Enforced symmetry breaking by invertible topological order

It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking by topological orders. In this work, we systematically study enforced breaking of a general finite group $G_f$ by a class of topological orders, namely 0D, 1D and 2D fermionic invertible topological orders. Mathematically, the symmetry group $G_f$ is a central extension of a bosonic group $G$ by the fermion parity group $Z_2^f$, characterized by a 2-cocycle $\lambda$ $\in H^2(G,Z_2)$. With some minor assumptions and for given $G$ and $\lambda$, we are able to obtain a series of criteria on the existence or non-existence of enforced symmetry breaking by the fermionic invertible topological orders. Using these criteria, we discover many examples that are not known previously. For 2D systems, we define the physical quantities to describe symmetry-enriched invertible topological orders and derive some obstruction functions using both fermionic and bosonic languages. In the latter case which is done via gauging the fermion parity, we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. We also study enforced breaking of the continuous group $SU_f(N)$ by 2D invertible topological orders through a different argument.


I. INTRODUCTION
Symmetry and topology play very important roles in modern physics.One of the most profound concepts in physics is spontaneous symmetry breaking [1].It is behind many physical phenomena, ranging from superconductivity and Bose-Einstein condensation in condensed matter systems to the unification of electromagnetic and weak forces in particle physics.For a long time, people believed that Landau symmetry-breaking theory could describe all possible phases and continuous phase transitions.The discovery of fractional quantum Hall effects (FQHE) [2] opened the door to the realm of topological phases of matter where interesting physics such as fractional charge and fractional statistics were uncovered [3].
Very recently, great effort has been made in the study of the interplay between symmetry and topology in quantum many-body systems.
While the mutual appreciation between topology and symmetry has resulted in many interesting physics, it is also known that they are not always compatible.For example, 2D Chern insulators with a nonzero Chern number must break the time reversal symmetry [54].It is manifested by the existence of chiral edge modes that reverse the direction under time reversal action and thereby cannot appear in time-reversal symmetric systems.In fact, time-reversal is broken in any chiral topological order.Such incompatibility can occur more generally in other systems and for other symmetries.We will refer to this phenomenon as enforced symmetry breaking (ESB) by topological orders.Of course, depending on one's viewpoint, it can also be called symmetry constraints on topological order.
To be possibly enforced to break by certain topological orders, there is a precondition on the symmetry: it must have a nontrivial action on the Hilbert space by its very definition.A few examples of these symmetries are: (1) the time-reversal symmetry T , under which the wave function must be complex-conjugated, (2) the mirror reflection R, under which the spatial orientation as- Z1 trivial sociated with the many-body wave function must be reversed, and (3) the fermion parity P f , under which the states containing an odd number of fermions must obtain a minus sign.More generally, a symmetry group G that contains P f , T , and/or R, also satisfies this precondition.For example, it was known that p x + ip y superconductor is incompatible with U f (1) particle number conservation, which is a nontrivial extension of the fermion parity group Z f 2 = {1, P f }.If this precondition is not imposed, then a symmetry can always be implemented as the identity operator, a rather trivial way, so that it is compatible with any topological order.So far, systematic exploration of ESB physics has not been done yet and a general framework is very much desired.
In this work, we study the ESB physics for a special class of topological orders, namely invertible topological orders (iTOs) [55].They are topological phases somewhat in between SPT phases and the topological orders that host anyons: they cannot be smoothly connected to the trivial product state even in the absence of symmetry, but they do not host anyon excitations.While it sounds exotic, all iTOs in low dimensions are known, including the 1D Majorana chain [56], 2D p x +ip y superconductors [57,58] (stacking even copies of which are topologically equivalent to integer quantum Hall states).See Table I for a summary of fermionic iTOs in low dimensions, which can all be realized in noninteracting fermionic systems.For bosonic systems, the only nontrivial invertible topological order is the 2D E 8 state.[59] They are called "invertible" because, for every state, there exists an inverse state such that stacking the two gives rise to the trivial state.The ESB physics for the bosonic E 8 state is simple, which is not compatible with all anti-unitary or orientation-reversing symmetries but compatible with all internal unitary symmetries.On the other hand, we will see that ESB physics for fermionic iTOs are extremely rich.
We will study 0D, 1D and 2D fermionic iTOs with a general finite group G f -in other words, symmetryenriched iTOs.Mathematically, the group G f is a central extension of a bosonic symmetry group G by the fermion parity symmetry Z f 2 .Different central extensions are characterized by non-trivial 2-cocycles λ ∈ H 2 (G, Z 2 ).However, we will also study an interesting example of the continuous symmetry group SU f (N ) in Sec.III D. The main result of this work is a set of criteria on whether a given G f is enforced to break by 0D, 1D, or 2D iTOs (summarized in Table II) and various examples that exhibit ESB physics.When deriving the 2D criteria, we de-velop the description of 2D symmetry-enriched fermionic iTOs using both fermionic and bosonic languages, with the latter achieved via gauging the fermion parity and turning the fermionic iTOs to bosonic SETs.In particular, we derive the formulas for the so-called obstruction functions, which are important components of the ESB criteria.It is unfortunate that in one of the cases, our explicit formula is not the most general.In the bosonic SET language, we also find that the obstruction functions are consequences of conditional anomalies, with the conditions inherited from the preconditions in the definition of the fermionic group G f .Characterization of symmetry-enriched fermionic iTOs involves two categories of quantities.The first category is a triplet (G, λ, ν), where G is the bosonic symmetry group, λ is a cocycle in H 2 (G, Z 2 ), and ν is an element of the iTO classification group (see Table I).The first two quantities G and λ determine the fermionic group G f , and the third quantity ν labels the iTO which can be viewed as a characterization of its intrinsic topological property.Even between these quantities, incompatibility may occur and lead to ESB phenomena.The ESB Criteria 1, 2 and 3 are of this type.The second category involves the quantities that describe symmetry enrichment.For examples, we define two quantities n 1 and n 2 (which are actually functions) in Sec.III A for 2D fermionic iTOs: n 1 characterizes if certain symmetry defects carry Majorana zero modes and n 2 characterizes how fermion parity conservation is achieved when defects fuse.Unlike the quantities in the first category which are given, they may vary.The symmetry enrichment quantities are well-defined only when they are compatible with the given quantities in the first category.Otherwise, if no compatible symmetry enrichment quantities exist, ESB occurs.The ESB Criteria 4, 5 and 6 are of this type.
The rest of this paper is organized as follows.In Sec.II, we study ESB by 0D and 1D fermionic iTOs.We set up our convention, derive the ESB criteria, and explore a few examples.Various 2D ESB criteria and examples are given in Sec.III.In particular, we develop a description of symmetry-enriched fermionic iTOs in Sec.III A using defects in the fermionic language.Also, an ESB example of SU f (N ) group is discussed in Sec.III D. We derive the 2D criteria in Sec.IV by transforming fermionic iTOs into bosonic SET states via gauging the fermion parity.The obstruction functions are obtained.We make a summary and discuss some potential generalizations in Sec.V. Appendices A, B and C contain some technical analyses.

II. ESB BY 0D AND 1D ITOS
To warm up, we discuss ESB by 0D and 1D fermionic iTOs in this section.We begin with the definition of symmetry group in fermionic systems.Then, we discuss ESB criteria, examples, and derivations of the criteria for 0D and 1D fermionic iTOs.
Let G f be the symmetry group of a general fermionic system.Throughout the paper, we assume all symmetries are internal and G f is finite, except in Sec.III D where a case of continuous symmetries will be discussed.In all fermionic systems, G f must contain a special element, the fermion parity P f , with P 2 f = 1.It must be preserved due to the requirement of locality.Moreover, P f should commute with all other symmetries.Let Z f 2 = {1, P f } be the subgroup formed by the fermion parity, which sits inside the center of G f .Mathematically, the relation between G f and Z f 2 is given by a short exact sequence where G is the quotient group G f /Z f 2 .For a given G, different G f 's are said to be different central extensions of G by Z f 2 .A central extension of G by Z f 2 is determined by a 2-cocycle λ ∈ H 2 (G, Z 2 ). 1 A brief review on group cohomology is given in Appendix A. A 2-cocycle is a function λ : G × G → Z 2 = {0, 1} that satisfies the condition: where g, h ∈ G, d is the coboundary operator, and "modulo 2" is implicitly assumed for addition. 2Two cocycles λ and λ are equivalent if λ = λ + dǫ and where 1 is the identity element of G.This convention will simplify many of our discussions.Given G and λ, the group G f can be constructed as follows: where 1 0 is the identity and 1 1 is the fermion parity P f .One can check that associativity of multiplication is guaranteed by the cocycle condition (2).Two fermionic groups G f and Gf , constructed from equivalent cocycles λ and λ respectively, are isomorphic.The isomorphism is given by g σ ↔ g σ+ǫ(g) , where g σ ∈ G f and g σ+ǫ(g) ∈ Gf .
In this work, we will use (G, λ) and G f interchangeably to describe symmetries in fermion systems. 3Nevertheless, we would like to emphasize that given any microscopic system, the symmetries g 0 and g 1 are physically distinct, corresponding to different operators (a.k.a.observables).So, one should keep in mind that a coboundary transformation ǫ(g) does have physical consequences.The group G may contain anti-unitary symmetries, such as time-reversal.To specify if a symmetry is unitary or anti-unitary, we need another piece of data, a group homomorphism where Z T 2 = {0, 1} (with group multiplication being addition modulo 2).The element g is unitary if s(g) = 0 and anti-unitary if s(g) = 1.It satisfies s(g) + s(h) = s(gh) (mod 2).The special case that s(g) = 0 for every g corresponds to that all symmetries are unitary.Since P f is unitary, we can make the extension s(g σ ) = s(g).
After the extension, s becomes a homomorphism from G f to Z T 2 .

B. ESB criterion for 0D iTO
In the usual sense, 0D systems neither host topological order nor support spontaneous symmetry breaking.So, it is not very interesting to study the phenomenon of ESB.However, there is indeed a sense of ESB as we explain below, exploring which helps to establish a few basic concepts.
One way to define 0D fermionic iTO is as follows.Consider 0D fermionic systems with a gapped unique ground state, so that they are invertible.Two systems are said to have the same topological order if the ground states can be smoothly deformed to each other by a fermionic unitary transformation(or bosonic unitary transformation with a Z f 2 symmetry).Under this definition, there are two inequivalent fermionic iTOs: those with the ground state being even under P f , and those with the ground state being odd under P f .Simple examples are the states |0 and c † |0 , respectively, where |0 is the vacuum in the Fock space and c † is a fermion creation operator.The latter is considered to be topologically non-trivial.We will refer to it as the "complex-fermion iTO".Now we ask if the complex-fermion iTO enforces certain symmetry group G f to break.It should be an explicit symmetry breaking, as there is no spontaneous symmetry breaking in 0D.Let |Ψ 0 be the ground state of any system that hosts the complex-fermion iTO.It follows from the definition that In the presence of a symmetry group G f , the ground state |Ψ 0 should form a one-dimensional representation of G f , with the condition (8) satisfied.Therefore, if P f = 1 in all its one-dimensional representations, G f is incompatible with the complex-fermion iTO.That is, G f is enforced to break by the complex-fermion iTO.This understanding can be used to derive a quantitative criterion on whether ESB occurs for a given symmetry group.Let G f be determined by the pair (G, λ), where λ ∈ H 2 (G, Z 2 ) is a 2-cocycle.We show that Criterion 1. G f is enforced to break by the complexfermion iTO, if and only if Derivation of the criterion will be deferred to Sec.II D. Here, we elaborate the criterion.
A 2-cocycle in where s(g) = 0, 1 denotes if g is unitary or anti-unitary, and K is the operation of complex conjugation with K[α] = α * .Two cocycles α and α are equivalent if they differ by a 2-coboundary: Different from Eq. ( 3), ǫ(g) can now be any U (1)-valued function.The equivalence classes [α] define the cohomology group H 2 (G, U T (1)) with the subscript "T " denoting the nontrivial action on U (1) values by the complex conjugation.

