# Average-exact mixed anomalies and compatible phases

Yichen Xu  Chao-Ming Jian  Department of Physics, Cornell University, Ithaca,
New York 14850, USA

###### Abstract

The quantum anomaly of a global symmetry is known to strongly constrain the
allowed low-energy physics in a clean and isolated quantum system. However,
the effect of quantum anomalies in disordered systems is much less understood,
especially when the global symmetry is only preserved on average by the
disorder. In this work, we focus on disordered systems with both average and
exact symmetries $A\times K$, where the exact symmetry $K$ is respected in
every disorder configuration, and the average $A$ is only preserved on average
by the disorder ensemble. When there is a mixed quantum anomaly between the
average and exact symmetries, we argue that the mixed state representing the
ensemble of disordered ground states cannot be featureless. While disordered
mixed states smoothly connected to the anomaly-compatible phases in clean
limit are certainly allowed, we also found disordered phases that have no
clean-limit counterparts, including the glassy states with strong-to-weak
symmetry breaking, and average topological orders for certain anomalies. We
construct solvable lattice models to demonstrate each of these possibilities.
We also provide a field-theoretic argument to provide a criterion for whether
a given average-exact mixed anomaly admits a compatible average topological
order.

###### Contents

  1. I Introduction
  2. II Review of the exact anomaly
    1. II.1 Anomalous symmetry actions on the boundary of SPT states
    2. II.2 Possible physical features of the exact anomaly
  3. III Disordered mixed state and AEM anomaly
    1. III.1 Mixed states and their symmetries
    2. III.2 Average SPT state in disordered systems
    3. III.3 Physical pictures of the AEM anomaly: fluctuation versus percolation
  4. IV SWSSB states compatible with the AEM anomaly
    1. IV.1 A solvable model with SWSSB
  5. V Disordered topological states compatible with the AEM anomaly
    1. V.1 Topological order and average topological order in disordered systems
    2. V.2 Review of anomaly-compatible TO in the clean limit
    3. V.3 TO and ATO compatible with the AEM anomaly
    4. V.4 Solvable model for symmetry-enriched TO and ATO
    5. V.5 Field-theoretical arguments: anomaly inflow in doubled Hilbert space
  6. VI Conclusion and outlook
  7. A Anomalous symmetry action from group cohomology
  8. B Construction of the symmetric infinite temperature mixed state
    1. B.1 The explicit form of the random Hamiltonian in Eq. (22)
    2. B.2 Derivation of Eq. (26)
  9. C The derivation of the unitary operators $\mathcal{V}$ and $\mathcal{W}$ via group extension and gauging on the lattice

##  I Introduction

Global symmetry and its quantum anomaly exert significant constraints on the
physics of quantum many-body systems. Originally introduced in the context of
quantum field theory Adler (1969); Bell and Jackiw (1969); Hooft (1980),
modern theoretical development has demonstrated the omnipresence of quantum
anomalies in various condensed matter systems. For example, on the boundary of
a symmetry-protected topological (SPT) state, anomalous symmetry actions are
the manifestation of the quantum anomaly induced by the topologically non-
trivial bulk Gu and Wen (2009); Chen et al. (2012, 2013); Else and Nayak
(2014); Kapustin and Thorngren (2014); Witten (2016); Tiwari et al. (2018);
Kawagoe and Levin (2021). Another class of example is given by the lattice
systems subject to the celebrated Lieb-Schultz-Mattis-Oshikawa-Hastings
(LSMOH) theorem Lieb et al. (1961); Oshikawa (2000); Hastings (2004) and
generalizations Cheng et al. (2016); Metlitski and Thorngren (2018); Cho et
al. (2017); Jian et al. (2018); Cheng and Seiberg (2023); Jiang et al. (2021);
Else and Thorngren (2020), where mixed quantum anomalies between the
crystalline symmetries (translation, crystal rotation, and etc) and the
internal symmetries (such as spin rotations) emerge at low-energies. Quantum
anomalies, once present, are topologically robust and strongly constrain the
allowed quantum phase in the IR. In particular, they forbid the quantum ground
states to be “trivially gapped” regardless of the details of the Hamiltonian.
The ground state compatible with the quantum anomaly must be either gapless,
symmetry-breaking, or topologically ordered.

The aforementioned situations mostly concern the pure ground states of clean
and isolated quantum systems. Meanwhile, physical systems that contain random
disorders or interact with an open environment are naturally described by
mixed (quantum) states. A mixed state is a probabilistic mixture of an
ensemble of pure states. In the case of disordered systems, the relevant mixed
state is formed by the ensemble of ground states in different disorder
configurations. From the symmetry perspective, in addition to exact symmetries
that leave each individual pure state invariant, a mixed state can also be
symmetric under average symmetries that map the pure states within the
ensemble to one another (A more formal definition of the exact and average
symmetries will be given later). Our understanding of the implication of
quantum anomalies of exact and/or average symmetries in mixed quantum states
is still in an early stage. Refs. Hastings, 2010; Fu and Kane, 2012; Kimchi et
al., 2018; Ma and Wang, 2023 investigated the disordered anomalous systems
with average spatial symmetries and found the anomaly (or, equivalently, the
disordered version of the LSMOH theorem) to impose significant constraints on
the low-energy physics still. For example, Ref. Kimchi et al., 2018 argued
that the ground-state ensemble of disordered magnets with spin 1/2 per
statistical unit cell cannot be constructed using the finite-depth quantum
circuit. For anomalies in open quantum systems Zhou et al. (2023); Hsin et al.
(2023); Kawabata et al. (2024); Lessa et al. (2024a); Wang and Li (2024), the
anomaly-induced constraints are mostly characterized by various entanglement
or dynamical aspects of the mixed state.

In contrast to the existing literature, our focus in this work is to identify
possible phases that are compatible with the mixed anomaly between an average
symmetry and an exact symmetry, dubbed the average-exact mixed anomaly (AEM
anomaly), in disordered systems. In these systems, each realization of the
disorders preserves the exact symmetry but breaks the average symmetry
explicitly. However, different disorder realizations are related to each other
by the action of the average symmetry, making the average symmetry preserved
by the entire disorder ensemble. Most crucially, apart from the possible
phases that smoothly connect to the clean limit, we discover mixed-state
phases that are compatible with the AEM anomaly but do not have clean-limit
counterparts. To demonstrate these new possibilities, we construct solvable
lattice models for each of these cases. These constructions can be easily
generalized to AEM anomalies of other global symmetry groups. We further use
field-theoretic arguments to understand what type of AEM anomalies allow these
new possibilities.

For clarity, we denote the entire global symmetry group by $G=A\times K$,
where $A$ is the average symmetry group and $K$ is the exact symmetry group by
our convention. We will subsequently use the term exact anomaly when the
entire symmetry group associated with the anomaly are exact symmetries.

Below, we list the AEM anomaly-compatible disordered phases that we have found
in this work and define their diagnostics via various forms of correlators of
the order parameters in the average density matrix over all the disorder
realizations..

  * •

Spontaneous symmetry breaking (SSB) states \- Analogous to the clean limit,
the AEM anomaly is compatible with a state that spontaneously breaks any part
of the symmetry group $G$ in the conventional sense. Such SSB states feature
long-ranged correlation functions in the disorder-averaged mixed density
matrix $\rho$

$\lim_{|x-y|\to\infty}\text{Tr}[\rho O(x)O^{\dagger}(y)]\sim O(1)$ (1)

for the order parameter $O(x)$ that transforms non-trivially under $G$.

  * •

Glassy states with strong-to-weak spontaneous symmetry breaking (SWSSB) \- In
Sec. IV, we show that there exists a different type of spontaneous symmetry-
breaking state, known as states with SWSSB, that is also compatible with the
AEM anomaly. In such a SWSSB state, the exact symmetry $K$ is spontaneously
broken down to an average symmetry, instead of being broken down completely to
a trivial group Lee et al. (2023); Ma et al. (2023); Lessa et al. (2024b). In
the open quantum systems where SWSSB were introduced, the exact/average
symmetries are also referred to as strong/weak symmetries and, hence, the name
strong-to-weak SSB. This recently proposed SWSSB possesses glassy orders that
can be diagnosed by an exponentially decaying correlator of an order parameter
$O(x)$ that transforms non-trivially under the exact symmetry $K$,

$\text{Tr}[\rho O(x)O^{\dagger}(y)]\sim e^{-|x-y|/\xi},$ (2)

but a non-vanishing long-range Rényi-2 correlator

$\lim_{|x-y|\to\infty}\frac{\text{Tr}\left[\rho O(x)O^{\dagger}(y)\rho
O(y)O^{\dagger}(x)\right]}{\text{Tr}\left(\rho^{2}\right)}\sim O(1).$ (3)

Note that another form for long-range correlator (known as the “fidelity
correlator”) has been proposed as an indicator for SWSSB Lessa et al. (2024b).
It is our working assumption that the Rényi-2 correlator can be used as the
diagnosis for SWSSB. We will comment on the fidelity correlator when suitable.

  * •

Topological and average topological orders \- For exact anomaly in spatial
dimensions $d\geq 2$, topological orders (TO) with anomalous symmetry actions
on the anyons offer a possible option for quantum phases compatible with the
anomalyWang and Senthil (2013); Kravec et al. (2015); Chen et al. (2015);
Metlitski et al. (2015); Wang and Levin (2017); Qi et al. (2019); Barkeshli et
al. (2019). For the AEM anomaly, we show that the presence of disorder in TO
leads to two options. One option is a disordered TO smoothly connected to the
clean limit-TO with the anomalous symmetry action. In some cases, there is a
new option, an average topological order (ATO), which can be thought of as a
TO but with partially lost quantum coherence caused by random anyonic
excitations. We will discuss the conditions for whether anomaly-compatible
ATOs is possible for the AEM anomaly of $G=A\times K$, and demonstrate our
result by constructing solvable lattice models in Sec. V.

  * •

Critical disordered states \- Between these disordered phases, there could
exist critical disordered phases that is compatible with the AEM anomaly. We
will briefly comment on their potential physical features in Sec. III.3, and
leave the study of these options to future works.

##  II Review of the exact anomaly

###  II.1 Anomalous symmetry actions on the boundary of SPT states

In this section, we review how a mixed anomaly of two exact global symmetries
$A$ and $K$ naturally arises at the boundary of SPT states in the clean limit
Else and Nayak (2014); Kawagoe and Levin (2021). This brief review will set
the stage for the discussion of the AEM anomalies in disordered systems in the
subsequent sections. We use $d$ to denote the boundary’s spatial dimension.
The corresponding bulk SPT hence lives in $d+1$ spatial dimensions or $(d+2)$
spacetime dimensions.

The SPT with exact symmetry $G=A\times K$ that leads to the desired exact
mixed anomaly on its boundary can be formulated via the decorated defect
construction of SPT states.Chen et al. (2014) This construction starts with a
system where $K$ is preserved, but $A$ is spontaneously broken. This system
can contain different configurations of topological defects, such as domain
walls, domain-wall junctions, etc, associated with the global symmetry $A$.
Each $d_{A}$ dimensional $A$-defect can be decorated by a non-trivial SPT
state in the same dimension protected by the symmetry $K$. If $K$ is trivial,
the “decoration” is simply a $U(1)$ phase that depends on the $A$-defect
configuration. Then, one can proliferate the defects of $A$ to restore the
full global symmetry $G$. The outcome of the proliferation is a $G$-SPT state
that corresponds to a non-trivial element in ${\cal H}^{d-d_{A}+1}[A,{\cal
H}^{d_{A}+1}[K,U(1)]]$ which, in light of the Künneth formula, is a part of
the cohomology classification ${\cal H}^{d+2}[A\times K,U(1)]$ of
$G$-SPTs.Chen et al. (2014).

On the boundary of this $G$-SPT, the bulk $A$-defects, together with the
decorated $K$-SPT, can terminate, causing the global symmetry $G$ to act
anomalously on the boundary. This is the manifestation of the (exact) mixed
anomaly between $A$ and $K$. To understand this anomaly, notice that each
$A$-defect that terminates on the boundary hosts a lower-dimensional anomaly
of $K$ originating from the boundary of the $K$-SPT state decorated onto the
$A$-defect. Consequently, the symmetry action of $K$ is conditioned under
$A$-spin configurations, implying that it cannot be strictly local. For a
finite group $G$, the explicit form of anomalous symmetry action can be
regularized on a lattice based on the element in the cohomology group ${\cal
H}^{d+2}[G,U(1)]$, see App. A which classifies the types of anomalies. The
regularized symmetry action is not “on-site”, i.e. not a tensor product of
actions on each lattice site, which is a consequence of the anomaly.

As an example useful in later discussions, we consider an anomaly of
$G=A\times K$ in $d=2$ where $A=K=\mathbb{Z}_{2}$. Its corresponding bulk SPT
state in 3+1$D$ is constructed by decorating each Ising domain wall of
$A=\mathbb{Z}_{2}$ by a 2+1$D$ SPT state protected by $K=\mathbb{Z}_{2}$,
known as the Levin-Gu stateLevin and Gu (2012). Therefore, in the 2+1$D$
anomaly on the boundary, each domain wall of $A$ hosts a 1+1$D$ $K$-anomaly.
The schematic form of the action of $K$ can be written down as follows. We
define Ising spins that transform under $A$ and $K$ as $\sigma$ and $\tau$,
respectively. Now, the $K$ action on a boundary domain wall $DW_{\sigma}$ of
$\sigma^{z}$ is inherited directly from the $\mathbb{Z}_{2}$ symmetry action
on the boundary of the Levin-Gu state:

$U_{K}\Big{|}_{DW_{\sigma}}=(-1)^{\frac{|DW_{\tau}\cap
DW_{\tau}|}{2}}\prod_{i\in DW_{\sigma}}\tau^{x}_{i},$ (4)

where $|DW_{\tau}\cap DW_{\tau}|$ is the number of intersection with the
$K$-domain walls, i.e. the domain wall of the $\tau$ spins, on the boundary
$A$-domain wall $DW_{\sigma}$. Meanwhile, the symmetry action of $A$ is
$U_{A}=\prod_{i}\sigma^{x}_{i}$. The explicit forms of $A$ and $K$ symmetry
actions regularized on a square lattice can be found in Sec. V.4 and Appendix
A.

The anomaly above corresponds to one of the $\mathbb{Z}_{2}$ elements in the
cohomology group ${\cal H}^{4}[A\times
K=\mathbb{Z}_{2}^{2},U(1)]=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ that classifies
the anomaly of $A\times K=\mathbb{Z}_{2}^{2}$ in 2+1$D$ (see App. A). The
other $\mathbb{Z}_{2}$ element in the cohomology classification represents
another anomaly where the role of $A$ and $K$ is exchanged, i.e. it arises on
the boundary of another 3+1$D$ SPT state where each domain wall of $K$ is
decorated with a $\mathbb{Z}_{2}$-SPT state protected by $A$. For clarity, we
denote these two mixed anomalies between $A=K=\mathbb{Z}_{2}$ as $\Omega_{1}$
and $\Omega_{2}$, respectively.

###  II.2 Possible physical features of the exact anomaly

We now summarize the physical features of anomaly-compatible phases in the
clean limit. The phases we are referring to in this section are all pure
states.

It is well-known that the exact anomaly is not compatible with a trivially
gapped ground state. Non-trivial quantum phases compatible with the (exact)
quantum anomaly includes:

  * •

SSB phase,

  * •

gapless phase,

  * •

symmetrically gapped and topologically ordered phase.

For a mixed anomaly between $A$ and $K$, it is compatible with a symmetry-
breaking state that breaks one of the two subgroups. For example, deep in the
SSB phase of $A$, creating an $A$-domain wall is energetically costly. As a
result, the symmetry action of $K$ is now essentially anomaly-free without the
presence of $A$-domain walls. Hence, the system can still be symmetric under
$K$.

There are two alternative scenarios where the global symmetry is not
spontaneously broken. One is a gapless state. Possible lattice Hamiltonians of
the gapless state can be constructed by symmetrizing the parent Hamiltonian of
a trivial product state using the lattice regularization of the anomalous
global symmetry actionsDupont et al. (2021); Tantivasadakarn et al. (2023a,
b), which usually works for anomalies in 1+1$D$Levin and Gu (2012); Zhang and
Levin (2023). In higher dimensions, numerical methods are usually needed to
determine whether a lattice Hamiltonian that is symmetric under the anomalous
symmetry action possesses a gapless low-energy spectrum.

