >
       ,

where V_ g forms a projective representation of H which can be chosen unitary by a suitable gauge of the
MPS <cit.>.  Injectivity of the MPS further implies that its
transfer operator
𝔼≡𝔼_ c_0^ c_0 has a unique fixed point 

    𝔼^N-ℓ-2=-1.2em
        < g r a p h i c s >

(w.l.o.g., we choose σ_R,σ_L≥0, and normalization 
implies tr[σ_Lσ_R]=1),
and using Eq. (<ref>), this implies that

    -1.2em
        < g r a p h i c s >
       .

Also, since [𝔼, V_ g⊗V̅_ g]=0,
uniqueness of the fixed point of 𝔼 implies that V_gσ_∙V_g^† = σ_∙, where ∙=L,R,
and the ordering of the indices of σ_∙ is chosen
accordingly.  With this, we can rewrite Eq. (<ref>) as 

    (ρ|gαℓρ→⟨gα̅^*⟩⟨gα⟩

where

    ⟨gα⟩ :=  -1.2em
        < g r a p h i c s >
       ,

and correspondingly ⟨gα̅^*⟩.  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 ⟨gα⟩, Eq. (<ref>). In order to
highlight the role played by the two layers, we will sometimes also write
⟨gα⊗g'α'⟩ :=⟨gα⟩, with g=(g,g'), α = (α,α').




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




We will now study the behavior of string order parameters ⟨gα⟩, Eq. (<ref>), with g∈ H 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. (<ref>), a physical
symmetry action U_ g can be replaced by a virtual symmetry
action V_ g, where the V_ k form a projective
representation of the symmetry group, i.e.,
V_ g V_ h=ω( g, h) V_gh, where
ω: H× H→U(1) is a so-called
2-cocycle – i.e., it satisfies ω( g, hk)ω( h,
k)=ω( g, h)ω(gh, k) due to associativity –
which, up to gauge choices V_ g∼ e^iϕ_ gV_ g is
classified by the second cohomology group H; this
discrete classification of the V_ g is what is underlying the
classification of symmetry-protected phases in one
dimensions <cit.>.

The 2-cocycle ω also encodes what happens when we commute V_ g and V_ h: 

    V_ g V_ h = ω( g,  h)  V_gh = 
    ω( g,  h)  V_hg  =
    ω( g,  h)/ω( h,  g) V_ h V_ g .

Here, ω( g, h)ω( h, g)=:ν_ h(
g) is called the slant product <cit.> of
ω with h; for abelian groups, it forms a one-dimensional
representation of H, ν_ h( g_1)ν_ h( g_2)=ν_
h( g_1 g_2) [

This can be seen using the cocycle conditions and the fact that the group
is abelian as follows:

    ν_ h( g_1)
    ν_ h( g_2)/ν_ h( g_1 g_2)   =ω( g_1, h)ω( g_2, h)
    	    ω( h, g_1 g_2)/ω( h, g_1)ω( h, g_2)
    	    ω( g_1 g_2, h) ω( h  g_1, g_2)/ω( h  g_1, g_2)
       =ω( g_1, h)ω( g_2, h)
    	    ω( h, g_1 g_2)
    	    ω( h  g_1, g_2)
    /ω( h,  g_1 g_2)
    	    ω( g_1, g_2)
    	    ω( h, g_2)
    	    ω( g_1 g_2, h)
       =ω( g_1, h)ω( g_2, h)
    	    ω( h, g_1 g_2)
    	    ω( h  g_1, g_2)
    /ω( h,  g_1 g_2)
    	    ω( h, g_2)
    	    ω( g_1, g_2 h)  
    	    ω( g_2, h)
    
       =ω( g_1, h)
    	    ω( h  g_1, g_2)
    /ω( h, g_2)
    	    ω( g_1, g_2 h)  
    
       =ω( g_1, h)
    	    ω( g_1  h, g_2)
    /ω( h, g_2)
    	    ω( g_1, h g_2)  
     =1  .

]. Note that we can always construct (non-unique) representations
γ and γ' of G such that
γ(g)γ'(g')=ν_ h((g,g')) for (g,g')∈ H: To
this end, let γ(g):=ν_ h((g,e)) for g∈ L, extend 
γ to a representation of g∈ K (formally, this corresponds to an 
induced representation), and define γ'(g):=ν_
h((g,g)/γ(g); finally, both γ and γ' can be extended
independently to representations of G.


We will now derive conditions on g and α under which 
⟨gα⟩  must be zero and demonstrate how in
the remaining cases, it can be made non-zero by an appropriate choice of
R_α, and we find that this is in one-to-one correspondence to
the inequivalent 2-cocycles, i.e., elements of H; the no-go
part part of this discussion has been first given in
Ref. <cit.> in the context of string order
parameters for SPT phases.
To this end, let us consider an MPS with a specific projective
representation V_ g with corresponding ω( g, h), and
consider a string order parameter 
gα evaluated in that MPS, 

    ⟨gα⟩ =  -1.2em
        < g r a p h i c s >
       .

We now insert a resolution of the identity
U_hU_h^† before R_α and use
R_αU_h = α( h) U_hR_α, which
gives

    < g r a p h i c s >

We thus find that for ⟨gα⟩ to be non-zero, it
must hold that α = ν_ g, the irreducible representation
obtained as the slant product of the 2-cocycle ω. Conversely, by
choosing  R_α such that 

    -1.2em
        < g r a p h i c s >

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

    < g r a p h i c s >

i.e., R_α transforms as α≡ν_ g on
H as required, and

    ⟨gα⟩ = 
    -1.2em
        < g r a p h i c s >
     = 1 .

It remains to see how R_α transforms under the action of the
full symmetry group G, and more specifically that the construction
can be generalized to any irrep α of G with restriction
α|_ H≡ν_ g; this,  together with how to
separate R_α into independent ket and bra actions, is
discussed in Appendix <ref>.

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
⟨gα⟩ can only be non-zero if α=ν_g,
and at the same time, it is always possible to set up the endpoint R_α of the string order parameter such that ⟨gα⟩ actually is non-zero.  






A string operator gα with
⟨gα⟩0 exists if and only if α(
h)=ν_ g( h) for all h∈ H, with α(
(h,h'))=α(h)α̅'(h'), g=(g,g'), and ν_
g( h)=ω( h, g)/ω( g, h), where ω is the
2-cocycle classifying the fixed point of the transfer operator.

Clearly, the same derivation for the other endpoint of the string,
⟨gα̅^*⟩, yields exactly the same
condition.



Note that Conditions <ref>
and <ref> together show that the “amount of
topological order” – this is, the number of anyons – is related to
“symmetry breaking gap” between ket and bra,  |K|/|L|, where H=K⊠ L: Deconfined anyons gα satisfy (g,g)∈
H, i.e., g∈ K, and (α,α)=ν_(g,g), which fixes
α on L and thus leaves |G|/|L| possibilities to extend it to
G, yielding a total of |K||G|/|L| deconfined anyons.  Out of those,
pairs gα and gkαβ are indistinguishable if
(gk,g)∈ H, i.e., k∈ L, and (αβ,α)=ν_(gk,g),
which fixes β on K, leaving |G|/|K| possible extensions; the
size of each set of indistinguishable anyons is thus |L||G|/|K|.  The
total number of anyons – the ratio of these numbers – is thus
(|K|/|L|)^2, and the total quantum dimension is |K|/|L|, the
“symmetry breaking gap” between ket and bra.




 §.§ Constraints from positivity




The condition that ⟨gα⟩0 iff α=
ν_ g (Condition <ref>) has been derived for
a general fixed point of MPO form.  However, as we have seen in
Sec. <ref>, 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 H (Condition <ref>).
As we will see now, positivity induces yet another constraint, namely on
the 2-cocycles realizable in the fixed point.

To this end, consider a positive fixed point ρ≥0 with an SPT
characterized by some 2-cocycle ω: H× H→ℂ, and consider some g', α, and α' such that
⟨(e,g')(α,α')⟩0.  Then, also for
the other endpoint 
⟨(e,g')(α,α')^*⟩0, and thus
[following Eq. (<ref>)] 

    0     < |(ρ|(e,g')(α,α')ℓρ|^2
    
       =
    |tr[ 
        eαℓρg'α'α̅'ℓρ]|^2
    
       =
    |tr[(√(ρ)eαℓ√(ρ))
         (√(ρ)g'α'α̅'ℓ√(ρ))
    ]|^2
    
       (*)≤tr[(√(ρ)eαℓ√(ρ))(…)^†]
        
       ×tr[(√(ρ)g'α'α̅'ℓ√(ρ))(…)^† ]
    
       =  (ρ|(e,e)(α,α)ℓρ
       ×
        (ρ|(g',g')(α',α')ℓρ ,

where we have used Cauchy-Schwarz in (*) [here, (…) denotes the
preceding term].  Following Eq. (<ref>), this
implies 
⟨(e,e)(α,α)⟩0, and thus (from
Condition <ref>)
α(h)=α(h)α̅(e)=ν_(e,e)((h,e)) ≡ 1 
for (h,e)∈ H.
At the same time, 
⟨(e,g')(α,α')⟩0 implies that 
α(h)=α(h)α̅'(e)=ν_(e,g')((h,e)), and thus

    1=ν_(e,g')((h,e)) = 
        ω((h,e),(e,g'))/ω((e,g'),(h,e)) ,

i.e.:


The projective representations of ket and bra symmetry actions must
commute,  

    V_(g,e) V_(e,g') = 
     V_(e,g') V_(g,e) ,

or ν_(g,e)( (e,g'))=1, where (g,e),(e,g')∈ H.





 §.§ 
Anyon condensation rules


Let us now show that the
Conditions <ref>–<ref>
exactly give rise to the anyon condensation rules mentioned in the
introduction:


  * 
Only self-bosons can condense.

  * 
Anyons become confined if and only if they have mutual non-bosonic
statistics with some condensed anyon.

  * 
Non-confined anyons which differ by a condensed anyon become
indistinguishable.