C. Examples
We illustrate 0D ESB by exploring a few examples.The simplest example is G = Z T 2 = {1, T }, where T is the time-reversal symmetry, and One can check that the two inequivalent cocycles are specified by a single quantity λ(T, T ) = 0 and 1, respectively.The cocycle with λ(T, T ) = 1 is non-trivial and corresponds to G f = Z 4 .One can show that (−1) λ(g,h) is non-trivial in H 2 (G, U T (1)).So, G f is enforced to break by the complex-fermion iTO, according to Criterion 1.This is actually the well-known Kramers theorem: for fermions with T 2 = P f , the odd-parity states must form a doublet.That is, a ground state satisfying (8) cannot be non-degenerate.
The simplest example of unitary symmetries is G f = D 8 , the dihedral group of order 8, with Z f 2 being the center of D 8 .More explicitly, let D 8 = {s n r m |n = 0, 1, m = 0, 1, 2, 3, r 4 = s 2 = 1, srs = r 3 , } and the fermion parity P f = r 2 .All irreducible representations of D 8 are listed in Table III.We observe that in all the 1D representations, P f = r 2 is represented by 1, which is inconsistent with the condition (8) for the complex-fermion iTO.Therefore, this D 8 is enforced to break by the complexfermion iTO.
The D 8 example can also be seen from Criterion 1.Nevertheless, let us use the criterion to study a family of examples, with D 8 being one of them.We take G to be a general Abelian group G = k i=1 Z Ni , with all symmetries being unitary.Without losing too much generality, we assume all N i are even.Then, the cohomology group where the indices i, j run in 1, 2, . . ., k, and the parameters p i , p ij = 0 or 1.To define w i (a, b) and w ij (a, b), let us denote the group elements as a = (a 1 , a 2 , ..., a k ) where where [n 1 + n 2 ] N = n 1 + n 2 (mod N ).Both w i and w ij only take a value 0 or 1.One can check λ(a, b) in the form (11) is indeed a 2-cocycle in H 2 ( i Z Ni , Z 2 ) for any choice of p i and p ij .It is useful to define the following quantities: where "modulo 2" is assumed, e i = (0, . . ., 0, 1, 0, . . ., 0) with only the ith entry being 1 and others being 0. These quantities have a nice property that they are invariant under coboundary transformations, so we will call them topological invariants for H 2 (G, Z 2 ) (see Appendix C 1 for more details).Inserting the explicit cocycle in (11), we obtain where i < j.By varying p i and p ij , the set {Ω i , Ω ij } can have 2 k(k+1)/2 distinct values, saturating the number |H 2 (G, Z 2 )| of cohomology classes.Therefore, it is a complete set of topological invariants and the representative cocycle in (11) ) (we drop the subscript T as we only consider unitary symmetries here).Under coboundary transformations of U (1)-valued functions, one can check that (−1) Ωi is not invariant, but (−1) Ωij remains a well-defined topological invariant.Moreover, (−1) Ωij is complete in the sense that it is enough to tell if (−1) λ(g,h) is trivial or non-trivial in H 2 (G, U (1)) [61].Therefore, (−1) λ(g,h) is non-trivial in H 2 (G, U (1)) if and only if any Ω ij = 1.For convenience, here and after, we call a 2-cocycle in H 2 (G, Z 2 ) as Type-I if all invariants Ω ij = 0, and otherwise we call it Type-II.Accordingly, the corresponding G f is enforced to break by the complex-fermion iTO if and only if λ is Type-II.
Take the simplest case G = Z 2 × Z 2 and let Ω 12 = 1.Then, ESB occurs for the corresponding G f .Depending on the values of Ω 1 and Ω 2 , we have two cases: (i) when

D. Derivation of 0D criterion
We prove an alternative statement that is equivalent to Criterion 1: G f is compatible with the complex-fermion iTO, if and only if (−1) λ(g,h) is trivial in H 2 (G, U T (1)).The equivalence is obvious.Derivation of this statement mainly involves the representation theory of groups.
Let us start with the "only if" direction.To show that, we assume that G f is compatible with the complexfermion iTO.Then, the ground state forms a onedimensional representation of G f .Let the representation be U (g σ )K s(g) for g σ ∈ G f , where U (g σ ) is a unitary operator and K is the operator of complex conjugation.In the special case that s(g) = 0 for all g ∈ G, it reduces to the usual unitary representation of a group.The operators satisfy which follows from the group multiplication law (5).To fulfill Eq. ( 8) and the requirement U (1 0 ) = 1, we have U (1 σ ) = (−1) σ .The convention (4) implies g 0 1 σ = g σ .So, we have Combining this equation with Eq. ( 16) and taking the shorthand notation U (g 0 ) = U (g), we immediately have It is a projective representation of G, where (−1) λ(g,h) ≡ α(g, h) is called a factor set.In general, the factor set of projective representations can be any 2-cocycle α(g, h) ∈ H 2 (G, U T (1)).A well known result is that if α is a nontrivial 2-cocycle, the representation cannot be 1D.The fact that we have a 1D representation implies that λ must be a trivial 2-cocycle in H 2 (G, U T (1)).Therefore, we have proven that if G f is compatible with the complexfermion iTO, (−1) λ(g,h) must be trivial in H 2 (G, U T (1)).To show the "if" direction, we explicitly construct a 1D representation of G f with Eq. ( 8) satisfied.If (−1) λ(g,h) is trivial in H 2 (G, U T (1)), it can be written as ǫ(g)K s(g) [ǫ(h)]/ǫ(gh), where ǫ(g) is some U (1)-valued function.Then, we take the representation to be One can check that it is a 1D representation of G f that satisfies both ( 16) and ( 8).This completes our proof.

E. ESB by 1D iTO
Invertible topological orders in 1D fermionic systems are classified by Z 2 .A representative of the nontrivial iTO is the famous Majorana chain, first discovered by Kitaev [56].So, we will refer to the nontrivial iTO as the "Majorana-chain iTO".The salient feature of the Majorana chain is that, when it is open, there exist robust zero modes at both ends, known as the Majorana zero modes (MZM).More specifically, it means the existence of Majorana operators γ l and γ r at the left and right ends, respectively, such that [γ l , H] = [γ r , H] = 0, where H is the Hamiltonian of the chain.Majorana operators are fermionic, self-adjoint, squared to 1, and guarantee a two-fold ground-state degeneracy.Intuitively, the latter means the MZM at each end carries a "fractional" Hilbert space of dimension √ 2. The degeneracy is topologically protected and cannot be lifted by any local perturbations that respect the fermion parity.Now consider a symmetry group G f , determined by the pair (G, λ) with λ ∈ H 2 (G, Z 2 ).We ask if G f is compatible with the Majorana-chain iTO.This question has already been answered in the seminal paper Ref. 60.In our language, the result of Ref. 60 can be stated as the following criterion: In other words, the Majorana-chain iTO is only compatible with G f = Z f 2 × G.This criterion holds regardless if G contains antiunitary symmetries.
To be self-contained, we briefly revisit the proof given in Ref. 60.Without loss of generality, systems hosting the Majorana-chain iTO can be viewed as a stack of 2n+ 1 Majorana chains with interaction between the chains allowed.For open boundaries, there are 2n + 1 Majorana operators at each end.Let γ 1 , γ 2 . . ., γ 2n+1 be those at the left end.Then, any local operator at the left end (leftlocal operator) can be written as a sum of products of the Majorana operators.An important feature of the algebra of left-local operators is that there exist and only exist two operators that commute with all left-local operators and that square to 1.The two operators are Z and −Z, with Z = i n γ 1 . . .γ 2n+1 and Z 2 = 1.The operator Z contains an odd number of Majorana operators so that P f Z = −ZP f .Note that the fermion parity P f is not left-local.Now consider symmetries in G f .For g σ ∈ G f , let U (g σ )K s(g) be the corresponding operator that acts on the whole low-energy Hilbert space that includes both left and right ends.Under the action of U (g σ )K s(g) , left-local operators remain left-local, with their algebraic structure preserved.In particular, Z can only be transformed to either Z or −Z.More specifically, where µ(g σ ) = 0, 1 specifies how Z transforms under the action of g σ .The specific µ(g σ ) depends on details, except µ(1 1 ) = 1 due to Then, between µ(g 0 ) and µ(g 1 ), one of them must be 0.
Let us pick out all the group elements with the µ value being 0. They are closed under multiplication and form a subgroup and accordingly G ′ is isomorphic to G. Hence, we have shown that the Majorana-chain iTO is compatible with G f only if it is a trivial extension of G by Z f 2 , i.e., λ is a trivial 2-cocycle in H 2 (G, Z 2 ).On the other hand, if λ is trivial, G f is always compatible with the Majorana-chain iTO -one simply represents all elements in G by the identity operator.This concludes the proof.

III. ESB BY 2D ITOS
Two-dimensional fermionic invertible topological phases are classified by Z.The iTO indexed by ν ∈ Z is exemplified by a stack of ν layers of p x + ip y superconductors (ν ≥ 0) or |ν| layers of p x − ip y superconductors (ν < 0).It is characterized by the chiral central charge c − = ν/2 of the gapless theory that lives on its edge.Physically, c − can be measured by the quantized thermal Hall conductance.We note that c − is odd under anti-unitary symmetries.If G f contains an anti-unitary symmetry, it is always incompatible with any nontrivial 2D iTOs.That is, any G f containing anti-unitary symmetries is enforced to break by any non-trivial fermionic iTOs.Hence, from now on, we will only consider G f of unitary symmetries.In this section, we will define the data to describe 2D symmetry-enriched iTOs and obtain some obstruction functions, after which we state the ESB criteria, followed with a few examples.Detailed derivations of ESB criteria will be given in Sec.IV.Section III D is a special subsection that discusses an ESB example with continuous symmetry group SU f (N ).

A. Symmetry-enriched fermionic iTOs
Different from usual 2D topological orders, iTOs do not support anyon excitations.Characterization of symmetry-enriched iTOs is more like that of SPT phases rather than SET phases.We will discuss two equivalent descriptions to characterize 2D fermionic symmetryenriched iTOs: (i) by studying properties of symmetry defects (i.e., static version of gauge fluxes) in the fermionic theory and (ii) by gauging the fermion parity Z f 2 and studying the resulting bosonic SETs of symmetry group G.In this section, we will use the former to define a set of data to describe 2D symmetry-enriched iTOs, as it is physically more intuitive.This description allows us to obtain two obstructions, O 2 and O 3 , which are important quantities in the ESB criteria.However, it is theoretically less mature than the latter, so we will use the second description to provide more detailed derivations of the ESB criteria in Sec.IV.
We define a triplet (ν, n 1 , n 2 ) to describe 2D symmetryenriched fermionic iTOs.The first quantity ν ∈ Z is defined above.It is a quantity associated with the intrinsic topology, which indexes the iTO and determines the chiral central charge c − = ν/2.Depending on whether ν is even or odd, the fermion-parity defect behaves differently: for ν being odd, it carries an odd number of Majorana zero modes (MZMs) [58], which cannot be completely annihilated by local perturbations; for ν being even, it carries an even number of MZMs, whose stability relies on the protection of other symmetries.A refined characterization is given by the dynamical fermionparity gauge flux, which is an anyon, denoted as v.According to Kitaev's 16-fold way [62], braiding statistics of the fermion-parity fluxes exhibit a 16-fold periodicity.In particular, the topological spin θ v = e iπν/8 .In fact, the ν = 16 fermionic iTO is topologically equivalent to the bosonic E 8 state stacked with a trivial fermionic insulator.The group G f acts the same as G on the the bosonic E 8 state, so they are always compatible -we simply set all elements in G f by the identity operator for the bosonic E 8 state.Since G f is compatible with a trivial fermionic insulator, we arrive at a conclusion: the ν = 16 iTO is compatible with any G f , i.e., no ESB will occur.Therefore, the phenomenon of ESB exhibits a 16-fold periodicity in ν for 2D iTOs.

n1 and O2
The quantity n 1 , a function G → Z 2 , describes the Majorana properties of symmetry defects: if n 1 (g) = 1, the g 0 defect carries an odd number of MZMs ; if n 1 (g) = 0, the g 0 defect carries an even (including zero) number of MZMs.The party of MZMs on the g σ defect is n 1 (g)+ νσ (mod 2).Parity of MZM numbers should respect the law of group multiplication g σ h τ = (gh) σ+τ +λ(g,h) , so we immediately obtain the following constraint where dn Here, modulo 2 is assumed in every equation.We observe that dn Eq. ( 22) can never hold, i.e., no valid n 1 (g) exists.This gives the first obstruction to a valid symmetry-enriched iTOs, which we simply call it the "O 2 obstruction".
It is easy to see that O 2 obstruction is trivial if and only if (i) ν is even, or (ii) ν is odd and λ is a coboundary.In Sec.IV B, we will derive the same result in the context of bosonic SETs.