For $d\geq 2$, the other anomaly-compatible phase is a topologically ordered
state, which will be reviewed in Sec. V.2.

##  III Disordered mixed state and AEM anomaly

###  III.1 Mixed states and their symmetries

Physically, a mixed state is a statistical ensemble of pure quantum states
$\left|\psi_{\alpha}\right\rangle$ labeled by some index $\alpha$, each with a
probability $p_{\alpha}$ that satisfies $\sum_{\alpha}p_{\alpha}=1$. We can
describe the mixed state using a density matrix

$\rho=\sum_{\alpha}p_{\alpha}\left|\psi_{\alpha}\right\rangle\left\langle\psi_{\alpha}\right|,$
(5)

so that the average value of an operator $O$ in the mixed state can be
conveniently written as $\langle O\rangle\equiv\text{Tr}[\rho
O]=\sum_{\alpha}p_{\alpha}\left\langle\psi_{\alpha}\right|O\left|\psi_{\alpha}\right\rangle$.

A mixed state can admit two distinct types of symmetries: exact and average
symmetries. Conceptually, a mixed state possesses an exact symmetry $K$ and an
average symmetry $A$ if each pure state in the statistical ensemble is
symmetric under $K$ (i.e.,
$U_{k}\left|\psi_{\alpha}\right\rangle=\left|\psi_{\alpha}\right\rangle$ for
each $k\in K$), and any two states related by symmetry $A$ have equal
probability (i.e. $p_{\alpha}=p_{\beta}$ if
$\left|\psi_{\beta}\right\rangle=U_{a}\left|\psi_{\alpha}\right\rangle$). More
generally, a mixed-state density matrix $\rho$ is symmetric under the exact
$K$ symmetry and average $A$ symmetry if

$\rho=U_{k}\rho=\rho U_{k}^{\dagger}=U_{a}\rho U_{a}^{{\dagger}}$ (6)

for all $k\in K$ and $a\in A$. Here, $U_{k}$ and $U_{a}$ are the operators
that implement the corresponding symmetry actions.

In this work, we are mainly interested in the case where the global symmetry
group is $G=A\times K$, i.e. the average and exact symmetries are independent
of each other, and their actions commute. 111More generally the global
symmetry group can be a group extension of $A$ by $K$ through the short exact
sequence  $1\to K\to G\to A\to 1,$ (7) since an average symmetry can be
extended to an exact symmetry. For example, there exists a mixed state with
global $G=\mathbb{Z}_{4}=\\{1,g,g^{2},g^{3}\\}$ symmetry, such that the
average symmetry is $g$ and $g^{3}$ while the exact symmetry is $g^{2}$. It
cannot be the other way round, since if $g^{2}$ is an average symmetry, $g$
has to also be an average symmetry.

There are two common settings where such $G$-symmetric mixed states occur. One
scenario concerns systems with random quenched disorders. The Hamiltonian of
such a disordered system can be written as $H(\\{h_{i}\\})$, where
$\\{h_{i}\\}$ parameterizes the strength of the independent random local
interactions in the vicinity of each lattice site $i$ and follows the
probability distribution $p(\\{h_{i}\\})$. The mixed state of interest in this
scenario is the ensemble of ground states of $H(\\{h_{i}\\})$ denoted by
$\left|\psi(h_{i})\right\rangle$. The disordered system is symmetric under $G$
if

  1. (i)

the Hamiltonian $H(\\{h_{i}\\}$ commutes the $K$ symmetry action for each
configuration $\\{h_{i}\\}$, namely

$\displaystyle U_{k}H(\\{h_{i}\\})U_{k}^{{\dagger}}=H(\\{h_{i}\\})$ (8)

for all $k\in K$,

  2. and
  3. (ii)

the disorder configurations related by the $A$ symmetry action share the same
probabilities distribution, namely

$\displaystyle U_{a}H(\\{h_{i}\\})U_{a}^{{\dagger}}=H(\\{h^{a}_{i}\\}),$ (9)

such that $p(\\{h_{i}\\})=p(\\{h^{a}_{i}\\})$ for all $a\in A$. Here,
$\\{h^{a}_{i}\\}$ is the configuration obtained from the symmetry action of
$a\in A$ on the configuration $\\{h_{i}\\}$.

Another setting for $G$-symmetric mixed states to occur is an open system.
When the $K$ symmetry charge is strictly preserved in the physical system and
the $A$ symmetry charge can be exchanged between the physical system and the
environment, the density matrix of the physical system can also satisfy the
condition Eq. (6). In the open system context, the exact and average symmetry
are also referred to as the strong and weak symmetries. In this work, we will
only focus on the mixed state arising from the previous setting of disordered
systems.

###  III.2 Average SPT state in disordered systems

Similar to the clean limit, a natural setting that gives rise to the AEM
anomaly of $G=A\times K$ is the boundary of the recently proposed average SPT
(ASPT) states protected by an average symmetry $A$ and an exact symmetry $K$Ma
and Wang (2023); Ma et al. (2023). Therefore, we briefly review the ASPT state
in disordered systems which helps our subsequent discussion of the AEM
anomaly.

A physical scenario of ASPT mixed states is a clean SPT state protected by
$G=A\times K$ exposed to random field disorder of the $A$ spins, i.e. the
local degrees of freedom that transform non-trivially under $A$. 222An
alternative origin for ASPT states is a pure SPT state protected by $G=A\times
K$ exposed to measurement or quantum decoherence whose quantum channel
possesses strong $K$ and weak $A$ symmetriesMa and Wang (2023); Zhang et al.
(2022); Ma et al. (2023); Lee et al. (2022).  The Hamiltonian for the
disordered ASPT state can be written as

$\displaystyle
H_{\text{ASPT}}=H_{\text{SPT}}+H_{\text{disorder}}(\\{h_{i}\\}),$ (10)

where $H_{\text{SPT}}$ denotes the parent Hamiltonian of the SPT state in the
clean limit, and $H_{\text{disorder}}$ contains a set of random “Zeeman field”
$\\{h_{i}\\}$ that polarizes the $A$ spins.

The physical picture of the ASPT mixed state is most pronounced in the strong
random field disorder limit, where each $A$ spin on lattice site $i$ is
polarized along the direction of the random field $h_{i}$ on the lattice
site333From the Imry-Ma argumentImry and Ma (1975); Aizenman and Wehr (1989,
1990) and its subsequent generalization to disorders in quantum systems
Greenblatt et al. (2009); Aizenman et al. (2012); Vojta (2019), if the spatial
dimension of the system is at or below the lower critical dimension $d_{c}$
($d_{c}=2$ for discrete symmetry and $d_{c}=4$ for continuous symmetry), any
spin orders associated with the SSB of $A$ in the clean limit will be
unfavorable with the presence of random field disorders that couples to the
order parameter of $A$.. Each ground state of the disordered Hamiltonian is
simply given by a configuration of (uncorrelated) $A$ spins pinned by the
random field $\\{h_{i}\\}$ where each defect in the $A$ spin configuration,
including the domain walls, the domain wall junctions, and etc, hosts a lower-
dimensional SPT state protected by the exact symmetry $K$.

With such a picture, we see that the exact symmetry group $K$ has to be non-
trivial in order for the ASPT to be topologically non-trivial. An “ASPT mixed
state” with only an average symmetry (i.e. $G=A$) is topologically trivial,
since the decoration on the $A$-defects are simply $U(1)$ phases in the clean
limit, which lose meaning in a mixed state without coherent superposition of
$A$-defects. Mathematically, this means that when calculating the cohomology
classification of $G$-ASPT using the Künneth formula ${\cal H}^{d+2}[A\times
K,U(1)]=\bigoplus_{D=0}^{d+2}{\cal H}^{d+2-D}[A,{\cal H}^{D}[K,U(1)]]$, the
element with $D=0$ no longer enters the classification.444Ref. Ma et al., 2023
pointed out that the element with $D=1$ also does not enter the classification
of disordered ASPT states, since the $K$ charges that are decorated onto the
point defects of $A$ can simply localize under disorders. However, as we will
see in Sec. IV, such a bulk is no longer an SPT state due to the fact that $K$
has been spontaneously broken down to an average symmetry when the $K$ charges
are localized. Therefore, it is still meaningful to discuss an AEM anomaly
that corresponds to the element with $D=1$.

###  III.3 Physical pictures of the AEM anomaly: fluctuation versus
percolation

Now that we have established the physical picture of the ASPT state, we can
discuss the AEM anomaly between the average symmetry $A$ and the exact
symmetry $K$. A way to understand this anomaly is to consider the boundary a
strongly disordered ASPT. Similar to the discussions in Sec. II.1, the mixed
anomaly is manifested in the way that, on each lower-dimensional $A$-defect
pinned by strong random-field disorders, the exact symmetry $k\in K$ acts in
an anomalous way (see Fig. 1). In a disordered setting, the $A$-defect
configurations are not coherently superposed, but simply mixed together
incoherently in the density matrix. As we will see below, this is the main
reason that gives rise to anomaly-compatible disordered phases beyond the
clean-limit possibilities.

Using such a decorated defect picture, we now discuss possible features of the
mixed state that are compatible with the AEM anomaly of $G=A\times K$.

We first consider the case where the exact symmetry $K$ is trivial. Recall
from Sec. III.2 that any ASPT mixed state protected solely by an average
symmetry $G=A$ is topologically trivial, we would expect that the anomaly of
purely average symmetry cannot prevent the system from becoming completely
trivial without any physical feature. Indeed, in the strong disorder limit,
the system can be simply a mixture of different quenched $A$-defect
configurations, whose density matrix is proportional to identity (i.e. an
infinite-temperature state): $\rho\propto\mathbbm{1}$. Such a disordered state
will always be symmetric under any average symmetry.

Moving on to the AEM anomaly of $G=A\times K$ with a non-trivial $K$. Due to
the presence of lower-dimensional $K$-anomalies decorated onto the
$A$-defects, each of them exhibits some non-trivial physical features. What
feature occurs depends on the details of interactions in the system. If the
disorder induces a strong localization effect, a possibility is that $K$ is
spontaneously broken on each $A$-defect in order to be compatible with the
anomaly of the exact symmetry $K$. The feature of this scenario can be
potentially detected using the correlators of the order parameters of $K$. In
fact, the behavior of the correlator depends on whether the $A$-defects
percolate or not. To see this, we consider the ground state
$\left|\psi_{\alpha}\right\rangle$ of some disorder realization $\alpha$ where
each $A$-defect is decorated with an SSB state of $K$. Since the $K$ is
spontaneously broken on each $A$-defect, we have

$O(x)\left|\psi_{\alpha}\right\rangle\sim
O(y)\left|\psi_{\alpha}\right\rangle$ (11)

for some local order parameter $O$ of $K$ only if $x$ and $y$ are located on
the same $A$-defect. If the defects of $A$ percolate, such a possibility is
almost 1 across the entire ensembleStauffer and Aharony (2018). Therefore, the
entire disordered state breaks $K$ spontaneously, which can be diagnosed by
the correlator of the order parameter $O(x)$ averaged over all possible
disorder realizations:

$\langle O(x)O^{\dagger}(y)\rangle\equiv\lim_{|x-y|\to\infty}\text{Tr}[\rho
O(x)O^{\dagger}(y)]\sim O(1).$ (12)

If the $A$-defects do not percolate, namely, if the SSB of the exact $K$
symmetry remains independent on each $A$-defect, such a correlator will
instead be short-ranged. We will show in Sec. IV that such a situation
indicates a breakdown of the exact symmetry to an average symmetry, i.e. an
SSWSB of $K$, in contrast to the usual SSB where $K$ is broken down to a
trivial group.

On the other hand, if there exists strong enough interaction between
$K$-spins, pairs of $A$-defects with conjugating $K$-anomalies can be “gapped
out” while preserving $K$ (see Fig. 1). However, forming such a pairing of all
the $A$ defects will favor a specific direction of $A$-spins. Therefore, when
the $A$-defects together with their $K$-anomalies are paired up under
interaction, we argue that the average symmetry $A$ will be spontaneously
broken.

Between these two limits, there could, in principle, exist a critical
disordered state that is compatible with the AEM anomaly. For example,
consider the case where each 1+1$D$ or higher dimensional $A$-defect hosts the
ground state of some gapless Hamiltonian that is compatible with the $K$
anomaly, and the $A$-defects percolate. Therefore, the gap of such a
disordered system should vanish in the thermodynamic limit for a typical
disorder realization, i.e. the system is gapless on average. The physical
picture here is conceptually similar to the quantum Hall plateau transition at
half-integer filling Chalker and Coddington (1988); Lee et al. (1993); Wang et
al. (2014) and the delocalized state at the Anderson localization transition
for Dirac fermions Morimoto et al. (2015); Fu and Kane (2012). We also note
that the correlator within a typical disorder realization is not necessarily
power-law in this case but the disordered average correlators are expected to
be power-law. One example is the random singlet state Fisher (1994); Liu et
al. (2018, 2020) that is compatible with the average LSM anomaly with an exact
spin-rotation symmetry $K=SO(3)$ and an average translation symmetry $A$,
where the spin-spin correlator decay as
$\langle\mathbf{S}(x)\cdot\mathbf{S}(y)\rangle\sim|x-y|^{-2}$ when averaging
over all disorder realizations, but
$\left\langle\psi_{\alpha}\right|\mathbf{S}(x)\cdot\mathbf{S}(y)\left|\psi_{\alpha}\right\rangle$
for a typical disorder realization $\alpha$ can be short-ranged. In fact, Ref.
Kimchi et al., 2018 proved in 1+1$D$ that, in a disordered mixed state that is
compatible with the average LSM anomaly, the average spin-spin correlator
cannot decay faster than the one for the random singlet phase without
spontaneously breaking the average translation symmetry.

We note that there also exist interactions that can lead to a symmetrically
gapped state with topological order or average topological order that is
compatible with the AEM anomaly, which we discuss in detail in Sec. V.2.

Figure 1: A typical sample in the strong random field disorder limit, where
the black arrows indicate the $A$ spins that are quenched and polarized by the
strong random fields. On each defect of $A$ marked by the black lines, the
exact symmetry $K$ acts anomalously. The red arrows indicates possible
interaction that can trivialize two conjugating $K$ anomalies, and the paired
$K$ anomaly carries a certain spin of $A$.

##  IV SWSSB states compatible with the AEM anomaly

We now consider the case with SWSSB where the $A$-defects do not percolate in
more detail. As alluded in Sec. III.3, even if each $A$-defect hosts an SSB
state of $K$ in this case, the two-point correlators of the order parameters
of $K$ in Eq. (12) will decay exponentially with relative distance, since
these two points have almost-zero probability of being on the same $A$-defect,
i.e.

$O(x)\rho\not\sim O(y)\rho.$ (13)

The physical picture here is that the $K$ symmetry breaking pattern fluctuates
independently at each $A$-defect, i.e., the system is a glassy state of
$K$-charges.