  §.§.§ Only self-bosons can condense.


Consider a condensed anyon gα, 
⟨(g,e)(α,1)⟩0. This requires
(g,e)∈ H=K⊠ L, i.e., g∈ L, and moreover
α(h)=ν_(g,e)((h,h')) ∀ (h,h')∈ H, and thus 
α(g)=ν_(g,e)((g,e))=1, i.e., gα is a self-boson.



  §.§.§ Anyons become confined if and only if they have mutual
non-bosonic statistics with some condensed anyon.


Let us first show that an unconfined anyon kβ, 
⟨(k,k)(β,β)⟩0, must have mutual
bosonic statistics with all condensed anyons gα,
⟨(g,e)(α,1)⟩0.
⟨(k,k)(β,β)⟩0 implies k∈ K and
β(h)β(h')=ν_(k,k)((h,h')) for (h,h')∈ H,
i.e., β(h)=ν_(k,k)((h,e)) for h∈ L. On the other hand,
⟨(g,e)(α,1)⟩0 implies (g,e)∈ H,
i.e., g∈ L, and α(h)=ν_(g,e)((h,h)) for h∈ K. We thus
have that

    α(k)β(g) = ν_(g,e)((k,k)) ν_(k,k)((g,e)) = 1 ,

since ν_ g( h)ν_ h( g)=1.

Conversely, consider a confined anyon kβ, 
⟨(k,k)(β,β)⟩=0: we will show that this
implies the existence of a condensed anyon gα, 
⟨(g,e)(α,1)⟩0, which has mutual
non-bosonic statistics, α(k)β(g)1, by explicitly
constructing such an anyon gα. 
⟨(k,k)(β,β)⟩=0 implies that either (i)
k∉ K or (ii) there exists (h,h')∈ H s.th.β(h)β(h')ν_(k,k)((h,h')).

Let us first consider case (i), k∉ K. Let g=e [thus (g,e)∈
H], and choose an irrep α of G s.th.α(h):=ν_(g,e)((h,h'))= ν_(e,e)((h,h'))≡ 1 for
(h,h')∈ H – this is, gα is condensed. On the other
hand, since k∉ K we can always choose α s.th. α(k)
1 (as the extension of the irrep from K to G is non-unique), and
thus, α(k)β(g) 1, i.e., the anyons have mutual non-bosonic
statistics.

Now consider case (ii): k∈ K, but there exists some (h_0,h_0')∈
H s.th. β(h_0)β(h_0')ν_(k,k)((h_0,h_0')).
Define g:=h_0h_0'^-1∈ L.  Since hermiticity implies
ν_(k,k)((h,e))=ν_(k,k)((e,h)) (as can be shown by
relating ν_(k,k) to the behavior of string order parameters)
and thus
ν_(k,k)((h,h))=ν_(k,k)((h,e))ν_(k,k)((e,h))=1, we have that

    β(g)     = β(g)β(h_0')β(h_0') 
    =β(h_0)β(h_0')
       ν_(k,k)((h_0,h_0')) 
    = ν_(k,k)((g,e)) ν_(k,k)((h_0',h_0')) 
       = ν_(k,k)((g,e)) .

Further, let α'(h'):=ν_(g,e)((e,h'))≡ 1 for
h∈ L, and extend it to the trivial irrep α'≡ 1 of G.
Then, α(h):=ν_(g,e)((h,h))/α'(h), h∈ K, can be
extended to an irrep α of G s.th.
α(h)=α(h)α'(h') = ν_(g,e)((h,h')) for all
(h,h')∈ H, i.e., gα is condensed. Finally,
α(k)β(g)ν_(g,e)((k,k)) ν_(k,k)((g,e)) = 1, i.e.,
gα and kβ have mutual non-bosonic statistics.




  §.§.§ Non-confined anyons which differ by a condensed anyon become
indistinguishable.


Let gα be condensed, i.e., g∈ L and
α(h)=ν_(g,e)((h,h')) ∀(h,h')∈ H, and
kβ unconfined, k∈ K and
β(h)β(h')=ν_(k,k)((h,h')) ∀(h,h')∈
H. Then, (gk,k)∈ H, and 

    α(h)β(h)β(h')   = ν_(g,e)((h,h'))ν_(k,k)((h,h')) 
    
       = ν_(h,h')((g,e))ν_(h,h')((k,k))
       = ν_(h,h')((gk,k)) 
        = ν_(gk,k)((h,h')) ,

i.e., the anyons kβ and gkαβ become
indistinguishable.




§ ANYON CONDENSATION IN D(ℤ_N) AND TWISTED DOUBLE MODELS



We will now show that in the case of cyclic groups, G=ℤ_N, this
allows for a full classification of all condensation patterns, and that
these condensation patterns give rise exactly to all twisted quantum
doubles D^ω_3(ℤ_M), where the twist ω_3 is given
by a 3-cocycle of ℤ_M [
Note that the same cannot hold for all abelian groups: Condensing from an
abelian group gives another abelian model, while twisting an abelian
model can give rise to non-abelian models <cit.>.
].
In what follows, we will write the
groups additively with neutral element zero, and addition is understood modulo
the order of the group.




 §.§ Allowed phases at the boundary


Let us first study the effect of the above conditions on the possible
SPT phases at the boundary, and thus the possible condensation patterns.
As we have seen, the symmetry G=ℤ_N×ℤ_N
of the transfer operator is broken down to a symmetry H=ℤ_qt⊠ℤ_q={(g,g+th) : g=0,…,qt-1, h=0,…,q-1}.
Let us now consider the restriction imposed by
Eq. (<ref>) on the second cohomology
classification of the projective representations of H,
H=ℤ_q <cit.>. To this end, given
an element n∈ H, n=0,…,q-1, we choose a projective
representation

    V_(g,g) = X^g V_(ht,0) = Z^hn ,

of H, g=0,…,qt-1, h=0,…,q-1, where X and Z are such that
ZX=μ XZ, μ=exp(2π i/q) (e.g. X a cyclic shift and Z a
diagonal q× q matrix), and where
V_(g+ht,g):=V_(g,g)V_(ht,0).  It is straightforward to check that
these yield q inequivalent (and thus all) projective representation,
e.g. by comparing the gauge-invariant commutator ω(
(t,0),(1,1))/ω( (1,1),(t,0)) = μ^n.  We now have that
V_(0,h't)=V_(h't,h't)V_((q-h')t,0) and thus
Eq. (<ref>) reads 

    V_(ht,0) V_(h't,h't)V_((q-h')t,0) = 
    V_(h't,h't)V_((q-h')t,0) V_(ht,0)

which using Eq. (<ref>) is equivalent to μ^hn·
h't=1, or (since h and h' are arbitrary) μ^nt = 1.
This is the case whenever nt is a multiple of q, i.e.,
n=kqgcd(t,q). Since at the same time, 0≤ n<q, we find that
k=0,1,…,gcd(t,q)-1. This is, out of the q
different SPT phases under the symmetry group H, only 
gcd(t,q) are allowed due to positivity constraints.





 §.§ Explicit construction of all twisted ℤ_t doubles
and completeness of classification for ℤ_N



We will now show that for cyclic groups, this classification is complete.
To this end, we will first show how to obtain all twisted quantum
doubles of ℤ_t by anyon condensation from a ℤ_N
double, and subsequently use this construction to derive explicit PEPS
models for all cases consistent with
Conditions <ref>–<ref>.




  §.§.§ Twisted doubles of ℤ_t from anyon condensation


In the following, we will describe how to construct all so-called
twisted quantum doubles D^ω_r(ℤ_t) of ℤ_t
by anyon condensation from a quantum double of some ℤ_N, and
derive the structure of the SPT at the boundary.  Here, ω_r≡ω_r(g,h,ℓ) is a so-called 3-cocycle, labelled by an element r of
third cohomology group H^3(ℤ_t,U(1))=ℤ_t, r=0,…,t-1. We will just state the corresponding results in the
main text and postpone the proofs to Appendix <ref>;
for more details on twisted double models and 3-cocycles, we refer the
reader to the appendix or Ref. <cit.>.

Let r=0,…,t-1 label an element of H^3(ℤ_t,U(1))=ℤ_t, set q=t/gcd(t,r), and let
N:=qt. We now define tensors with non-zero elements

    M(a) := √(q/t)   
        < g r a p h i c s >
     ω(a,g,h-g_0)  ,
    
     
    N(a) := √(q/t)   
        < g r a p h i c s >
    ω(a,g,h-g_0)
 
for a=0,…,t-1.  Here, the range of the thick vertical indices is
0,…,t-1 and that of the thin (horizontal and vertical) indices is
0,…,q-1, and h=0,…,q-1, g=0,…,t-1, and
g_0=g mod q. Depending on the index, variables are understood
modulo t or q.  The  ω≡ω_r is defined by

    ω(a,g,d)  = exp[
        2π i r d/t^2(a+g-(a+g) mod t)
    ]   ,

where there is no modular arithmetics in the exponential except for
the mod t.
Eqs. (<ref>,<ref>) determine the
amplitude of all non-zero elements of M(a) and N(a), respectively, while all
tensor elements inconsistent with the labels of the indices are zero.

The PEPS tensor for the model is now defined as 

    A =t/q^2∑_a  -3em
        < g r a p h i c s >
      ,

where the inner legs correspond to the physical and the outer legs to the
virtual indices. As we show in
Appendix <ref>, the PEPS defined by this
model describes a twisted ℤ_t double model with twist ω.
As also shown in the appendix, A satisfies A^†=A=A^2, which implies
that AA^†=A and thus the transfer operator is of the form

    < g r a p h i c s >
      ,

where the second equality holds since

    < g r a p h i c s >
     δ_ab  .

We thus find that the left and right fixed points of the transfer operator
are again described by the same tensor.  As it turns out, for any fixed
a they describe an injective MPS, and thus, the boundary exhibits t
symmetry broken sectors labelled by a=0,…,t-1.