Definition of n2
Now we focus on the case that ν is even; we will comment on the odd ν case at the end of Sec.III A 3. In this case, O 2 (g, h) is always trivial and n 1 is well-defined.Then, we move on to define the third quantity n 2 .Mathematically, it is a function, n 2 : G × G → Z 2 , subject to certain ambiguities and conditions that we will describe.The physical definition is slightly involved, depending on whether the relevant symmetry defects carry MZMs or not (see Fig. 1).Below, we will refer to the defects that carry MZMs as Majorana defects, and otherwise as non-Majorana defects.
Let us start with simplest situation that n 1 (g) = n 1 (h) = 0 for the group elements g and h.Both g σ and h σ defects are Abelian and non-Majorana.Due to the existence of the local fermion f , g σ defects come in two types: g 0 σ and g 1 σ .(We use g σ to denote both group elements and defects.)The two defects are related to each other by fusing the fermion f : Nevertheless, the choice of which defect is g 0 σ is a convention.One can set up the convention as follows.Since the defects are non-Majorana, we choose a local fermion (a) FIG. 1. Four different situations to define n2(g, h) for even ν.Black dots represent non-Majorana defects, and red dots represent Majorana defects.The dashed lines are branch cuts needed to insert static gauge fluxes, the other end of which are not shown.In every case, we fuse and then re-split the defects using an operator W , and the operator W is chosen in a way that produces the states depicted here (see the main text for discussions).In (b) and (c), Z0 is a Majorana operator associated with some other defect.
parity operator, denoted as P (g) with P (g) 2 = 1 for g 0 defects. 4If P (g)|Ψ = |Ψ , where |Ψ is the state that contains a g 0 defect, we call this defect g 0 0 ; if P (g)|Ψ = −|Ψ , we call it g 1 0 .The local fermion parity operator P (g) is ambiguous up to a sign.If one instead uses P (g) = −P (g) for setting up the convention, the two notations g 0 σ and g 1 σ will be swapped.Since ν is even, fermion-parity defects are also non-Majorana.We denote the two types as 1 0 1 and 1 1 1 , conventionally determined by a local fermion parity operator P (1 1 ).We will denote the vacuum as 1 0 0 and the fermion f as 1 1 0 , and also refer to them as "defects" by abusing notation.Obviously, they are determined by measuring the actual fermion parity P f .For convenience, let P (1) be either P (1 1 ) or P f , measuring which allows us to determine a defect 1 α σ .They satisfy the fusion rule where α, β, σ, ρ = 0, 1 and modulo 2 is implicitly taken for addition.The piece σρν/2 implies different fusion rules of fermion-parity defects for ν = 4k + 2 and ν = 4k, which is a well-known result (see e.g.Ref. [62]).That being said, we define a general g α σ defect as The fusion outcome on the right is unique so it is a good definition.We remark that fusion rules of defects are non-commutative and the order is important in (26).We A configuration of four defects.When the fermionparity defect 1 0 1 moves across the branch cut of the g0 Majorana defect (along the red dashed line), it turns into 1 1  1 .
always view a g α σ defect as a composite of g 0 0 and 1 α σ defects, with 1 α σ sitting on the right.We now consider fusing two non-Majorana defects.Fusing g 0 0 and h 0 0 defects must give rise to one of the two defects, (gh) 0 λ(g,h) or (gh) 1 λ(g,h) .Let us denote the possible superscript as n 2 (g, h).Then, Figure 1(a) shows the corresponding image.The quantity n 2 (g, h) depends on the symmetry-enriched iTO.Let |Ψ be the state containing the initial g 0 0 and h 0 0 defects, and let W be the operator that does the fusion and splitting associated with (27).Then, Eq. ( 27) can be understood as follows: by properly choosing W , we first require and then take a measurement of P (1) on the state W |Ψ such that It means that we choose an operator W such that it turns g 0 0 and h 0 0 into (gh) 0 0 , and n 2 (g, h) is defined by measuring P (1).In general, the fusion and splitting can be done by other operators, say W ′ that turns the original defects into (gh) n2(g,h) 0 and 1 0 λ(g,h) .However, no matter which operator is used, the total fermion parity must be conserved.That is, the equation XP (g)P (h)|Ψ = (−1) n2(g,h) P (gh)P (1)X|Ψ , (30) holds for either X = W or X = W ′ .Then, the quantity n 2 (g, h) must be the same, so it is well defined.In Fig. 1(a), we have used W instead of W ′ for illustration.If we change the convention P (g) → (−1) ǫ(g) P (g) (with the convention of P (1) fixed), the quantity n 2 (g, h) undergoes the following transformation Mathematically, it is a coboundary transformation.Accordingly, n 2 (g, h) is well-defined only up to the coboundary ambiguity (31).It is equivalent to use general defects g α 0 and h β 0 to define n 2 (g, h).Next, we consider the situation in Fig. 1(b).In this case, n 1 (g) = 1 and n 1 (h) = 0, i.e., g 0 defects are Majorana while the h 0 defects are non-Majorana.The Majorana g 0 defects come in only one type, which we simply denote as g 0 .In this case, we cannot define a local fermion parity operator.Instead, as discussed in Ref. [60] and also Sec.II E, there exists a local involutory fermionic unitary operator Z that commutes with all local operators.Similar to the local fermion parity operator, Z is also ambiguous up to a sign.Since Z is very like the usual Majorana operator, we will simply refer to it as the Majorana operator.For every g with n(g) = 1, we choose a Majorana operator Z(g), out of the two available, for the g 0 defect.Once this convention is set, we can define a fermion parity operator out of these Majorana operators.Consider 2n Majorana defects, (g j ) 0 with j = 1, . . ., 2n.There is a 2 n dimensional degenerate Hilbert space associated with these defects.We define the fermion parity as P = i n j Z(g j ).It is clear that change of the convention Z(g j ) → −Z(g j ) for any defect will result in a sign flip of P.
Fusing g 0 and a fermion-parity defect 1 α 1 , we obtain a g 1 defect, which is also a Majorana defect.Regardless of α, we always have g 0 × 1 α 1 = g 1 .One may want to have a "split view" of g 1 as in (26).However, such a splitting is not unique at the level of fusion rules.One needs to define it carefully.We will not define the splitting of an individual defect here, but instead will define a splitting combined with a fusion of two defects below, for our our purpose of defining n 2 (g, h).
We are now ready to define n 2 (g, h) in Fig. 1(b).Fusing g 0 and h 0 0 defects gives rise to a (gh) λ(g,h) defect, which is a Majorana defect as n 1 (gh) = n 1 (g) + n 1 (h).We further split it into (gh) 0 and 1 n2(g,h) λ(g,h) , with n 2 (g, h) determined using a similar procedure as in the first situation.Let us describe the procedure, which is slightly different, as Majorana operators are involved.Since a single Majorana operator does not form a Hilbert space, let us assume that there is an auxiliary Majorana operator Z 0 , at somewhere not close to the two defects.Let |Ψ be the initial state containing g 0 and h 0 0 , and let W be the operator associated with the whole fusing and splitting process.Then, similar to the first situation, we define n 2 (g, h) as follows.By properly choosing W , we first require and then we take a measurement of P (1) on the state W |Ψ , The latter measurement defines n 2 (g, h).That is, we require the final state contains (gh) 0 and 1 n2(g,h) λ(g,h) defects after fusion and splitting, as shown in Fig. 1(b).W is a fusion and splitting operator, so it should be bosonic to preserve fermion parity.Also it acts only near the defects.Because Z 0 is away from the defects, we must have W Z 0 = Z 0 W .Therefore, Z 0 in Eq. ( 32) can be readily removed, so that the definition of n 2 (g, h) is independent of Z 0 .Like in the first situation, we may choose another operator W ′ that does a different fusion and splitting.However, the value of n 2 (g, h) cannot be changed for fixed Z(g), P (h), Z(gh) and P (1) due to total fermion parity conservation.If we change our conventions of Z(g), P (h) and Z(gh), the same ambiguity (31) results.
The situation in Fig. 1(c) is similar to Fig. 1(b).In the last situation of Fig. 1(d), we have both n 1 (g) = n 1 (h) = 1.The fusion outcomes are non-Majorana defects.In this case, the two Majorana operators Z(g) and Z(h) are enough to form a Hilbert space, so no auxiliary Majorana operator is needed.We define n 2 (g, h) by first requiring and then measuring P (1) on the state W |Ψ where |Ψ is the initial state and W is a properly chosen fusion and splitting operator associated.One can similarly argue that n 2 (g, h) is well defined.The ambiguity on n 2 (g, h) due to change of convention is the same as above.This completes our definition of n 2 (g, h).Let us summarize that, for each g 0 defect, we have chosen a local operator Also, we have chosen a local fermion parity operator P (1) for determining defect 1 α σ .These local operators allow us to define the fermion parity in the low-energy Hilbert space associated with defects, which in turn allows us to define n 2 (g, h).As the local operator is subject to an ambiguity L(g) → −L(g), the quantity n 2 (g, h) is defined up to the transformation (31).
3. Associativity of defect fusion.For simplicity, we do not distinguish Majorana and non-Majorana defects in the diagrams.If some defects are non-Majorana, we have dropped the superscript and they should mean g 0 0 , h 0 0 , or k 0 0 .The quantity ∆(g, h, k) = n2(g, h) + λ(g, h)n1(k), where the term λ(g, h)n1(k) comes from the passing of 1 λ(g,h) across the branch cut of k0.