To probe the symmetry-breaking order on each $A$-defect, notice that the
following similarity holds between the two density matrices with order
parameter of $K$ acting at two different spatial locations:

$O(x)\rho O^{\dagger}(x)\sim O(y)\rho O^{\dagger}(y),$ (14)

regardless of the positions $x$ and $y$. This is because the $K$ charges are
now independently condensed on both the $A$ defects on which $x$ and $y$ are
located555$O(x)\rho O^{\dagger}(x)$ is not similar to $\rho$, since the entire
state ensemble in $\rho$ share a definitive symmetry charge of $K$ different
that of $O(x)\rho O^{\dagger}(x)$.. As proposed by Refs. Lee et al., 2023; Ma
et al., 2023; Lessa et al., 2024b, the relations in Eqs. (13) and (14)
indicate that the exact symmetry $K$ has been spontaneously broken down to an
average symmetry, rather than being broken down completely to a trivial group.
Such a pattern of symmetry breaking is called strong-to-weak spontaneous
symmetry breaking (SWSSB). To quantify the similarity suggested by (14), a
convenient choice to calculate is the normalized overlap between these two
density matrices:

$\langle O(x)O^{\dagger}(y)\rangle_{2}\equiv\frac{\text{Tr}[O(x)\rho
O^{\dagger}(x)O(y)\rho O^{\dagger}(y)]}{\text{Tr}[\rho^{2}]},$ (15)

which is the Rényi-2 version of the two-point charge correlator of $K$. In
short, the SWSSB phase is characterized by a finite value of
$\lim_{|x-y|\to\infty}\langle O(x)O^{\dagger}(y)\rangle_{2}$ and short-range
correlated $\langle O(x)O^{\dagger}(y)\rangle$666As pointed out by Ref. Lessa
et al., 2024b, a characterization of the similarity in Eq. (14) that is more
robust against perturbations via strongly symmetric local quantum channels is
through the fidelity between these two density matrices:
$\displaystyle\langle O(x)O^{\dagger}(y)\rangle_{2}^{F}\equiv F[O(x)\rho
O^{\dagger}(x),O(y)\rho O^{\dagger}(y)],$ (16) where
$F(\rho,\sigma)=\left[\text{Tr}(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}})\right]^{2}$
is the fidelity between two density matrices $\rho$ and $\sigma$. The two
correlators are generally inequivalent but behave qualitatively similarly in
diagnosing SWSSB order deep inside this phase..

We note that the definition in Eq. (15) is related to the Edwards-Anderson
order parameter of spin glass Parisi (1983). To see this, we plug in the
density matrix with the diagonal basis
$\rho=\sum_{\alpha}p_{\alpha}\left|\psi_{\alpha}\right\rangle\left\langle\psi_{\alpha}\right|$
and assume that any two disorder realizations $\alpha$ and $\beta$ have zero
overlap even with a local charge excitation, i.e.
$\left\langle\psi_{\alpha}\right|O(x)O^{\dagger}(y)\left|\psi_{\beta}\right\rangle\approx
0$. We then have

$\langle
O(x)O^{\dagger}(y)\rangle_{2}\approx\frac{\sum_{\alpha}p_{\alpha}^{2}|\left\langle\psi_{\alpha}\right|O(x)O^{\dagger}(y)\left|\psi_{\alpha}\right\rangle|^{2}}{\sum_{\alpha}p_{\alpha}^{2}},$
(17)

which matches the Edwards-Anderson order parameter if $O$ is taken to be the
Ising spin operator and the probability for each disorder realization $\alpha$
is proportional to $p_{\alpha}^{2}$.777The fidelity correlator of SWSSB,
$\langle O(x)O^{\dagger}(y)\rangle_{2}^{F}$, will be simplified to the
Edwards-Anderson order parameter where each disorder realization $\alpha$ has
probability $p_{\alpha}$, instead of $p_{\alpha}^{2}$.

###  IV.1 A solvable model with SWSSB

To demonstrate a 1+1D SWSSB state that is compatible with the AEM anomaly, we
construct a physical disordered model to realize an SWSSB of the exact
symmetry $K$ that is compatible with the 1+1$D$ AEM anomaly of $G=A\times K$
between $A=K=\mathbb{Z}_{2}$. The anomalous symmetry we focus on can be
defined on a 1$D$ lattice as the following non-on-site actions (see App. A for
the derivation of the anomalous symmetry actions on the lattice starting from
the group cohomology that characterizes the anomaly).

$U_{A}=\prod_{i=1}^{N}\tau^{x}_{i+\frac{1}{2}},\
U_{K}=\prod_{i=1}^{N}C\left(\sigma^{z}_{i},\tau^{z}_{i-\frac{1}{2}}\tau^{z}_{i+\frac{1}{2}}\right)\sigma^{x}_{i},$
(18)

where the $\sigma$ spins are located on the lattice sites and $\tau$ spins are
on the bonds (see Fig. 2), and $N$ is the number of lattice sites.
$C(A,B)=e^{\frac{\mathrm{i}\pi}{4}(1-A)(1-B)}$ is the controlled unitary
between two Pauli operators $A$ and $B$. We also assume a periodic boundary
condition $\sigma^{z}_{N+1}=\sigma^{z}_{1}$ and
$\tau^{z}_{N+\frac{1}{2}}=\tau^{z}_{\frac{1}{2}}$. One can verify that
$U_{K}^{2}=1$. From the expression of $U_{K}$, the anomaly is manifested in
the way that each $\tau^{z}$ domain wall hosts a half charge of $K$:
$U_{K}\Big{|}_{\tau^{z}_{i-\frac{1}{2}}\tau^{z}_{i+\frac{1}{2}}=-1}=\sigma^{z}_{i}\sigma^{x}_{i}=\mathrm{i}\sigma^{y}_{i}$,
so that locally on each domain wall $U_{K}$ squares to $-1$.

In the clean limit (where both $A$ and $K$ are exact), various anomaly-
compatible states can be constructed using the following HamiltonianChen et
al. (2014)

$\displaystyle H_{\mathbb{Z}_{2}^{2}}=$ (19)
$\displaystyle-\left[\sum_{i}(J_{\sigma}\sigma^{x}_{i}+J_{\tau}\tau^{x}_{i+\frac{1}{2}})+U_{K}\sum_{i}(J_{\sigma}\sigma^{x}_{i}+J_{\tau}\tau^{x}_{i+\frac{1}{2}})U_{K}^{-1}\right]$
$\displaystyle=-\sum_{i}\left[J_{\tau}\tau^{x}_{i+\frac{1}{2}}\left(1+\sigma^{z}_{i}\sigma^{z}_{i+1}\right)+J_{\sigma}\sigma^{x}_{i}\left(1+\tau^{z}_{i-\frac{1}{2}}\tau^{z}_{i+\frac{1}{2}}\right)\right].$

The ground state of the Hamiltonian can be tuned by $J_{\sigma}/J_{\tau}$.
When $J_{\sigma}/J_{\tau}\gg 1$ (or $\ll 1$), the Hamiltonian favors a
ferromagnetic ground state of $\tau^{z}$ (or $\sigma^{z}$) while
$\sigma^{x}_{i}=1$ (or $\tau^{x}_{i}=1$) on each site, which spontaneously
breaks $A$ (or $K$). These two SSB states are separated by a critical point at
$J_{\sigma}=J_{\tau}$, at which the Hamiltonian is gapless and is exactly
solvable by mapping to a fermionic chain.

To construct a disordered state with SWSSB of $K$, we need to design a random-
bond type of disorder that enforces a ferromagnetic state of $\sigma^{z}$ in a
“random basis”, while also explicitly breaking the symmetry $A$ down to an
average symmetry. The random-basis ferromagnetic can be achieved by defining
the following unitary operator that randomly rotate the basis for $\sigma^{z}$

$W(\\{h_{i},l_{i}\\})=\prod_{i}V_{i}^{\frac{1-h_{i}}{2}}\left(\tau^{z}_{i+\frac{1}{2}}\right)^{\frac{1-l_{i}}{2}},$
(20)

where
$V_{i}=\sigma^{x}_{i}C\left(\sigma^{z}_{i},\tau^{z}_{i-\frac{1}{2}}\tau^{z}_{i+\frac{1}{2}}\right)$
and $h_{i},l_{i}=\pm 1$ with equal probabilities denotes random flips required
for a SWSSB state. The unitary $W$ is symmetric under $K$ and $A$ since

$\displaystyle
U_{A}W(\\{h_{i},l_{i}\\})U_{A}^{{\dagger}}=\left(\prod_{i}l_{i}\right)W(\\{h_{i},l_{i}\\}),$
$\displaystyle
U_{K}W(\\{h_{i},l_{i}\\})U_{K}^{{\dagger}}=W(\\{h_{i},l_{i}\\}).$ (21)

Note that the random basis rotation $W(\\{h_{i},l_{i}\\})$ randomly adds to
the system a symmetry charge $l_{i}=\pm 1$ of $A$ with equal probabilities in
every unit cell. Using this random basis rotation, we construct a disordered
Hamiltonian:

$H(\\{h_{i},l_{i}\\})\equiv
W(\\{h_{i},l_{i}\\})H^{Y}_{\mathbb{Z}_{2}^{2}}W^{-1}(\\{h_{i},l_{i}\\}),$ (22)

where

$\displaystyle H^{Y}_{\mathbb{Z}_{2}^{2}}=$
$\displaystyle-\sum_{i}\left[J_{\tau}\tau^{y}_{i+\frac{1}{2}}\left(1+\sigma^{z}_{i}\sigma^{z}_{i+1}\right)\right.$
(23)
$\displaystyle\quad\quad\quad\quad\quad\left.+J_{\sigma}\sigma^{x}_{i}\left(1+\tau^{z}_{i-\frac{1}{2}}\tau^{z}_{i+\frac{1}{2}}\right)\right]$

is obtained from $H_{\mathbb{Z}_{2}^{2}}$ by modifying the $J_{\tau}$
interaction so that it explicitly breaks $A$. The ensemble of random
Hamiltonians still preserves $A$ on average:

$U_{A}H(\\{h_{i},l_{i}\\})U_{A}^{-1}=H(\\{h_{i},-l_{i}\\}).$ (24)

Since $V_{i}$ commutes with other $V_{j}$ and $\tau^{z}_{j+\frac{1}{2}}$, each
of the randomly transformed Hamiltonian $H(\\{h_{i},l_{i}\\})$ still consists
of local terms (see Appendix B for its explicit form). Physically, the
randomly rotated $J_{\tau}$ term mimics a random bond Ising-type interaction
between two nearby spins that is symmetric under $K$ but explicitly breaks
$A$.

We now solve for the mixed-state density matrix for the ensemble of ground
states of $H(\\{h_{i},l_{i}\\})$. First, the ferromagnetic ground state of
$H^{Y}_{\mathbb{Z}_{2}^{2}}=H(\\{h_{i}=1,l_{i}=1\\})$ in the limit
$J_{\sigma}/J_{\tau}\ll 1$ reads

$\displaystyle\left|\psi_{0}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|\uparrow\uparrow\dots\right\rangle_{\sigma}+\left|\downarrow\downarrow\dots\right\rangle_{\sigma}\right)\otimes\left|+_{y}+_{y}\dots\right\rangle_{\tau},$
(25)

where the $\sigma$ spins are in the superposition of two ordered
configurations so that $\left|\psi_{0}\right\rangle$ is overall charge-neutral
under $K$. Energetically, we can show that the state
$\left|\psi_{0}\right\rangle$ with the combination
$\left(\left|\uparrow\uparrow\dots\right\rangle_{\sigma}+\left|\downarrow\downarrow\dots\right\rangle_{\sigma}\right)$
has a lower energy than its counterpart with
$\left(\left|\uparrow\uparrow\dots\right\rangle_{\sigma}-\left|\downarrow\downarrow\dots\right\rangle_{\sigma}\right)$
in a finite-size system. This can be seen via a degenerate perturbation theory
in the subspace of the two degenerate ground states
$\\{\left|\uparrow\uparrow\dots\right\rangle_{\sigma},\left|\downarrow\downarrow\dots\right\rangle_{\sigma}\\}$,
by treating the $J_{\sigma}$ term in the Hamiltonian
$H^{Y}_{\mathbb{Z}_{2}^{2}}$ as a perturbation. Later, we will use
$\left|\psi_{0}\right\rangle$ to construct an enssemble of disordered ground
states of $H(\\{h_{i},l_{i}\\})$ within the charge-neutral sector of $K$.
Conceptually, fixing the charge sector is analogous to a standard finite-
temperature canonical thermal ensemble with a fixed charge.

Using $\left|\psi_{0}\right\rangle$, the ensemble of disordered ground states
of $H(\\{h_{i},l_{i}\\})$ can then be described by the following density
matrix

$\displaystyle\rho_{\mathbb{Z}_{2}^{2}}$
$\displaystyle=\frac{1}{2^{2N}}\sum_{h_{i},l_{i}}W(\\{h_{i},l_{i}\\})\left|\psi_{0}\right\rangle\left\langle\psi_{0}\right|W^{-1}(\\{h_{i},l_{i}\\})$
(26) $\displaystyle=\frac{\mathbbm{1}+U_{K}}{2^{2N}}.$

The detailed derivation can be found in Appendix B. The ensemble of disordered
ground state can be regarded as an infinite-temperature canonical ensemble of
$K$ charges: every state in the ensemble is overall charge-neutral under $K$
since both $\left|\psi_{0}\right\rangle$ and $W(\\{h_{i},l_{i}\\})$ are
$K$-symmetric. It is straightforward to check that $\rho_{\mathbb{Z}_{2}^{2}}$
has an exact $K$ symmetry and an average $A$ symmetry. The SWSSB of the exact
symmetry $K$ in $\rho_{\mathbb{Z}_{2}^{2}}$ can be detected by the correlators

$\frac{\text{Tr}[\rho_{\mathbb{Z}_{2}^{2}}\sigma^{z}_{i}\sigma^{z}_{j}\rho_{\mathbb{Z}_{2}^{2}}\sigma^{z}_{i}\sigma^{z}_{j}]}{\text{Tr}[\rho_{\mathbb{Z}_{2}^{2}}^{2}]}=1\text{
and }\text{Tr}[\rho_{\mathbb{Z}_{2}^{2}}\sigma^{z}_{i}\sigma^{z}_{j}]=0$ (27)

for any $i\neq j$.888The fidelity correlator
$\langle\sigma^{z}_{i}\sigma^{z}_{j}\rangle_{2}^{F}$ for
$\rho_{\mathbb{Z}_{2}^{2}}$ is also 1:
$\langle\sigma^{z}_{i}\sigma^{z}_{j}\rangle_{2}^{F}=F(\sigma^{z}_{i}\rho_{\mathbb{Z}_{2}^{2}}\sigma^{z}_{i},\sigma^{z}_{j}\rho_{\mathbb{Z}_{2}^{2}}\sigma^{z}_{j})=F\left(\frac{\mathbbm{1}-U_{K}}{2^{2N}},\frac{\mathbbm{1}-U_{K}}{2^{2N}}\right)=1$

Interestingly, by increasing $J_{\sigma}/J_{\tau}$ , the random Hamiltonian in
Eq. (22) interpolates between SWSSB state of $K$ and a SSB state of $A$, as
shown in the phase diagrom in Fig. 2. To see this, we start from the clean
limit ferromagnetic state
$\left|\psi_{1}\right\rangle=\frac{1}{\sqrt{2}}\left|++\dots\right\rangle_{\sigma}\otimes\left(\left|\uparrow\uparrow\dots\right\rangle_{\tau}+\left|\downarrow\downarrow\dots\right\rangle_{\tau}\right)$
in the limit $J_{\sigma}/J_{\tau}\gg 1$ (we pick the superposition of $\tau$
spins for the same reason as in Eq. (25)). In the presence of disorder
$\\{h_{i},l_{i}\\}$, the ensemble of ground states is still an SSB state of
$A$:

$\displaystyle\rho_{\mathbb{Z}_{2}^{2}}^{\prime}=$
$\displaystyle\frac{1}{2^{2N}}\sum_{h_{i},l_{i}}W(\\{h_{i},l_{i}\\})\left|\psi_{1}\right\rangle\left\langle\psi_{1}\right|W^{-1}(\\{h_{i},l_{i}\\})$
(28) $\displaystyle=$
$\displaystyle\left|++\dots\right\rangle\left\langle++\dots\right|_{\sigma}$
$\displaystyle\otimes\frac{1}{2}\left(\left|\uparrow\uparrow\dots\right\rangle\left\langle\uparrow\uparrow\dots\right|_{\tau}+\left|\downarrow\downarrow\dots\right\rangle\left\langle\downarrow\downarrow\dots\right|_{\tau}\right).$

We leave the nature of the phase boundary between the $A$-SSB state and
$K$-SWSSB state to future studies.

Figure 2: (a) The 1$D$ lattice realization of the mixed anomaly between
$A=Z=\mathbb{Z}_{2}$.(b) Schematic phase diagram of the random Hamiltonian in
Eq. (22).

##  V Disordered topological states compatible with the AEM anomaly

In this section, we consider the possible topologically ordered (TO) state and
average topologically ordered (ATO) state that are compatible with the AEM
anomaly of $G=A\times K$ in spatial dimension $d=2$.