The tensor A has a ℤ_qt symmetry with generator
S_(i_1,i_2),(j_1,j_2) =
δ_i_1+1,j_1δ_i_2+1,j_2ω(1,i_1,i_2-i_1) (with i_1,
j_1 mod t and i_2, j_2 mod q), which follows
from the local condition

    -0.3em
        < g r a p h i c s >
      ,

where U_ij≡ U(a)_ij = δ_ijω(1,a,i). 
Together with its “twin” equation

    -1.8em
        < g r a p h i c s >
      ,

V_ij≡ V(a)_ij = e^iϕ_aω(1,a-1,i)δ_i,j-1,
Eq. (<ref>) allows us to verify that the symmetry
ℤ_qt×ℤ_qt in the fixed point
[Eq. (<ref>)] is broken to ℤ_qt⊠ℤ_q, with generators G_1=S⊗ S^†
and G_2=S^t⊗. The element n∈ℤ_qt⊠ℤ_q=ℤ_q labelling the virtual symmetry
action – determined by the commutation relation of the virtual
representations P_1(a)=U(a)V(a+1) and P_2(a)=∏_i=a^a+t-1U(i)
of G_1 and G_2 – is given by n=rqt.

Overall, given t and r=0,…,t-1, and q=t/gcd(t,r), we
thus have constructed a PEPS with bond dimension qt and virtual ℤ_qt symmetry which describes a twisted ℤ_t quantum double
with twist r∈H^3(ℤ_t,U(1))=ℤ_t. In the fixed
point of the transfer operator, the symmetry is broken down to 
ℤ_qt⊠ℤ_q, and the cocycle n∈ℤ_qt⊠ℤ_q=ℤ_q characterizing the
virtual symmetry action in the fixed point is given by n=rqt.



  §.§.§ 
Completeness of the
Conditions <ref>–<ref>


Let us now show that this construction allows us to obtain PEPS models for
cyclic G for any case compatible with
Conditions <ref>–<ref>.
Concretely, those conditions imply that given a virtual symmetry ℤ_N in the tensor, it can be broken down to any ℤ_qt⊠ℤ_q symmetry where qt|N  (“|” denotes “divides”), and
furthermore, the label n of the cocycle characterizing the fixed points
must be a multiple of qgcd(q,t),
n=kqgcd(q,t).

To this end, define q':=gcd(q,t), and let α=q/q',
n'=n/α=kq/αgcd(q,t)=k=0,…,q'-1. Next, let
β=gcd(n',q'), and define n”=n'/β and q”=q'/β. With
γ=αβ, we then have that q=γ q” and n=γ n”.
Since q”|q' and q'|t, x:=t/q” is integer.    Then, the construction for the
twisted double with t̃=t=xq” and twist r̃=xn” described in
the preceding section yields 

    q̃ = t/gcd(r̃,t) 
        = xq”/gcd(x n”,xq”)  
        = q”/gcd(n”,q”) 
        = q” ,

and the 2-cocycle of the fixed point is characterized by 
ñ = r̃q̃t = xn”q”xq” = n”.

We thus know how to create a model with parameters t, q”=q/γ,
and n”=n/γ; let us denote its tensor by A^i_k_1,k_2,k_3,k_4, with
k_s=0,…,q”t-1, and the generator of the ℤ_q”t symmetry
by S; w.l.o.g., we choose a basis |k_s) such that S=∑
|k+1)(k|.  We will now show how from this model, we can create
a PEPS with parameters t, q, and n, and overall symmetry ℤ_M with M=qt.  (In a second step, we will then generalize this to any
ℤ_N with qt|N.) To this end, we extend the bond space to a
M=γ (q”t)-dimensional space, k_s(ℓ_s,k_s),
ℓ_s=0,…,γ-1, and construct the new tensor Ã by
tensoring each virtual index of A independently with an equal weight
superposition of all |ℓ_s), i.e., 

    Ã^i_(ℓ_1,k_1),(ℓ_2,k_2),(ℓ_3,k_3),(ℓ_4,k_4)
    = A^i_k_1,k_2,k_3,k_4

As the generator of the ℤ_M symmetry we choose the regular
representation in ℤ_M with basis |ℓ q”t+k), i.e., S̃:|ℓ,k)↦|ℓ+⌊ k/q”t⌋,k+1); since each ℓ_s
index is in a uniform superposition ∑ |ℓ_s), S̃ acts
exactly as S on the non-trivial degrees of freedom k_s of Ã,
while leaving ∑ |ℓ_s) invariant.  The resulting tensor has thus a
ℤ_M symmetry which is broken to ℤ_γ
q”t⊠ℤ_γ q”=ℤ_qt⊠ℤ_q in the fixed point, with q:=γ q”.  The element
n∈ℤ_γ q”t⊠ℤ_γ
q”=ℤ_γ q” is determined by the commutation phase of
the virtual representations of the two generators S̃⊗S̅̃̅ and S̃⊗ in the fixed point
MPS, which equal those of S⊗S̅ and S⊗,  and
which is thus exp[2π i nγ q”] = exp[2π i
n”q”]; we therefore have n≡γ n”, as claimed.

To obtain the most general case, we still need to show how to go from a
ℤ_M to a ℤ_N symmetry (with M|N) which in the fixed
point is broken down to at least ℤ_N×ℤ_N, and
possibly further. To this end, let σ:=N/M, 
denote the original tensor again by A^i_k_1,k_2,k_3,k_4 with
k_s=0,…,M-1, extend the indices as (k_s,ℓ_s) with
ℓ_s=0,…,σ-1, and define the new tensor 

    Ã^i_(k_1,ℓ_1),(k_2,ℓ_2),(k_3,ℓ_3),(k_4,ℓ_4)
    = A^i_k_1,k_2,k_3,k_4δ_ℓ_1=ℓ_2=ℓ_3=ℓ4 ,

where δ_ℓ_1=ℓ_2=ℓ_3=ℓ4=1 if all ℓ_s are equal,
and zero otherwise. Further, define S̃ = S^1/σ⊗∑_ℓ=0^σ-1|ℓ+1⟩⟨ℓ| (with addition modulo
σ). Clearly, S̃ generates a representation of ℤ_M
(which is faithful if S was faithful). Further, the additional degrees
of freedom labelled by ℓ yield two independent GHZ states (i.e.,
correlated block-diagonal structures) in ket and bra level in the fixed
point, which are cyclicly permuted by the action of S̃: The
ℤ_N×ℤ_N symmetry is thus at least broken to
ℤ_M×ℤ_M, with the model in each symmetry broken
sector described by the original PEPS, and the ℤ_M symmetry
action generated by S̃^σ = S⊗.  

Together, this concludes the construction of an explicit example for all
cases consistent with
Conditions <ref>–<ref>.








§ EXAMPLE: CONDENSATION OF D(ℤ_4) AND THE DOUBLE SEMION
MODEL



In the following, we will discuss some examples for anyon condensation in
doubles D(ℤ_N).  As a warm-up, we will start with the Toric Code
model D(ℤ_2), and then discuss in detail the possible
condensations in D(ℤ_4), where we will see how condensing a dyon
– corresponding to a non-trivial SPT at the boundary – can give rise to
the doubled semion model which cannot be described as a double model of a
group.

Given the double D(ℤ_N), its excitations gα are
labelled by group elements g=0,…, N-1 and irreps α=exp(2π i
k/N), k=0,…,N-1, where α(g)≡α^g. (We will again
write the group additively with neutral element 0.) The
self-statistics for a half-exchange of two gα particles is
α^g, and the phase acquired through the full exchange of
gα and hβ is given by α^hβ^g. Fusing
particles gα and hβ results in
g+hαβ.

As derived in Sec. <ref>, in order
for a particle gα to condense, it must be have bosonic
self-statistics, i.e., α^g=1.  This leads to the identification
of gα with the vacuum 01, and subsequently to the
identification all pairs hβ and h+gβα.
Moreover, all particles hβ which braid non-trivially with
gα, α^hβ^g1, become confined.  





 §.§ 
Warm-up: Condensation of the Toric Code




Let us start by considering the Toric Code model 𝔻(ℤ_2).
It has four particles: The vacuum ∅ = 01, the
magnetic particle m=11, the electric particle
e=0-1, and the fermion f≡e×m
= 1-1; they can be visualized in a two-dimensional grid with g
and α as row and column labels, respectively,
Fig. <ref>a. e and m (marked red)
have bosonic self-statistics α^g and can therefore condense.
Fig. <ref>b illustrates the condensation of the
e particle: e is identified with the vacuum (indicated by
connected dots), and since both m and f have
non-trivial mutual statistics α^hβ^g with e (as
α=-1, g=0 for e and h=1 for m, f),
they become confined (indicated by grayed out boxes).

Let us now study the condensation in terms of the symmetry of the transfer
operator.  We have G=ℤ_2×ℤ_2. The possible
symmetry breaking patterns H=K⊠ L are given by H=ℤ_2⊠ℤ_2, H=ℤ_2⊠ℤ_1, and H=ℤ_1⊠ℤ_1, respectively. This is
shown in Fig. <ref>c, where the horizontal layers are
arranged according to their “ket-bra symmetry breaking gap” |K|/|L|
corresponding to the number of anyons in the model, and the arrows
point in the direction of decreased symmetry.  

Let us now consider the three possibilities case by case.


  * H=ℤ_2⊠ℤ_1: This is the topological case. On
the one hand, we have following Cond. <ref> that
⟨gα⊗gα⟩0 for all g and
α, since (g,g)∈ H and 
the restriction of (α,α) to H is
(α,α)((h,h))=α(h)α̅(h)=1=ν_(h,h) [as
H is trivial]; this is, all particles gα are
unconfined. On the other hand,
⟨gα⊗g'α'⟩=0 whenever either
g g' [as (g,g')∉ H] or αα' [as then
(α,α')((h,h))≢1]; this is, no particles are condensed.