Derivation of O3
Before showing the condition imposed on n 2 (g, h), we discuss a property regarding braiding between Z(g) and fermion parity defects.For concreteness, consider the defect configuration in Fig. 2.There are four defects: a g 0 Majorana defect, its anti-defect g 0 (which may be ḡ0 or ḡ1 depending on λ), a fermion-parity defect 1 0 1 and its anti-defect 1 0 1 (which may be 1 0 1 or 1 1 1 depending on ν).Imagine adiabatically braiding the 1 0 1 defect across the branch cut between the g 0 and g 0 defect.We claim that 1 0 1 turns into 1 1 1 after passing through the branch cut.One way to understand this is through the bosonic SET language discussed in Sec.IV: when 1 0 1 passes through the branch cut, the state is acted by g, whose action is permutation of two defects 1 0 1 and 1 1 1 (v and vf in the notation of Sec.IV).Here we consider a slightly different view.Let us assume that the braiding process makes a full loop.Then, we retract the loop back to the position of the fermion parity defect.The whole process is equivalent to an action of the fermion parity on the disk whose edge is the loop.Let this action be the operator U (P f ).Since Z(g) is inside this disk and it is a fermionic operator, we must have Z(g)U (P f ) = −U (P f )Z(g).More specifically, let |Ψ be the initial state before braiding and retracting.Then Accordingly the fermion parity of the pair g 0 and g 0 flips a sign after the braiding and retracting process.Since the total fermion parity must be conserved and the local parity of 1 0 1 is untouched, the defect 1 0 1 must flip to 1 1 1 .If the g 0 defect is non-Majorana, the local fermion parity of 1 0 1 remains.Stating it compactly, we have: a 1 α σ defect turns into 1 α+σn1(g) σ , when passing through the branch cut of a g 0 defect.
With the above preparation, we now prove the condition that n 2 (g, h) should satisfy.It follows from the associativity of defect fusion (Fig. 3).Consider fusing three defects, g 0 , h 0 and k 0 .If n 1 (g) = 0, we pick the g 0 0 defect; if n 1 (g) = 1, there is only a Majorana defect g 0 to pick (similarly for h 0 and k 0 defects).There are two ways that we can fuse the defects, as shown through the two paths in Fig. 3.Most steps only make use of the definition of n 2 (g, h) in Fig. 1 under a properly chosen fusion and splitting operator W , except the step associated with the red dashed line in the lower path that uses the property in Fig. 2. At the end of both paths, we have two fermion-parity defects (or simply 1, f ).Further fusing the two defects according to Eq. ( 25) and requiring the fermion parity to be equal, we obtain where "modulo 2" is assumed for the addition.We define the function where the higher cup product ∪ 1 is reviewed in Appendix A. Then, Eq. ( 38) can be compactly written as where we have adjusted a few irrelevant minus signs to match the definitions.According to the mathematical properties of ∪ and ∪ 1 products, O 3 is a cocycle in H 3 (G, Z 2 ).It may be a nontrivial cocycle.However, the right-hand side of Eq. ( 40) is always a coboundary.Accordingly, for given ν, n 1 and λ, there might be no solution of n 2 to Eq. ( 40), hence leading to an obstruction to have a valid n 2 .Once O 3 is trivial, Eq. ( 40) is a condition that n 2 should satisfy.Different solutions, subject to the coboundary transformation (31), correspond to different symmetry-enriched fermionic iTOs.Note that for ν = 4k, the λ ∪ 1 λ term in O 3 vanishes (after taking modulo 2).This is an important difference between ν = 4k + 2 and ν = 4k.Finally, we comment that O 3 obstruction is absent for odd ν's.First of all, the triviality of O 2 requires λ to be a coboundary.Then, let λ(g, h) = ǫ(g)+ǫ(h)−ǫ(gh).One may use the same way as above to define n 2 .However, let us use the defects g ǫ(g) , instead of g 0 , for picking the local fermion parity operator P (g) or Majorana operator Z(g).Then, in the definition of n 2 , we always have Note that 1 h) , i.e., there is no fermionparity defect involved.A simple check of the fusion process in Fig. 3 shows that O 3 obstruction is absent.A summary of possible obstructions is shown in Table IV.

B. ESB criteria
We have defined the triplet (ν, n 1 , n 2 ) for the description of 2D symmetry-enriched fermionic iTOs.In principle, there is another piece of data, corresponding to stacking bosonic SPT states.However, that data is irrelevant to our study of ESB physics, so we will not discuss it.We have also derived the O 2 and O 3 obstruction functions, by checking the consistency between the MZM composition rule or fermion parity conservation and the defect fusion rules.In fact, there is another level of obstruction, denoted as O 4 , which follows from the consistency between symmetry action on local Hilbert spaces of defects and defect fusion rules.We will not derive O 4 in the fermionic language in this work, but instead turn to the bosonic SET language and discuss it in Sec.IV.A symmetry group G f is enforced to break by a fermionic iTO of index ν, if there are no valid n 1 and n 2 that make O 2 , O 3 and O 4 all trivial, i.e., if there are no valid 2D symmetry-enriched fermionic iTOs.In this section, we will state and elaborate the ESB criteria, and illustrate the criteria via various examples.Derivations of the criteria, including re-derivation of O 2 and O 3 and the derivation of O 4 in bosonic SET language, will be given in Sec.IV.

odd ν
We first consider 2D fermionic iTOs with ν being odd.Consider symmetry ground G f , determined by the pair (G, λ).We show that Criterion 3. G f is enforced to break by the 2D fermionic iTOs with odd ν, if and only if λ is a non-trivial 2-cocycle in H 2 (G, Z 2 ).In other words, the odd-ν iTOs are only compatible with This criterion can be easily seen from the O 2 obstruction alone.For odd ν, we have G and there is always a valid symmetry-enriched iTO -the one with all symmetries in G represented by the identity operator.
One may notice that this criterion is the same as Criterion 2 for the Majorana-chain iTO.It is not a coincidence.One can understand this connection by dimensional reduction.Imagine we insert a pair of fermion parity defects in a cylindrical geometry (Fig. 4).There are two MZMs located at the two ends when ν is odd.If we reduce the cylinder to a 1D system, it becomes a 1D Majorana-chain iTO.It is important to note that P f is in the center of G f so that G f remains the symmetry group of the effective 1D system.Then, if we apply Criterion 2, the "only if" direction of Criterion 3 results.One may extend this method to argue if a general g defect is allowed to carry MZMs.One subtlety is that inserting a g defect will break the symmetry group G f down to the centralizer C g of g in G f .If C g is a nontrivial extension by Z f 2 , then g cannot carry MZM as it contradicts with Criterion 2. However, if C g is a trivial extension of Z f 2 , we cannot give a definite answer.

ν = 4k + 2
For even ν, the obstruction O 2 is always trivial and n 1 is a cocycle in H 1 (G, Z 2 ).We need to consider O 3 and O 4 .Here, we discuss the ESB criterion for ν = 4k + 2. Given a symmetry G f determined by (G, λ), we claim that Criterion 4. G f is enforced to break by the 2D fermionic iTOs with ν = 4k + 2, if one of the two situations occurs: (i) the quantity ), for all n 1 that make O 3 a coboundary and for all n 2 that satisfy O 3 = dn 2 .We conjecture that there is no other situation such that G f is enforced to break for ν = 4k + 2.
A few comments are in order.First, the quantities λ, n 1 , n 2 and O 3 are valued in {0, 1}.We repeat the definitions of the cup products for convenience: where again additions are defined modulo 2.More details on cohomology operations are reviewed in Appendix A. According to properties of cup products, O 3 is always a (trivial or nontrivial) 3-cocycle in H 3 (G, Z 2 ) as long as . Second, we do not have the most general expression of O 4 yet.However, we know that it is a functional of λ, n 1 , n 2 , and ν.It can be defined only if O 3 is a coboundary.We also know that it should be a U (1)-valued 4-cocycle in H 4 (G, U (1)) from our knowledge of bosonic SET physics.In the special case that n 1 = 0 and O 3 = λ ∪ 1 λ is trivial, O 4 has the following expression: where " d" is used to denote the usual coboundary operation but without taking modulo 2, and additions in the exponents do not assume modulo 2 either.We will argue in Sec.IV that a more general (but still incomplete) expression of O 4 can be obtained by stacking fermionic iTOs with fermionic SPT states.While we do not have the general expression of O 4 , the O 3 obstruction alone is enough to give rise to many ESB examples.In particular, we give a complete answer to ESB phenomena that come from the O 3 obstruction for general finite Abelian groups G = i Z Ni .This is discussed in Appendix C 2. A few simple examples will be discussed in Sec.III C 1.
), for all n 1 that make O 3 = λ∪n 1 a coboundary and for all n 2 that satisfy O 3 = dn 2 .We conjecture that there is no other situation such that G f is enforced to break for ν = 8k + 4.
Regarding the expression of O 4 , we will argue in Sec.IV E that a general and complete expression can be obtained by a stacking trick.Here, we give a simpler version of the formula for the case that n 1 = 0: and where both λ and n 2 are valued in {0, 1}, "modulo 2" is not taken for d, and additions elsewhere are taken to be "modulo 4".
), for all n 1 that make O 3 = λ∪n 1 a coboundary and for all n 2 that satisfy O 3 = dn 2 .We conjecture that there is no other situation such that G f is enforced to break for ν = 16k + 8.
Compared to the ν = 8k + 4 case, the difference lies in the expression of O 4 .We will argue in Sec.IV E that a general and complete expression of O 4 can be obtained by stacking fermionic SPTs and using the SPT formulas.Here, we give a simpler version of the formula for the case that n 1 = 0: Note that if one sets ν = 16k + 8 in (45) and considers that dλ = 0 (mod 2), then Eq. ( 44) reduces to Eq. (46).The above special expression of O 4 allows us to obtain a general result for finite Abelian G.We show that for finite Abelian group G, ESB will never occur at ν = 8k, regardless of λ.That is, if G f is determined by a finite Abelian group G and an arbitrary λ ∈ H 2 (G, Z 2 ), it will never be enforced to break by the ν = 8k fermionic iTOs.(The case of ν = 16k has been generally discussed at the beginning of Sec.III A.) To see that, we first notice that the piece (−1) λ∪λ or (−1) n2∪n2 has appeared in the study of 2D fermionic SPT states [20].It was known that this piece is always a trivial 4-cocycle for finite Abelian G [63].One may use the topological invariants in (54) to directly check this fact.Therefore, O 4 reduces to (−1) n2∪λ .Then, we can choose n 2 = 0 such that O 4 is trivial.That is, if n 1 = n 2 = 0, the obstruction functions O 3 and O 4 are both trivial, which gives rise to a valid symmetry-enriched iTO.
Therefore, to look for ESB by ν = 16k + 8 fermionic iTOs, we must go to non-Abelian G.It is usually not easy to check if O 4 is a trivial or non-trivial cocycle for non-Abelian group G, even for the expression in (46).Fortunately, we will argue in Sec.III D that SU f (8n) (n being any positive integer) will be enforced to break by 2D fermionic iTOs with index ν = 16k + 8, through a different argument.That argument actually applies to iTOs with any even ν.
Finally, we remark that although ESB will never occur for ν = 16k, there are still nontrivial obstruction functions.These obstruction functions were obtained previously in the study of SPT states.[24][25][26]