###  V.1 Topological order and average topological order in disordered systems

Before discussing the compatibility with the AEM anomaly, it is necessary to
clarify the notion of TO (and ATO) in disordered systems. Similar to the clean
limit, we define TO in a disordered 2+1$D$ system as an SSB state of a set of
(emergent) exact discrete 1-form symmetries, where each 1-form symmetry is
associated with an Abelian anyon in the TO. The TO can be diagnosed by the
expectation value of the 1$D$ loop operator $O_{\gamma}$ that is charged under
the 1-form symmetry, which should decay no faster than perimeter law when
averaged over all disorder realizations: $\langle
O_{\gamma}\rangle\equiv\text{Tr}(\rho O_{\gamma})\sim
e^{-\alpha|\gamma|}$Hastings and Wen (2005), where $\gamma$ represents the
loop and $|\gamma|$ is its length. Such a disordered state can be smoothly
connected to the clean-limit topological order. On the contrary, in the
unbroken phase of the 1-form symmetry, such an expectation value decays with
the area $S_{\gamma}$ that $\gamma$ encircles: $\langle O_{\gamma}\rangle\sim
e^{-\beta S_{\gamma}}$ with some constant $\beta$. These two types of
behaviors of $\langle O_{\gamma}\rangle$ are commonly known as “perimeter law”
and “area law”, respectively.

Generalizing the interpretation of TO as an SSB state of 1-form symmetry, we
can define the ATO state in disordered systems as an SWSSB state of 1-form
symmetrySohal and Prem (2024); Ellison and Cheng (2024); Zhang et al. (2024).
Similar to the diagnosis of SWSSB states of a 0-form symmetry in Sec. IV, the
ATO exhibits an area law in the ensemble average of the loop operator.
However, the exact 1-form symmetry is not completely broken down to a trivial
group, which is signaled by the perimeter law of the Rényi-2 loop expectation
value999A fidelity-based loop expectation value can be similarly defined as
$\langle O_{\gamma}\rangle_{2}^{F}\equiv F(\rho,O_{\gamma}\rho
O_{\gamma}^{\dagger})$:

$\langle O_{\gamma}\rangle_{2}\equiv\frac{\text{Tr}[\rho O_{\gamma}\rho
O_{\gamma}^{\dagger}]}{\text{Tr}[\rho^{2}]}\sim e^{-\alpha|\gamma|}.$ (29)

In contrast, a topologically trivial state should exhibit an area law for both
$\langle O_{\gamma}\rangle$ and $\langle O_{\gamma}\rangle_{2}$. We caution
that our diagnosis of ATO via $\langle O_{\gamma}\rangle$ and $\langle
O_{\gamma}\rangle_{2}$ should be treated as a working assumption for this
paper. The optimal diagnosis of ATOs in general disordered and open systems
will be left for future studies.101010There is an important caveat in
generalizing such a definition to topologically ordered mixed state in open
quantum systemsFan et al. (2023); Bao et al. (2023); Su et al. (2023, 2024);
Li and Mong (2024); Sohal and Prem (2024); Ellison and Cheng (2024). In these
cases the loop operator expectation value $\langle O_{\gamma}\rangle$ will
always decay in perimeter law across the error threshold of the TO, beyond
which the error-corrupted TO cannot be connected to the clean TO via local
quantum channels, and should hence be regarded as an ATOSang et al. (2023).
SSB of 1-form symmetry is tricky to define in these cases using loop operator
as an order parameter, see Zhang et al. (2024).

A concrete example of ATO is given by the disordered toric-code state (or
$\mathbb{Z}_{2}$ gauge theory) on a 2$D$ square lattice Tsomokos et al.
(2011), whose Hamiltonian reads

$H_{\text{ATO}}(\\{h_{l}\\})=-g_{\mathcal{A}}\sum_{i}\mathcal{A}_{i}-g_{\mathcal{B}}\sum_{p}\mathcal{B}_{p}-\sum_{i}h_{l}a^{z}_{l}.$
(30)

Here $\mathcal{A}_{i}=\prod_{i\in\partial l}a^{x}_{l}$ and
$\mathcal{B}_{p}=\prod_{l\in p}a^{z}_{l}$ stabilize the clean toric-code
ground state $\left|\text{TC}\right\rangle$, which satisfies
$\mathcal{A}_{i}\left|\text{TC}\right\rangle=\mathcal{B}_{p}\left|\text{TC}\right\rangle=\left|\text{TC}\right\rangle$.
$i,l,p$ label the sites, links, and plaquettes of the square lattice.
$\\{h_{l}\\}$ are independent random variables drawn from a bimodal
distribution $h_{l}=\pm h$ with equal probabilities $p=1/2$.

Without disorders, the Hamiltonian possesses a magnetic 1-form symmetry
generated by $U_{m}^{(1)}(\gamma)=\prod_{l\in\gamma}a^{z}_{l}$ and an electric
1-form symmetry
$U_{e}^{(1)}(\tilde{\gamma})=\prod_{l\in\tilde{\gamma}}a^{x}_{l}$, where
$\tilde{\gamma}$ denotes a loop on the dual square lattice. The subscripts of
the 1-form symmetry actions reflect the fact that the line operator of the $m$
anyon, the excitation associated with the plaquette violation
$\mathcal{B}_{p}=-1$, is charged under $U_{m}^{(1)}(\gamma)$ if $\gamma$
encircles $p$. Similar, the line operator of the $e$ anyon, the excitation
associated with a vertex violation $\mathcal{A}_{i}=-1$, is charged under
$U_{e}^{(1)}(\gamma)$.

In the presence of weak disorders in the limit $h\ll
g_{\mathcal{A},\mathcal{B}}$, the disordered Hamiltonian in Eq. (30) is in the
TO phase, since the disorder strength never exceed the anyonic gap of $e$ and
$m$. Therefore, we expect the disordered state can be smoothly connected to
the clean limit toric code ground state $\left|\text{TC}\right\rangle$.

We now show that the ATO phase corresponds to the case where
$g_{\mathcal{A}}\ll h\ll g_{\mathcal{B}}$Tsomokos et al. (2011); Zhang et al.
(2024). Since the disorder strength is within the gap of the $m$ anyon,
$U_{m}^{(1)}$ is still an exact symmetry for $H_{\text{ATO}}$. Therefore, for
each disorder realization $\\{h_{l}\\}$, the ground state
$\left|\\{h_{l}\\}\right\rangle$ is still stabilized by $\mathcal{B}_{p}$
since disorder strength is within the gap of the $m$ anyons. However, the
stabilizer $\mathcal{A}_{i}$ will be randomly violated by the disorder. In
other words, we have an ensemble of incoherently exited $e$ anyons due to
disorder. We can write down a schematically “fixed point” density matrix for
the ATO (up to normalization):

$\displaystyle\rho_{\text{ATO}}\propto$ $\displaystyle\sum_{s_{i}=\pm
1}\left|\mathcal{A}_{i}=s_{i},\mathcal{B}_{p}=1\right\rangle\left\langle\mathcal{A}_{i}=s_{i},\mathcal{B}_{p}=1\right|$
(31) $\displaystyle=$
$\displaystyle\prod_{p}\frac{\mathbbm{1}+\mathcal{B}_{p}}{2}.$

One can confirm that such a disordered state is an ATO with SWSSB of
$U^{(1)}_{m}$ using the loop order parameter of $U^{(1)}_{m}$,
$O_{\tilde{\gamma}}=\prod_{l\in\tilde{\gamma}}a^{x}_{l}$:

$\displaystyle
U^{(1)}_{m}(\gamma)\rho_{\text{ATO}}=\rho_{\text{ATO}}U^{(1)}_{m}(\gamma)=\rho_{\text{ATO}}$
(32) $\displaystyle\text{Tr}[\rho_{\text{ATO}}O_{\tilde{\gamma}}]=0,\
\frac{\text{Tr}[\rho_{\text{ATO}}O_{\tilde{\gamma}}\rho_{\text{ATO}}O_{\tilde{\gamma}}]}{\text{Tr}[\rho_{\text{ATO}}^{2}]}=1$

for any loop $\gamma$ and $\tilde{\gamma}$ on the original and dual lattice,
respectively. The first line confirms that the $U^{(1)}_{m}$ is an exact
symmetry of $\rho_{\text{ATO}}$ and the second line shows the expected
behaviors of the correlations function for an ATO.111111The fidelity loop
expectation value $\langle O_{\tilde{\gamma}}\rangle_{2}^{F}$ is also 1.

Another ATO with randomly excited $m$ anyons can be similarly constructed by
disorders of $a^{x}$ instead of $a^{z}$ in $H_{\text{ATO}}$.

In the rest of the section, we will mainly focus on a special type of 2+1$D$
TO: the deconfined phase in the gauge theory with a finite Abelian gauge
group, denoted by $N$, where the physical picture of the disordered toric code
generalizes. The anyons in such a TO are point excitations that correspond to
the gauge charge and gauge flux121212For a discrete gauge theory in $(d+1)D$,
the gauge flux excitation has spacial dimension $d-2$.. The gauge flux anyons
form a fusion group $N$, and the gauge charges form a fusion group
$\hat{N}=\text{Hom}[N,U(1)]$ that is dual to $N$. The discrete gauge theory
possesses two sets of 1-form symmetries in the deconfined phase: the
“magnetic” 1-form symmetry generated by the Wilson loops, and the “electric”
1-form symmetry generated by the ’t Hooft loops, which are both spontaneously
broken due to deconfinement. An ATO can be obtained from the clean deconfined
phase by randomly exciting either gauge charges or gauge fluxes through
disorder, while the other sector remains coherent. For example, if the
disorder randomly excites the gauge charges, the electric 1-form symmetry is
explicitly broken down to an average symmetry. Meanwhile, the magnetic 1-form
symmetry is in a SWSSB state. That is because the symmetry generator of the
electric 1-form is the order parameter of the magnetic 1-form symmetry. Since
the density matrix still possesses an average electric 1-form symmetry, the
Rényi-2 expectation value of order parameter for the magnetic 1-form symmetry,
$\langle O_{\gamma}\rangle_{2}$, is hence non-vanishing. The disordered toric
code state is one example of such a scenario. Similarly, an ATO with randomly
excited gauge fluxes can also be obtained in this way, which is an SWSSB state
of the electric 1-form symmetry.

###  V.2 Review of anomaly-compatible TO in the clean limit

We now briefly review the construction of the topologically ordered (TO) state
that are compatible with the exact anomaly of $G$ in $(d+1)D$ where $d\geq 2$.
To construct such a gauge theory, we first need to extend the symmetry group
$G$ to a larger group $G^{\prime}$ via the short exact sequence

$1\to N\to G^{\prime}\to G\to 1,$ (33)

where $N$ is a discrete Abelian group and a normal subgroup of $G^{\prime}$:
$G=G^{\prime}/N$. Such a group extension is characterized by a map
$\mathfrak{e}:G\times G\to N$ in the second cohomology group,
$\mathfrak{e}\in{\cal H}^{2}(G,N)$. Refs. Wang et al., 2018; Tachikawa, 2020;
Wen and Potter, 2023 show that if one is able to find a group extension to
$G^{\prime}$ through a finite group $N$ that trivializes the anomaly, the
gauge theory with gauge group $N$ is then compatible with the $G$-anomaly.
Furthermore, when $G$ is finite, such a group $N$ always exists. 131313We note
that for some exact anomalies, there is an obstruction of having an anomaly-
compatible topologically ordered state, see Refs Gaiotto et al., 2017; García-
Etxebarria et al., 2017; Cordova and Ohmori, 2019; Cheng et al., 2023; Yang
and Cheng, 2023; Kapustin and Sopenko, 2024.

After the anomaly is trivialized via the group extension to $G^{\prime}$, the
TO can be obtained by gauging the normal subgroup $N$, which brings the global
symmetry back to $G$. The TO is the deconfined phase of the $N$ gauge theory,
both the gauge charge and gauge flux anyons transform non-trivially under
symmetry. The gauge charge of $N$ carries a projective representation of $G$,
which is inherited from the structure of group extension
$n=\mathfrak{e}(g_{1},g_{2})$. In addition, the anomaly also assigns a
projective representation of $G$ to the gauge flux, which can be represented
by an element in the second group cohomology ${\cal H}^{2}_{\rho}(G,\hat{N})$,
where $\rho$ is the symmetry action of $G$ on the anyons in the $N$-gauge
theoryBarkeshli et al. (2019); Cheng et al. (2017). A TO where its anyons
transform non-trivially under global symmetry is called a symmetry-enriched
TO. In general, if a TO carries the anomaly of $G$, the way the anyons
transform under the global symmetry cannot be realized in an anomaly-free
2+1$D$ systemBarkeshli et al. (2019); Chen et al. (2015).

For example, we consider symmetry-enriched TO states that are compatible with
the 2+1$D$ exact anomalies $\Omega_{1,2}$ of $G=A\times K$ with
$A=K=\mathbb{Z}_{2}$ discussed in Sec. II.1. To trivialize the anomaly
$\Omega_{1}$, we pick $N=\mathbb{Z}_{2}$ and extend the global symmetry group
$G$ by

$\displaystyle\mathfrak{e}(ak^{i},a^{j}k)=n,$ (34)

where $a,k,n$ are the non-trivial group elements of $A,K,N$ and $i,j=0,1$ (all
the other situations are mapped to the trivial group element in $N$). The
extended global symmetry group becomes $G^{\prime}=D_{4}$.141414To be clear,
$D_{4}$ stands for the dihedral group with 8 elements. Such a group is
sometimes denoted as $D_{8}$. We then gauge $N$ in $G^{\prime}=D_{4}$, which
yields a $\mathbb{Z}_{2}$ gauge theory enriched by $G=A\times
K=\mathbb{Z}_{2}^{2}$. From $\mathfrak{e}$, the $e$ anyon carries a projective
representation between $A$ and $K$, and $m$ anyon transforms as a half charge
of $K$. We denote such a symmetry-enriched TO state as $e_{AK}m_{K}$, similar
to the notation used in Ref. Wang and Senthil, 2013 where the subscripts label
how the anyon transforms under $A\times K$. Alternatively, the anomaly can be
trivialized via a group extension to $G^{\prime}=A\times
K^{\prime}=\mathbb{Z}_{2}\times\mathbb{Z}_{4}$, in which case
$\mathfrak{e}^{\prime}(a^{i}k,a^{j}k)=n$. After gauging $N$, the global
symmetry $G$ fractionalizes as $e_{K}m_{AK}$. These two symmetry-enriched TOs
are essentially the same after relabelling the $e$ and $m$ anyons.

For the other anomaly $\Omega_{2}$, which is equivalent to $\Omega_{1}$ but
with the roles of $A$ and $K$ interchanged, one can similarly extend the
global symmetry to $G^{\prime}=D_{4}$ or $G^{\prime}=A^{\prime}\times
K=\mathbb{Z}_{4}\times\mathbb{Z}_{2}$ via $N=\mathbb{Z}_{2}$. Gauging $N$, we
arrive at two equivalent symmetry enriched TOs $e_{AK}m_{A}$ and
$e_{A}m_{AK}$.

###  V.3 TO and ATO compatible with the AEM anomaly

We are now ready to discuss disordered TO and ATO enriched by $G=A\times K$
that are compatible with the AEM anomaly of $G$ in 2+1$D$. It is easy to see
that if a clean TO enriched by the exact symmetry $G$ is compatible with the
$G$-anomaly in the clean limit, weakly disordering it within the TO phase will
certainly give a disordered state that is compatible with the AEM anomaly.
Therefore, the most important question here is whether a strong enough
disorder is capable of driving the TO into an ATO by randomly exciting some
anyons in the TO without causing spontaneous symmetry breaking of either the
exact symmetry $K$ or the average symmetry $A$.

If an anyon $e$ in the TO transforms projectively under some subgroup $H$ of
$G$, creating $e$ anyon excitations will break $H$. Therefore, in the clean
limit, coherently proliferating such an anyon leads to either SSB of $H$ or
some gapless degrees of freedom. In a disordered system where $e$ anyons are
incoherently excited, $H$ can still be an average symmetry of the disordered
states, since different symmetry-breaking states caused by $e$ anyon
excitations can transform into each other under the action of $H$.
Furthermore, if $H=A$, incoherently exciting $e$ will no longer cause any
symmetry breaking now that $A$ is already an average global symmetryMa et al.
(2023). Therefore, in the clean TO that carries the exact anomaly of
$G=A\times K$, if there exists an anyon $e$ that transforms non-trivially only
under the subgroup $A$, the AEM anomaly of $G=A\times K$ can instead be
carried by an ATO where $e$ can be incoherently excited. In contrast, if all
anyons in the TO transform non-trivially under $K$, we cannot have an anomaly-
compatible ATO without SSB or SWSSB of the (0-form) symmetry $K$.

Now consider the 2+1$D$ discrete gauge theory constructed in Sec. V.2 that
carries the exact anomaly. Since the charge and flux sectors are dual to each
other, we mainly focus on the possible ATO with incoherently excited gauge
charges. Recall that the projective representation carried by the gauge charge
is inherited from the group extension $\mathfrak{e}$. Therefore, if the
$G$-anomaly can be trivialized by a group extension that only involves $A$,
the gauge charges will transform projectively only under $A$. Consequently,
they can be randomly excited without breaking the average symmetry $A$ or the
exact symmetry $K$ in a disordered system. Our argument can be summarized as
the following:

An ATO compatible with an AEM anomaly of $G=A\times K$ exists if there exists
a group extension via a finite Abelian group $N$ of the form $\displaystyle
1\to N\to G^{\prime}=A^{\prime}\times K\to G=A\times K\to 1$ (35) that
trivializes the exact anomaly of $G$.

We expect that a similar result holds in higher dimensions, where a deconfined
phase of the discrete gauge theory of $N$ with randomly excited gauge charges
is compatible with the AEM anomaly if the group extension in Eq. (35) exists.
We also note that such a condition is sufficient but not necessary for having
an anomaly-compatible ATO.

As an example, we discuss possible TO and ATO that are compatible with the two
AEM anomalies of $G=A\times K=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ in 2+1$D$,
which are obtaining from “disordering” the exact anomalies $\Omega_{1,2}$
(which are introduced in II.1). These two AEM anomalies will still be referred
to as $\Omega_{1,2}$. Recall from Sec. V.2 that the exact anomaly $\Omega_{1}$
can be trivialized by extending to $G^{\prime}=A\times
K^{\prime}=\mathbb{Z}_{2}\times\mathbb{Z}_{4}$, but not to a group
$G^{\prime}$ of the structure $A^{\prime}\times K$. Therefore, it is
impossible to construct an ATO via the discrete gauge theory construction
mentioned above. In Sec. V.5, we will use a field theory perspective to
further argue that there is, in fact, no ATO compatible with the AEM anomaly
of $\Omega_{1}$. Indeed, in the symmetry-enriched TO that carries the exact
anomaly of $\Omega_{1}$, both the $e$ and $m$ anyons transform non-trivially
under $K$, so none of them are allowed to be randomly excited without symmetry
breaking. In contrast, the exact anomaly of $\Omega_{2}$ can be extended to
$G^{\prime}=A^{\prime}\times K=\mathbb{Z}_{4}\times\mathbb{Z}_{2}$. Hence,
there exists an ATO that is compatible with the AEM anomaly of $\Omega_{2}$.
Indeed, in the symmetry-enriched TO $e_{A}m_{AK}$ that carries the exact
anomaly of $\Omega_{2}$, the $e$ anyon now only transforms under the average
symmetry $A$. Therefore, an ATO can be obtained by randomly exciting the $e$
anyons. We will confirm these results using a solvable lattice model in Sec.
V.4.

Figure 3: In the $e_{AK}m_{K}$ state, a pair of $e$ anyons created by an open
Wilson line is charged under $K$ when the line intersects with an $A$ domain
wall.

A useful way to understand the obstruction of having an ATO that is compatible
with the AEM anomaly of $\Omega_{1}$ is to consider an open Wilson line in the
$N=\mathbb{Z}_{2}$-gauge theory that excites a pair of $e$ anyons in the
$e_{AK}m_{K}$ state in a typical disordered sample where $A$-spins are pinned
by the random-field disorder (see Fig. 3). Since $e$ carries a projective
representation between $A$ and $K$, when the open Wilson line crosses a domain
wall of $A$, it creates a local $K$-charge (which will be clarified further in
the subsequent subsections). In the entire disorder ensemble of $A$, domain
walls appear randomly in each disorder realization. Thus, anyonic excitations
will introduce random $K$ charges to each sample depending on the number of
$A$ domain walls that cross the $e$ modulo anyon lines $2$. As such, the exact
symmetry $K$ is broken down to an average symmetry if one allows random $e$
anyon excitations. A similar argument in the clean limit implies that
proliferating the $e$ anyon will induce SSB of $K$ (to a trivial group).

We note that for an ATO, one cannot solely use the symmetry fractionalization
pattern of the anyons that have not been randomly disordered in determining
whether the ATO, by itself, is anomalous or not. The distinction needs to be
exposed by creating a coherent anyonic excitation on top of the disordered
state. Similarly, one cannot simply tell if a fully ordered SSB state is
anomalous without considering the effect of domain-wall excitations. As an
example, the AEM anomaly of $\Omega_{2}$ can be carried by the ATO obtained
from incoherently exciting the $e$ anyons in the $e_{A}m_{AK}$ state. The only
symmetry fractionalization in the ATO that remains meaningful is $m_{AK}$.
Such a fractionalization pattern is also realizable by an ATO derived from a
TO enriched by anomaly-free global symmetries, in which the $e$ anyon
transforms as the linear representation of both $A$ and $K$ and the $m$ anyon
carries the projective representation between $A$ and $K$ (see Ref. Lu and
Vishwanath, 2016 and also Appendix C).

Figure 4: A lattice realization of the 2+1$D$ anomaly of
$G=\mathbb{Z}_{2}^{2}$. Here, each lattice site (black dot) hosts two flavors
of Pauli spins denoted by $\sigma$ and $\tau$. Each plaquette is labeled by
$p$, whose four corners are denoted by $p_{1,2,3,4}$ in counterclockwise
order. To extend symmetry to $D_{4}$ by $N$, we further introduce $\mu$ spins
on each lattice site. The $\mathbb{Z}_{2}$ gauge field of $N$ lives on bonds
between lattice sites marked by red squares.

###  V.4 Solvable model for symmetry-enriched TO and ATO

To bolster our arguments above, we explicitly construct an exactly solvable
lattice model of the clean limit symmetry-enriched TO $e_{AK}m_{K}$ that
carries the exact anomaly of $\Omega_{1}$ of $G=A\times K=\mathbb{Z}_{2}^{2}$
in 2+1$D$, and discuss its fate against various forms of disorders that can
potentially drive it towards the ATO state. To discuss the AEM anomaly
$\Omega_{2}$ in the same lattice representation, in this subsection, we
redefine the AEM anomaly of $\Omega_{2}$ as the disordered version of the
exact anomaly of $\Omega_{1}$ but with the subgroups $A$ being exact and $K$
being average. This is equivalent to the original definition of the AEM
anomaly of $\Omega_{2}$.

The anomalous symmetry actions of $A$ and $K$ for the exact anomaly
$\Omega_{1}$ can be represented on a square lattice with two flavors of Pauli
spins $\sigma_{i}$ and $\tau_{i}$ on the lattice sites:

$U_{A}=\prod_{i}\sigma^{x}_{i},\
U_{K}=\prod_{p}C(-\tau^{z}_{p_{1}}\tau^{z}_{p_{3}},\tau^{z}_{p_{2}}\tau^{z}_{p_{4}},\sigma^{z}_{p_{3}})\prod_{i}\tau^{x}_{i},$
(36)

where $C(A,B,C)=e^{\frac{\mathrm{i}\pi}{8}(1-A)(1-B)(1-C)}$ is the controlled-
control operator between three Pauli operators $\rho,\sigma$ and $\tau$.
$p_{1,2,3,4}$ denote the four lattice sites that forms the plaquette $p$ in
the counterclockwise order, see Fig. 4. These symmetry representations can be
directly obtained from group cohomology, see Appendix A.

Now, we begin the construction of the symmetry-enriched TO in the clean
limit.Recall from Sec. V.2 that the exact anomaly $\Omega_{1}$ can be
trivialized by extending the global symmetry to $G^{\prime}=D_{4}$ or
$\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ via group extension by
$N=\mathbb{Z}_{2}$. To gauge $N$ and obtain the anomaly-compatible TO, we
introduce the gauge field of $N$ on the links $l$ of the square lattice (see
Fig. 4), whose Pauli operators are denoted by $a^{x,z}_{l}$. As shown in
Appendix C, after symmetry extension and gauging, the wave function of the
symmetry-enriched TO can take the following form: the $N$-gauge theory, whose
wave function is denoted by $\left|a\right\rangle$, is coupled to the “matter
fields” $\sigma$ and $\tau$ via a set of unitary operators

$\displaystyle\left|\psi\right\rangle=\mathcal{V}\mathcal{W}\left(\left|\sigma,\tau\right\rangle\otimes\left|a\right\rangle\right),$
(37)

where $\left|\sigma,\tau\right\rangle$ represents a possible state for the
$\sigma$ and $\tau$ spins. As far as the symmetry action is concerned, the
unitaries $\mathcal{V}$ and $\mathcal{W}$ describe the coupling between the
$\mathbb{Z}_{2}$ gauge field and the “matter fields” $\sigma$ and $\tau$,
which gives rise to the projective representation carried by the gauge charge
and flux. Their explicit forms are

$\displaystyle\mathcal{V}=\prod_{p}C(\tau^{z}_{p_{1}},\tau^{z}_{p_{1}}\tau^{z}_{p_{2}},a^{z}_{p_{2}p_{3}})C(\tau^{z}_{p_{1}},\tau^{z}_{p_{1}}\tau^{z}_{p_{4}},a^{z}_{p_{4}p_{3}}).$
(38)

and

$\displaystyle\mathcal{W}=\prod_{l}C\left(a^{x}_{l},\sigma^{z}_{l_{1}}\sigma^{z}_{l_{2}},\tau^{z}_{l_{1}}\right).$
(39)

Here, $l_{1,2}$ are two sites of the bond $l$, and $l_{1}$ is always below or
to the left of $l_{2}$. $p_{i}p_{j}$ means the edge connecting the two sites
$p_{i}$ and $p_{j}$. The derivations of $\mathcal{V}$ and $\mathcal{W}$ can be
found it App. C.

In the $\mathcal{V}\mathcal{W}$-transformed frame, the global symmetries are

$\displaystyle U_{A}^{\prime}$
$\displaystyle\equiv\mathcal{VW}U_{A}\mathcal{W}^{-1}\mathcal{V}^{-1}=\prod_{i}\sigma^{x}_{i},$
(40) $\displaystyle U_{K}^{\prime}$
$\displaystyle\equiv\mathcal{VW}U_{K}\mathcal{W}^{-1}\mathcal{V}^{-1}$
$\displaystyle=\prod_{p}C\left(\prod_{l\in\partial
p}a^{z}_{l},\tau^{z}_{p_{1}}\right)\prod_{i}C\left(\prod_{i\in\partial
l}a^{x}_{l},\sigma^{z}_{i}\right)\tau^{x}_{i}.$

These symmetry representations capture the fractionalization pattern
$e_{AK}m_{K}$ of the global symmetries. The $N$ charge (the $e$ anyon), which
corresponds to a local violation of the Gauss law at the site $i$ by
$\mathcal{A}_{i}=\prod_{i\in\partial l}a^{x}_{l}=-1$, leads to a local
projective representation between $A$ and $K$ represented by
$U^{\prime}_{A}|_{i}=\sigma^{x}_{i}$ and
$U^{\prime}_{K}|_{i}=\sigma^{z}_{i}\tau^{x}_{i}$. A $\pi$-flux of $N$ (the $m$
anyon) at the plaquette $p$ with $\mathcal{B}_{p}=\prod_{l\in\partial
p}a^{z}_{l}=-1$ carries the half charge of $K$ as indicated by
$U^{\prime}_{k}|_{p_{1}}=\tau^{z}_{p_{1}}\tau^{x}_{p_{1}}=\mathrm{i}\tau^{y}_{p_{1}}$.
Furthermore, the fact that the $e$ anyon line is charged under $K$ when
crossing a domain wall of $A$ can now be explicitly shown using the symmetry
action. Assigning $\sigma^{z}=1$ on one side of the domain wall and
$\sigma^{z}=-1$ on the other side, the non-on-site part of the symmetry action
of $K$ in Eq. (40) now simply acts on the domain wall:

$\prod_{i}C\left(\prod_{i\in\partial
l}a^{x}_{l},\sigma^{z}_{i}\right)\overset{DW_{\sigma}}{\longrightarrow}\prod_{l\in
DW_{\sigma}}a^{x}_{l}.$ (41)

Meanwhile, the open Wilson line that creates a pair of $e$ anyons can be
represented by $W(\gamma)=\prod_{l\in\gamma}a^{z}_{l}$. Therefore, we have

$U_{K}^{\prime}W(\gamma)\left(U_{K}^{\prime}\right)^{-1}=(-1)^{|\gamma\cap
DW_{\sigma}|}W(\gamma),$ (42)

where $|\gamma\cap DW_{\sigma}|$ represents the number of times the Wilson
line $\gamma$ intersects with $DW_{\sigma}$.

Since the anomaly-compatible TO under discussion is the deconfined phase of
the $N$ gauge theory, the gauge sector of the state is in the state
$\left|a\right\rangle=\left|\text{TC}\right\rangle$, so that
$\mathcal{A}_{i}\left|a\right\rangle=\mathcal{B}_{p}\left|a\right\rangle=\left|a\right\rangle$.
The wave function of the entire system now reads

$\left|\psi_{e_{AK}m_{K}}\right\rangle=\mathcal{VW}\left(\left|+\right\rangle_{\sigma}\otimes\left|+\right\rangle_{\tau}\otimes\left|\text{TC}\right\rangle\right),$
(43)

where we have taken
$\left|+\right\rangle_{\sigma}=\bigoplus_{i}\left|\sigma^{x}_{i}=1\right\rangle$
and similarly for $\left|+\right\rangle_{\tau}$ to ensure the $A\times K$
symmetry of the system. To set the stage for the discussion of the effect of
disorders, we write down an exactly solvable parent Hamiltonian for
$\left|\psi_{e_{AK}m_{K}}\right\rangle$:

$\displaystyle H_{e_{AK}m_{K}}=$
$\displaystyle-\mathcal{VW}\left[\sum_{i}\sigma^{x}_{i}\left(1+\mathcal{A}_{i}\right)+\sum_{p}\tau^{x}_{p_{1}}\left(1+\mathcal{B}_{p}\right)\right.$
(44)
$\displaystyle\left.+g_{\mathcal{A}}\sum_{i}\mathcal{A}_{i}+g_{\mathcal{B}}\sum_{p}\mathcal{B}_{p}\right]\mathcal{W}^{-1}\mathcal{V}^{-1}.$

We now discuss possible disorders that can drive the system towards the ATO.
For convenience, we will work in the $\mathcal{VW}$-transformed frame in which
symmetries are represented by $U_{A,K}’$. We first consider $\Omega_{2}$,
since we have argued for the possibility of an ATO by randomly exciting the
$m$ anyon in the $e_{AK}m_{K}$ state (as a reminder, $K$ is the average
symmetry for $\Omega_{2}$). From the clean limit Hamiltonian in Eq. (44), we
introduce the following disorders that randomly excite the $m$ anyon and
explicitly break $K$ down to an average symmetry:

$\displaystyle H_{\Omega_{2}}=$ $\displaystyle
H_{e_{AK}m_{K}}-\sum_{l}h^{(a)}_{l}a^{x}_{l}+\sum_{i}h_{i}\tau^{z}_{i}$ (45)
$\displaystyle\approx$
$\displaystyle-\mathcal{VW}\left\\{\sum_{l}h^{a}_{l}a^{x}_{l}C(\tau^{z}_{i_{1}(l)},\tau^{z}_{i_{2}(l)})+\sum_{i}\left[h_{i}\tau^{z}_{i}\right.\right.$
$\displaystyle\left.\left.+\sigma^{x}_{i}\left(1+\mathcal{A}_{i}\right)+g_{\mathcal{A}}\mathcal{A}_{i}\right]+g_{\mathcal{B}}\sum_{p}\mathcal{B}_{p}\right\\}\mathcal{W}^{-1}\mathcal{V}^{-1},$

where $h^{(a)}_{l}=\pm h^{(a)}$ and $h_{i}=\pm h$ both with probabilities
$\frac{1}{2}$ and in the second line we have taken the strong disorder limit
$h^{(a)},h\gg 1$ and $g_{\mathcal{A}}>h^{(a)}>g_{\mathcal{B}}$.
$\tau^{z}_{i_{1,2}(l)}$ are the two spins entangled with $a^{z}$ through
$\mathcal{V}$. In the second line, we have omitted the
$\tau^{x}_{p_{1}}\left(1+\mathcal{B}_{p}\right)$ term in $H_{\Omega_{2}}$
since it does not commute with the strong $\tau^{z}$ disorder terms. Because
the original symmetries $U_{A,K}$ do not involve the gauge field, the random
$a^{x}$ disorder is always symmetric. In the $\mathcal{VW}$-transformed frame,
the coefficient of the random $a^{x}$ disorder,
$h^{a}_{l}C(\tau^{z}_{i_{1}(l)},\tau^{z}_{i_{2}(l)})$, is still a random
number $\pm h^{(a)}$ with probabilities $\frac{1}{2}$ for any quenched values
of $\tau^{z}$. As such, the gauge sector will be been driven to an ATO whose
fixed-point density matrix is

$\displaystyle\rho_{\text{ATO}}^{\prime}=\prod_{i}\frac{\mathbbm{1}_{a}+\mathcal{A}_{i}}{2}.$
(46)

Collecting the ground states for every disorder realization, the density
matrix of the ensemble of ground states can be written as

$\rho_{\Omega_{2}}=\mathcal{VW}\left(\left|+\right\rangle\left\langle+\right|_{\sigma}\otimes\frac{\mathbbm{1}_{\tau}}{2^{N_{\tau}}}\otimes\rho_{\text{ATO}}^{\prime}\right)\mathcal{W}^{-1}\mathcal{V}^{-1},$
(47)

which has an exact $A$ symmetry and average $K$ symmetry (without SSB or
SWSSB).

Next, we move on to the AEM anomaly of $\Omega_{1}$. As argued in Sec. V.3, it
is impossible to obtain an anomaly-compatible ATO for the AEM anomaly of
$\Omega_{1}$ by randomly exciting either the $e$ or $m$ anyon, since both of
them transforms non-trivially under $K$. Randomly exciting any of them induces
SWSSB of the exact symmetry $K$. To see this in the lattice model, we
construct the following Hamiltonian where $e$ is randomly excited by a random
field disorder of $a^{z}$ and $A$ is broken down to an average symmetry
($h_{l}^{(a)}$ and $h_{i}$ follows the same distribution as in Eq. (45)):

$\displaystyle H_{\Omega_{1}}=$ $\displaystyle
H_{e_{AK}m_{K}}-\sum_{l}h^{(a)}_{l}a^{z}_{l}+\sum_{i}h_{i}\sigma^{z}_{i}$ (48)
$\displaystyle\approx$
$\displaystyle-\mathcal{VW}\left\\{\sum_{l}h^{(a)}_{l}a^{z}_{l}C(\sigma^{z}_{l_{1}}\sigma^{z}_{l_{2}},\tau^{z}_{l_{1}})\right.$
$\displaystyle\left.+\sum_{i}\left(h_{i}\sigma^{z}_{i}+g_{\mathcal{A}}\mathcal{A}_{i}\right)\right.$
$\displaystyle\left.+\sum_{p}\left[g_{\mathcal{B}}\mathcal{B}_{p}+\tau^{x}_{p_{1}}\left(1+\mathcal{B}_{p}\right)\right]\right\\}\mathcal{W}^{-1}\mathcal{V}^{-1}.$

We again consider the strong disorder limit
$g_{\mathcal{B}}>h^{(a)}>g_{\mathcal{A}}$ and $h,h^{(a)}\gg 1$, so that the
term $\sigma^{x}_{i}\left(1+\mathcal{A}_{i}\right)$ in $H_{e_{AK}m_{K}}$ can
be omitted. We notice that the random $a^{z}$ disorder now depends on whether
these is a $\sigma^{z}$ domain wall located between $l_{1}$ and $l_{2}$. Away
from the domain wall, the $a^{z}$ disorder drives the gauge sector to an ATO
where the density matrix is schematically
$\prod_{p}\frac{\mathbbm{1}+\mathcal{B}_{p}}{2}$. However, on the $\sigma^{z}$
domain wall, the term becomes $h_{l}^{(a)}a^{z}_{l}\tau^{z}_{l_{1}}$. Since
$h^{(a)}\gg 1$, such an interaction on the $A$ domain wall aligns (or anti-
aligns) $\tau^{z}_{l_{1}}$ with $a^{z}_{l}$ on the $\sigma^{z}$ domain wall in
the $\mathcal{VW}$-transformed frame whenever $l\in DW_{\sigma}$. In fact,
this is a direct consequence of the fact that the $e$ anyon line is charged
under $K$ when it intersects an $A$ domain wall (see Fig. 3 and also Eq.
(42)), which in this case is created by the $a^{z}$ disorders.

To confirm the SWSSB of $K$ after the $e$ anyons in such a disordered state,
we calculate the Rényi-2 $K$ charge correlator using the density matrix
$\rho_{\Omega_{1}}$ of the disordered ground states of $H_{\Omega_{1}}$:

$\displaystyle\frac{\text{Tr}[\rho_{\Omega_{1}}\tau^{z}_{i}\tau^{z}_{j}\rho_{\Omega_{1}}\tau^{z}_{i}\tau^{z}_{j}]}{\text{Tr}[\rho_{\Omega_{1}}^{2}]}=\frac{\text{Tr}[\rho_{\Omega_{1}}^{\mathcal{VW}}\tau^{z}_{i}\tau^{z}_{j}\rho_{\Omega_{1}}^{\mathcal{VW}}\tau^{z}_{i}\tau^{z}_{j}]}{\text{Tr}[\rho_{\Omega_{1}}^{2}]}$
(49)
$\displaystyle\approx\left(\frac{1}{2}\right)^{2}\frac{\text{Tr}[\rho_{\Omega_{1}}^{\mathcal{VW}}a^{z}_{l_{i}}a^{z}_{l_{j}}\rho_{\Omega_{1}}^{\mathcal{VW}}a^{z}_{l_{i}}a^{z}_{l_{j}}]}{\text{Tr}[\rho_{\Omega_{1}}^{2}]}\approx\frac{1}{4},$

where
$\rho_{\Omega_{1}}^{\mathcal{VW}}=\mathcal{WV}\rho_{\Omega_{1}}\mathcal{V}^{-1}\mathcal{W}^{-1}$
and $l_{i,j}$ are the bonds on the $\sigma^{z}$ domain wall near the sites $i$
and $j$. The first equality in the second line approximately holds since each
site $i$ has probability $\frac{1}{2}$ of being on the domain wall of
$\sigma^{z}$, in which case $\tau^{z}_{i}$ will be energetically locked to
either $a^{z}_{l_{i}}$ or $-a^{z}_{l_{i}}$ depending on the sign of
$h^{(a)}_{l}$. Away from the domain wall, the $\tau$ spins are polarized in
$\tau^{x}=1$ by the term $\tau^{x}_{p_{1}}\left(1+\mathcal{B}_{p}\right)$ in
the Hamiltonian $H_{\Omega_{1}}$. Therefore, they do not contribute to the
Rényi-2 correlator. In the second line, we use the fact that
$a^{z}_{l_{i}}a^{z}_{l_{j}}$ commutes the density matrix of the ATO.

The disordered lattice model with random $m$ anyon excitations can be
similarly constructed, and such a model in the strong disorder limit will also
exhibit SWSSB of $K$.

###  V.5 Field-theoretical arguments: anomaly inflow in doubled Hilbert space

The possibility of an anomaly-compatible ATO enriched by the global symmetries
can be discussed using an field-theoretical approach via coupling the
disordered system to background gauge fields of the global symmetries.

In the clean limit, the mixed anomaly $\Omega_{1}$ can be probed by background
$\mathbb{Z}_{2}$ gauge fields of $A$ and $K$. Since the global symmetry
$G=A\times K$ is anomalous, the partition function of the 2+1$D$ system is
only gauge-invariant when the system lives on the boundary of a 3+1$D$ bulk
$M_{4}$, whose response to the gauge fields is:

$Z_{\Omega_{1}}\sim\exp\left(\mathrm{i}\pi\int_{M_{4}}\mathfrak{a}\cup\mathfrak{k}\cup\mathfrak{k}\cup\mathfrak{k}\right),$
(50)

where $\mathfrak{a},\mathfrak{k}\in H^{1}(M_{4},\mathbb{Z}_{2})$ are
$\mathbb{Z}_{2}$ background gauge fields of $A$ and $K$, respectively. The
action on the exponent of Eq. (50), called the anomaly inflow, is the response
of the 3+1$D$ SPT state to external $A$ and $K$ gauge fields (after the gapped
matter field of the SPT state is integrated out). To see this, we show that
the action matches the decorated defect construction: suppose the action lives
in a cartesian product space $M_{4}=S_{1}\times M_{3}$. An domain wall of $A$
can be created by threading a “$\pi$-flux” of $A$ through $S_{1}$, which is
achieved by modifying the background gauge field
$\mathfrak{a}\to\mathfrak{a}+\Delta\mathfrak{a}$, such that
$\int_{S_{1}}\Delta\mathfrak{a}=1$. Such a modification generates a new factor
$\exp\left(\mathrm{i}\pi\int_{M_{3}}\mathfrak{k}\cup\mathfrak{k}\cup\mathfrak{k}\right)$,
which describes a 2+1$D$ Levin-Gu state probed by the $\mathbb{Z}_{2}$ gauge
field $\mathfrak{k}$Kapustin and Thorngren (2014); Wen (2015); Wang et al.
(2015). Similarly, the inflow of the exact anomaly $\Omega_{2}$ can be
described by the following partition function

$Z_{\Omega_{2}}\sim\exp\left(\mathrm{i}\pi\int_{M_{4}}\mathfrak{a}\cup\mathfrak{a}\cup\mathfrak{a}\cup\mathfrak{k}\right).$
(51)

Meanwhile, the TO can be viewed as an SSB state that is compatible with the
mixed anomaly in 2+1$D$ between a pair of 1-form $\mathbb{Z}_{2}$ symmetries
$\mathbb{Z}_{2e}^{(1)}\times\mathbb{Z}_{2m}^{(1)}$ represented by
$U^{(1)}_{e,m}$. This 1-form anomaly is captured by the action in the 3+1$D$
bulk $M_{4}$ as

$\exp\left(\mathrm{i}\pi\int_{M_{4}}\mathfrak{b}_{e}\cup\mathfrak{b}_{m}\right),$
(52)

where $\mathfrak{b}_{e,m}\in H^{2}(M_{4},\mathbb{Z}_{2})$ are 2-form
background gauge fields of $\mathbb{Z}_{2e,m}^{(1)}$, respectively. In the
$e_{AK}m_{K}$ state, the $e$ anyon carries a mutual projective representation
between $A$ and $K$, thus the background gauge field of
$\mathbb{Z}_{2e}^{(1)}$ can be identified with the background gauge fields of
$A$ and $K$ as $\mathfrak{b}_{e}=\mathfrak{a}\cup\mathfrak{k}$. Similarly,
$\mathfrak{b}_{m}=\mathfrak{k}\cup\mathfrak{k}$.Benini et al. (2019); Hsin and
Lam (2021) We then see that the two anomaly inflows in Eq. (50) and Eq. (52)
agrees with each other. In the lattice model, these identifications are made
explicit through the unitaries $\mathcal{V}$ and $\mathcal{W}$ in Eqs. (38)
and (39).

We now use the field-theoretical formalism to discuss TO and ATO states that
are compatible with the AEM anomalies $\Omega_{1,2}$. To write down the
anomaly inflow of the AEM anomaly, we utilize the doubled Hilbert space
formalism by mapping the density matrix
$\rho=\sum_{\alpha}p_{\alpha}\left|\psi_{\alpha}\right\rangle\left\langle\psi_{\alpha}\right|$
of the system to a quantum state
$\left|\rho\right\rangle\rangle=\sum_{\alpha}p_{\alpha}\left|\psi_{\alpha}\right\rangle_{L}\otimes\left|\psi_{\alpha}\right\rangle_{R}$
through the Choi–Jamiołkowski isomorphismJamiołkowski (1972); Choi (1975). The
Hilbert space is then doubled, and so does the exact symmetry $K$, which now
becomes a pair of independent symmetries $K_{L}$ and $K_{R}$, which acts only
on the left and right subspace of the doubled Hilbert space (the action of
$K_{R}$ needs to be complex conjugated). This means that the background gauge
field for $K_{L,R}$, $\mathfrak{k}_{L,R}$, can also be independently
introduced. In order for the doubled state to correspond to a physical density
matrix (that is Hermitian), it should automatically be symmetric under the
$\mathbb{Z}_{2}^{*}$ anti-unitary symmetry that swaps the left and right
Hilbert spaces with complex conjugation. Since the left/right symmetries are
also permuted by $\mathbb{Z}_{2}^{*}$, the exact symmetry $K$ after the
Choi–Jamiołkowski isomorphism becomes

$(K_{L}\times K_{R})\rtimes\mathbb{Z}_{2}^{*}.$ (53)

Therefore, the response to the external gauge field $\mathfrak{k}_{L,R}$
should be symmetric under
$\mathbb{Z}_{2}^{*}:\mathfrak{k}_{L}\leftrightarrow\mathfrak{k}_{R}$.
Meanwhile, the average symmetry $A$ acts simultaneously on both subspaces,
i.e. there only exists a single diagonal $A$ gauge field $\mathfrak{a}$. From
these symmetry perspective, the anomaly inflows of the AEM anomalies of
$\Omega_{1,2}$ in the doubled Hilbert space can be written as (the superscript
D denotes that the inflow is in doubled Hilbert space)

$Z_{\Omega_{1}}^{\text{D}}\sim\exp\left[\mathrm{i}\pi\int_{M_{4}}\left(\mathfrak{a}\cup\mathfrak{k}_{L}\cup\mathfrak{k}_{L}\cup\mathfrak{k}_{L}-\mathfrak{a}\cup\mathfrak{k}_{R}\cup\mathfrak{k}_{R}\cup\mathfrak{k}_{R}\right)\right]$
(54)

and

$Z_{\Omega_{2}}^{\text{D}}\sim\exp\left[\mathrm{i}\pi\int_{M_{4}}\left(\mathfrak{a}\cup\mathfrak{a}\cup\mathfrak{a}\cup\mathfrak{k}_{L}-\mathfrak{a}\cup\mathfrak{a}\cup\mathfrak{a}\cup\mathfrak{k}_{R}\right)\right].$
(55)

We now discuss the compatibility between the ATO and the AEM anomaly using
such a formalism. The ATO with incoherently excited $e$ anyons is an SSB state
compatible with the AEM anomaly between an average 1-form symmetry
$\mathbb{Z}_{2e}^{(1)}$ and exact 1-form symmetry $\mathbb{Z}_{2m}^{(1)}$,
whose anomaly inflow in the doubled Hilbert space can be written as

$Z_{\text{ATO}}^{\text{D}}\sim\exp\left[\mathrm{i}\pi\int_{M_{4}}\left(\mathfrak{b}_{e}\cup\mathfrak{b}_{m,L}-\mathfrak{b}_{e}\cup\mathfrak{b}_{m,R}\right)\right].$
(56)

Comparing Eq. (56) with Eq. (54) and Eq. (55), we see that the AEM anomaly
inflow of $\Omega_{2}$ can be matched by the inflow of ATO if one assigns
$\mathfrak{b}_{e}=\mathfrak{a}\cup\mathfrak{a}$ and
$\mathfrak{b}_{m,L/R}=\mathfrak{a}\cup\mathfrak{k}_{L/R}$ in Eq. (56). This
confirms that the AEM anomaly of $\Omega_{2}$ can be carried by an ATO, where
$e$ is randomly excited and $m$ still carries a projective representation of
$A$ and $K$. In contrast, the inflow of the AEM anomaly of $\Omega_{1}$ in Eq.
(54) cannot be matched by Eq. (56), which indicates the impossibility of
having an anomaly-compatible ATO state.

We expect that the field-theoretical approach using the anomaly inflow in
doubled Hilbert space can be easily generalized to higher dimensional AEM
anomalies.

The possibility of an SWSSB state of $K$ to be compatible with $\Omega_{1,2}$
can also be understood using a similar AEM anomaly inflow argument starting
from Eq. (54) and (55). In the doubled Hilbert space, a $K$-SWSSB state
corresponds to an SSB state where the two symmetries $K_{L,R}$ are
spontaneously broken down to a diagonal symmetry that transforms the $K$-spins
in the two Hilbert spaces simultaneously. Therefore, the response of a
$K$-SWSSB state to the external gauge fields $\mathfrak{k}_{L,R}$ should
schematically be

$Z_{\text{$K$-SWSSB}}^{\text{D}}\sim\delta(\mathfrak{k}_{L}-\mathfrak{k}_{R}).$
(57)

The physical picture of the partition function is that there should be no
response to the fluctuation that corresponds to the difference between the two
Hilbert spaces. Consequently, the AEM anomalies will have a trivial inflow to
the bulk, because Eqs. (54) and (55) vanishes after imposing the consequence
of the $K$-SWSSB $\mathfrak{k}_{L}=\mathfrak{k}_{R}$. Therefore, the SWSSB
state of $K$ is compatible with both AEM anomalies in Eqs. (54) and (55).

##  VI Conclusion and outlook

In this work, we have explored various disordered phases that are compatible
with the AEM anomaly of $G=A\times K$. We utilized a decorated-defect picture
in the strong disorder limit that allowed us to physically understand the
interplay between the constraint imposed by the quantum anomaly of the
remaining exact symmetry and the localization effect of the disorder. With
loosened constraints on global symmetry (with a part it being average
symmetry) and quantum coherence, we found phases that have no clean limit
analogs. One example is the glassy state with SWSSB, which is physical argued
as a direct consequence of the lack of defect percolation of the average
symmetry. We further constructed exactly solvable lattice models for SWSSB
state that is compatible with a 1+1$D$ mixed anomaly.

We then discussed another possible AEM anomaly-compatible disordered phase
intrinsic to mixed states, the ATO. In contrast to the SWSSB state, the ATO is
not always compatible with the AEM anomaly. We argued for a sufficient
condition for an AEM anomaly to be carried by an ATO, motivated by the
construction of the anomaly-compatible TO in the clean limit. Namely, if the
anomaly of the global symmetry group $G=A\times K$ can be trivialized by
extending to a larger global symmetry group of the form
$G^{\prime}=A^{\prime}\times K$ by a discrete group $N$, the $N$-charge in the
topologically ordered $N$-gauge theory can be freely exited without breaking
either the average symmetry $A$ or the exact symmetry $K$. This leads to an
option for an anomaly-compatible ATO, i.e. an ensemble of random $N$-gauge
charge excitations. On the contrary, if such an extension is impossible,
exciting an $N$-charge induces SWSSB of the exact symmetry $K$, and hence, an
anomaly-compatible symmetric ATO is impossible to construct in this way.
Utilizing the Choi-Jamiołkowski map of the density matrix, we further
bolstered our results using a field-theoretical argument of anomaly inflow in
doubled Hilbert space. The cases where ATO is possible correspond to when the
inflow of the AEM anomaly between the exact and average 1-form symmetries
carried by the ATO can be matched to the AEM anomaly of 0-form global
symmetry.

We conclude with some open questions and comments.

  * •

Anomaly of a purely exact symmetry in disordered systems \- Throughout this
work, we have mainly focused on disordered systems whose global symmetry has a
direct product structure between exact and average symmetry, and the mixed
anomaly is always between these two subgroups. It is possible that all the
disorder realizations preserve the entire global symmetry group, in which case
we have a disordered state that is compatible with the exact anomaly. Such a
scenario is recently discussed in Ref. Lessa et al., 2024a, which proves that
the exact anomaly constrains the multi-partite separability of the mixed
state. However, this work does not emphasize on the possible physical feature
of the mixed state that are compatible with the exact anomaly.

Gathering intuitions from our work, an SWSSB state of the entire global
symmetry should be an option for a mixed state compatible with the exact
anomaly. Once the exact symmetry is spontaneously broken down to an average
symmetry, the anomaly imposes no further constraint (recall from Sec. III.3
that an anomaly of an entirely average symmetry is trivial). For example, a
$G$-symmetric infinite temperature state can always be compatible the exact
anomaly of $G$, whose density matrix can be mathematically constructed by
projecting the infinite temperature state to a fixed charge sector of $G$.
Denoting such a projecting operator as $P_{G}$, the density matrix is simply
proportional to it: $\rho\propto P_{G}$. Since the infinite temperature state
always explicitly breaks the entire symmetry group $G$ down to an average
symmetry, the projected density matrix will likely represent an SWSSB state of
$G$.

Meanwhile, based on our arguments in Sec. V.5, it seems impossible that any
ATO without SWSSB can be compatible with an anomaly of a purely exact
symmetry.

  * •

Anomaly of non-trivially extended global symmetries \- Another scenario we
have not yet covered is when the global symmetry is not a simple direct
product of the average and the exact symmetry. One special case of this is
when the global symmetry group is mathematically a central extension of the
average symmetry $A$ by the exact symmetry $K$. Ref. Ma et al., 2023 has shown
that such a global symmetry setting can give rise to intrinsically average SPT
states, which have no clean limit analog. Consequently, the boundary of them
would host a corresponding AEM anomaly of such a global symmetry. A similar
situation is a mixed anomaly of a global symmetry that is non-trivially
extended from an average spacial symmetry by an exact internal
symmetryThorngren and Else (2018); Else and Thorngren (2019). The physical
consequence of these anomalies remains to be explored.

  * •

Disordered phase transitions between anomaly-compatible phases \- Throughout
the work, we mainly focused on the strong-disorder regime in arguing for the
existence of various anomaly-compatible phases that are intrinsic to mixed
states. Meanwhile, the nature of the phase transition between two disordered
anomaly-compatible phases remains unclear. Since the phases we find in this
work are all symmetry-breaking in some sense (with SSB or SWSSB of 0-form or
higher-form symmetries), direct continuous phase transitions between them
would be phase transitions between two different symmetry-breaking order. For
example, we showed in Sec. IV.1 that, both an SSB state of $A$ and an SWSSB
state of $K$ can be realized using a single disordered Hamiltonian. If such a
transition is continuous, it opens up the possibility of a disordered
generalization of the deconfined quantum critical point (DQCP)Senthil et al.
(2004), a transition between two different 0-form symmetry-breaking phases.

  * •

Entanglement constraint versus physical feature of the AEM anomaly \- Another
important open direction is to connect various disordered AEM-anomaly-
compatible phases with the quantum-information-theoretic constraints imposed
by AEM anomaly, as proposed by Refs. Kimchi et al., 2018; Lessa et al., 2024a;
Wang and Li, 2024. In light of Ref. Kimchi et al., 2018, it would be
meaningful to construct various anomaly-compatible disordered mixed states on
the level of quantum states using symmetric quantum channels with the starting
point being a trivial product state, in addition to their physical origin as
ground states of disordered Hamiltonians. Such a construction should help
understand better the quantum-information-theoretic constraint imposed by the
AEM anomaly and their relation to physical observables.

Acknowledgements \- YX and CMJ thank Yimu Bao, Zhen Bi, Meng Cheng, Tarun
Grover, and Jun Ho Son for helpful discussions. YX thanks Zhen Bi, Zhu-Xi Luo,
Cenke Xu, Carolyn Zhang, and Jian-Hao Zhang for collaboration on a related
topic. YX acknowledges support by the NSF through the grant OAC-2118310. CMJ
is supported by Alfred P. Sloan Foundation through a Sloan Research
Fellowship. This research was supported in part by grant NSF PHY-2309135 to
the Kavli Institute for Theoretical Physics (KITP), where part of this work
was performed.

##  Appendix A Anomalous symmetry action from group cohomology

In this appendix we will obtain the anomalous form of symmetry action directly
from group cohomology that classifies the anomaly and also the corresponding
SPT state in one higher dimension. SPT states protected by $G$ in $D$ space-
time dimensions is classified by the cohomology group ${\cal
H}^{D}[G,U(1)]$Chen et al. (2013). The elements in the cohomology group are
represented by $D$-cocycles $\omega_{D}:G^{D}\to U(1)$ that maps $D$
independent elements of $G$ to a $U(1)$ phase, and satisfies the cocycle
condition that its differential map is trivial:

$1=(d_{D}\omega_{D})(g_{1},g_{2},\dots,g_{D+1})\equiv\omega_{D}^{s(g_{1})}(g_{2},\dots,g_{D+1})\prod_{i=1}^{D}\omega_{D}^{(-1)^{i}}(g_{1},\dots,g_{i-1},g_{i}g_{i+1},g_{i+2},\dots)\omega_{D}^{(-1)^{D+1}}(g_{1},\dots,g_{D})$

where $s(g)=-1$ when $g$ is time-reversal or other anti-unitary symmetry, and
otherwise $s(g)=1$. Two cocycles $\omega_{D}$ and $\omega_{D}^{\prime}$ are
equivalent if they differ only by a co-boundary:

$\frac{\omega_{D}(g_{1},\dots,g_{D})}{\omega_{D}^{\prime}(g_{1},\dots,g_{D})}=d_{D-1}w_{D-1}(g_{1},\dots,g_{D}),$
(58)

where $w_{D-1}:G^{D-1}\to U(1)$ is a $(D-1)$-cocycle. For later convenience,
we define the cochain $\nu_{D}:G^{D+1}\to U(1)$, so that

$\nu_{D}(1,g_{1},g_{2},\dots,g_{D})=\omega_{D}(g_{1},g_{1}^{-1}g_{2},g_{2}^{-1}g_{3},\dots,g_{D-1}^{-1}g_{D}),$
(59)

and

$\displaystyle\nu(1,g_{1},\dots,g_{D})=\nu^{s(g)}(g,gg_{1},\dots,gg_{D})$ (60)

for any $g\in G$. The cochain satisfies a simpler cocycle condition

$1=d_{D}\nu_{D}(g_{0},g_{1},\dots,g_{D+1})=\prod_{i=0}^{D+1}\nu_{D}^{(-1)^{i}}(g_{0},\dots,g_{i-1},g_{i+1},\dots,g_{D+1}).$
(61)

The wave function of the corresponding SPT state can be directly constructed
from the $D$-cocycle $\omega_{D}$. The $D-1$ dimensional space is first
triangularized into simplices, and a branching rule is assigned so that one
can assign an order of the $D$ vertices in the simplex $S$ as
$S_{1},S_{2},\dots,S_{D}$. Each simplex is also assigned with an orientation
$\epsilon_{S}=\pm 1$. The local Hilbert space on each lattice site $i$ is
spanned by a basis form by all the group elements,
$\\{\left|g_{i}\right\rangle|g\in G\\}$. The symmetry acts locally as
$U_{g}\left|g_{i}\right\rangle=\left|gg_{i}\right\rangle$ for any $g\in G$.
The SPT wave function can be constructed by a linear superposition of all $g$
configurations:

$\left|\psi_{\text{SPT}}\right\rangle\sim\sum_{\\{g_{i}\\}}\prod_{s}\nu^{\epsilon_{S}}_{D}(1,g_{S_{1}},g_{S_{2}},\dots,g_{S_{D}})\bigotimes_{i}\left|g_{i}\right\rangle.$

The wave function is symmetric under $G$ actions, since any symmetry action
$U_{g}\left|\psi_{\text{SPT}}\right\rangle$ on the basis vectors
$\left|g_{i}\right\rangle$ in $\left|\psi_{\text{SPT}}\right\rangle$ can be
absorbed into the factors of the simplices, and each factor now becomes

$\displaystyle\nu_{D}^{s(g)\epsilon_{S}}(1,g^{-1}g_{S_{1}},\dots,g^{-1}g_{S_{D}})$
$\displaystyle=\nu_{D}^{\epsilon_{S}}(g,g_{S_{1}},\dots,g_{S_{D}})$ (62)
$\displaystyle=\nu_{D}^{\epsilon_{S}}(1,g_{S_{1}},\dots,g_{S_{D}})\prod_{i=1}^{D}\nu_{D}^{\epsilon_{S}(-1)^{i}}(1,g,g_{S_{1}},\dots,g_{S_{i-1}},g_{S_{i+1}},\dots,g_{S_{D}}),$

where we have used the cocycle condition in the last equality. On an infinite
system, the extra factors in the product above will be cancelled by nearby
simplices that share the $D-2$ dimensional simplex with $S$, hence the entire
product is invariant. However, on an open lattice, the symmetry action will
leave extra factors on the lattice boundary, so that the wave function will no
longer be symmetric:

$U_{g}\left|\psi_{\text{SPT}}\right\rangle\sim\sum_{\\{g_{i}\\}}\prod_{S\in\partial}\nu_{D}^{\epsilon_{\mathfrak{S}}}(1,g,g_{\mathfrak{S}_{1}},\dots,g_{\mathfrak{S}_{D-1}})\prod_{s}\nu^{\epsilon_{s}}_{D}(1,g_{s_{1}},g_{s_{2}},\dots,g_{s_{D}})\bigotimes_{i}\left|gg_{i}\right\rangle.$
(63)

Here $S\in\partial$ means that one $D-1$ dimensional face $\mathfrak{S}$ of
the simplex $S$ is on the lattice boundary. Hence, the symmetry action on the
spins on the boundary can be effectively written as

$U_{g}\bigotimes_{i\in\partial}\left|g_{i}\right\rangle=\prod_{\mathfrak{S}}\nu_{D}^{\epsilon_{\mathfrak{S}}}(1,g,g_{\mathfrak{S}_{1}},\dots,g_{\mathfrak{S}_{D-1}})\bigotimes_{i\in\partial}\left|gg_{i}\right\rangle=\prod_{\mathfrak{S}}\omega_{D}^{\epsilon_{\mathfrak{S}}}(g,g^{-1}g_{\mathfrak{S}_{1}},g_{\mathfrak{S}_{1}}^{-1}g_{\mathfrak{S}_{2}},\dots,g_{\mathfrak{S}_{D-2}}^{-1}g_{\mathfrak{S}_{D-1}})\bigotimes_{i\in\partial}\left|gg_{i}\right\rangle$
(64)

The anomalous symmetry actions on the boundary of various SPT states in the
main text can be directly obtained from the result above. For example, for the
Levin-Gu state protected by $\mathbb{Z}_{2}=\\{I,g\\}$, the only non-trivial
map of the 3-cocycle is $\omega_{3}(g,g,g)=-1$, while any other combination of
three group elements is mapped to $1$. If one defines
$\left|I\right\rangle=\left|\sigma^{z}=1\right\rangle$ and
$\left|g\right\rangle=\left|\sigma^{z}=-1\right\rangle$, the symmetry action
on the 1+1$D$ boundary can be obtained from Eq. (64) as (see can simply set
$\epsilon_{\mathcal{S}}=1$ here)

$U_{g}\bigotimes_{i=1}^{N_{\partial}}\left|\sigma^{z}_{i}=n_{i}\right\rangle=\prod_{i=1}^{N_{\partial}}\omega_{3}\left(g,gg^{\frac{1-n_{i}}{2}},g^{\frac{1-n_{i}}{2}}g^{\frac{1-n_{i+1}}{2}}\right)\bigotimes_{i=1}^{N_{\partial}}\left|\sigma^{z}_{i}=-n_{i}\right\rangle=\prod_{i=1}^{N_{\partial}}C(n_{i},n_{i}n_{i+1})\bigotimes_{i=1}^{N_{\partial}}\left|\sigma^{z}_{i}=-n_{i}\right\rangle.$
(65)

Hence
$U_{g}=\prod_{i=1}^{N_{\partial}}C(\sigma^{z}_{i},\sigma^{z}_{i}\sigma^{z}_{i+1})\sigma^{x}_{i}$,
which agrees with the decorated defect result in Sec. II.1.

The symmetry action of the $G=\mathbb{Z}_{2}^{2}$-anomalies $\Omega_{1,2}$ in
2+1$D$ can also be obtained in this way. Denoting the non-trivial element of
$A$ and $K$ by $a$ and $k$, the anomaly $\Omega_{1}$ is characterized by the
following 4-cocycle (for $\Omega_{2}$ the role of $a$ and $k$ is simply
reversed):

$\omega_{4}(a^{i}k,a^{j}k,a^{l}k,ak^{m})=-1$ (66)

for any $i,j,l,m=0$ or $1$, while all the other cocycles are equal to $1$. To
write down the anomalous symmetry actions, we define two sets of Pauli spins
$\sigma$ and $\tau$, so that
$\left|a^{i}k^{l}\right\rangle=\left|\sigma^{z}=1-2i\right\rangle\otimes\left|\tau^{z}=1-2l\right\rangle$.
Since the values of the cocycles are $\pm 1$, the orientation does not matter,
and the symmetry actions on a triangular lattice can be written as

$\displaystyle
U_{A}\bigotimes_{i}\left|\sigma^{z}_{i}=m_{i}\right\rangle\otimes\left|\tau^{z}_{i}=n_{i}\right\rangle=$
$\displaystyle\left|\sigma^{z}_{i}=-m_{i}\right\rangle\otimes\left|\tau^{z}_{i}=n_{i}\right\rangle,$
(67) $\displaystyle
U_{K}\bigotimes_{i}\left|\sigma^{z}_{i}=m_{i}\right\rangle\otimes\left|\tau^{z}_{i}=n_{i}\right\rangle=$
$\displaystyle\prod_{\triangle_{s}}\omega_{4}(k,kk^{\frac{1-n_{s_{1}}}{2}},k^{\frac{1-n_{s_{1}}}{2}}k^{\frac{1-n_{s_{2}}}{2}},a^{\frac{1-m_{s_{2}}}{2}}a^{\frac{1-m_{s_{3}}}{2}})\bigotimes_{i}\left|\sigma^{z}_{i}=m_{i}\right\rangle\otimes\left|\tau^{z}_{i}=-n_{i}\right\rangle$
$\displaystyle=$
$\displaystyle\prod_{\triangle_{s}}C(-n_{s_{1}},n_{s_{1}}n_{s_{2}},m_{s_{2}}m_{s_{3}})\bigotimes_{i}\left|\sigma^{z}_{i}=m_{i}\right\rangle\otimes\left|\tau^{z}_{i}=-n_{i}\right\rangle$
$\displaystyle=$
$\displaystyle\left[\prod_{\triangle_{s}}C(\tau^{z}_{s_{1}},\tau^{z}_{s_{1}}\tau^{z}_{s_{2}},\sigma^{z}_{s_{2}}\sigma^{z}_{s_{3}})\prod_{i}\tau^{x}_{i}\right]\bigotimes_{i}\left|\sigma^{z}_{i}=m_{i}\right\rangle\otimes\left|\tau^{z}_{i}=n_{i}\right\rangle.$

To recover the convenient forms of anomalous symmetry actions in Eq. (36), we
combine controlled unitaries of two nearby triangles with opposite
orientations (see Fig. 5). Rearranging the controlled unitaries in each
plaquette, it is straightforward to verify that the non-on-site factor of
$U_{K}$ in Eq. (67) is equal to the one in Eq. (36).

In a similar fashion, the 1+1$D$ $\mathbb{Z}_{2}^{2}$ anomaly discussed in
Sec. IV.1 can also be obtained from the non-trivial 3-cocycle
$\omega_{3}(a^{i}k,a^{j}k,ak^{l})=-1$ for any $i,j,l=0,1$ using two sets of
Pauli spins $\sigma$ and $\tau$.

Figure 5: The triangular lattice in writing down the anomalous symmetry action
of $U_{K}$. The black arrows indicate the branching rule of each triangular
plaquette. Two nearby triangles (marked in green) can be combined so that the
controlled unitaries can be written on a square lattice in Eq. (36).

##  Appendix B Construction of the symmetric infinite temperature mixed state

In this appendix, we carry out the calculation in the first line of Eq. (26)
and the explicit form of the random Hamiltonian in Eq. (22).

###  B.1 The explicit form of the random Hamiltonian in Eq. (22)

We now compute the random unitary conjugation in Eq. (22) explicitly. It is
easy to see that the $J_{\sigma}$ terms are invariant under the action of
$W(\\{h_{i},l_{i}\\})$. Meanwhile, the $J_{\tau}$ terms transform under
conjugation of $V_{i}$ as

$V_{i}\tau^{y}_{i+\frac{1}{2}}\left(1+\sigma^{z}_{i}\sigma^{z}_{i+1}\right)V_{i}^{-1}=\tau^{y}_{i+\frac{1}{2}}\left(-\sigma^{z}_{i}+\sigma^{z}_{i+1}\right)=-V_{i+1}\tau^{y}_{i+\frac{1}{2}}\left(1+\sigma^{z}_{i}\sigma^{z}_{i+1}\right)V_{i+1}^{-1}.$
(68)

Other conjugations by $V$ can be similarly computed. Using the fact that the
unitary $V_{i}$ commutes with $V_{j}$ for any sites $i$ and $j$, and
conjugating by $\tau^{z}$ only flips the sign of the $J_{\tau}$ term, we
collect all the conjugation results and arrive at the explicit form of the
disordered Hamiltonian:

$H(\\{h_{i},l_{i}\\})=-\sum_{i}\left[J_{\tau}l_{i}\frac{1+h_{i}h_{i+1}}{2}\tau^{y}_{i+\frac{1}{2}}\left(1+\sigma^{z}_{i}\sigma^{z}_{i+1}\right)+J_{\tau}l_{i}\frac{h_{i+1}-h_{i}}{2}\tau^{y}_{i+\frac{1}{2}}\left(\sigma^{z}_{i}-\sigma^{z}_{i+1}\right)+J_{\sigma}\sigma^{x}_{i}\left(1+\tau^{z}_{i-\frac{1}{2}}\tau^{z}_{i+\frac{1}{2}}\right)\right].$
(69)

We see that $H(\\{h_{i},l_{i}\\})$ is essentially a random-bond Ising model
whose bond-interaction is charged under $A$. Therefore, the bond-randomness
breaks $A$ down to an average while preserving $K$ exactly.

###  B.2 Derivation of Eq. (26)

We now derive the density matrix that describes the ensemble of disordered
ground states of $H(\\{h_{i},l_{i}\\})$. Using the definition in Eq. (20), the
pure ferromagnetic state $\left|\psi_{0}\right\rangle$ is transformed into

$\displaystyle W(\\{h_{i},l_{i}\\})\left|\psi_{0}\right\rangle=$
$\displaystyle\prod_{i}\left[\sigma^{x}_{i}C\left(\sigma^{z}_{i},\tau^{z}_{i-\frac{1}{2}}\tau^{z}_{i+\frac{1}{2}}\right)\right]^{\frac{1-h_{i}}{2}}\left(\tau^{z}_{i+\frac{1}{2}}\right)^{\frac{1-l_{i}}{2}}\frac{1}{\sqrt{2}}\left(\left|\uparrow\uparrow\dots\right\rangle_{\sigma}+\left|\downarrow\downarrow\dots\right\rangle_{\sigma}\right)\otimes\left|+_{y}+_{y}\dots\right\rangle_{\tau}$
(70) $\displaystyle=$
$\displaystyle\frac{\bigotimes_{i}\left|\sigma^{z}_{i}=h_{i}\right\rangle\left|\tau^{y}_{i+\frac{1}{2}}=l_{i}\right\rangle+\bigotimes_{i}\left|\sigma^{z}_{i}=-h_{i}\right\rangle\left|\tau^{y}_{i+\frac{1}{2}}=l_{i}h_{i}h_{i+1}\right\rangle}{\sqrt{2}}$
$\displaystyle=$
$\displaystyle\frac{\mathbbm{1}+U_{K}}{\sqrt{2}}\bigotimes_{i}\left|\sigma^{z}_{i}=h_{i}\right\rangle\left|\tau^{y}_{i}=l_{i}\right\rangle.$

In the last equality we have used the definition of $U_{K}$ from Eq. (18).
Collecting all possible values of $h_{i},l_{i}=\pm 1$, we have

$\rho_{\mathbb{Z}_{2}^{2}}=\frac{\mathbbm{1}+U_{K}}{\sqrt{2}}\frac{\mathbbm{1}}{2^{2N}}\frac{\mathbbm{1}+U_{K}}{\sqrt{2}}=\frac{\mathbbm{1}+U_{K}}{2^{2N}}.$
(71)

There is an easier derivation by noticing that the clean ferromagnetic state
is stabilized these following $2N$ unitaries:
$\sigma^{z}_{i}\sigma^{z}_{i+1}$, $\tau^{z}_{i+\frac{1}{2}}$, and $U_{K}$.
Unitaries in $W(\\{h_{i},l_{i}\\})$ randomly flip the eigenvalues of every
local stabilizer between $\pm 1$, but keep $U_{K}=1$. Hence, the eventual
density matrix is proportional to the projection operator
$\frac{\mathbbm{1}+U_{K}}{2}$.

##  Appendix C The derivation of the unitary operators $\mathcal{V}$ and
$\mathcal{W}$ via group extension and gauging on the lattice

As outlined in Sec. V.2 and Sec. V.4, the TO that carries the exact anomaly
$\Omega_{1}$ of $G=A\times K=\mathbb{Z}_{2}^{2}$ can be constructed by first
extending the global symmetry to $G^{\prime}=D_{4}$ via the short exact
sequence $1\to N=\mathbb{Z}_{2}\to D_{4}\to G=\mathbb{Z}_{2}^{2}\to 1$, after
which the $G$-anomaly will be trivialized. This means that the $D_{4}$
symmetry should act in a completely on-site way, which can be represented
using three sets of Pauli operators on a 2$D$ square lattice in the following
way:

$\tilde{U}_{A}=\prod_{i}\sigma^{x}_{i},\
\tilde{U}_{K}=\prod_{i}C(\mu^{x}_{i},\sigma^{z}_{i})\tau^{x}_{i},\
\tilde{U}_{N}=\prod_{i}\mu^{x}_{i}.$ (72)

The group formed by the three symmetry actions is indeed $D_{4}$, since
$(\tilde{U}_{A}\tilde{U}_{K})^{2}=\tilde{U}_{N}$. Our goal is to gauge the
normal subgroup $N$ and recover the anomalous symmetry actions of $G$ in Eq.
(36).

As a warm-up, we first review the usual way that the $\mathbb{Z}_{2}$ global
symmetry is gauged on the lattice. This will result in an anomaly-free
$\mathbb{Z}_{2}$ gauge theory enriched by $G=A\times K=\mathbb{Z}_{2}^{2}$,
where the $e$ anyon (the gauge charge of $N$) carries a projective
representation of $A$ and $K$. To this end, we introduce the $\mathbb{Z}_{2}$
gauge field of $N$ on each link $l$ of the lattice, whose Pauli operators are
denoted by $a_{l}^{x,z}$ (see Fig. 4). The local gauge constraint is
implemented via the Gauss law, whose form on the lattice reads

$\mu^{x}_{i}\prod_{i\in\partial l}a^{x}_{l}\equiv\mu^{x}_{i}\mathcal{A}_{i}=1$
(73)

for every lattice site $i$. The local $N$ charge excitation is gauge-invariant
only if being attached by a Wilson line:
$\mu^{z}_{i}\left(\prod_{l\in\gamma_{ij}}a^{z}_{l}\right)\mu^{z}_{j}$, where
$\gamma_{ij}$ is a line that connects sites $i$ and $j$. In the deconfined
phase, each plaquette has vanishing $\mathbb{Z}_{2}$ gauge flux, so that
$\mathcal{B}_{p}\equiv\prod_{l\in p}a^{z}_{l}=1$ for every $p$. This means
that we can effectively replace $\mu^{z}_{l_{1}}\mu^{z}_{l_{2}}$ by
$a^{z}_{l}$, where $l_{1,2}$ are the two neighboring lattice sites connected
by $l$.

We now gauge $N$ in the extended symmetry action in Eq. (72) following the
recipe above. Replacing $\mu^{x}_{i}$ in $U_{K}$ using the Gauss law, we
arrive at the following $\mathbb{Z}_{2}^{2}$ global symmetry action

$\tilde{U}_{A}\to\prod_{i}\sigma^{x}_{i},\
\tilde{U}_{K}\to\prod_{i}C\left(\prod_{i\in\partial
l}a^{x}_{l},\sigma^{z}_{i}\right)\tau^{x}_{i}=\mathcal{W}\prod_{i}\tau^{x}_{i}\mathcal{W},$
(74)

where the unitary operator $\mathcal{W}$ is given by Eq. (39). Therefore, in
the $\mathcal{W}$-transformed frame, the global symmetry is completely on
site: the global symmetry action of $K$ is simply $\prod_{i}\tau^{x}_{i}$, and
$U_{A}=\prod_{i}\sigma^{x}_{i}$ is symmetric under conjugation of
$\mathcal{W}$. The $\mathbb{Z}_{2}$ gauge theory of $N$ in the
$\mathcal{W}$-transformed frame is therefore decoupled from the matter fields
$\sigma$ and $\tau$.

To obtain the anomalous symmetry action $U_{K}$ in Eq. (36), the gauging of
$N$ needs to be done in a “twisted” wayTachikawa (2020), so that the $N$
charge is tied to the local configuration of the $\sigma$ and $\tau$ spins.
More explicitly, the Gauss law of the twisted gauging should be enforced as

$\mu^{x}_{i}\prod_{i\in\partial
l}a^{x}_{l}=C(-\tau^{z}_{p_{1}}\tau^{z}_{p_{3}},\tau^{z}_{p_{2}}\tau^{z}_{p_{4}}),$
(75)

where the plaquette $p$ is located at the bottom-left of the site $i$.
Substituting the relation above into $\tilde{U}_{K}$ in Eq. (72), we indeed
recover the anomalous symmetry action of $K$ in Eq. (36):

$\tilde{U}_{K}\to\prod_{i}C\left(\prod_{i\in\partial
l}a^{x}_{l},\sigma^{z}_{i}\right)\prod_{p}C(-\tau^{z}_{p_{1}}\tau^{z}_{p_{3}},\tau^{z}_{p_{2}}\tau^{z}_{p_{4}},\sigma^{z}_{p_{3}})\prod_{i}\tau^{x}_{i}=\mathcal{W}U_{K}\mathcal{W}^{-1}.$
(76)

Finally, we note that the twisted Gauss law in Eq. (75) can actually be
“untwisted” using the unitary operator $\mathcal{V}$ defined in Eq. (38):

$\displaystyle\mathcal{V}\left(\mu^{x}_{i}\prod_{i\in\partial
l}a^{x}_{l}\right)\mathcal{V}^{-1}$
$\displaystyle=\mu^{x}_{i}\prod_{i\in\partial l}a^{x}_{l}\times
C(\tau^{z}_{p_{1}},\tau^{z}_{p_{1}}\tau^{z}_{p_{2}})C(\tau^{z}_{p_{1}},\tau^{z}_{p_{1}}\tau^{z}_{p_{4}})C(\tau^{z}_{p_{2}},\tau^{z}_{p_{2}}\tau^{z}_{p_{3}})C(\tau^{z}_{p_{4}},\tau^{z}_{p_{4}}\tau^{z}_{p_{1}})$
(77) $\displaystyle=\mu^{x}_{i}\prod_{i\in\partial l}a^{x}_{l}\times
C(-\tau^{z}_{p_{1}}\tau^{z}_{p_{3}},\tau^{z}_{p_{2}}\tau^{z}_{p_{4}})=1.$

Therefore, in the $\mathcal{V}\mathcal{W}$-transformed frame, the gauge sector
is completely decoupled from the matter fields $\sigma$ and $\tau$, and the
wave function of the entire system should be a product state of the form in
Eq. (37).

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