  * H=ℤ_1⊠ℤ_1: This is the trivial phase in
which e=0-1 is condensed (and thus m=11
is confined). Firstly, since all symmetries are broken, 
⟨gα⊗g'α'⟩=0 whenever g0 or
g'0, which implies that m and
f=e×m=1-1 are confined. On the other
hand, 
⟨0α⊗0α'⟩0, since
(0,0)∈ H and (α,α') restricted to H is trivially
the identity, and thus equals ν_(0,0).  


  * H=ℤ_2⊠ℤ_2: This is another trivial phase, in
which m=11 is condensed and e=0-1
is confined. Firstly, note that while ℤ_2⊠ℤ_2=ℤ_2, we have that qt=q=2 and gcd(t,q)=1,
i.e., only the trivial cocycle is allowed due to positivity. We have that 
⟨g1⊗g'1⟩0 since
(g,g')∈ H and (1,1)=ν_(g,g'), implying that m
is condensed.  On the other hand,
⟨gα⊗g'α'⟩=0 whenever
α1 or α' 1, since (α,α')(
(g,g'))=α(g)α'(g')≢1 for some (g,g')∈ H,
i.e., e and f are confined.






 §.§ Condensation of D(ℤ_4)




Let us now turn to our second example, the double D(ℤ_4). The
anyon table is given in Fig. <ref>a; here, we find three
bosons (marked red), namely 2e, 2m, and the dyon
d=2e×2m. It is straighforward to work out
the particle tables obtained by condensation: While condensing
2e or 2m leads to two inequivalent toric codes,
condensing d – as shown in Fig. <ref>b – leads
to the so-called double semion model, with particles s and
s̅, which have (anti-)semionic self-statistics g^α=±
i, and which fuse with themselves to the vacuum and with each other to
the non-trivial boson b=0-1≡21. The double
semion model is not a regular double model but can be obtained by twisting
D(ℤ_2) with a non-trivial 3-cocycle of ℤ_2, and is
thus the simplest example of a twisted model obtained by condensing a
regular double.

Let us now study the possible symmetry breaking pattern H of
D(ℤ_4), shown in Fig. <ref>c. We find six
possibilities.



  * H = ℤ_4⊠ℤ_1.
This is the D(ℤ_4) phase; the discussion is analogous to the
case 1 for the Toric Code in Sec. <ref> above.


  * H = ℤ_2⊠ℤ_1.  This is a toric code phase
in which the 2e particle has been condensed.  We have that
⟨gα⊗g'α'⟩=0 unless
(g,g')∈ H, i.e., g=g'=0 or g=g'=2, which implies that
1* and 3* are all confined, and 2* is
uncondensed. On the other hand, ⟨gα⊗gα'⟩0 iff α(g)α'(g)=1 (as
there is only a trivial cocycle), and thus 1-1 is condensed, and
0i≡0-i and 21≡2-1 form the
electric and magnetic particle of the Toric Code, respectively.


  * H = ℤ_4⊠ℤ_2.
This is the first case with non-trivial H=ℤ_2, and
thus exhibits two distinct condensed phases with identical symmetry
breaking pattern.

The phase with trivial cocycle corresponds to a Toric Code phase in which
the 2m particle has been condensed. 
First, ⟨gα⊗g'α'⟩=0 
whenever α^hα'^h'≢1 for some (h,h')∈
H, i.e. unless α=α'=±1, and thus *± i are
confined, while *-1 is not condensed. On the other hand, 
⟨g±1⊗g'±1⟩0  iff (g,g')∈ H: Thus, 21 condenses, and 11≡31 and
0-1≡2-1 form the new magnetic and electric particles,
respectively.

Let us now turn towards the phase with non-trivial cocycle. As we will
see, it corresponds to a double semion model with the condensation pattern
indicated in Fig. <ref>.  It is straightforward to check
that for the non-trivial cocycle of H=ℤ_2,
ν_(g,g')((h,h'))=i^gh(-i)^g'h' (e.g., by checking it on the
generators).  Then, ⟨gα⊗g'α'⟩=0 whenever
α^hα'^h'≢ν_(g,g')((h,h')) for
some (h,h')∈ H, i.e. unless α=± i^g and α'=±
i^g' (with the identical choice of ±). This implies that all
g± i^g+1 are confined, and only anyons gi^g can
condense. Since ⟨gα⊗01⟩0
in addition requires (g,0)∈ H, we find that it is 2-1
which condenses.





  * H = ℤ_1⊠ℤ_1.
This is a trivial phase where all e particles have been
condensed; it is fully analogous to 
case 2 for the Toric Code in Sec. <ref>.


  * H = ℤ_2⊠ℤ_2.
This is a trivial phase where 2e and 2m have been
condensed.  We have that ⟨gα⊗g'α'⟩=0  unless g,g'∈{0,2}, i.e., 1* and
3* have been confined. It is also zero unless
α^gα'^g'=1 for all g,g'∈{0,2} (there is only the
trivial cocycle), and thus, *± i is confined as well. For all
remaining cases, ⟨gα⊗g'α'⟩0, and thus, all other particles are
condensed.


  * H = ℤ_4⊠ℤ_4.
This is a trivial phase where all m particles have been
condensed; it is fully analogous to 
case 3 for the Toric Code in Sec. <ref>. Note that
there is again only the trivial cocycle.









 §.§ Numerical study




In the following, we provide numerical results on different topological
phases which can be obtained through condensation from a D(ℤ_4)
double model, and the transitions between them. To this end, we have
constructed a three-parameter family interpolating between different fixed
point models, including the D(ℤ_4) phase, both Toric Code
phases, the double semion phase, and a trivial phase.  Here, we will limit
ourselves to a brief overview of the results; an in-depth discussion of
the specific wavefunction family considered as well as the numerical
methods used, together with additional results, will be presented
elsewhere <cit.>.






Let us start by introducing the family of tensors used: 

    A(θ_DS,θ_TC,ℤ_2,θ_TC)=
    
        < g r a p h i c s >
    .

Here, the four outside legs correspond to the virtual indices, while the
four inside legs are the physical indices.  The rings (and the green dots)
describe MPOs all of which mutually commute: 


  * The outermost black ring is the MPO of the D(ℤ_4) quantum
double, ∑_g U_g^⊗ 4, U_g=X^g, with X the generator of
the regular representation of ℤ_4.

  * The red ring describes a deformation towards the MPO projector for the
ℤ_4⊠ℤ_2 double semion model, where 

    < g r a p h i c s >
    =(X^2)^i(Z^2)^i+j,
       i,j=0,1 ,

with Z the generator of the diagonal representation of ℤ_4,
and
< g r a p h i c s >
=
diag(coshθ_DS2, sinhθ_DS2).  For θ_DS=∞ (and
θ_TC,ℤ_2=θ_TC=0), this gives the
double semion MPO, while for θ_DS=0, it acts trivially. 


  * The blue ring describes a deformation towards the H=ℤ_2⊠ℤ_1
Toric Code, where

    -1.2em
        < g r a p h i c s >
    =
    δ_ijexp((-1)^iθ_TC,ℤ_2Z^2), i,j=0,1 .

For θ_TC,ℤ_2=∞, this projects the 
D(ℤ_4) MPO onto a ℤ_2 subgroup and thus yields the Toric
Code, while for θ_TC,ℤ_2=0, it acts trivially.


  * Green circles
< g r a p h i c s >
=exp(θ_TCX^2)
describe a deformation towards a H=ℤ_4⊠ℤ_2
Toric Code phase: For θ_TC=∞, this enhances the
symmetry of the D(ℤ_4) MPO to H=ℤ_4⊠ Z_2,
while for θ_TC=0, it once again acts trivially.

The two Toric Code constructions correspond to the two ways of embedding a
“normal” ℤ_2⊠ℤ_1⊂ℤ_2×ℤ_2 Toric Code into a ℤ_4×ℤ_4
symmetry described in Sec. <ref>.
Note that since all projectors commute with each other, their order does
not matter.














We have studied the phase diagram of this family using infinite Matrix
Product States (iMPS) to approximate the fixed point of the transfer
operator, by iteratively applying the transfer operator and truncating the
bond dimension to some given χ, keeping translational symmetry. From
the resulting fixed point iMPS, we can then [using
Eq. (<ref>)] immediately compute the order parameters
for condensation, ⟨gα⊗01⟩, and
deconfinement, ⟨gα⊗gα⟩,
respectively, allowing us to distinguish the different topological phases
and map out the phase diagram.  The condensation and deconfinement order
parameters also allow us to study the nature of the phase transitions.
Notably, this gives us non-zero order parameters, and thus critical
exponents β, for both sides of a condensation-driven phase
transition: in the uncondensed phase, the deconfinement order parameter is
non-zero, while in the condensed phase, the condensate fraction is
non-zero. Note that we use the string operator corresponding to
excitations in the fixed point wavefunction to measure the order parameter
throughout the phase diagram; this is in exact analogy to the use of order
parameters in conventional phase transitions.  In addition to that, we can
further characterize the phase transition by looking at the scaling of the
correlation length ξ, which we can extract either from the fixed point
iMPS, or from the finite-size transfer operator and a finite size scaling
(note though that this length does need not be equal to the physical
correlation length, as it includes e.g. certain anyon-anyon correlation
functions).

In order to understand the structure of our three-parameter family,
Eq. (<ref>), we have computed the different
condensation and deconfinement order parameters along the three
hyperplanes for which one θ_∙=0; the resulting phase diagram
is shown in Fig. <ref>.  We find that the system exhibits
all phases encoded by the three MPOs in
Eq. (<ref>), as well as a trivial phase with
H=ℤ_2⊠ℤ_2, which can be understood
analytically in the limit where two of the θ_∙ are taken to
infinity. As expected, the family thus exhibits phase transitions related
to the condensation of anyons from D(ℤ_4) to Toric Code and
Double Semion, and from either to the trivial phase; more notably, though,
the family also exhibits direct phase transitions between the Toric Code
and the Double Semion model, which are not related by anyon condensation.

