C. Examples
In this subsection, we give a few examples of ESB by 2D fermionic iTOs.For odd-ν iTOs, any G f with λ being a non-trivial 2-cocycle is enforced to break.For ν = 16k + 8, we do not have an example of finite groups.However, in Sec.III D, we give an example of Lie group.So, we discuss the cases of ν = 4k + 2 and ν = 8k + 4 below.
We will consider examples with G being Abelian and focus on those ESB solely due to O 3 obstructions for ν = 4k + 2. We will discuss some general setup regarding group cohomology of finite Abelian groups and topological invariants (quantities that are invariant under coboundary transformations).Then, we will specialize to G = Z 2 × Z 2 and Z 2 × Z 4 .Discussions on general Abelian G as well as some discussions on O 4 are given in Appendix C.
Recall that λ ∈ H 2 (G, Z 2 ) can be generally parametrized in Eq. ( 11) and the topological invariants {Ω i , Ω ij } in (14) are complete for finite Abelian group G = k i=1 Z Ni (we assume N i being even without loss of generality).We note that Ω i , Ω ij are valued in {0, 1}, and Ω ij = Ω ji , Ω ii = 0. Also, a general 1-cocycle n 1 ∈ H 1 (G, Z 2 ) can be parametrized as where q i = 0 or 1, v i (a) = a i (mod 2), and a = (a 1 , . . ., a k ) is an integer vector to denote group elements of G.The cohomology group H 1 (G, Z 2 ) = 2 k and different choices of {q i } exhaust all cohomology classes of 1-cocycles.We show in Appendix C that one can define a complete set of topological invariants for 3-cocycles in H 3 (G, Z 2 ).Given u ∈ H 3 (G, Z 2 ), the topological invariants can be defined as follows.Let χ a (b, c) = u(a, b, c) − u(b, a, c) + u(b, c, a), then where e i is the ith generator of G and "modulo 2" is again assumed.We remark that Ξ ij and Ξ ji are independent, while Ξ ijk is a fully antisymmetric tensor.For the special 3-cocycle Accordingly, for given {Ω i , Ω ij } (i.e., given λ), the corresponding G f is enforced to break if there is no solution of {q i } to the equations Ξ ij = Ξ ijk = 0.This result applies to an arbitrary finite Abelian group.If there are solutions, then one needs to further check the O 4 obstruction.Now we specialize to G = Z 2 ×Z 2 .In this case, Eq. ( 49) becomes where Ξ = (Ξ 11 , Ξ 22 , Ξ 12 , Ξ 21 ) T .We ask if there is any choice of Ω 1 , Ω 2 and Ω 12 such that there is no q 1 and q 2 making Ξ = 0.It turns out that when Ω 1 = Ω 2 = Ω 12 = 1, there is indeed no solution.In this case, the fermionic symmetry group G f = Q 8 , the quaternion group with its center identified as Z f 2 .Therefore, G f = Q 8 is enforced to break by ν = 4k + 2 fermionic iTOs.
Similarly, for G = Z 2 × Z 4 , Eq. ( 49) becomes It is not hard to check that there is ESB by ν = 4k + 2 iTOs for two cases: (1) Ω 1 = 0 and Ω 2 = Ω 12 = 1, ( 2) 2. ν = 8k + 4 For ν = 8k + 4, we consider examples with G = (Z 2 ) n .Again, we use Eq. ( 11) to parameterize λ ∈ H 2 (G, Z 2 ) and use the topological invariants {Ω i , Ω ij } in ( 14) to characterize it.We use (47) to parameterize n 1 ∈ H 1 (G, Z 2 ).In the current case, evaluating the topological invariants in (48) for To limit the possible solutions {q i }, we will constrain ourselves to the case that Ω i = 1 for all i.Then, we have Ξ ii = q i .The only solution to Ξ ii = 0 is q i = 0 for all i, i.e, n 1 = 0.This makes the special expression of O 4 in (44) applicable.In this case dn Therefore, ESB will occur if O 4 in ( 44) is nontrivial for all possible n 2 .Like λ in (11), we will parameterize n 2 as follows where x i and x ij can also take 0 or 1.Using these parametrization, O 4 in ( 44) can be parametrized by the integers p i , p ij and x i , x ij .It can be proven that for Abelian G, (−1) n2∪n2 is a trivial U(1)-valued 4-cocycle.Therefore, we only need to consider the rest part of O 4 , which fortunately is linear in x i , x ij .For Abelian group G = i Z Ni , one efficient way to check whether a 4-cocycle χ(a, b, c, d) ∈ H 4 (G, U (1)) is nontrivial or not is to evaluate the following set of topological invariants [61]: n=1 χ e l ,ei (e j , ne j )χ e l ,ej (e i , ne i ), e iΘ ijk,l = χ e l ,ei (e k , e j ) χ e l ,ei (e j , e k ) where χ a,b (c, d) is two-steps slant product of χ(a, b, c, d) over a and b recursively and we denote the group element of G as a = i n i e i with e i as the generator of subgroup Z Ni .The sufficient and necessary condition that χ(a, b, c, d) is a trivial 4-cocycle is that all these topological invariants are equal to one.
We consider two cases: G = Z n 2 with n = 3 and n = 4.We will show that the former case does not lead to ESB but the latter might do.First, we consider n = 3.For G = (Z 2 ) 3 , there are many different λ.Assisted by computer, we can show that for none of these λ's, O 4 is nontrivial for all n 2 .Here, we specifically focus on the case λ = i w i + i<j w ij , the calculation of which helps for the n = 4 case.We need to consider eight independent topological invariants -six e iΘ i,l with i = l and two e iΘ ij,l with i = j = l.To look for ESB, it is equivalent to ask whether there exists at least one set of x i and x ij suc that or equivalently Θ i,l = 0, (mod 2π) Θ ij,l = 0, (mod 2π).( 56) Recall that we can ignore n 2 ∪ n 2 in O 4 .So, these equations are a set of (modular) linear equations respect to x i and x ij and can be solved straightforwardly.It turns out that there exsit only two solutions, that is Therefore, there exists n 2 such that all the invariants are trivial.In other words, there is no ESB for G = (Z 2 ) 3 with λ = i w i + i<j w ij .Now we consider the case n = 4 and λ = i w i + i<j w ij .Now there are in total 21 independent topological invariants -twelve e iΘ i,l , eight e iΘ ij,l and one e iΘ ijk,l .First, we find out the solution space of x i and x ij such that all the 20 topological invariants e iΘ i,l and e iΘ ij,l are equal to one.Similar to the case n = 3, we find that there are only two such solutions of x i , x ij : (1) for all i and j.However, neither of the two solutions make the last topological invariant e iΘ ijk,l to be 1.In other words, there is no such n 2 that all the 21 topological invariants are equal to 1. Therefore, we find an example of ESB by 2D fermionic iTOs with ν = 8k + 4: In this subsection, we consider 2D fermionic iTOs with a continuous symmetry group SU f (N ) (N being even).The main motivation is to look for examples of ESB by fermionic iTOs with ν = 16k + 8.It turns out that this example is quite neat and gives rise to ESB for all even ν's, when N varies.Our argument in this example does not follow other parts of this work.It is a generalization of an argument given in Ref. 64 for SU f (2).
Let us first explain some basic structures of SU f (N ).It is the usual SU (N ) group, and the subscript "f " denotes that the fermion parity group Z f 2 is embedded.To be concrete, let us use a representation of N × N matrices.
The group SU f (N ) has a center Z f N , represented by the diagonal matrices e in2π/N I, with n = 0, 1, . . ., (N −1) and I being the N × N identity matrix.We consider even N , and Z f 2 is identified to the group {I, −I}.An important fact that we will use is that there exists a U f (1) subgroup and the center Z f N is its subgroup.Such a group is not unique.One of the choices is U f (1) = {U θ |0 ≤ θ < 2π} and where the last diagonal term is e i(1−N )θ and other diagonal terms are e iθ .It is easy to check that U 2π/N generates the center and the fermion parity is U π .In short, we have . Now we consider an SU f (N )-enriched 2D fermionic iTO with even ν.First, with respect to U f (1), we can define a Hall conductance σ H .It is known that, for fermionic iTOs with even ν, we have The "modulo 8" is due to the E 8 state which does not contribute to σ H but changes the chiral central charge c − = ν/2 by 8. Next, we compute σ H in two ways.On one hand, by Laughlin's argument, σ H is equal to the charge Q 2π accumulated on an adiabatically inserted 2π flux.On the other hand, we may also adiabatically insert N copies of 2π/N fluxes.Let us denote the defect corresponding to a 2π/N flux as v, and Q v is the U f (1) charge carried by v.The total charge carried by the v defects should be equal to Q 2π .Then, we have In general, Q v is fractional, because v is a defect instead of a local excitation.We argue that in the presence of SU f (N ), the charge Q v must be an integer.Let us first give a simple argument and then justify it more rigorously.Recall that U 2π/N lies in the center of SU f (N ).Accordingly, inserting a 2π/N flux (the v defect) does not break SU f (N ).So, v must carry a projective representation of SU f (N ).However, SU f (N ) is a connected and simply connected compact Lie group, which does not support nontrivial projective representations.Accordingly, v carries a linear representation of SU f (N ), which is also a linear representation of the subgroup U f (1).That means, v must carry an integer charge of U f (1).The linear representation carried by v is generally irreducible due to energy consideration.
Two clarifications are needed to make the above argument justified.First, from our understanding of SETs, symmetry fractionalization goes beyond projective representations.Technically speaking, we need to replace H 2 (G, U (1)) which classifies projective representations with H 2 (G, A), where A is an Abelian group formed by Abelian anyons under fusion.However, for G = SU f (N ), it can be shown mathematically that both H 2 (G, U (1)) and H 2 (G, A) are trivial.Since v is a defect, not an anyon, readers may still not be convinced.There is another way to see this: one can gauge the Z f N center and turn the fermionic iTO into a bosonic topological order enriched by the quotient group P SU (N ) = SU f (N )/Z f N .In this way, v becomes a true anyon.Symmetry fractionalization on v is classified by H 2 (P SU (N ), A), whre A is the fusion group associated with the bosonic topological order whose structure depends on σ H . Using the universal coefficient theorem, one can show that H 2 (P SU (N ), A) = Tor{H 2 (P SU (N ), U (1)), A}, where Tor is a cohomological operation and H 2 (P SU (N ), U (1)) = Z N .Regardless of what group Tor{H 2 (P SU (N ), U (1)), A} is, what is important to us is the physical meaning: Tor picks out those projective representations of P SU (N ) that are compatible with the fusion group A. This interpretation has been widely used for SO(3), which is equal to SU f (2)/Z f 2 (see, e.g., Ref. [65]).We believe it is applicable generally.Accordingly, symmetry fractionalization on v can all be characterized by projective representations of P SU (N ), before considering the compatibility to fusion rules.Finally, it is well known that projective representations of P SU (N ) are simply linear representations of SU f (N ).So, it goes back to our argument above.
Second, the irreducible linear representation of SU f (N ) carried by v in the fermionic picture is usually multi-dimensional.Then, when we insert multiple 2π/N fluxes, they may stay in different states inside the irreducible linear representation, as these states are energetically degenerate.However, U f (1) charges carried by different states in an irreducible representation of SU f (N ) can only differ by a multiple of N .This is again because is the center of SU f (N ), so that all states in an irreducible representation of SU f (N ) must carry the same Z f N charge.This is equivalent to say that U f (1) charges can only differ by a multiple of N .Accordingly, even if this subtlety is taken into account, Eq. ( 59) is relaxed to which is enough for our purpose.With these clarifications, we now combine ( 58) and (60).We immediately have where n is any integer and "gcd" stands for greatest common divisor.Equation ( 61) must be satisfied by all symmetry-enriched fermionic iTOs.If it cannot be satisfied, that means the symmetry group SU f (N ) is enforced to break by the iTO.More explicitly, we have 1.SU f (2l) is enforced to break by 2D fermionic iTOs with ν = 4k + 2; 2. SU f (4l) is enforced to break by 2D fermionic iTOs with ν = 8k + 4; 3. SU f (8l) is enforced to break by 2D fermionic iTOs with ν = 16k + 8; where k, l are any positive integers.For ν = 16k, ( 61) is always satisfied, which is consistent to the 16-fold periodicity of ESB physics argued above.

IV. DERIVATION OF 2D ESB CRITERIA
In this section, we give alternative derivations of O 2 and O 3 obstructions, and derive the O 4 obstruction and 2D ESB criteria.Compared to Sec.III A where we define n 1 and n 2 using defects in the fermionic language, the main strategy here is to gauge the fermion parity group Z f 2 and turn the fermionic iTO into a bosonic topological order.
Let us denote the resulting bosonic topological order as C ν , for the fermionic iTO of index ν.Detailed properties of C ν can be found in Ref. 62.When ν is odd, C ν contains three anyons: where f corresponds to the original fermion and σ is a non-Abelian fermion-parity flux of quantum dimension When ν is even, C ν contains four anyons: where v and vf are two Abelian fermion-parity fluxes satisfying the fusion rule v × f = vf .The even-ν cases can be further distinguished by the fusion rule of fermionparity fluxes, where k is an integer.In all cases, the topological spin of a fermion parity flux is θ σ = e iνπ/8 or θ v = θ vf = e iνπ/8 , and the mutual statistics between f and fluxes is Hence, there is a 16-fold periodicity in ν.
With Z f 2 gauged, there remains a global symmetry group G. Hence, we obtain an SET state of topological order C ν and symmetry group G.We will see that it is not an arbitrary SET, but one with certain symmetry properties fixed by the 2-cocycle λ ∈ H 2 (G, Z 2 ).Studying such a conditional SET leads to new consistency relations that do not exist in unconditional SETs.These conditional consistency relations, which we call conditional anomalies, give rise to the O 2 and O 3 obstructions which otherwise do not exist.(There is also an Õ3 obstruction in general bosonic SETs, but O 3 is different from, although related to, Õ3 .)Together with an O 4 obstruction of bosonic SETs, they can be used to derive the ESB criteria associated with the original fermionic iTOs.

A. Basics of SETs
Before studying our specific SETs, we review the basics of the general theory of bosonic SET states.For more details, one can refer to Ref. 49.The description of SETs involves several layers of data which we briefly explain below one by one.
First, a bosonic topological order C contains anyons 1, a, b, . . ., where 1 is the trivial anyon.Mathematically, C is described by a unitary modular tensor category (UMTC).[62] Physically, anyons are characterized by their fusion and braiding properties.They follow a set of fusion rules a × b = c N c ab c, where the fusion coefficient N c ab is a non-negative integer.There exists a unique anyon ā, namely the anti-particle of a, such that N 1 aā = 1.Two important quantities of each anyon a are the quantum dimension d a and topological spin θ a .For Abelian anyons, d a = 1 and θ a is its self-statistics.Associated with each N c ab = 0 there is a vector space V ab c , called the fusion or splitting space, whose dimension is N c ab .A key quantity that relates different fusion spaces is the so-called F symbol, which is an isomorphism between fusion spaces of three anyons If one exchanges two anyons, it corresponds to another isomorphism, the R symbol, R ab c : V ab c → V ba c .Every C processes a set of topological symmetries, which form a group denoted as Aut(C).We only consider those that are unitary and orientation-preserving.A topological symmetry is an invertible map from C to itself.It contains two parts: (i) a permutation of anyons a → a ′ ≡ ϕ(a), (63) and (ii) an action on the fusion space where is a phase factor and |a, b; c ∈ V ab c .For simplicity, we have assumed N c ab = 0 or 1, as all our cases satisfy this assumption.Topological symmetries keep all the data of C invariant, e.g., One can see that if u ab c is multiplied by γ a γ b /γ c , equations (65) remain unchanged.Such a transformation is called a natural isomorphism.It is regarded as an equivalence relation between different sets of {u ab c }. Group multiplication of Aut(C) in the aspect of u ab c is upto natural isomorphisms.It is believed that {u ab c } is fixed up to natural isomorphisms once anyon permutation in (63) is given, which is indeed the case in our examples.
Second, consider a system with the microscopic symmetries forming a group G.If it is unbroken at low energy, G must be mapped into Aut(C).This is characterized by a group homomorphism ρ : G → Aut(C).That is, every g ∈ G is associated with a topological symmetry, denoted as ρ g .We denote the anyon permutation ρ g (a) = g a for short.For anyon permutations, ρ is an exact group homomorphism.On the other hand, the action on states in fusion space is upto natural isomorphisms: where κ g,h is a natural isomorphism.Let u g (a, b, c) be the phase factor associated with ρ g 's action on fusion spaces.Then, the explicit expression of κ g,h is where ḡ is a short-hand notation for g −1 .Since κ g,h is a natural isomorphism, we can decompose it as The quantity β a (g, h) is subject to two kinds of ambiguities: (i) the decomposition ( 68) is not unique and a shift With β a (g, h), we define an important quantity By definition, Ω a is a coboundary in H 3 ρ (G, U (1)) for every a.Also, associativity of ρ g can be used to show that where Õ3 (g, h, k) is an Abelian anyon in C and M a,b is the mutual statistical phase between a and an Abelian anyon b.Moreover, Õ3 (g, h, k) is a 3-cocycle in H 3 ρ (G, A), where A denotes the group of Abelian anyons in C.However, it is important to note that Õ3 (g, h, k) might be a nontrivial cocycle.The ambiguities in β a (g, h) induce coboundary transformations in Õ3 (g, h, k).Accordingly, only the cohomology class [ Õ3 ] in H 3 ρ (G, A) matters.Third, other than topological actions, symmetries in G also act on local degrees of freedom around each anyon.Consider a state that contains a set of anyons {a i }, which are spatially well-separated.The overall action of g ∈ G is a combination of local and topological actions: where g is a local unitary operator, supported in the neighborhood of anyon a i , and ρ g is the topological action in the fusion space of {a i } which can be decomposed into those in (64).The local operators form a projective representation where g U (i) ) for every a.The local unitary operator U (i) g has a phase ambiguity.At the same time, ρ g also has a phase ambiguity due to natural isomorphisms.The two ambiguities shall be correlated such that R g R h = R gh .It was shown in Ref. [49] that a nice quantity to look at is the ratio where w(g, h) is an Abelian anyon in A ⊂ C. It is equivalent to say that whenever N c ab = 0. Accordingly, gauge transformations of η a (g, h) and β a (g, h) shall be correlated such that (75) always holds.Combining (69), ( 70), ( 73), (74) and that η a (g, h) is 2-cocycle, one can show that Õ3 (g, h, k) = dw(g, h, k) where x stands for the anti-particle of anyon x.This implies that Õ3 (g, h, k) should be a 3-coboundary in H 3 ρ (G, A) to give rise to a valid w(g, h).If Õ3 (g, h, k), obtained from its definitions (69) and ( 70), is a nontrivial 3-cocycle, Eq. ( 76) can never have a solution for w(g, h), implying that the form of symmetry action in (71) does not hold.Then, it is said that the topological action ρ has a symmetry localization obstruction.
When the obstruction Õ3 (g, h, k) is trivial, one then look for w(g, h) that satisfies (76).It is not hard to see that given w(g, h) a solution, w(g, h)t(g, h) is also a valid solution if t(g, h) ∈ H 2 ρ (G, A).At the same time, w(g, h) is subject to an ambiguity w(g, , where ζ(g) is an arbitrary Abelian anyon in A. This ambiguity is due to an independent phase shift that we can perform on U (i) g or equivalently on η a (g, h) for a fixed β a (g, h).Then, different solution classes [w] to (76) are said to describe different symmetry fractionalization classes.They are related to each other by a cohomology class [t] ∈ H 2 ρ (G, A).
Once w(g, h) and β a (g, h) are given, the projective phase factor η a (g, h) is determined by ( 73) and (74).Fourth, for a given symmetry action ρ ∈ Aut(C) and a symmetry fractionalization class described by w(g, h), there are additional consistency conditions to satisfy.For an SET to be valid, one shall be able to insert symmetry defects (i.e., couple to a background gauge field) and these defects shall form a fusion category.It was shown in Ref. [33] that given ρ and ω, one can construct a quantity Õ4 out of them.It is a 4-cocycle in H 4 (G, U (1)) and subject to coboundary transformations.For the defects to form a consistent fusion theory, it is required the cohomology class [ Õ4 ] is trivial.If it is nontrivial, we say there is an Õ4 obstruction.The general expression Õ4 is complicated.A simple one with a trivial ρ is given by F w(h,k),w(g,hk),w(ghk,l) F w(h,k),w(hk,l),w(g,hkl) × F w(g,h),w(k,l),w(gh,kl) F w(g,h),w(gh,k),w(ghk,l) F w(k,l),w(h,kl),w(g,hkl) F w(k,l),w(g,h),w(gh,kl) , where the R and F symbols are associated with Abelian anyons in C. If [ Õ4 ] is trivial, one can study the fusion theory of defects, which involves additional F symbols among the defects.It was shown that different defect theories can be obtained by twisting the defect F symbols with a 3-cocycle ). Physically, it corresponds to stack an SET with a 2D SPT state which is indeed classified by ).In this work, we will not explore the data α 3 as it is irrelevant to the phenomenon of enforced symmetry breaking.
To summarize, there are two obstruction classes: (i) existence of a valid symmetry fractionalization class [w] requires [ Õ3 ] to be trivial; (ii) a consistent defect theory further requires [ Õ4 ] to be trivial.A valid bosonic SET state requires both obstruction classes to be trivial.