We have studied a number of these phase transitions in more detail; in the
following, we illustrate our findings through a few examples, and refer
the reader for a more detailed analysis to Ref. <cit.>.
First, we have studied the phase transitions in the
θ_TC,ℤ_2=0 plane.  Fig. <ref> shows
the order parameters along line (I) in Fig. <ref>, which
describes a D(ℤ_4) to Toric Code transition. Since we have an
analytical mapping to the 2D Ising model for this line, it can serve as a
benchmark, and we find indeed very good agreement with the analytic
predictions.  Further study suggests the existence of an analytical
mapping for the entire θ_TC,ℤ_2=0 plane; for the
θ_DS=0 plane, the critical exponents still match those of
the 2D Ising model, though the existence of an exact mapping is unclear.
On the other hand, the transitions in the θ_TC=0 plane
seem to belong to  a different universality class.  As an example,
Fig. <ref> shows the transition along the line (II) in
Fig. <ref>, for which we find critical exponents
ν=1.05(7) for the correlations in the fixed point of the transfer
operator, and β_+=0.04(1) and β_-=0.23(4) for the anyon
condensation and deconfinement order parameters, respectively; notably,
the critical exponent β is different on the two sides of the
transition. We observe that the critical exponents β_± change
continously as we move along the transition line in the
θ_DS=1 plane towards the θ_TC,ℤ_2 plane, ultimately reaching β_±=1/8; a detailed discussion
will be given elsewhere <cit.>.

Finally, let us turn towards the direct Toric Code – Double Semion
transition, previously only studied with exact diagonalization and on
quasi-1D systems <cit.>, whose nature is yet
to be resolved.  As one would assume that interactions generally give
rise to condensation of excitations, one expects that an interpolation
between the two models would typically drive the Toric Code through some
condensation transition, either into a trivial or a more complex phase
[such as the D(ℤ_4) model], and from there through another
condensation-driven transition to the Double Semion model, and a direct
transition would at least require some fine-tuning of interactions.

We can identify one such fine-tuned transition between the (H=ℤ_4⊠ℤ_2) Toric Code and Double Semion phase in our phase
diagram in the θ_TC,ℤ_2=0 plane at
(θ_DS^c,θ^c_TC)=
(12ln(1+√(2)),  12ln(1+√(2)));
this is a multi-critical point adjacent to all four phases which goes away
as one perturbs away from θ_TC,ℤ_2=0, separating
the Toric Code from the Double Semion phase.  Fig. <ref>a
shows the transition through this point along line (III) in
Fig. <ref>, and we find that it is a second order phase
transition, driven by two “counterpropagating” condensation and
de-condensation transitions, thus preserving the total number of anyons;
like all transitions in that plane, it is again in the 2D Ising universality
class.  Note however that this is a phase transition between two phases
with an identical H=ℤ_4⊠ℤ_2 symmetry at the
boundary,  and therefore corresponds to an SPT phase transition at the
boundary in the absence of symmetry breaking, and can therefore only be
detected by string order parameters rather than conventional local order
parameters.  Note however that it has been shown that in certain cases
string order parameters can be mapped to local order parameters through a
duality mapping <cit.>.

As it turns out, there is another way of obtaining a direct phase
transition between the H=ℤ_4⊠ℤ_2 Toric Code
and Double Semion phase, namely by interpolating between the on-site
transfer operators A^† A of the two fixed point models, rather
than the tensors A themselves. Since such an interpolation 𝔼(θ)=θ A_0^† A_0 + (1-θ) A_1^† A_1 yields a
positive semidefinite 𝔼(θ)≥0, we can construct a
continuous path A(θ) of PEPS tensors by decomposing
𝔼(θ)=A(θ)^† A(θ).  This interpolation
yields again a direct transition between the two phases, and a thorough
analysis of the order parameters, shown in Fig. <ref>b,
gives compelling evidence that the phase transition is first order.  Thus,
in order to understand the nature  of a generic Toric Code – Double
Semion phase transition (given it can even be realized in a robust way)
requires further study. In this context, it is an interesting question
whether imposing specific symmetries on the system allows one to
generically obtain a direct transition between these two phases, rather
than requiring fine-tuning of the interactions.




§ CONCLUSIONS AND OUTLOOK


In this paper, we have studied anyon condensation in Projected Entangled
Pair State models, and have derived conditions governing the condensation and
confinement of anyons.  In order to do so, we have related the behavior of
anyons to string order parameters and thus symmetry protected order
in the fixed point of the transfer operator, this is, the entanglement
spectrum of the system.  We have derived four conditions: Two characterize
the possible symmetry breaking and SPT phases consistent with positivity
of the entanglement spectrum, while the other two related these symmetry
breaking and SPT patterns to the condensation and confinement of anyons.
Specifically, we found that there are topological phases which cannot be
distinguished through their symmetry breaking pattern, but solely through
the SPT structure of their entanglement spectrum, and which describe
phases not related by anyon condensation.  For the case of cyclic groups,
this classification allowed to construct all twisted doubles by condensing
non-twisted double models.

We have exemplified our discussion with the ℤ_4 quantum double,
which can give rise to both Toric Code and Double Semion phases which form
an example of phases with identical symmetry breaking pattern but
inequivalent SPT order in the entanglement spectrum. We have also provided
numerical results for the phase diagram and the phase transitions of the
model.  To this end, we have used that the concepts developed in this paper
allow us to measure order parameters for condensation and deconfinement
and thus extract critical exponents for the order parameter. In
particular, we found that this model can realize direct phase transitions
between the Toric Code and Doubled Semion models which are not related by
anyon condensation, and for which we found both first and second order
transitions.

A natural question is the interpretation of symmetry broken and SPT phases
in the fixed point of the transfer operator in terms of physical
properties of the entanglement spectrum and/or edge
physics <cit.>: Symmetries U_gρ
U_g^†=ρ imply that the entanglement spectrum ρ is
block-diagonal, i.e., it originates from a symmetric Hamiltonian. An
additional single-layer symmetry U_gρ=ρ implies that the density
operator must live in the trivial irrep sector, while a broken symmetry
and the resulting dependence on distant boundary conditions implies the
existence of a non-local anomalous term in the entanglement Hamiltonian
which depends on distant boundaries and encodes a topological
superselection rule <cit.>.  The implications of SPT order
on the entanglement spectrum, on the other hand, are much less clear, and
it would be very interesting to identify the features of the entanglement
spectrum which would allow to distinguish e.g. Toric Code and Doubled Semion order.

It is likely that our results generalize to the case of non-abelian
groups, and beyond that to general Matrix Product Operator
symmetries <cit.>. An obstacle is that the one-to-one
correspondence between string order parameters and SPTs breaks
down <cit.>: While it is known that non-abelian
SPTs are still characterized by group cohomology, we have used SPT phases
to classify the behavior of string order parameters rather than the other
way around, and are thus looking for a classification of the
behavior of non-abelian string order parameters instead. Let us note, however, that a major
simplification might come from the fact that for non-abelian double
models, the irrep at the end of a string must be an irrep of its
normalizer, so it might well be possible that the problem can be
abelianized to an extent which allows to yet again relate it to SPT order.
A related question is the generalization of our results to the
case of non-hermitian transfer operators, or even PEPS which
encode a corresponding global symmetry in a non-trivial way.  In that
case, string order parameters are evaluated between non-identical left and
right fixed points, and the analogy to expectation values in physical
states, and thus the correspondence of string order parameters with SPT
phases, breaks down; for instance, it is not even clear whether the
projective symmetry representation for pairs of left and right fixed
points must be equal.


Finally, the maybe most important question, which goes far beyond the
scope of this work, is a rigorous justification of our main technical
assumption, namely that the structure of the fixed point space of a
transfer operator for a PEPS in a gapped phase is well described by Matrix
Product Operators.  While this is well motivated due to the short-range
nature of the correlations in the system, and is well-tested numerically
through numerous PEPS simulations using contraction schemes which model the
boundary as an MPO, it has withstood rigorous assessment up to now. A
better understanding of this question would lead to a
number of important insights regarding the structure of gapped phases, the
nature of the entanglement spectrum, or the convergence of numerical
methods, just to name a few. 















We acknowledge helpful conversations with 
M. Barkeshli,
N. Bultinck,
M. Marien, 
B. Sahinoglu,
C. Xu,
and B. Yoshida.
This work has received support by the EU through the FET-Open project QALGO
and the ERC Starting Grant No. 636201 (WASCOSYS), 
the DFG through Graduiertenkolleg 1995,
and the Jülich Aachen Research Alliance (JARA) through JARA HPC
grant jara0092 and jara0111.






§ CONSTRUCTION OF EXPLICIT ENDPOINTS 



In this appendix, we provide an explicit construction for all anyons
gα which are either condensed or deconfined following
Condition <ref>, i.e. α|_
H=ν_ g.  To this end, we proceed in two steps: First, we
generalize the construction of Eq. (<ref>) to obtain
R_α which transform as irreps α of G
rather than only H.  Second, we show that for the case of
condensation, g=(g,e) and α=(α,1), and for the case of
deconfinement, g =(g,g) and α=(α,α), these
R_α allow to construct actual anyons, i.e., single-layer
endpoints, for which ⟨gα⟩0; this is exactly
what is also required in
Section <ref>, where we derive the
anyon condensation rules from
Conditions <ref>–<ref>.



 §.§ Construction of R_α for irreps of G


Let g∈ H, and α an irrep of G such that
α|_ H=ν_ g. The idea of
Eq. (<ref>) was to use injectivity of the MPS tensor
M to define R_α such that 

    -3em
        < g r a p h i c s >
      .

The tensor M describes one symmetry-broken sector (with residual
symmetry group H) only. In order to construct some R_α which transforms as an irrep of G, we therefore first
need to construct an MPS which does not break the symmetry.  To this end,
choose representants f_𝔞∈ G of every
symmetry-broken sector 𝔞∈ G/ H, such
that 

    G=⊕_𝔞∈ G/ H f_𝔞 H ;

by starting from the generators of the quotient group G/ H, it is
possible to pick f_𝔞 such that f_𝔞𝔟= f_𝔞 f_𝔟.  Now define

    < g r a p h i c s >

and ℳ^i = ⊕_𝔞 M^i_𝔞; clearly,
ℳ^i is block-injective (i.e., injective on the space of
block-diagonal matrices).  Given k∈ G, there is a unique
decomposition k= f_𝔞 h, h∈ H, and thus 

    -2.6em
        < g r a p h i c s >
      ;

this is, the virtual action of U_ k is 

    𝒱_ k:=(⊕ V_ h)Π_𝔞 ,

where Π_𝔞 permutes the blocks by virtue of
𝔟↦𝔞^-1𝔟; note that 𝒱_ k forms a projective representation of G (the trivial
induced projective representation induced by V_ h). Now define 

    𝒲 := ⊕_𝔟α( f_𝔟) V_ g ,

and choose R_α such that

    < g r a p h i c s >

– this is always possible since 𝒲 is block-diagonal and ℳ is
block-injective. (We use a thick line to indicate the larger “direct
sum” virtual space.) We now have that

    𝒱_ k𝒲𝒱_ k^†    = 
    [(⊕ V_ h)Π_𝔞]
        [⊕_𝔟α( f_𝔟) V_ g]
        [Π_𝔞^†(⊕ V^†_ h)]
    
        = 
    [(⊕ V_ h)]
        [⊕_𝔟'α( f_𝔞𝔟') V_ g]
        [(⊕ V_ h^†)]
    
        = 
    ν_ g( h)[⊕_𝔟'α( f_𝔞𝔟') V_ g]
    = 
    ν_ g( h)α( f_𝔞) 𝒲
       = 
    α( k) 𝒲 ,

where we have used ν_ g( h)=α( h). It immediately
follows that

    < g r a p h i c s >

i.e., R_α indeed transforms as the irrep α of
G.




 §.§ Explicit construction of condensed anyons


Let us now show that we can explicitly construct condensed anyons
gα: Given a R_(α,1) for which
⟨(g,e)(α,1)⟩0, we show how to construct
a single-layer anyon (i.e., an endpoint to a string of g's transforming
like α) with non-zero expectation value 
⟨gα⊗e1⟩0, where the endpoint
in the bra layer is trivial. To this end, we start by decomposing

    R_(α,1) = ∑ X^s_α⊗Y̅^s_1 ,

where X_α and Y_1 transform like α and trivially,
respectively.  Since R_(α,1) gives a non-zero expectation value
⟨(g,e)(α,1)⟩0, there must be at least
one s_0 for which this also holds; we thus obtain a separable
endpoint X^s_0_α⊗Y̅^s_0_1≡
X_α⊗Y̅_1 with 
⟨(g,e)(α,1)⟩0; however, Y_1 can still
be different from the identity.  In order to make the endpoint in the bra
layer entirely trivial, we use that

    < g r a p h i c s >

since A is G-injective (and C is G-invariant), and thus, 

    < g r a p h i c s >

with the endpoint 

    < g r a p h i c s >

for the condensed anyon gα.

Note that a simple application of Cauchy-Schwarz yields that any condensed
anyon is also deconfined.




 §.§ Explicit construction of deconfined anyons


Similar to the preceding section, in this scenario we start from some R_(α,α) s.th. 
⟨(g,g)(α,α)⟩0, corresponding to a
deconfined anyon gα, and want to construct identical
endpoints Z_α for the ket and bra layer such that 
⟨gα⊗gα⟩0.
We again start by decomposing 

    R_(α,α) = ∑ X^s_α⊗Y̅^s_α .

Let us define the shorthand ⟨ X_α⊗ Y_α⟩:=
⟨gα⊗gα⟩, where S⊗S̅ has endpoints X⊗ Y.  Now pick s_0 such that ⟨
X_α⊗Y̅_α⟩≡⟨ X^s_0_α⊗Y̅^s_0_α⟩ 0.  If 
⟨ X_α⊗X̅_α⟩ 0 or
⟨ Y_α⊗Y̅_α⟩ 0, we can choose
Z_α:=X_α (or Z_α:=Y_α), and have found the desired
non-vanishing identical ket and bra endpoint ⟨ Z_α⊗
Z_α⟩0.  Let us now consider the case where
both are zero. Let ϕ such that ⟨ X_α⊗ e^-iϕY̅_α⟩>0, and define Z_α := X_α+e^iϕY_α. Then, 

    ⟨ Z_α⊗Z̅_α⟩   =
    ⟨ X_α⊗X̅_α⟩ + 
    ⟨ Y_α⊗Y̅_α⟩
             +⟨ X_α⊗ e^-iϕY̅_α⟩ + 
    ⟨ e^iϕ Y_α⊗X̅_α⟩
       = 2Re ⟨ X_α⊗ e^-iϕY̅_α⟩ >
    0 ,

thus again yielding identical endpoints Z_α for ket and bra with
non-vanishing expectation value.





§ GENERALIZATION TO DRESSED ENDPOINTS



Let us now show that the no-go results of
Conditions <ref>–<ref>
derived in Sec. <ref> equally hold for general
endpoints; the explicit construction for any endpoint compatible with all
the conditions has already been provided in
Appendix <ref>. Let us recall that a general anyon
is of the form 

    -0.8cm
        < g r a p h i c s >
     .

For deriving the no-go results, we generally need to consider joint
ket-bra objects; we thus define

    -0.55cm
        < g r a p h i c s >
    -0.55cm
        < g r a p h i c s >
     .

The generalization of Eq. (<ref>), describing a general
string order parameter for a ket and bra anyon pair, evaluated in a pair
of fixed points ρ_ c and ρ_ c', is thus
of the form

    O_ c^ c':=
        -1.3cm
        < g r a p h i c s >
      .

Just as in Sec. <ref>, a central role will be played by
the (mixed) transfer operator 𝔼̃_ c^ c'; we
will therefore analyze its structure in detail in the following.  



 §.§ Structure of 𝔼̃_ c^ c'


The major complication as compared to the discussion in
Section <ref> is that for an M≡ M^ c
describing an injective MPS ρ which is a fixed point of the
transfer operator, the tensor 

    D:=  -1.6em
        < g r a p h i c s >

describing the MPS obtained after applying 𝕋 only needs to be
proportional to ρ, with a possibly size-dependent
proportionality constant.  This has two
consequences <cit.>: First, D can
consists of several diagonal blocks D_s,s, s=1,…,S each of which
describes a copy of the original MPS, i.e., D_s,s=M̃^i_s,  where
each M̃^i_s is equal to M^i up to a block-dependent gauge
transform, M̃^i_s = γ_s X_s M^i X_s^-1 with some
left-invertible
X_s.  Second, there can in addition be off-diagonal blocks D_s,t
coupling blocks s and t, which however—up to reordering of
blocks—must be upper triangular, i.e., D_s,t≡ 0 if s>t.  This
implies that any product D^i_1D^i_2⋯ D^i_L can contain each
off-diagonal block D^i_s,t at most once, and in particular contains
only a finite number of off-diagonal blocks. W.l.o.g., we will assume that
D is normalized such that the largest |γ_s|=1, with
the normalization of the M^i as before.

Let us now consider what this implies when taking large powers (𝔼̃^ c'_ c)^K, K→∞. In that
case, there will be large contiguous blocks of the form F_s:=∑ D_s^i,
c'⊗M̅^i, c= γ_s∑ X_s M_s^i,
c'X_s^-1⊗M̅^i, c (specifically, there will be at least
one block with length at least K/S), which will therefore converge to
the fixed point of the corresponding original transfer operator 𝔼^ c'_ c, up to normalization and a gauge transform.  In
particular, this implies for c c' that (𝔼̃_ c^ c')^K decays exponentially in K.  For c= c',
pick the largest contiguous block F_s within (𝔼̃_
c^ c)^K, and notice that it converges to a rank-1 projector onto
its non-degenerate leading eigenvectors, which therefore transform
trivially under the group action. Since 𝔼̃_ c^
c commutes with the symmetry action, further applications of
𝔼̃_ c^ c to this rank-1 projector do not
change the irrep label of the fixed point.  (The symmetry actions on the
different blocks are related by the corresponding gauge transform X_s
and label irreducible representations in the same way; note that we only
care about the symmetry action on the bond degree of freedom of the MPS to
the extent they are related to order parameters, i.e., the symmetry action
on the “physical” degrees of freedom.) (𝔼̃^ c_
c)^K will generally be a sum over terms in which different
blocks F_s with |γ_s|=1 converge to their fixed point, and thus,
(𝔼̃^ c_
c)^K→∑_iσ_R^iσ_L^i, where all
σ_∙^i transform trivially under the symmetry.  (Though it is
not relevant in what follows, it is worth noting that terms containing more
than one block F_s which converges to the fixed point cannot appear in
any expectation value, since their weight grows linearly with the system
size, whereas in the normalization only single blocks can show up.)






 §.§ Application to dressed endpoints



Let us verify that the modified expectation value Eq. (<ref>)
satisfies the same Conditions as before.

Condition <ref> is only about the symmetry
breaking pattern (and does not involve anyon strings), and is thus
entirely unaffected.

The off-diagonal terms in the expectation value Eq. (<ref>)
again vanish, since the corresponding large power of the off-diagonal
transfer operator will decay as the largest eigenvalue of the mixed
transfer operator 𝔼^ c_ c', and thus faster than the
diagonal terms, as we will see. 

Also, from Eq. (<ref>) one can immediately infer that the
expectation value is independent of the fixed point chosen, using again
the same argument as in Eq. (<ref>).

Next, in analogy to Eq. (<ref>), let us consider what
happens when we separate a pair of anyons. If g∉ H, we again
obtain a mixed transfer operator and thus the corresponding expectation
value vanishes, yielding
Condition <ref>.
If g∈ H, we can again move the symmetry action to the bond degree
of freedom of the MPS, and are thus left with 

    -1.3em
        < g r a p h i c s >
      .

As discussed above, we have that the fixed point space of the transfer
operator is of the form

    -1.3em
        < g r a p h i c s >

where the σ_∙^i transform trivially under the symmetry
action.  While the endpoints don't decouple any more, we still have that 
the expectation value of Eq. (<ref>)
converges to an average over products of expectation values

    -1.4em
        < g r a p h i c s >
       .

We can now follow the same reasoning as before: Using that 

    < g r a p h i c s >
       ,

we have that 

    -2em
        < g r a p h i c s >
     ,

which shows that Eq. (<ref>) can only be
non-vanishing if α( h)=ν_ g( h), yielding
Condition <ref>.  Note that the converse –
that there exists a suitable R_α whenever the condition is
satisfied – has already been shown in
Appendix <ref>.

Finally, the proof of Condition <ref> does
not make explicit reference to the form of the anyons, but just uses the
restrictions on ⟨gαℓ⟩,
Eq. (<ref>), obtained in
Condition <ref>.





§ 
REALIZATION OF ALL TWISTED ℤ_T DOUBLE MODELS THROUGH
CONDENSATION FROM D(ℤ_N)




In this Appendix we will discuss PEPS tensors which describe
ℤ_t twisted quantum
doubles <cit.> 
 with twist r ∈ [0,t-1].  We will
show that the PEPS tensors have symmetry ℤ_N, where N=qt and
q=t/gcd(t,r). We will explicitly construct the fixed points of
the transfer matrix and show that their residual symmetry is given by
ℤ_qt×ℤ_q and that their entanglement structure corresponds to
the second cohomology class rq/t.


We start of with defining a right and left handed building blocks for a matrix product operator, referred to as MPO tensors, M(a)_ij and N(a)_ij, i.e. for each a∈[0,t-1] and i,j∈[0,q-1] we define a qt× qt dimensional matrix,. The non-zero matrix elements of M(a)_g_0h_0 are given by [M(a)_g_0h]_{g,h},{g+a,h+a}  = √(q/t)ω(a,g,h-g_0)δ_g_0≡ g, with g∈[0,t-1] and g_0,h∈[0,q-1]. Here ω is a 3-cocycle which we define below and δ_g_0≡ g is unity if g_0 = g mod q, zero otherwise. Note that the subscripts g+a and h+a can be greater than t and q, respectively. Here and in the following we will implicitly use modulo t or q when calculating indices which can only take values smaller than t or q respectively. We will use subscript 0 to distinguish between a variable modulo t and q if both are used in the same equation, i.e. as in the definition of [M(a)_g_0h]_{g,h},{g+a,h+a} for g. The left and right handed MPO tensors are related by N(a)_h_0g_0 = M(a)_g_0h_0 (bar denotes complex conjugation). The non-zero values of M(a)_g_0h and N(a)_hg_0 can also be depicted graphically by:

    M(a)_g_0h =√(q/t)   
        < g r a p h i c s >
     ω(a,g,h-g_0)  , 
    
     N(a)_hg_0 = √(q/t)   
        < g r a p h i c s >
     ω(a,g,h-g_0)  ,

where at the r.h.s. g can take any value satisfying g_0 = g mod q. The horizontal (red, dotted) legs correspond to the indices of M(a) and N(a) , the vertical legs correspond to the indices of the matrices M(a)_g_0h_0  and M(a)_g_0h_0. The thick leg (t dim) and the thin leg (q dim) together form a qt dimensional space. The thick edges of the box indicate its orientation and is also used to distinguish M from N. The 3-cocycle ω is defined by:

    ω(a,g,d)  = exp[2π i r d/t^2(a+g-⌊ a+g ⌋)]    ,

where ⌊·⌋ denotes modulo t and r∈ [0,t-1]
specifies the specifies the class of the 3-cocycle. Note that this gauge
differs (by a co-boundary) from the one defined in
Ref. propitius:phd-thesis. This cocycle has the following
invariant ω(a,g,d) = ω(a,g,d+q), and satisfies the following
cocycle condition: 
    ω   (g_1,g_2,g_3)ω(g_1,g_2+g_3,g_4)ω(g_2,g_3,g_4) 
       = ω(g_1+g_2,g_3,g_4)ω(g_1,g_2,g_3+g_4)   .

for any set of g_i's. We use two copies of N(a) and M(a) to construct the map A(a) = ∑_ijkl M(a)_ij⊗ M(a)_jk⊗ N(a)_kl⊗ N(a)_li. The PEPS tensor is linear combination of these maps: A =t/q^2∑_a A(a) which can be graphically represented by 

    ∑_a
        < g r a p h i c s >
      .

The inner legs correspond to a physical site and the outer four groups of two legs correspond to the four auxiliary sites.



 §.§ 
Twisted Quantum double

The above defined PEPS tensor can also be obtained by starting from MPO
tensors for the twisted double defined in Ref. bultinck:mpo-anyons:

    M(a)_gh =   
        < g r a p h i c s >
     ω(a,g,h-g)  ,
    
     N(a)_hg =   
        < g r a p h i c s >
     ω(a,g,h-g)  .

All legs (thick) correspond to a t dimensional space. As indicated by expression (<ref>) these MPO tensors can also be used to create a PEPS tensor. We will now discuss two unitaries which can be used to relate the state described by the above MPO tensors to the state described by the MPO tensors given by Eqs. (<ref>) and (<ref>). First consider U acting on a t^5 dimensional space with non-zero matrix entries U_ijpqm,ijklm = δ_p,k-(x-x_0)δ_q,l-(m-m_0), where x = i+l-j. Acting with this unitary on the following physical sites

    U
        < g r a p h i c s >
      ,

gives a state whose reduced density matrix for the physical legs indicated by a thin leg has support on only a q dimensional space, i.e. spanned by the first q vectors of the computational basis. Acting with multiple copies of this unitary (one for each PEPS link) one can effectively reduce the dimension of the on-site Hilbert space from t^8 to (qt)^4. Note that the order does not matter since U acts diagonally on overlapping sites. In the following three steps we successively reduce the entanglement space  indicated by the green arrow: 

    < g r a p h i c s >
      .

The first reduction is valid since these indices are also coupled through the remaining 6 MPO tensors (we could actually have removed this link completely). In the second and third step we make use of the invariance of the 3-cocylce ω(a,g,d) = ω(a,g,d+q). The MPO tensor labeled by *, only depends on the indicated index modulo q. Applying this reduction to all plaquettes almost gives the model arising from the PEPS defined in Eq. (<ref>), except for the entanglement space between the upper and left MPO tensors of the PEPS tensor still being t-dimensional. A second unitary Ũ will reduce this entanglement. It acts on a t^3q dimensional space as:

    < g r a p h i c s >
      ,

 and has non-zero matrix entries Ũ_{g_1+kq,g_2+kq,g_3,g_4},{g_1,g_2,g_3,g_4} = F(⌊ g_2+kq-g_3⌋/q,⌊ g_2-g_3⌋/q) ω(g_3,kq,g_4-g_3) ω(g_1,kq,g_4-g_3), where again ⌊·⌋ denotes modulo t and F(a,b) = √(q/t)exp[2abqπ i /t]. Ũ is unitary since F is a Fourier transform in the difference between second and third index (mod q). Before applying this unitary the reduced density matrix ρ_23 of the state, corresponding to the sites labeled by 2 and 3 in the above equation, is a maximally mixed state, whereas after applying this unitary, ρ_23 has Schmidt rank q. The corresponding 3-cocycles in the  definition for Ũ ensure that after disentangling, the MPO tensors labeled by * in the above equation, still have the right phase factor. Indeed we have that Ũω(a+h-h',h',g-h)ω(a,h,g-h)|h',h,h,g⟩ = ∑_kα_k|h'+kq,h+kq,h,g⟩ where α_k is given by:

    α_k =     ω(h',kq,g-h)ω(a+h-h',h',kq+g-h)·
       ω(h,kq,g-h)ω(a,h,kq+g-h)
    
     =    ω(a+h,kq,g-h)ω(a+h-h',h'+kq,g-h)·
       ω(a+h-h',h',kq)·
       ω(a+h,kq,g-h)ω(a,h+kq,g-h)ω(a,h,kq)·
    
    =   ω(a,h+kq,g-h)ω(a+h-h',h'+kq,g-h)  ,

for any integer a. The second equation follows after applying the cocycle condition, Eq. (<ref>), twice: once with g_1 = a+h-h', g_2= h', g_3=kq and g_4=g-h and once with g_1 = a, g_2= h, g_3=kq and g_4=g-h. The first and third equality follow from the invariance ω(a,g,d) = ω(a,g,d+q).



 §.§ Properties

We will shows that the tensor constructed is a projector: A^† = A^2 = A. Both properties can be studied on the level of the MPO tensors M(a) and N(a). First of we have that M(-a)_ij and M(a)_ij^† are related by a gauge transformation:

    M(a)_ij^† = ∑_kl Q(a)_ikM(-a)_klQ(a)_jl  .

The matrix entries of Q(a) are given by the cocycle defined in Eq. (<ref>): Q(a)_ij = ω(-a,a,i)δ_i+a_0,j. The above relation follows from the cocycle condition: non-zero entries on the l.h.s. are ω(a,g,j-i) (for matrix indices [{g+a,j+a},{g,j}] with g_0=i), the corresponding matrix entries on the r.h.s. are ω(-a,a,i)ω(-a,a+g,j-i)ω(-a,a,j). These are equal by Eq. (<ref>) using g_1=-a, g_2=a, g_3=g and g_4=j-i, and from the fact that in the chosen gauge for ω we have that ω(0,g_3,g_4) = 1. The same equation can also be derived for N(a). Since Q obeys ∑_jQ(a)_ijQ(a)_kj = δ_ik it follows that the tensor A is Hermitian.



The product of two MPO tensors M(a) and M(b) is related to the MPO
tensor M(a+b) by a gauge transformation:  ∑_mn
Z(a,b)_i,mn (M(b)_mk· M(a)_nl) = √(q/t)∑_j
M(a+b)_ij Z(a,b)_j,kl where Z(a,b) is a q×
q^2 matrix whose non-zero entries are given by Z(a,b)_i,kl =
ω(a,b,i)δ_i,kδ_i+b_0,l. This equation can also be
represented graphically as:

    < g r a p h i c s >
      .

 This relation follows again from the cocycle condition, Eq. (<ref>): i.e. one can verify that ω(a,b,i)ω(b,g,k-i)ω(a,b+g,k-i) = ω(a+b,g,k-i)ω(a,b,k), being the entry-wise equation for the above relation for matrix entries [{g,k}, {g+a+b,k+a+b}] with g_0=i. The zipper Z obeys  ∑_kl Z(a,b)_i,klZ(a,b)_j,kl =  δ_ij and ∑_iZ(a,b)_i,kl Z(a,b)_i,mn = δ_mkδ_nlδ_m+b_0,n which can be represented graphically as:

    < g r a p h i c s >
      ,
    
        < g r a p h i c s >
      .

Note that the product of two zippers, given by Eq. (<ref>), is not equal to identity but rather equal to a projector. These equations are used in showing that A^2=A. To see this one first uses zippers Z(a,b) to simplify the product A(b)A(a) to q^2/t^2 A(a+b) which can best be graphically explained:

    < g r a p h i c s >

In the first equation one used Eq. <ref> to insert two zippers. Although this product of zippers is a projector rather than identity, this equation is still valid since the support of this projector contains the image of the product of two MPO tensors. Eq. <ref> is used to move one of the two zippers along the string of MPO tensors after which Eq. <ref> is used to remove the two zippers. Using this equation one can show that A^2  = t^2/q^4∑_abA(b)A(a) = 1/q^2∑_ab A(a+b) = t/q^2∑_c A(c) = A.  This motivates the pre-factor of t/q^2 in the definition of A.


The last property of the PEPS-tensor we will discuss in this section is that the corresponding transfer-matrix can be constructed from the MPO tensors: T = ∑|r_a)(l_a|, where |r_a) and |l_a) are given by

    |r_a)    = ∑_{i_n} M(a)_i_1i_2⊗ M(a)_i_2i_3⊗…⊗ M(a)_i_Li_1  ,
    
    |l_a)    = ∑_{i_n} N(a)_i_1i_2⊗ N(a)_i_2i_3⊗…⊗ N(a)_i_Li_1  .

The crucial step in deriving this statement is that the product of two MPO building blocks reduces to a delta function: Tr M(a)_ijN(b)_nm^T  = δ_abδ_inδ_jm. Or graphically:

    < g r a p h i c s >
     δ_ab  .

This motivates the factor of √(q/t) in the definition of the MPO tensors. Moreover, it can be used to show that the left and right eigenvectors are orthogonal: (l_a|r_b) ∝δ_ab. Using the above equation one can graphically derive the fixed points of T as follows:

    < g r a p h i c s >
      .







 §.§ Symmetries


In this section we show that the PEPS tensor A has a ℤ_N symmetry, where N=qt (hence the transfer matrix T has a ℤ_N×ℤ_N symmetry), and that the fixed points of the the transfer matrix break this symmetry to ℤ_qt×ℤ_q. Moreover, the remaining symmetry acts projectively on the auxiliary space of the fixed points (being the MPO-string space). To derive these statements we introduce a unitary S which relates M(a) and M(a+1) up to a gauge transformation: M(a)_ij S = ∑_klU(a)_ikM(a+1)_klU(a)_jl. Similarly, by combining this with Eq. (<ref>) it follows that  S^† M(a) is related to M(a-1) as S^† M(a)_ij = ∑_klV(a)_ikM(a-1)_klV(a)_jl where V(a) = Q(a)U(-a)Q(-a+1). Both equations can be represented graphically: 
 
    < g r a p h i c s >
      and
        < g r a p h i c s >
      .

The symmetry S is defined by S_(i_1,i_2),(j_1,j_2) = δ_i_1+1,j_1δ_i_2+1,j_2ω(1,i_1,i_2-i_1) and the gauge transformation U is defined by U(a)_ij = δ_ijω(1,a,i). Note that S is independent of a. The equations follows from the cocycle condition ω(1,a+g,d)ω(a,g,d) = ω(1,a,g)ω(1+a,g,d)ω(1,a,g+d).


To see that S is a generator of ℤ_N we evaluate S^t. It is
a diagonal matrix,  (S^t)_(i,i+d),(i,i+d) = ∏_j=1^t ω(1,j,d)
= exp[2π idr/t] = exp[2π
id/qr/gcd(t,r)], whose matrix entries are q-th roots
of unity, which are moreover primitive if gcd(d,q)=1 (for example,
d=1). Thus S^N = 1 and N is the smallest exponent for which this is
the case. Both S and S^† are symmetries of the tensor A and of
the transfer matrix. They are not symmetries of the fixed points  |r_a)
and |l_a). Only the global action of S⊗ S^† and the global
action of S^t⊗𝕀 are symmetries of the fixed points, and
they generate the group  ℤ_qt×ℤ_q. Their
action on an MPO tensor is given by S^† M(a)_ijS =
∑_klP_1_ikM(a)_klP_1_jl and  M(a)_ijS^t =
∑_klP_2_ikM(a)_klP_2_jl The corresponding gauge
transformations are P_1(a) = U(a)V(a+1) and P_2(a) =
∏_i=a^a+t-1 U(i). The later of these two gauge transformations is
most easily analyzed since U(a) is diagonal: P_2(a)_nn =
∏_m=a^a+t-1ω(1,m,n) = exp[-2π i
n/qr/gcd(t,r)]. Hence P_2(a) is independent of a and
is (up to a permutation) the generalized Pauli Z matrix in
ℤ_q. The non-zero matrix entries of the other gauge
transformation, being P_1(a)_n,n+1 are all equal, independent of n,
due to:

    ω(1,a,n)ω(-a-1,a+1,n)
          ·ω(1,-a-1,1+a+n)ω(a,-a,a+1+n)
    
    =   ω(1,a,n)ω(-a,a+1,n)
          ·ω(1,-a-1,a+1)ω(a,-a,a+1+n)
    
    =   ω(1,a,n)ω(a,-a,a+1)ω(1,-a-1,a+1)ω(a,1,n)
    
    =   ω(a,-a,a+1)ω(1,-a-1,a+1)  .

Here we have used the cocycle condition twice and in the last step we use that in our choice of gauge for ω we have that ω(a,1,n) = ω(1,a,n). Hence, up to a phase P_1 is a shift operator which upon conjugation by P_2 gives rise to a phase

    P_1P_2P_1^† P_2^†  = exp[-2 π i/qr/gcd(t,r)]   .

Thus together P_1 and P_2 generate a projective representation of ℤ_qt×ℤ_q, and r/gcd(t,r)=rq/t specifies the corresponding second cohomology class. 


This family of examples saturates all possible boundary theories of ℤ_N invariant PEPS models satisfying the conditions stated in the main text, in which the diagonal symmetry is maximal. In the general case (Condition 1) the residual symmetry is ℤ_qt×ℤ_q where qt is a merely a divisor of N, instead of qt=N. However, by increasing the dimension of the auxiliary space with a factor of x=N/(qt) one could simply add extra trivial symmetry to the PEPS tensor which would imply extra symmetry of the transfer matrix. The fixed points will break this extra symmetry because they do not have support on the added auxiliary space, and hence the residual symmetry is still ℤ_qt×ℤ_q.





10

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hastings:arealaw
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cirac:mpdo-rgfp
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Note1
Note that this does not imply that the fixed point space is actually spanned by
  the "026A30C ρ _c). 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.

rispler:peps-symmetrybreaking
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pollmann:symprot-1d
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chen:spt-order-and-cohomology
X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen,
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propitius:phd-thesis
M. de Wild Propitius,
Topological interactions in broken gauge theories,
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Note2
This can be seen using the cocycle conditions and the fact that the group is
  abelian as follows: 
    ν _h(g_1) ν _h(g_2)/ν _h(g_1g_2)   =ω (g_1,h)ω
      (g_2,h) ω (h,g_1g_2)/ω (h,g_1)ω
      (h,g_2) ω (g_1g_2,h) +.1667emω (hg_1,g_2)/ω (hg_1,g_2)
       =ω (g_1,h)ω
      (g_2,h) ω (h,g_1g_2) ω (hg_1,g_2) /ω (h, g_1g_2)
      ω (g_1,g_2) ω (h,g_2) ω (g_1g_2,h)
       =ω (g_1,h)ω (g_2,h) ω (h,g_1g_2) ω (hg_1,g_2) /ω (h, g_1g_2) ω (h,g_2) ω
      (g_1,g_2h) ω (g_2,h) 
       =ω (g_1,h) ω (hg_1,g_2) /ω (h,g_2) ω (g_1,g_2h) 
       =ω
      (g_1,h) ω (g_1 h,g_2) /ω (h,g_2)
      ω (g_1,hg_2)  =1  .
.

pollmann:spt-detection-1d
F. Pollmann and A. M. Turner,
Phys. Rev. B 86, 125441 (2012), arXiv:1204.0704.

Note3
Note that the same cannot hold for all abelian groups: Condensing from an
  abelian group gives another abelian model, while twisting an abelian model
  can give rise to non-abelian models <cit.>.

iqbal:preparation
M. Iqbal et al.,
in preparation .

zanardi:overlap-curvature
P. Zanardi, P. Giorda, and M. Cozzini,
Phys. Rev. Lett. 99, 100603 (2007), quant-ph/0701061.

gu:fidelity-phasetransitions
S.-J. Gu,
Int. J. Mod. Phys. B 24, 4371 (2010), arXiv:0811.3127.

morampudi:z2-phase-transition
S. C. Morampudi, C. von Keyserlingk, and F. Pollmann,
Phys. Rev. B 90, 035117 (2014), arXiv:1403.0768.

duivenvoorden:stringorder-symbreaking
K. Duivenvoorden and T. Quella,
Phys. Rev. B 88, 125115 (2013), arXiv:1304.7234.

yang:peps-edgetheories
S. Yang et al.,
Phys. Rev. Lett. 112, 036402 (2013), arXiv:1309.4596.