B. O2 and symmetry fractionalization condition
We now apply the above general theory to our case, the topological order C ν enriched by symmetry group G.In this subsection, we consider the case of odd ν.In this case, there is no nontrivial topological symmetry, i.e., Aut(C ν ) = Z 1 .Then, no symmetries in G permute anyons.It is known that the Õ3 obstruction is always trivial without anyon permutations.We can take the gauge u ab c = 1, β a (g, h) = 1, such that Õ3 (g, h, k) = 1.make some important comments here before proceeding.The fermion f is a local excitation in the original iTO, so the fusion space V f f 1 is not topological but instead local.Then, one should impose the condition that u f f 1 = 1 after Z f 2 is gauged.This condition was previously discussed in Ref. [66].It is a necessary condition for establishing an inverse mapping from the bosonic SET back to the fermionic iTO.With this condition imposed, natural isomorphisms on f can only be γ f = ±1.Then, the quantity β f (g, h) in the decomposition (68) shall also take a value +1 or −1.Under an appropriate gauge choice, we find that we can set β f (g, h) = 1 for all C ν 's (the cases of even ν's are discussed in Appendix B).In fact, we postulate that the most general case is β f (g, h) = γ f (gh)/[γ f (g)γ f (h)], and a shift β f (g, h) → β f (g, h)ν f (g, h) is not allowed in general (see a discussion below).The O 2 and O 3 obstructions in Sec.III A are correctly recovered only under this assumption.We believe that it is due to the locality of f in the original fermionic systems, such that (78) below describes the absolute symmetry fractionalization on f and disallows a shift β f (g, h) → β f (g, h)ν f (g, h).We will take β f (g, h) = γ f (gh)/[γ f (g)γ f (h)] as an assumption in this work, and we refer the readers to recent works [67,68] on general fermionic SET states for more systematic discussions.
Symmetry fractionalization is characterized by an Abelian anyon w(g, h) ∈ A ⊂ C ν .Again, it is special in our SET as it originates from a fermionic theory with symmetry group G f determined by G and a 2-cocycle λ ∈ H 2 (G, Z 2 ).From our study of 0D iTOs (Sec.II D), we know that the fermion f obeys a projective representation of G, with the projective factor being (−1) λ(g,h) .Accordingly, when Z f 2 is gauged, the local action of G on f shall be associated with a projective phase factor It is an important condition, constraining the possible symmetry fractionalization classes [w].In fact, for odd ν, we show below that the condition (78) is so strong that it is actually not compatible with any symmetry fractionalization pattern when λ is a nontrivial cocycle in H 2 (G, Z 2 ).We remark that due to (75) and the condition that natural isomorphism γ f = ±1, gauge transformations on η f (g, h) are also constrained with values ±1.This makes sense, as the right-hand side of Eq. ( 78) only has a ±1 ambiguity from coboundary transformations of λ(g, h).We also remark that the condition (78) implies that we interpret g ∈ G as g 0 ∈ G f .Generally speaking, g ∈ G corresponds to the coset {g 0 , g 1 } in the quotient group G f /Z f 2 .We now show that C ν with an odd ν is incompatible with (78) when λ is a non-trivial cocycle.Combining Eqs. ( 73), ( 74) and (78), we obtain As w(g, h) = 1 or f , the left-hand side must be equal to 1.Under the assumption that That is, Eq. (79) has no solution w(g, h), if λ is a nontrivial 2-cocycle in H 2 (G, Z 2 ).Instead, for even ν, the fermion-parity fluxes are Abelian so that w(g, h) has choices other than 1 and f , making solutions of Eq. ( 79) to exist even if λ is a non-trivial cocycle (see Sec. IV C).Accordingly, we recover the obstruction O 2 (g, h) = νλ(g, h), first discussed in Sec.III A. We remark that if we allow with {ν a (g, h)} respecting anyon fusion, the O 2 obstruction becomes ν[λ(g, h) + ν f (g, h)].Since ν f (g, h) may be a non-trivial 2-cocycle in general, it is inconsistent with the result from Sec. III A. This justifies the assumption on β f (g, h).We note that O 2 does not exist in usual SETs.It exists in our SETs due to the condition (78).We will call it a conditional obstruction or anomaly.The obstruction O 2 alone is enough to establish Criterion 3, which is complete for the ESB physics in all odd-ν iTOs.So, we will move on to the even-ν cases below.However, it does not mean that there is no other obstructions for the odd-ν cases.The Õ4 obstruction of bosonic SETs will generally be even if λ is trivial.
C. O3 and O4 for ν = 4k + 2 We now consider even ν, by starting with ν = 4k + 2. Different from the odd-ν case, the group of topological symmetries Aut(C ν ) = Z 2 for even ν. 5The nontrivial one is associated with the permutation v ↔ vf .The symmetry action on C ν is given by a group homomorphism Equivalently, , the symmetry g permutes v and vf ; if n 1 (g) = 0, it does not.In other words, the anyon permutation is given by For n 1 to be valid, it requires the Õ3 obstruction to be trivial.We verify this explicitly in Appendix B that Õ3 is indeed trivial for even ν.In fact, we show in Appendix B that Õ3 (g, h, k) = 1 under certain gauge choice.The quantity n 1 in (80) is the same n 1 defined in Sec.III A that characterizes the Majorana zero mode on the g 0 and g 1 defects (for even ν, if g 0 carries MZM, so is g 1 ).This correspondence has already been known before in the context of fermionic SPT phases [20].To see that, imagine inserting a g defect in the bosonic SET and winding a v anyon around defect.If n 1 (g) = 1, v becomes vf after winding around the g defect, implying that an anyon vf × v = f is absorbed into the defect.That is, a g defect has the ability to absorb f and it must carry a Majorana zero mode in the original fermionic iTO.
After the topological symmetry action is specified by n 1 , we now discuss the local symmetry action characterized by the symmetry fractionalization w(g, h) ∈ A = C ν .
Again, w(g, h) is subject to the condition (79).However, different from the odd-ν case, w(g, h) can now be any anyon in C ν = {1, f, v, vf }.For convenience, we take β f (g, h) = 1 and make a comment on general β f (g, h) later.Then, for (79) to be satisfied, w(g, h) must take the following form where n 2 (g, h) = 0, 1 is any 2-cochain in C 2 (G, Z 2 ).This makes η f (g, h) = (−1) λ(g,h) .Note that (82) is already a restricted form of w(g, h), since λ(g, h) is required to be a cocycle due to the original fermionic iTO.We claim that the quantity n 2 (g, h) is the same n 2 that we define in the fermionic language.Intuitively, n 2 (g, h) = 0 or 1 represents whether f appear in the symmetry fractionalization anyon w(g, h).More importantly, we show below that it satisfies the same condition dn 2 = O 3 as the one in Sec.III A.
A few remarks on gauge transformations are in order.First, w(g, h) is defined up to an ambiguous anyon dζ(g, h) = ρ g [ζ(h)]ζ(g)ζ(gh).In general, ζ(g) = v a(g) f b(g) , with arbitrary a(g), b(g) = 0, 1.However, to make a one-to-one correspondence between bosonic SETs and fermionic iTOs, we need to put some restrictions on ζ(g).Recall that ζ(g) originates from an extra phase ambiguity of the local operator U (i) g near anyon a i after (75) is satisfied.When a(g) = 1, we will have if the anyon a i = f .This means, it switches g 0 and g 1 in the fermionic language.We shall not allow such a transformation as 0 and g 1 are distinct symmetries in G f , as discussed in Sec.II A. In other words, we do not allow coboundary transformation on λ(g, h) and g ∈ G has been taken to be g 0 ∈ G f .The remaining ambiguity, ζ(g) = f b(g) , brings a coboundary transformation n 2 (g, h) → n 2 (g, h) + db(g, h).Accordingly, distinct solutions n 2 (g, h) to Eq. ( 86) are defined up to a coboundary db(g, h) in B 2 (G, Z 2 ).Then, given n 2 (g, h) being a solution, n 2 (g, h) + t(g, h) is a distinct solution if t(g, h) is a non-trivial cocycle in H 2 (G, Z 2 ).Second, we have taken the gauge β f (g, h) = 1 and Õ3 = 1.Under other gauges, one may expect Õ3 to be any possible anyons in A = {1, f, v, vf }.However, this contradicts with the fact that the right-hand side of (84) can only be 1 or f .Here, we show that Õ3 can only be 1 or f .In general, we have g) .Then, inserting this into the definition (69) of Ω a (g, h, k) and taking a = f , we find Ω f (g, h, k) = (−1) d 2 σ(g,h,k) = 1. (87) Then, Õ3 (g, h, k) must be 1 or f according to the definition (70).It is not hard to check that gauge transformations on β v (g, h) result in a shift in n 2 (g, h), which is general any way.So, O 3 in (85) is the general form.Third, it is straightforward to show that a coboundary transformation on λ does not change the cohomology class of It is expected because coboundary transformations on λ correspond to isomorphisms of G f and O 3 obstructions should be the same for isomorphic G f 's.
For certain choices of n 1 , the obstruction O 3 may be trivial, allowing a valid n 2 and thereby a valid symmetry fractionalization.In these cases, we need to further consider the Õ4 obstruction.As discussed in Sec.IV A, the general expression of Õ4 obstruction in bosonic SETs with non-trivial anyon permutations is complicated.In principle, for the current specific topological order C ν that is simple enough, one may try to solve the consistency equations of defects to obtain the general Õ4 .Here, we avoid this calculation and instead derive Õ4 for the case where λ ∪ 1 λ is a trivial coboundary, using the stacking trick to be discussed in Sec.IV D. From now on, we will drop the "˜" on O 4 to imply that it is specific for the topological order C ν .To use the stacking trick, we will need two parts, O A 4 and O B 4 .Here we give O A 4 .
Consider the case that there is no anyon permutation under G, i.e., n 1 (g) = 0 for every g.Assuming that O 3 = λ ∪ 1 λ is trivial, there exists n 2 (g, h) that satisfies O 3 = λ ∪ 1 λ = dn 2 .The symmetry fractionalization is then given by the anyon w(g, h) in (82).Since there is no anyon permutation, the formula (77) of Õ4 is applicable.We still need the F and R symbols of C ν .It is known that F and R symbols of general Abelian topological orders can be parametrized as follows [34]: where an integer vector x = (x 1 , x 2 , ...) with x i = 0, 1, ..., N i − 1 denotes an Abelian anyon, Φ i is the selfstatistical phase of the ith generating anyon, K ij is the mutual statistical phase between the ith and jth generating anyons, and [x i + y i ] = x i + y i (mod N i ).The anyons form a group A = i Z Ni under fusion.For ν = 4k + 2, we have A = Z 4 and Φ 1 = νπ/8 and K 11 = νπ/4.Under this notation, the symmetry fractionalization anyon w(g, h) in (82) becomes ω(g, h) = λ(g, h) + 2n 2 (g, h).
Inserting it into the formula (77) of Õ4 , we obtain where we have added a superscript "A" to imply that it holds only for n 1 = 0 and trivial λ ∪ 1 λ.
A more general case with n 1 = 0 (but still with a trivial λ ∪ 1 λ) can be obtained using the stacking trick in Sec.IV D. As explained there, O 4 is given by adding the O A 4 in Eq. ( 89), and the O B 4 [n 1 , n 2 ] in Eq. (90).For the most general case that even λ ∪ 1 λ is nontrivial (but with a trivial O 3 = n 1 ∪ λ + λ ∪ 1 λ), it requires a formula of O A 4 that works for nonzero n 1 , which unfortunately we do not have yet.

D. Stacking trick
We now describe the stacking trick.Given ν, the obstruction O 4 for cases with arbitrary choices of n 1 and n 2 can be computed using the following stacking trick.First, we need the formula for O 4 for a special choice of to indicate its dependence on ν.In this work, such special choice is always constructed with n A 1 = 0 to avoid the complication of anyon permutation, and such O A 4 is given in Eqs.(89), ( 97) and (100), for cases of ν = 4k+2, 8k+4 and 16k + 8, respectively.In the case of ν = 4k + 2, n A 2 is one of the solutions to dn 2 = O 3 = λ ∪ 1 λ.(That means, if λ ∪ 1 λ is nontrivial, we will not have n A 2 so that we cannot use the stacking trick.)The latter two cases will We will also denote n 1 = n B 1 and n 2 = n B 2 , but both can vary.
Stacking the two anomalous states together, their O 4 obstructions also add up.This is because the two states A and B can be realized on the surface of 3D bulk states that are bosonic SPT ] is a nontrivial cocycle for all possible combinations of n 1 and n 2 .
We notice that the 2-cochain n 2 describing the stacked anomalous cocycle may not be n A 2 + n B 2 , as stacking two SPT states with nontrivial n B 1 produces a twist on the total n 2 .However, such twists are irrelevant to our goal of finding all possible iTOs for a given ν.For fixed n A 1 and n A 2 , by varying n B 1 and n B 2 , we shall be able to exhaust all possible valid symmetry-enriched iTOs.

E. O3 and O4 for ν = 4k
We now consider the O 3 and O 4 obstructions for ν = 4k and derive ESB Criteria 5 and 6.The discussion is in parallel to the above subsection, so we will be brief in places that are not very different.
In this case, the topological order is again denoted as To begin, we define the quantity n 1 : G → Z 2 , which determines the same permutation as in (81).Again, it has the physical meaning of whether g 0 and g 1 defects carry Majorana zero modes in the original fermionic iTOs.Similarly to the ν = 4k + 2 case, we show in Appendix B that the symmetry localization anomaly Õ3 is trivial, regardless of the choice of n 1 .Moreover, there exists a gauge that β f (g, h) = 1 and Õ3 = 1.
Next, symmetry fractionalization w(g, h) is again given in the form (82) in the gauge β f (g, h) = 1, so that the conditions (78) and (79) are satisfied.The general fusion rules for ν = 4k are which is different from (83).With the form of w(g, h) in (82), the anyon permutation (81) and the gauge Õ3 (g, h, k) = 1, the obstruction condition (76) becomes: Accordingly, we define O 3 for ν = 4k as General properties of the cup product guarantee O 3 to be a cocycle in H 3 (G, Z 2 ).Then, Eq. (92) becomes where we take v and f as the first and second generators of Z 2 × Z 2 .More explicitly, by inserting the values of Φ and K and Eq. ( 88) into the expression (77), we have For our special choice of n A 2 = 0, we further have where a ∪ b and a ∪ 1 b are the (higher) cup product of two cochains.For ν = 16k + 8, the difference lies in the F and R symbols.They are given by the parameters: Φ = (π, π) , K = 0 π π 0 (98) Plugging Φ and K into (77), we have Then, for our special choice of n A 1 = n A 2 = 0, we have To summarize, we can write the expressions of O 4 with n 1 = 0 compactly as This expression applies to all ν's.

V. DISCUSSIONS
To summarize, we have studied the phenomenon of enforced symmetry breaking (ESB) by 0D, 1D and 2D fermionic invertible topological orders (iTOs).We have obtained a set of criteria for asserting the existence or non-existence of ESB for finite groups, and illustrated the ESB physics with examples in all cases.We give both fermionic and bosonic descriptions (via gauging fermion parity) of symmetry-enriched fermionic iTOs.
It is very interesting to generalize the current study to ESB by general 2D fermionic topological orders, i.e., those with anyon excitations.For example, our dimensional reduction argument in Sec.III B 1 can be easily used to show that G f from a nontrivial cocycle λ(g, h) ∈ H 2 (G, Z 2 ) is incompatible with those fermionic topological orders that support both Majorana and non-Majorana vortices after gauging the fermion parity (e.g., the SO(3) 3 fermionic topological order).Also, it is interesting to study enforced breaking of anti-unitary symmetries, such as time reversal.In this work, we only study 2D fermionic iTOs which are chiral, so that enforced breaking of time-reversal is obvious.However, for non-chiral fermionic topological orders, it is a challenging problem.Moreover, ESB might also exist for 3D fermionic topological order, and it would be very interesting to explore such examples.
For ESB phenomenon to occur, it is required that, by definition, the symmetries should have some nontrivial action in the Hilbert space, such as non-trivial extension by fermion parity or time reversal symmetry.In other words, there are conditions imposed at the very beginning.It has been clearly seen in our exploration of bosonic SETs that the obstruction functions O 2 and O 3 are due to the imposed conditions.We have named them conditional anomalies.The conditions on the bosonic SETs in our study follow from the conditions in the original fermionic iTOs.However, generally speaking, conditions on properties (such as symmetry fractionalization) of topological orders may be imposed by other reasons.For example, in the study of spin liquids, the spinon should always carry a half-integer spin in systems with an odd number of spin-1 2 s per unit cell [70] , and this indeed causes ESB phenomenon for certain topological orders [71].Therefore, a general study on conditional anomalies or conditional obstructions are very important and helpful.Finally, we conjecture that the novel concept of ESB can also be defined for gapless systems and might have important implications in fundamental physics, such as CP violation problem in Standard Model.We will leave these potential directions for future study.
Note added.Some preliminary results, regarding 0D and 1D ESB physics and 2D ESB physics from O 2 and O 3 obstructions, were previously reported in the Workshop on Strongly Correlated Systems held in January 2020 in Shenzhen, China, by one of the authors (C.W.).While we are preparing for the manuscript, we become aware of the work Ref. [72] which studies 2D symmetry-enriched fermionic iTOs and derives the general classification, including the most general form of the O 4 obstruction function.However, Ref. [72] does not study the phenomenon of enforced symmetry breaking.We also become aware of the works Refs.[67,68,73] which study the general theory of fermionic SET phases.the F and R symbols given by Now we consider the topological symmetries of C ν , which form the group Aut(C ν ).Recall from Sec. IV A that a topological symmetry contains two pieces of data: (1) a permutation of anyons x → x ′ = ϕ(x) and (2) an action in the fusion space ϕ(|x, y; z ) = u x ′ y ′ z ′ |x ′ , y ′ ; z ′ .For the current C ν , there is one and only one nontrivial permutation: φ where Θ(x) = 0 if x = 0, Θ(x) = 1 if x = 1, 2, 3, and x is the antiparticle of anyon x.Together with the trivial topological symmetry, we have Aut(C ν ) = Z 2 = {1, φ}.
We still need to specify the action of the nontrivial permutation φ in fusion spaces, i.e., to specify the phase factor u xy z .Since z is uniquely determined by x and y in Abelian topological orders, let us denote the action as where x ′ = φ(x).The condition that F symbol is invariant under φ action (see Eq. ( 65) in the main text) leads to where we have used φ(x) = x.Plugging the F symbol above into the last piece, we have By inserting (B3), we can simplify this expressions to where in the second equality, we have used the fact that [x + y] − y = x (mod 2).Comparing (B5) and (B8), we find that one choice of u(x, y) is u(x, y) = f (x, y) = e i πν 2 xΘ(y) .(B9) A general solution u(x, y) that satisfies (B5) can be expressed as u(x, y) = f (x, y)w(x, y), where w(x, y) satisfies δw = 1, i.e., it is a 2-cocycle in Z 2 (Z 4 , U (1)).
Next, we consider a group homomorphism from G to Aut(C ν ), namely If ρ g is nontrivial in Aut(C ν ), the symmetry g ∈ G acts as φ in (B4), with u(x, y) fixed in (B9); otherwise it acts as a natural isomorphism, which we fix to be u xy z = 1 for any x, y, z.More specifically, for a state |x, y; z , we have ρ g (|x, y; z ) = U g ( g x, g y)| g x, g y; g z , (B11) where g x := ρ g (x) denotes the anyon permutation and U g (x, y) can be unified as U g (x, y) = e i πν 2 n1(g)xΘ(y) , (B12) where n 1 (g) = 0 or 1, for ρ g being trivial or nontrivial in Aut(C ν ) respectively.Since ρ a group homomorphism, we have dn 1 (g, h) = n 1 (g) + n 1 (h) − n 1 (gh) = 0, (mod 2).where we used the property that xΘ(y) = ḡxΘ( ḡy) (mod 2) holds for any g.Accordingly, the phase factor β x (g, h) defined in (68) can be set to 1, making the Õ3 obstruction defined in (69) and ( 70) to be 1.This concludes the ν = 4k + 2 case.
We can now assert that O 3 is trivial if and only if Ξ ij = Ξ ijk = 0 for all i, j, k.Recall that Ω i , Ω ij (i.e., λ) are given in our problem to generate the fermionic symmetry group G f .For given {Ω i , Ω ij }, the condition Ξ ij = Ξ ijk = 0 is a set of equations with {q i } being the unknowns.Then, G f is enforced to break if these equations have no solutions.That is, ESB of G f occurs if O 3 is nontrivial regardless of n 1 .
For ν = 4k, there is always a solution to Ξ ij = Ξ ijk = 0: q i = 0 for all i, i.e., a trivial n 1 .So, O 3 obstruction alone cannot lead to ESB.For ν = 4k + 2, we do have situations that ESB can occur.With straightforward analysis on equations in (C17), we find a few examples of ESB, which are summarized in Table VI.Two general properties of (C17) are: (1) If Ω ij = 0 for all i and j and Ω i is nonzero for at least one i, say Ω I = 0 of a fixed I, the requirement Ξ iI = 0 gives q i = 0 for all i.That is, there is one and only one solution n 1 = 0.

A result for ν = 4k + 2
We have only considered O 3 obstruction above.In this subsection, we consider O 4 in a special case.We show the following corollary for finite Abelian groups.
Corollary 1.For 2D fermionic iTOs with ν = 4k + 2 and a finite Abelian group G, ESB can not occur if λ is a type-I 2-cocycle in H 2 (G, Z 2 ).
Recall that we have defined type-I 2-cocycles in H 2 (G, Z 2 ) as those with Ω ij = 0 for all i, j.If Ω i = 0 for all i, i.e., λ = 0, then ESB can never occur.If at least one Ω i is nonzero, then, according to the result at the end of Appendix C 2, we must have q i = 0 for all i to have a vanishing O 3 .To show that ESB does not occur, it remains to check if O 4 vanishes when n 1 = 0 and λ is type-I.
In that case, the O 4 obstruction formula in (89) is applicable.For type-I λ, we can write it as λ = i p i dv i /N i .It leads to two consequences: dλ = 0, and which means the piece e iνπλ∪λ/8 is a U (1)-valued 4coboundary.Then, one can easily check that n 2 = 0 makes the O 4 in (89) a U (1)-valued 4-coboundary.Accordingly, we find a solution that n 1 = n 2 = 0 such that both O 3 and O 4 vanish.Hence, there is no ESB and the corollary is proven.

3. ν = 8k + 4
Now we consider the case ν = 8k + 4. As discussed in Sec.III A 3, when ν = 4k, the O 3 obstruction reduces to O 3 = λ ∪ n 1 .Then, O 3 obstruction alone cannot give rise to enforced symmetry breaking, because there always exists the case n 1 = 0 such that O 3 = 0. Accordingly, ESB may occur only if both O 3 and O 4 are considered.We claim that Criterion 5. G f is enforced to break by the 2D fermionic iTOs with ν = 8k + 4, if a quantity

4. ν = 16k + 8
In the case ν = 16k + 8, again both O 3 and O 4 need to be taken into account to support ESB.The statement of ESB criterion is the same as in Criterion 5: Criterion 6. G f is enforced to break by the 2D fermionic iTOs with ν = 16k + 8, if a quantity

TABLE I .
Invertible topological orders of fermionic systems in low dimensions.

TABLE III .
Character table of the dihedral group D8.The first row shows the conjugacy classes and the next five rows show the characters of the five irreducible representations of D8.The fermion parity P f is identified with the elment r 2 .dim 1 r 2 {r, r 3 } {s, sr 2 } {sr, sr 3

TABLE IV .
O2, O3 or O4 obstructions for 2D fermionic iTOs with index ν.The mark "×" means that the corresponding obstruction is always trivial, and " " means it may be nontrivial.Note that O3 is meaningful only if O2 is trivial, and O4 is meaningful only if both O2 and O3 are trivial.
) follows from the fact that Q 4 is actually a 4-cocycle in H 4 (G, Z 4 ).To see that, first, it is straightforward to check that 2n 2 ∪(n 2 +λ) is Z 4 -valued 4-cocycle, following the general properties of cup products.Second, the pieceP(λ) ≡ λ ∪ λ + dλ ∪ 1 λ is mathematically known as the Pontryagin square of a Z 2 -valued cocycle λ.It is known that P(λ) is a Z 4 -valued cocycle (see Appendix A).Accordingly, Q 4 is a 4-cocycle in H 4 (G, Z 4 ).This special expression of O 4 allows us to discover a few ESB examples for ν = 8k + 4 fermionic iTOs with a finite Abelian group G, which we will discuss in Sec.III C 2.
We note that (44) and (43) are the same but ν takes different values.

TABLE V .
[26]tions of quantities in the stacking trick.F [n A 2 , n B 2 , n1, ν] is some functional that depends on n A 2 , n B 2 , n1 and ν To use the stacking trick, we necessarily require O A 3 to be a trivial cocycle, so that we have a special solution of n A 2 , which together with λ determines O A 4 .Next, to obtain all possible fermionic iTO states with index ν and generic choices of n 1 and n 2 , we stack the above chiral iTO state with a generic nonchiral fermionic SPT state B, which may also be anomalous with another O B 4 obstruction.The state B is characterized by a1cocycle n 1 ∈ H 1 (G, Z 2 ) and a 2-cochain n 2 ∈ C 2 (G, Z 2 ), satisfying dn 2 = λ ∪ n 1 .Its anomaly is given by[26] states characterized by cocycles O A Then, if O 3 is a nontrivial cocycle, there is no solution n 2 (g, h) to Eq. (94).Nevertheless, the obstruction O 3 alone cannot guarantee ESB to occur: the case that n 1 = 0 makes O 3 = 1, which always allows a solution n 2 (g, h).So, we will need to move on to O 4 to look for ESB physics.Gauge transformations on w(g, h) and Õ3 are similar to the ν = 4k + 2 case.That is, gauge transformations on w(g, h) make the solutions n 2 (g, h) to Eq. (94) distinct up to a coboundary db(g, h) in B 2 (G, Z 2 ), and Õ3 can only be 1 or f in general.Hence, the O 3 obstruction in Sec.III A is recovered.We now use the stacking trick in Sec.IV D to derive O 4 .The obstruction O B 4 is already given in (90), so we only need O A 4 here.First, we notice that since n 1 = 0 always makes O 3 vanish, we can choose n A 1 = 0 and n A 2 = 0 for the special state A. This greatly simplifies our problem.Below we describe O A 4 for different ν's.For ν = 8k + 4, explicit parametrizations of the F and R symbols are given by (88) where now x = (x 1 , x 2 ) labels Z 2 × Z 2 anyons, and similarly for y, z.The selfstatistics Φ i and the mutual statistics K ij of the generating anyons are Φ

TABLE VI .
Examples of enforced symmetry breaking due to O3 obstruction, for ν = 4k + 2, obtained by solving conditions Ξij = Ξ ijk = 0. We have used the topological invariants {Ωi, Ωij } to label λ ∈ H 2 (G, Z2).If a topological invariant is not shown, its value is arbitrary.GESB occurs if the topological invariants of λ satisfy: