Lieb-Schultz-Mattis type theorems for Majorana models with discrete symmetries

We prove two Lieb-Schultz-Mattis type theorems that apply to any translationally invariant and local fermionic $d$-dimensional lattice Hamiltonian for which fermion-number conservation is broken down to the conservation of fermion parity. We show that when the internal symmetry group $G^{\,}_{f}$ is realized locally (in a repeat unit cell of the lattice) by a nontrivial projective representation, then the ground state cannot be simultaneously nondegenerate, symmetric (with respect to lattice translations and $G^{\,}_{f}$), and gapped. We also show that when the repeat unit cell hosts an odd number of Majorana degrees of freedom and the cardinality of the lattice is even, then the ground state cannot be simultaneously nondegenerate, gapped, and translation symmetric.

In 1961, Lieb, Schultz, and Mattis (LSM) proved a theorem on the low-lying excited states of the nearestneighbor antiferromagnetic quantum spin-1/2 chain 1 . Accordingly, a quantum spin chain with an odd number of spin-1/2 degrees of freedom per repeat unit cell that is simultaneously translation and SO(3)-spin-rotation symmetric cannot realize a gapped and symmetric ground state in the thermodynamic limit.
Since its original formulation, the LSM theorem has been generalized in a number of ways including extensions to higher dimensions, other global continuous symmetry groups, and different kinds of spatial symmetries  . For instance, it has been understood that the SO(3) spin-rotation symmetry is not an essential requirement for an LSM constraint. In fact, LSM-type theorems for U (1) number-conserving Hamiltonians have been established in arbitrary dimensions. These theorems state that systems with non-integer filling fraction ν, defined as the average number of particles per unit cell, cannot have a translationally invariant, nondegenerate, and short-range entangled ground state [2][3][4][5][6]8,9,17 . Similar constraints have also been worked out for numberconserving Hamiltonians that have non-symmorphic or magnetic space group symmetries 10,11,15,23 .
A number of LSM-type theorems pertaining to discrete internal symmetries combined with crystallographic symmetries have also been worked out 11,15,[23][24][25][26][27] . In the context of spin-chains with discrete symmetries, LSMtype theorems were proved by Ogata et al. in Refs. [26][27][28]. They found that a translationally-invariant spin chain with half-integer spin at each site that possesses either time-reversal or Z 2 × Z 2 -rotation symmetries (rotations by π around two axis, say x and z), cannot have a nondegenerate, gapped, and symmetric ground state. The proof of this statement is based on the fact that a nondegenerate gapped ground state of a local Hamiltonian satisfies the so called split property 29,30 . LSMtype no-go theorems are then derived for states satisfying the split property by using operator algebra techniques and Gelfand-Naimark-Segal (GNS) construction. Alternatively, similar no-go constraints are obtained in Refs. 31 and 32 within the framework of matrix product states (MPS) [33][34][35] . These derivations are based on the fact that one can approximate 36 the nondegenerate gapped ground states of local Hamiltonians by injective MPS 37,38 . There are two hypothesis common to many of these LSM-type theorems with discrete symmetries. It is presumed that there exist local (on-site) degrees of freedom that span a local Hilbert space and realize a nontrivial projective representation of the global symmetry. It is also presumed that the global Hilbert space is obtained by postulating that the local degrees of freedom commute when separated in space. The resulting LSM-type theorems are applicable to bosonic systems. Generalizations to bosonic quantum systems in arbitrary dimension with crystallographic symmetries and general discrete Abelian symmetries have been proposed using the notion of lattice homotopy 25 .
There exists similarities between Hamiltonians obeying LSM-type constraints and the boundary modes of a symmetry-protected-topological (SPT) insulator. An SPT insulator has a nondegenerate, symmetric, and gapped ground state when periodic boundary conditions are imposed. When open boundary conditions are imposed, the effective low-energy quantum Hamiltonian governing the dynamics of the boundary modes of an SPT insulator supports a ground state that is either (i) gapless, (ii) symmetry-broken, (iii) or topologically ordered if the boundary is no less than two dimensional 24,[39][40][41][42][43] . From a low-energy perspective, both the effective boundary Hamiltonian of an SPT insulator and the bulk Hamiltonian satisfying an LSM-type constraint display quantum anomalies 18,44,45 . For the former case, the quantum anomaly is typically that for a global symmetry that acts locally on the boundary [46][47][48][49][50] . In the latter case, there is a mixed quantum anomaly between a global symmetry that acts locally and a spatial symmetry such as translation. These parallels led to the formulation of so-called weak-SPT-LSM-type theorems 19,44,[51][52][53][54] . In particular, Ref. 52 has conjectured an LSM-type constraint for d-dimensional fermionic lattice Hamiltonians with the help of a bulk-boundary correspondence. The fermionic d-dimensional lattice Hamiltonian is interpreted as the low-energy effective theory of a fermionic d + 1-dimensional lattice Hamiltonian that is gapped but supports midgap boundary states such that (i) they can be localized at each site of the ddimensional lattice (ii) where they span a local fermionic Fock space. The parent fermionic (d+1)-dimensional lattice Hamiltonian is an example of a weak-SPT fermionic insulator. LSM-like constraints on the ground states of the d-dimensional lattice Hamiltonian are inherited from the symmetries that protect the boundary states of the (d + 1)-dimensional lattice Hamiltonian.
As compared to LSM-type theorems for bosonic and U (1)-charge-conserving fermionic Hamiltonians, LSMtype theorems for fermionic Hamiltonians without any U (1)-conserving symmetries are much less explored 49,52 . These LSM-type constraints would be relevant for any long-range superconducting order with fully broken SU (2)-spin-rotation that coexists with some additional long-range order. Such fermionic Hamiltonians always admit a formulation in terms of Majorana degrees of freedom. To the best of our knowledge, there are no proofs of LSM-type constraints relevant to fermionic lattice Hamiltonians with translation symmetry and some discrete internal symmetry (such as time-reversal symmetry, say) for which fermion-number conservation is broken down to the conservation of fermion parity.
In the present work, we state and prove two LSM-type theorems. They apply to translationally invariant lattice Hamiltonians acting on a fermionic Fock space spanned by Majorana degrees of freedom. The lattice is embedded in d-dimensional Euclidean space. For Theorem 1, there also exists a global symmetry associated to a symmetry group G f that can be realized locally, i.e., the number of Majorana degees of freedom in a repeat unit cell of the latttice is even. We prove within the framework of fermionic MPS (FMPS) that whenever the Majorana degrees of freedom within a single repeat unit cell realize a nontrivial projective representation of G f , then the lattice Hamiltonian cannot have a nondegenerate, gapped, and symmetric ground state that can be described by an even-or odd-parity injective FMPS (for d > 1 we must assume that G f is Abelian and all its elements are represented by unitary operators), in agreement with the conjecture made by Cheng in Ref. 52 when the fermion number is conserved. For Theorem 2, it is only assumed that the repeat unit cell supports an odd number of Majorana modes, the cardinality of the lattice is even, and translation symmetry holds. It then follows that the ground state cannot be simultaneously nondegenerate, gapped, and translation symmetric.
The rest of the paper is organized as follows. We introduce the main results of the present work in the form of Theorems 1 and 2 in Sec. II. We present an overview of the internal symmetry group G f and its projective representations in Sec. III. We introduce the framework for FMPS and present a FMPS-based proof of Theorem 1 in Sec. IV when the dimension d of space is one. Theorem 2 when d = 1 is then proved by making use of Theorem 1. An independent proof of Theorem 2 is given for any dimensions d of space in Sec. V. A weaker version of Theorem 1 for any d ≥ 1 is also provided in Sec.. V. The latter proof is based on symmetry twisted boundary conditions on twisted lattices. Finally, we collect several examples in Sec. VI and conclude the main body of the paper with a summary in Sec. VII. We present details about group cohomology, FMPS construction and proof of Theorem 1 in Appendices A, B and C, respectively.

II. MAIN RESULTS
The notion of a global fermionic symmetry group G f with a local action plays a central role in this paper. What we have in mind with this terminology is any lattice model obtained by discrete translations of a repeat unit cell. The same even integer number n = 2m of Majorana degrees of freedom (flavors), i.e., a local fermionic Fock space of dimension 2 m , is attached to any repeat unit cell. The fermion parity can then be defined for any repeat unit cell. Even though the fermion parity is generically not conserved locally, it must be conserved globally. Hence, the symmetry group G f necessarily contains as a subgroup the cyclic group of two elements generated by the fermion parity. The action of any other element from the symmetry group G f can be defined locally, i.e., its action is represented by the same polynomial on the algebra generated by the Majorana operators within the repeat unit cell for any repeat unit cell of the lattice. As with the fermion parity, this symmetry element need not be conserved locally but must be conserved globally. A translation along the basis that generates the lattice relates different repeat unit cells. Similarly, any crystalline symmetry lies outside of the symmetry group G f .
The motivation for Theorem 1 is the following. Given is a local lattice Hamiltonian H that is symmetric under the group G trsl × G f , where G trsl denotes the group of lattice translations. Assume that the ground state |Ψ is symmetric under both G trsl and G f , i.e., the ground state can change by no more than a multiplicative phase factor (combined with complex conjugation when the symmetry is represented by an antinunitary operator). Assume that a gap separates the ground state from all excited states. Are there sufficient and necessary conditions for |Ψ to be degenerate?
A sufficient condition for all energy eigenvalues to be degenerate is that at least one generator from G trsl and one element g from G f are represented globally by operators that commute projectively, i.e., when passing the translation operator from the left to the right of the operator representing globally the element g, a multiplicative phase factor that cannot be gauged to unity arises. Another sufficient condition is that g is represented globally by an operator that squares to minus the identity (Kramers' theorem). However, none of these conditions are necessary for the ground state to be degenerate in the thermodynamic limit. Even if all operators representing G trsl and all global operators representing G f commute pairwise and if all global operators representing G f square to unity, a gapped and symmetric ground state might still be degenerate in the thermodynamic limit.
To develop an intuition for this last claim, we consider a one-dimensional lattice with the topology of a ring and assume that G f is an Abelian group such that any element g ∈ G f is represented globally by the unitary operator U g . We denote with |exp(iK), exp(iU g ) a gapped many-body ground state of H that is a si-multaneous eigenstate with the eigenvalue exp(iK) of the unitary operator T 1 representing a translation by one repeat unit cell and of U g with eigenvalue exp(iU g ). We denote with T 1,h |exp(iK), exp(iU g ) the many-body state obtained from |exp(iK), exp(iU g ) by translating all lattice repeat unit cell by one to the right and by acting locally with the local unitary operator representing h ∈ G f on the last repeat unit cell of the chain. The energy expectation value of T 1,h |exp(iK), exp(iU g ) is no less than the ground-state energy. A sufficient condition for T 1,h |exp(iK), exp(iU g ) to be orthogonal to |exp(iK), exp(iU g ) is that the product U g T 1,h differs from the product T 1,h U g by a multiplicative phase exp(iχ g,h ) = 1 that cannot be gauged away. If the energy expectation value of the state T 1,h |exp(iK), exp(iU g ) that is orthogonal to the ground state |exp(iK), exp(iU g ) converge to the ground-state energy in the thermodynamic limit, we have shown that the groundstate is degenerate even when T 1 and U h commute for all g ∈ G f . Proving rigorously this last claim is difficult. However, this claim is plausible when the lattice Hamiltonian is local and the ground state is gapped, for the gapped ground state |exp(iK), exp(iU g ) is then short-range entangled and so is T 1,h |exp(iK), exp(iU g ) . Their energy expectation values should only differ by a term of order zero in the number of repeat unit cells. This energy difference thus vanishes in the thermodynamic limit.
Theorem 1. Any one-dimensional lattice Hamiltonian that is local, admits a global fermionic symmetry group G f with a local action, and is invariant under all lattice translations cannot have a nondegenerate, gapped, and (G f and translation symmetric) symmetric ground state that can be described by an even-or odd-parity injective FMPS if G f is realized by a nontrivial projective representation on the local Fock space.
Theorem 2. A translationally invariant and local Majorana Hamiltonian with an odd number of Majorana degrees of freedom per repeat unit cell cannot have a nondegenerate, gapped, and translationally invariant ground state. Comment 1. The thermodynamic limit is implicit in both theorems. Comment 2. Theorem 1 is only predictive when G f is realized by a nontrivial projective representation on the local Fock space. When G f is a Lie group its projective representation on the local Fock space can be trivial. If so, Theorem 1 is not predictive. However, one can use complementary arguments such as the adiabatic threading of a gauge flux to decide if the ground state is degenerate. It is when G f is a finite group that the full power of Theorem 1 is unleashed. Comment 3. Theorem 1 is proved within the FMPS framework in Sec. IV. A weaker form of Theorem 1 holds in any dimension if it is assumed that G f is Abelian and can be realized locally using unitary operators. The weaker version of Theorem 1 that is valid in any dimension is proved using tilted and twisted boundary conditions in Sec. V. Comment 4. Theorem 2 applies in any dimension of space.

III. PROJECTIVE REPRESENTATIONS OF SYMMETRIES OBEYED BY MAJORANAS
Theorems 1 and 2 relate the quantum dynamics obeyed by Majoranas to the symmetries they obey. The smallest symmetry group associated to Majoranas originates from the conservation of the parity (evenness or oddness) of the total number of fermions. This symmetry is associated to a cyclic group of order two that we shall denote with Z F 2 . Other symmetries are possible, say timereversal symmetry or spin rotation symmetry. All such additional symmetries define a second group G. The first question to be answered is how many different ways are there to marry into a group G f the intrinsic symmetry group Z F 2 of Majoranas with the model-dependent symmetry group G. This problem in known in group theory as the central extension of G by Z F 2 . It delivers a family of distinct equivalence classes with each equivalence class [γ] in one-to-one correspondence with the cohomology group H 2 G, Z F 2 . This result is motivated in Sec. III A.
Once a representative symmetry group G f has been selected from [γ] ∈ H 2 G, Z F 2 , its representation on the Fock space spanned by all the local quantum degrees of freedom, a set that includes Majorana operators, must be constructed. Hereto, there are many possibilities. Their enumeration amounts to classifying the inequivalent projective representations of the group G f . All the inequivalent projective representations of any one of the groups G f obtained in Sec. III A are in one-to-one correspondence with the cohomology group H 2 G f , U (1) c . This result is motivated in Sec. III B. The computation of H 2 G f , U (1) c is done in Sec. III B and Sec. VI.
Given two projective representations, a third one can be obtained from a graded tensor product as is explained in Sec. III C. We shall describe the stacking rules used to construct nontrivial projective representations.
A. Marrying the fermion parity with the symmetry group G For quantum systems built out of an even number of local Majorana operators, it is always possible to express all Majorana operators as the real and imaginary parts of local fermionic creation or annihilation operators. The parity (evenness or oddness) of the total fermion number is always a constant of the motion. If F denotes the operator whose eigenvalues counts the total number of local fermions in the Fock space, then the parity operator (−1) F necessarily commutes with the Hamiltonian that dictates the quantum dynamics, even though F might not, as is the case in any mean-field treatment of superconductivity.
We denote the group of two elements e and p Z F 2 . . = {e, p | e p = p e = p, e = e e = p p} , (3.1) whereby e is the identity element and we shall interpret the quantum representation of p as the fermion parity operator. It is because of this interpretation of the group element p that we attach the upper index F to the cyclic group Z 2 . In addition to the symmetry group Z F 2 , we assume the existence of a second symmetry group G with the composition law · and the identity element id. We would like to construct a new symmetry group G f out of the two groups G and Z F 2 . Here, the symmetry group G f inherits the "fermionic" label f from its center Z F 2 . One possibility is to consider the Cartesian product with the composition rule The resulting group G f is the direct product of G and Z F 2 . However, the composition rule (3.2b) is not the only one compatible with the existence of a neutral element, inverse, and associativity. To see this, we assume first the existence of the map whereby we impose the conditions γ(id, g) = γ(g, id) = γ(g −1 , g) = γ(g, g −1 ) = e, (3.3b) for all g ∈ G and for all g 1 , g 2 , g 3 ∈ G. Second, we define G f to be the set of all pairs (g, h) with g ∈ G and h ∈ Z F 2 obeying the composition rule One verifies the following properties. First, the order within the composition h 1 h 2 γ(g 1 , g 2 ) is arbitrary since Z F 2 is Abelian. Second, conditions (3.3b) and (3.3c) ensure that G f is a group with the neutral element (id, e), the inverse (g −1 , h −1 ), and the center (id, Z F 2 ), i.e., the group G f is a central extension of G by Z F 2 . Third, the map γ can be equivalent to a map γ of the form (3.3a) in that they generate two isomorphic groups. This is true if there exists the one-to-one map induced by the map such that the condition In other words, γ and γ generate two isomorphic groups if the identity holds for all g 1 , g 2 ∈ G. This group isomorphism defines an equivalence relation. We say that the group G f obtained by extending the group G with the group Z F 2 through the map γ splits when a map (3.4b) exists such that κ(g 1 · g 2 ) · γ(g 1 g 2 ) = κ(g 1 ) · κ(g 2 ) (3.7) holds for all g 1 , g 2 ∈ G, i.e., G f splits when it is isomorphic to the direct product (3.2). The task of classifying all the non-equivalent central extensions of G by Z F 2 through γ is achieved by enumerating all the elements of the second cohomology group H 2 G, Z F 2 , see Appendix A. We define an index [γ] ∈ H 2 G, Z F 2 to represent such an equivalence class, whereby the index [γ] = 0 is assigned to the case when G f splits.

B. Projective representations of the group G f
We denote with Λ a d-dimensional lattice with j ∈ Z d labeling the repeat unit cells. We are going to attach to Λ a Fock space on which projective representations of the group G f constructed in Sec. III A are realized. This will be done using four assumptions. Assumption 1 We attach to each repeat unit cell j ∈ Λ the local Fock space F j . This step requires that the number of Majorana degrees of freedom in each repeat unit cell is even. It is then possible to define the local fermion number operatorf j and the local fermion-parity operatorp We assume that all local Fock spaces F j with j ∈ Λ are "identical", in particular they share the same dimensionality D. This assumption is a prerequisite to imposing translation symmetry. Assumption 2 Each repeat unit cell j ∈ Λ is equipped with a representationû j (g) of G f through the conjugation of any operatorô j acting on the local Fock space F j . The representation (3.9a) of g ∈ G f can either be unitary or antiunitary. More precisely, let be a homomorphism. We then have the decomposition is a unitary operator with the fermion parity ρ(g) ∈ {0, 1} ≡ Z 2 acting linearly on F j and K denotes complex conjugation on the local Fock space F j . Accordingly, the homomorphism c(g) dictates if the representation of the element g ∈ G f is implemented through a unitary operator [c(g) = 1] or an antiunitary operator [c(g) = −1]. Finally, we always choose to represent locally the fermion parity p ∈ Z F 2 by the Hermitian operatorp j , Assumption 3 For any two elements g, h ∈ G f [to simplify notation, g • γ h ≡ g h for all g, h ∈ G f ], whereby e = g g −1 = g −1 g denotes the neutral element and g −1 ∈ G f the inverse of g ∈ G f , we postulate the projective representation whereby the identity operator acting on F j is denoted 1 1 D and the function must be compatible with the associativity in G f , i.e., for all g, h, f ∈ G f 55 . The u(1)-valued function φ satisfying (3.11b) is an example of a 2-cocycle. Given the neutral element e ∈ G f , a normalized 2-cocycle obeys the additional constraint φ(e, g) = φ(g, e) = 0 (3.11c) for all g ∈ G f . Two 2-cocycles φ(g, h) and φ (g, h) are said to be equivalent if they can be consistently related through a map [a u(1) gauge transformation] ξ : as follows. The equivalence relation φ ∼ φ holds if the transformationû between the 2-cocycles φ(g, h) associated to the projective representationû(g) and the 2-cocycles φ (g, h) associated to the projective representationû (g) 56 . In particular,û is equivalent to an ordinary representation (a trivial projective representation) if φ (g, h) = 0 for all g, h ∈ G f . Any φ ∼ 0 is called a coboundary. For any coboundary φ there must exists a ξ such that (3.14) The space of equivalence classes of projective representations is obtained by taking the quotient of 2-cocyles (3.11b) by coboundaries (3.14). The resulting set is the second cohomology group H 2 G f , U (1) c , which has an additive group structure. Appendix A gives more details on group cohomology. Assumption 4 We attach to Λ the global Fock space F Λ by taking the appropriate product over j of the local Fock spaces F j . This means that we impose some algebra on all local operators differing by their repeat unit cell labels. Example 1, any two local fermion number operatorsf j andf j must commute for any two distinct repeat unit cell j = j ∈ Λ. The total fermion and fermion-parity numbers are respectively. Example 2, Any two Majorana operators labeled by j = j ∈ Λ must anticommute. If the filling pattern is smoothly tuned (through an on-site potential whose magnitude is color coded, say) so that both circles in an repeat unit cell have the same filling, then the permutation of the left and right circles within all repeat unit cells is a symmetry. One may then choose a smaller repeat unit cell, a square centered about one circle only. (c) Image of (a) under the permutation of the empty and filled circle within all repeat unit cells.
Example 3, The algebrâ u j (g)û j (g ) = (−1) ρ(g) ρ(g )û j (g )û j (g) (3.17) holds for any distinct j = j ∈ Λ and any g, g ∈ G f because of Eq. (3.9d). We then define the operator that implements globally on the Fock space F Λ the operation corresponding to the group element g ∈ G f . Theorem 1 refers to nontrivial projective representations of the symmetry group G f constructed in Sec. III A. We are going to characterize the projective representations constructed in Sec. III B by two indices [ν] and [ρ] and show that a projective representation of the symmetry group G f is nontrivial if and only if either of these two indices is nontrivial (i.e., not the neutral element of some underlying Abelian group).
Theorem 1 presumes the existence of a local projective representation of the symmetry group G f . This is only possible if the local Fock space F j defined in Sec. III B is spanned by an even number of Majorana operators. This hypothesis precludes a situation in which a fermion number operator is well defined globally but not locally, for example when the lattice Λ is made of an even number of repeat unit cells, but a repeat unit cell is assigned an odd number of Majorana operators. (This can happen upon changing the parameters governing the quantum dynamics as is illustrated in Fig. 1.) We introduce the index µ = 0, 1 to distinguish both possibilities. The case µ = 0 applies when the number of local Majorana operators at site j ∈ Λ is even, in which case the number of repeat unit cells in Λ is any positive integer. The case µ = 1 applies when the number of local Majorana operators at site j ∈ Λ is odd, in which case the number of repeat unit cell in Λ must necessarily be an even positive integer. The triplet of indices allows to treat Theorem 1 and 2 together, as we are going to explain. A similar set of three indices also appears in the classification of one-dimensional fermionic SPT phases. This is not an accident, for fermionic SPT phases can also be classified in terms of the projective representations realized by the global symmetries after projection on the boundaries 57-61 .

Indices [ν] and [ρ]
Theorem 1 is only predictive once it is established that a projective representation of G f is non trivial. We recall that (i) the group G f is a central extension of the group G by Z F 2 through the map γ defined in Eq. (3.3) and (ii) only the equivalence classes [γ] ∈ H 2 G, Z F 2 deliver non-isomorphic groups. Choosing an element [γ] ∈ H 2 G, Z F 2 specifies G f . In turn, a projective representation of G f is specified by choosing an element φ from the equivalence class [φ] ∈ H 2 G f , U (1) c where the 2-cocycle φ was defined in Eq. (3.11). By convention, a projective representation of G f is trivial if φ ∼ 0, i.e., is the trivial (i.e., neutral) element of H 2 G f , U (1) c equipped with the addition as group composition.
Deciding if a projective representation of G f is non trivial amounts to calculating H 2 G f , U (1) c . This task is facilitated by the fact that H 2 G f , U (1) c can be related to C 2 G, U (1) c , H 2 G, Z F 2 , and H 1 G, Z 2 , (see Appendix A for definitions of these sets) as was shown in the physics literature by Turzillo and You in Ref. 60. When the equivalence class [γ] ∈ H 2 (G, Z F 2 ) is trivial, [γ] = 0, the set C 2 G, U (1) c reduces to the second cohomology group H 2 G, U (1) c as we shall explain.
To understand this claim, we consider first the simpler case when the central extension G f of the group G by Z F 2 splits, i.e., when according to Sec. III A. If so, we can apply the Kuenneth formula for group cohomology, Equation (3.22) states that the Abelian group H 2 G f , U (1) c is the direct product between the Abelian group H 2 G, U (1) c and the Abelian group The cohomology group H 2 (G, U (1) c ) is obtained by restricting the domain of definition of the function φ in Eq. (3.11) to the domain of definition G × G. We will reserve the letter ν to denote such a function. An element of the Abelian group H 2 (G, U (1) c ) with the addition as group composition is the equivalence class [ν] with the neutral element The presence of the cohomology group H 1 (G, Z 2 ) on the right-hand side of Eq. (3.22) can be understood as follows. Choose h in Eq. (3.10b) to be the fermion parity p (by the inclusion map). We then havê u(g)û(p) = e iφ(g,p)û (g p),û(p)û(g) = e iφ(p,g)û (p g).

(3.24)
Because p belongs to the center of G f , g p = p g implies thatû (g)û(p) = e i[φ(g,p)−φ(p,g)]û (p)û(g). (3.25) Because the eigenvalues of the fermion parity operator are 0 or 1, φ(g, p) − φ(p, g) = n π (3.26) for some integer n, i.e., the projective representation of the parity operatorû(p) either commutes (n even) or anticommutes (n odd) with the projective representation u(g) of any element of G. Hence, we may define the map (3.32) The local representation of G f over F This representation satisfies the composition rulê where the phase φ(g, h)∈ [0, 2π) is The first two terms arise due to the composition rule of the two representationsû (1) j andû (2) j . The last term encodes the fact that representation (1) of element h and representation (2) of element g anticommute if both of them have odd fermion parity. Equation (3.33c) can be decomposed into the stacking relations Here, ρ 1 ρ 2 is the cup product of the two 1-cocycles ρ 1 and ρ 2 (see Appendix A for the definition of the cup product). Turzillo and You in Ref. 60 have shown that a similar relationship holds in the context of SPT phases of matter. The graded tensor product also allows to stack two projective representations (0, 0, µ 1 ) and (0, 0, µ 2 ) with µ 1 = 1 mod 2 and µ 2 = 1 mod 2. The result is a projective representation of the form [ν], [ρ], 0 since it is then possible to define a local Fock space.

IV. MAJORANA LSM THEOREM IN ONE DIMENSION
In this section, we sketch the proofs for Theorems 1 and 2 in 1D space using the machinery of fermionic matrix product states (FMPS). We relegate some intermediate steps and technical details to Appendix C. We begin with a definition of FMPS in Sec. IV A (further background can be found in Appendix B. and Refs. 58-60, and 63).
The strategy that we follow is to prove that so-called injective FMPS are only compatible with a trivial projective representation of the symmetry group G f discussed in Sec. III. The main steps of the proof for Theorems 1 and 2, the first main results of this paper, are provided in Sec. IV B and IV C, respectively. We close by discussing parallels with the SPT phases in Sec. IV D.

A. Fermionic Matrix Product States
Consider a one-dimensional lattice Λ ∼ = Z N . At the repeat unit cell j = 1, · · · , N , the local fermion number operator is denotedf j and the local Fock space of dimension D j is denoted F j ∼ = C D j . We define with |ψ σ j , σ j = 1, · · · , D j , (4.1a) an orthonormal basis of F j such that The fermion parity eigenvalue of the basis element |ψ σ j is thus denoted (−1) |σ j | with |σ j | ≡ 0, 1. The local Fock space F j admits the direct sum decomposition where, given p = 0, 1, j . . = span |ψ σ j , σ j = 1, · · · , D j |σ j | = p .
(4.2b) One verifies that dim F To construct the Fock space F Λ for the lattice Λ, we demand that the direct sum (4.2) also holds for F Λ . This is achieved with the help of the Z 2 tensor product ⊗ g . This tensor product preserves the Z 2 -grading structure. We define the reordering rule on any two basis elements |ψ σ j and |ψ σ j of F j and F j for any two distinct sites j ∈ Λ and j ∈ Λ, respectively. The rule (4.3) guarantees that states are antisymmetric under the exchange of an odd number of fermions on site j with an odd number of fermions on site j while symmetric otherwise. We then define the fermionic Fock space F Λ for the lattice Λ to be As the parity |σ j | of the state |ψ σ j can be generalized to the parity |σ| of the state |Ψ σ through the action of the global fermion number operator the Fock space (4.4) inherits the direct sum decomposition (4.2a), Any state |Ψ ∈ F Λ has the expansion with the expansion coefficient c σ ∈ C. Such a state is homogeneous if it belongs to either F Λ , in which case it has a definite parity |Ψ| ≡ 0, 1. From now on, we assume that all local Fock spaces are pairwise isomorphic, i.e., This assumption is needed to impose translation symmetry below. We describe the construction of two families of states that lie in F The 2 × 2 grading that is displayed is needed to represent the Z 2 grading in Eq. (4.6) as will soon become apparent. The anticommuting matrices P and Y belong to the set Mat(2M, C) of all 2M × 2M matrices. This set is a 4M 2dimensional vector space over the complex numbers 64 . For any σ j = 1, · · · , D with j ∈ Λ we choose the matrices with the help of which we define the matrices and from Mat(2M, C). Observe that Eq. (4.10c) is a special case of Eq. (4.10b). For any σ j = 1, · · · , D with j ∈ Λ, the matrix P commutes (anticommutes) with A (p) σ j when |σ j | = 0 (|σ j | = 1), for both p = 0, 1. In contrast, the matrix Y commutes with A for all σ j = 1, · · · , D with j ∈ Λ. We are ready to define the FMPS. We define states with either periodic boundary conditions (PBC) or antiperiodic boundary conditions (APBC) denoted by the parameter b = 0 or 1, respectively. They are and for any choice of the matrices (4.10b) and (4.10c), respectively, and with the basis (4.4) of the Fock space F Λ . The following properties follow from the cyclicity of the trace and from the fact that Y is traceless. (4.14b) σ j }; b changes by a multiplicative phase under a translation by one repeat unit cell. Indeed, one verifies that where T b is the generator of translation by one repeat unit cell with boundary conditions b = 0, 1 and k ∈ Z. σ j } for p = 0, 1, respectively. For example, the similarity transformation , · · · , D, j = 1, · · · , N, (4.16) with U any matrix that commutes with P leaves the trace unchanged. Another example occurs if there exists a nonvanishing matrix Q = Q 2 ∈ Mat(2M, C) such that Indeed, one verifies that, for any matrix Z ∈ Mat(2M, C), Eq. (4.17) implies the identity D are decomposable into a block diagonal form. It can be shown that a sufficient condition on the D matrices A (0) 1 , · · · , A (0) D to prevent that they share the same reducible form is to demand that there exists an integer 1 ≤ ≤ N such that the vector space spanned by the D matrix products is Mat(2M, C) 65 . More precisely, for any A∈Mat(2M, C), it is possible to find D coefficients a In order to restrict the redundancy in the choice of the matrices (4.10) that enter the FMPS (4.13), we make the following definitions.
The need to distinguish the definitions of injectivity for even-and odd-parity FMPS stems from the fact that for an odd-parity FMPS the matrix Y commutes with A D . In other words, Y is in the center of the algebra closed by products of A D . Injectivity requires this center to be generated by 1 1 2M and Y , i.e., the algebra closed by products of matrices A For the center to be generated by no more than 1 1 2M and Y , the product of G 1 , · · · , G D must close a simple algebra of M × M matrices, which is precisely the Definition 2. The following properties of FMPS are essential to the proofs of Theorems 1 and 2.
of 2M × 2M matrices span Mat(2M, C) for any injective even-parity FMPS. The D products G σ 1 · · · G σ of M × M matrices span Mat(M, C) for any ≥ injective oddparity FMPS. Property 5. If two sets of matrices {A σ j } generate the same injective FMPS, there then exists an invertible matrix U and a phase ϕ U ∈ [0, 2π) such that 58 for any σ j = 1, · · · , D, and for p = 1. Here, the phase ϕ U is needed to compensate for the possibility that the matrix U anticommutes with P or Y . We also observe that the index σ j that labels the local fermion number is preserved under the conjugation by U . The transformation (4.20) that leaves an injective FMPS invariant is called a gauge transformation.
With the formalism introduced in Secs. III and IV A, we restate Theorem 1 as follows: If an even-or odd-parity injective FMPS is translation-invariant and symmetric under a projective representation of the symmetry group G f defined in Sec. III, then the projective representation of G f must have a trivial second group cohomology class [φ] = 0. We recover Theorem 1 by negating this statement.

B. Proof of Theorem 1
Our strategy is inspired by the study of injective bosonic MPS assumed to be G f invariants made by Tasaki in Ref. 32. For the fermionic case, we shall distinguish the cases of even-and odd-parity FMPS, as each case demands distinct conditions for injectivity. For the case of even-parity injective FMPS, we shall establish the following identity between any matrix A ∈ Mat(2M, C) and a given norm preserving W ∈ Mat(2M, C) that is induced by a projective representation of the symmetry group G f . There exists a phase δ ∈ [0, 2π) and a nonvanishing positive integer such that holds for all = , +1, +1, · · · and all A∈Mat(2M, C). This is only possible if which obviously holds when A is the identity matrix 1 1 2M . For the case of odd FMPS, we shall establish the same identity as (4.21) for any matrix A ∈ Mat(2M, C) that commutes with matrix Y , i.e., Y is in the center of the algebra spanned by such matrices A. Theorem 1 will follow from the interpretation of the condition δ = 0 as the projective representation of G f defined in Sec. III to have trivial second group cohomology class.

Case of even-parity injective FMPS
We start from the even-parity injective FMPS Let g be an element from G f be represented by the operator U (g) as defined in Sec. III B.
On the one hand, we have the identity On the other hand, we have the identity is an eigenstate of the norm-preserving operator U (g), as it should be if G f is a symmetry. By the assumption of injectivity, the matrices A (0) σ j (g) and e iη(g;b)/N A (0) σ j are related by a similarity transformation (4.20), i.e., there exists an invertible matrix U (g) and a phase ϕ for any σ j . We massage Eq. (4.25) into Consider a second element h ∈ G f asides from g ∈ G f . We can use the relation (4.26a) with g replaced by the composition g h. We can also iterate the relation (4.26a) by evaluating the composition ) that relates the normalized 2-cocycle defined in Eqs. (3.10) and (3.11) to the phase (4.26a), (ii) the map represented by for any σ j = 1, · · · , D, where We are going to make use of the injectivity of the FMPS a second time after massaging Eq. (4.27c) into for any σ j = 1, · · · , D. For any integer = 1, 2, · · · , iteration of Eq. (4.28) gives . Hence, we arrive at the identity for any A ∈ Mat(2M, C), which implies, in turn, that W (g, h) belongs to the center of the algebra spanned by monomials j=1 A (0) σ j . For even-parity FMPS, this center is one-dimensional as it is generated by the unit matrix 1 1 2M . In particular, we can choose A = 1 1 2M for which and, therefore, [φ] = 0.

Case of odd-parity injective FMPS
The odd-parity FMPS differs from the even-parity FMPS in that the D products A (1) σ for any ≥ span a subalgebra of Mat(2M, C) with the center spanned by 1 1 2M and Y . This difference is of no consequence until reaching the odd-parity counterpart to Eq. (4.25). However, for the odd-parity conterpart to Eq. (4.25) multiplication of U (g) from the left by any element from the center generated by 1 |a(g)| 2 + |b(g)| 2 = 1, (4.32) leaves Eq. (4.25) unchanged. To fix this subtlety, we replace U (g) in Eq. (4.25) by U (0) (g) which is defined by With this change in mind, all the steps leading to Eq. (4.27) for the even-parity case can be repeated for the odd-parity case. The analogue to the even-parity coboundary condition (4.31b) then follows, thereby completing the proof of Theorem 1.

C. Proof of Theorem 2
Theorem 1 presumes the existence of a local fermionic Fock space, i.e., of an even number of Majorana degrees of freedom per repeat unit cell. This hypothesis precludes translation invariant lattice Hamiltonians with odd number of Majorana operators per repeat unit cell such as Here, the Hermitan operators {γ j =γ † j } obey the Majorana algebra γ j ,γ j = 2δ jj , j, j = 1, · · · , 2M, (4.34b) and the total number 2M of repeat unit cell is an even integer. Hamiltonian H K realizes the critical point between the two topologically distinct phases of the Kitaev chain. In the continuum limit, it describes a helical pair of Majorana fields and has a gapless spectrum. Motivated by this example, we now prove a separate LSM constraint on Majorana lattice models with an odd Majorana flavors per repeat unit site. We use Theorem 1 for the proof.
Let n ≥ 0 be an integer and be the spinor made of 2m + 1 Majorana operators. Let the Hamiltonian H be local and translationally invariant. We write which is the sum of two copies of Hamiltonian (4.36). The repeat unit cell labeled by j = 1, · · · , 2M now contains two Majorana spinors labeled by α = 1, 2. Hamiltonian (4.37) thus acts on a Fock space which is locally spanned by an even number of Majorana flavors. At each site j = 1, · · · , 2M one can define a local fermionic Fock space. Since there is no coupling between the two copies α = 1, 2 of Majorara spinors, H inherits from H the nondegenerate gapped ground state Since at each site j, there is no term coupling the two copiesχ j , H is invariant under any local permutation The local representation of the fermion parity operator is Under the transformation (4.38a), the local fermion parity operator P j acquires the phase (−1) 2n+1 = −1. Therefore, the symmetry transformation (4.38a) anticommutes with P j . This anticommutation relation implies a nontrivial second group cohomology class [φ] = 0 of G f , independent of the group of onsite symmetries of Hamiltonian (4.36) 67 . Therefore, by Theorem 1 Hamiltonian H cannot have a nondegenerate gapped ground state. This is in contradiction with the initial assumption that Hamiltonian (4.36) has the nondegenerate gapped ground state |Ψ 0 . We observe that dimensionality d of space played no role in the proof of Theorem 2 until Theorem 1 was used. Hence, Theorem 2 holds for any d if Theorem 1 holds for any d.
One can interpret Theorem 2 as the inability to write down an injective FMPS for the ground state of translationally invariant Hamiltonians with an odd number of Majorana flavors per repeat unit cell. This is because one cannot define the matrices A σ j as there is no well-defined Fock space at site j to begin with.

D. LSM constraints and classification of 1D fermionic SPTs
In order to prove Theorem 1, we have shown that δ(g, h; b) defined in Eq. (4.27a) vanishes. Consequently, W (g, h; b) defined in Eq. (4.27d) must be proportional to the unit matrix 1 1 2M . If so, the similarity transformations V (g; b), V (h; b), and V −1 (g h; b) that enter W (g, h; b) must also realize a projective representation of G f . This observation allows us to draw a bridge to the classification of 1D fermionic SPT phases.
It is known that group cohomology classes corresponding to representations of G f induced by similarity transformations V (g) classify bosonic 24 encodes the information about the projective representations of the symmetry group G. The index [ρ] encodes the algebra between the representations of a group element g ∈ G and the fermion parity p ∈ Z F 2 . Finally, the index µ ∈ {0, 1} characterizes the total fermion parity of the SPT ground state, or equivalently the parity of the total number of boundary Majorana modes.
Although the same cohomology group H 2 G f , U (1) c appears in the classification of 1D LSM-type constraints and 1D SPT phases, they have a different origin. For the 1D LSM-type constraints, H 2 G f , U (1) c arises when classifying the projective representations of G f on a local Fock space. For the 1D SPT phases, H 2 G f , U (1) c arises when classifying the boundary projective representations of global symmetries.

V. MAJORANA LSM THEOREMS IN HIGHER DIMENSIONS
In this section, we extend Theorem 1 to any dimension d of space when the symmetry group G f is Abelian and all elements g ∈ G f are represented by unitary operators. Our method is inspired by the one used recently in Ref. 70 for quantum spin Hamiltonians.
Consider a d-dimensional lattice Λ with periodic boundary conditions in each linearly independent directionμ =1, · · · ,d such that Λ realizes a d-torus. Let each repeat unit cell be labeled as j and host a local fermionic Fock space F j that is generated by a Majorana spinorχ j with 2n componentsχ j,l , l = 1, · · · , 2n. The fermionic Fock space attached to the lattice Λ is denoted by F Λ . We impose the global symmetry corresponding to the central extension G f of G by Z F 2 as defined in Sec. III A, whereby G f is assumed to be Abelian. We also impose translation symmetry. If the d-dimensional lattice Λ has Nμ repeat unit cell in theμ-direction and thus the cardinality the translation group is By assumption, the combined symmetry group is the Cartesian product group The representation of the translation group (5.2) is generated by the unitary operator Tμ whose action on the Majorana spinors is along theμ-direction (eμ is a basis-vector along theμdirection). Imposing periodic boundary conditions implies Any translationally-and G f -invariant local Hamiltonian acting on F Λ can be written in the form whereĥ j is a local Hermitian operator centered at an arbitrary repeat unit cell j. More precisely, it is a finiteorder polynomial in the Majorana operators centered at j that is also invariant under all the non-spatial symmetries, i.e.,ĥ for any g ∈ G f . Instead of extracting spectral properties of Hamiltonian H pbc directly, we shall do so with the family of Hamiltonians indexed by g ∈ G f and given by is a G f -symmetric and local Hermitian operator and T1(g) is the "g-twisted translation operator" to be defined shortly. We shall derive LSM-like constraints for H tilt twis (g) and then explain why those LSM-like constraints also apply to H pbc . To this end, we will explain what is meant by the upper index "tilt" for tilted and the lower index "twis" for twisted and how H tilt twis (g) and H pbc differ.
A. Case of a d = 1-dimensional lattice As a warm up, we first consider the one-dimensional case, i.e., Λ ∼ = Z N . We impose two assumptions in addition to those previously assumed. These are that every element in G f is unitarily represented (Assumption 5) and that G f is an Abelian group (Assumption 6). These two assumptions were superflous when proving Theorem 1 using injective FMPS in Sec. IV. This drawback is compensated by the possibility to extend the proof that follows to any dimension d of space.
Twisted boundary conditions are implemented by defining the symmetry twisted translation operator T1(g) .
. =v 1 (g) T1 (5.7a) through its action . We then consider any Hamiltonian of the form (5.6) where the operatorĥ tilt 1 in Eq. (5.6) is nothing but the operatorĥ j in Eq. (5.5a) with Λ restricted to a one-dimensional lattice. Such a twisted boundary condition is equivalent to coupling the Majorana operators to a background Abelian gauge field with a holonomy g ∈ G f around the spatial cycle. The effect of turning on such a background field is that it delivers the operator algebra (see Appendix D) The same algebra with ρ(g) ≡ 0 for all g ∈ G f was obtained by Yao  If χ(g, h) mod 2π is nonvanishing, one-dimensional representations of (5.8) are not allowed. The ground state of any Hamiltonian of the form (5.6) is either degenerate or spontaneously breaks the symmetry in the thermodynamic limit. We have rederived the LSM Theorem 1 for the Abelian group G f that is represented unitarily when twisted boundary conditions apply.
If we assume that the choice of boundary conditions cannot change the ground-state degeneracy when all excited states are separated from the ground states by an energy gap, then the LSM Theorem 1 applies to all boundary conditions compatible with translation symmetry that are imposed on the one-dimensional chain Λ and, in particular, to Hamiltonians of the form (5.5) with Λ restricted to a one-dimensional lattice that obey periodic boundary conditions. A necessary condition for this assumption to hold is that all correlation functions between local operators decay sufficiently fast, a condition known to be an attribute of any Hamiltonian with gapped ground states 72 .

FIG. 2.
Example of a path that visits all the sites of a two-dimensional lattice with the topology of a torus.
We emphasize that, in rederiving Theorem 1, we have taken (i) the group G f to be Abelian and (ii) representationû j (g) to be unitary for all g ∈ G f . There exist several challenges in relaxing both of these assumptions. When the group is taken to be non-Abelian, one cannot consistently define a twisted Hamiltonian (5.6) that is invariant under both global symmetry transformations U (h) and symmetry twisted translation operators T1(g) without imposing stricter constraints on local operatorŝ h tilt 1 than Eq. (5.6). The challenges with imposing antiunitary twisted boundary conditions with the group element g ∈ G f are the following. First, complex conjugation is applied on all the states in the Fock space F Λ . This means that Hamiltonian (5.5) can differ from Hamiltonian (5.6) through an extensive number of terms when c(g) = −1, in which case it is not obvious to us how to safely tie some spectral properties of Hamiltonian (5.6) and Hamiltonian (5.5). Second, not all representations of the group G f are either even or odd under complex conjugation, in which case conjugation of T1(g) by U (h) −1 need not result anymore in a mere phase factor multiplying T1(g) when c(g) = −1. In view of this difficulty with interpreting antiunitary twisted boundary conditions, we observe that the FMPS construction of LSM-type constraints is more general than the one using twisted boundary conditions.
We now assume that Λ is a d > 1-dimensional lattice. We would like to generalize the twisted boundary conditions (5.7) obeyed by the Majorana operators to arbitrary spatial dimensions. There is no unique way for doing so. In what follows, we construct a group of translations G tilt trsl that is cyclic. This is achieved by imposing tilted or sheared boundary conditions. After construct-ing G tilt trsl , we twist the boundary conditions in a particular way using the local representations of elements of the on-site (internal) symmetry group G f . The operators representing translations on the lattice with tilted and twisted boundary conditions may not commute with the operators representing elements of G f , even though all elements of G tilt trsl commute with all elements of G f by assumption (5.3). When this is so, the representation of G tilt total = G tilt trsl × G f is necessarily larger than one dimensional, in which case the ground states are either degenerate or the symmetry group G tilt total = G tilt trsl × G f is spontaneously broken.
Our strategy is to construct the counterpart of Eqs. (5.7) and (5.8). To this end, we are going to trade the translation symmetry group (5.2), which is a polycyclic group when d > 1, for the cyclic group and define the combined symmetry group The intuition underlying the construction of the tilted translation symmetry group G tilt trsl is provided by Fig. 2. As a set, the elements of G tilt trsl can be labeled by the elements of G trsl , namely However, as a group we would like to label the elements of G tilt trsl as those of the cyclic group with |Λ| elements, i.e., This is achieved by carefully choosing the group composition for the elements (5.11), i.e., by iterating d−1 central extensions.
Step 1. We consider Z N1 generated by t1 and extend it by Z N 2 generated by t 2 through the map for any a, b = 1, · · · , N1, to obtain Z N1 N 2 the group of translations on the tilted lattice restricted to R 2 . Here, the notation [a + b] n is used to denote addition modulo n. This group extension can be summarized by the short exact sequence and is labeled by the extension classes Using this extension class and the standard expression for group composition in an extended group, we may identify t1 as the generator of Z N1 N 2 .
Step 2. We consider Z N1 N 2 generated by t1 and extend it by Z N 3 generated by t 3 through the map the group of translations on the tilted lattice restricted to R 3 . This group extension can be summarized by the short exact sequence and is labeled by the extension classes Using this extension class and the standard expression for group composition in an extended group, we may identify t1 as the generator of Z N1 N 2 N 3 .
Step d − 1. We consider Z N1···Nd −1 generated by t1 and extend it by Z Nd generated by t d through the map for any a, b = 1, · · · , N1 · · · Nd −1 , to obtain Z N1··· Nd the group of translations on the tilted lattice Λ. This group extension can be summarized by the short exact sequence and is labeled by the extension classes Using this extension class and the standard expression for group composition in an extended group, we may identify t1 as the generator of Z N1 ···Nd . At the quantum level, we represent the cyclic group (5.9) by replacing Eq. (5.4) with where tμ(j) is the action of the cyclic group G tilt trsl on the repeat unit cell j ∈ Λ. The cyclicity of G tilt trsl ≡ Z N1 ···Nd ≡ Z |Λ| is enforced by With this convention, ( T1) a with a = 1, · · · , |Λ| represents all the elements (t1) a with a = 1, · · · , |Λ| of G tilt trsl . Equation (5.5) takes the form holds for any g ∈ G f . The locality of the polynomial h tilt 1 is no longer manifest when comparing the integers that now label the local Majorana operators inĥ tilt 1 . The locality ofĥ tilt 1 is inherited from the fact thatĥ j is local whileĥ tilt 1 is nothing but a mere rewriting ofĥ j in the cyclic representation of j ∈ Λ.
Because of the cylicity of G tilt trsl ≡ Z N1 ···Nd ≡ Z |Λ| and of its quantum representation, we can adapt the definition (5.7) for the twisted translation operator when d = 1 to that when d > 1. We define for any g ∈ G f with c(g) = +1 the generator of twisted translation on any Majorana operator labeled by j ∈ Λ. Here, I ≡ (1, · · · , 1) ∈ Λ, N = (N1, · · · , Nd) ∈ Λ, and j = (n1, · · · , nd) with nμ = 1, · · · , Nμ. One verifies that these twisted translation operators satisfy the twisted operator algebra We have extended Theorem 1 to any spatial dimension d. As discussed at the end of Sec. IV C, if Theorem 1 holds for any d, then so does Theorem 2. It is nevertheless instructive to provide an alternative proof of Theorem 2 for any spatial dimension d without relying on Theorem 1.
We consider a d-dimensional lattice Λ such that at each repeat unit cell labeled by j ∈ Λ, there exists a Majorana spinorχ j with 2n + 1 componentsχ j,l , l = 1, · · · , 2n + 1. To have a well-defined total Fock space on lattice Λ, we set the total number of sites |Λ| in the lattice to be even. On lattice Λ, we impose the tilted translation symmetry group G tilt trsl defined in Eq. (5.12). Let T1 be the representation of the generator of the cyclic group G tilt trsl with the action (5.16) on the Majorana spinorsχ j . In terms of the Majorana spinorsχ j , the total fermion parity operator P has the representation P . .
Conjugation of the fermion parity operator P by the tilted translation operator T ∞ delivers where we arrived at the last equality by noting that |Λ| is an even integer. The factor (−1) |Λ|−1 arises since each spinorχ j consists of an odd number of Majorana operators. The nontrivial algebra (5.21) implies that the ground state of any Hamiltonian that commutes with P , the gnerators of the tilted translation group, and the generators of G f is either degenerate or spontaneously breaks translation or G f symmetry. If one assumes that the degeneracy of the ground states when gapped is independent of the choice made for the boundary conditions, we reproduce Theorem 2. We note that the algebra (5.21) was shown in Ref. 49 for a one dimensional Majorana chain and interpreted as the existence of Witten's quantum-mechanical supersymmetry 73 .

VI. EXAMPLES
All our results apply to any central extension G f of the group G by the group Z F 2 associated to the fermion parity. Establishing LSM-type conditions requires (i) constructing a projective representation of G f and (2) verifying which one of the group cohomology classes [φ] ∈ H 2 G f , U (1) c is realized by this projective representation. We have shown how the group cohomology classes We shall choose the central extension G f of the group G by the group Z F 2 to be the split Abelian group Their group cohomology is reviewed in Appendix A.
Given any one of these groups, we shall define a global fermionic Fock space F Λ = F 0 ⊕ F 1 and construct a projective representation that realizes the indicator obeying the Majorana (Clifford) algebra χ j,a ,χ j ,a = 2δ j,j δ a,a , j, j ∈ Λ, a, a = 1, · · · , n. (6.1b) The indicator µ takes the value 0 when n = 2m is an even integer, in which case the cardinality |Λ| of the lattice Λ is any positive integer and we may always define the fermionic creation and annihilation operatorŝ  respectively. In order to define the operation of complex conjugation K on both the local and global Fock space, we define j ,2b −1 (6.3a) for any pair of complex number z, w ∈ C (for any j, j ∈ Λ and b, b = 1, · · · , m) and This implies the transformation law Kχ j,2b−1 K = +χ j,2b−1 , Kχ j,2b K = −χ j,2b , (6.3c) for any j ∈ Λ and b = 1, · · · , m. The indicator µ takes the value 1 when n = 2m + 1 is an odd integer, in which case the cardinality |Λ| of the lattice Λ must be an even positive integer. For notational symplicity, we shall assume that Λ is bipartite, that is the disjoint union of two interpenetrating sublattices Λ A and Λ B such that all nearest neighbors of the sites in Λ A belong to Λ B and vice versa. We impose on Λ the topology of a torus. We define the fermionic creation and annihilation operatorŝ . =χ j,a + iχ (j+µ),a 2 , (6.4a) where j ∈ Λ A , a = 1, · · · , 2m+1, and µ is any fixed basis vectors spanning Λ. The global fermionic Fock space F Λ is then n j,a = 0, 1,ĉ j,a |0 = 0 .
In order to define the operation of complex conjugation K on the global Fock space, we define K zĉ † j,a + wĉ j ,a K . . = z * ĉ † j,a + w * ĉ j ,a (6.5a) for any pair of complex number z, w ∈ C (for any j, j ∈ Λ A and a, a = 1, · · · , 2m + 1) and This implies the transformation law Kχ j,a K = +χ j,a , Kχ (j+µ),a K = −χ (j+µ),a , (6.5c) for any j ∈ Λ A and a = 1, · · · , 2m + 1. Both for µ = 0, 1, we shall assume a quantum dynamics governed by Hamiltonians of the form (5.5) where h j in Eq. (5.5b) is a finite-order polynomial in the Majorana operators. The order of each monomial entering this polynonmial is necessarily even for Z F 2 to be a symmetry group. The Hamiltonian is noninterating if the order of h j is two, interacting otherwise. The finiteness of the order guarantees locality. We also introduce the notion of range of h j which is the largest separation between the lattice indices of the Majorana operators present in h j . If the range vanishes, then Hamiltonian (5.5) is the sum over |Λ| commuting Hermitian operators, in which case the spectrum of Hamiltonian (5.5) is obtained by diagonalizing h j .
For the split Abelian group G f = G×Z F 2 with G = Z T 2 , we find that only the projective representations of the group algebra that belong to the trivial group cohomology class can be realized by noninteracting fermions. Any projective representation of the group algebra that belong to a nontrivial group cohomology class prohibits bilinear terms in the fermions in any G f -symmetric Hamiltonian of the form (5.5). Such intrinsically interacting Hamiltonians are quantum perturbations of classical Ising-type Hamiltonians.
For the split Abelian group we find that any projective representation of the group algebra that belongs to a nontrivial group cohomology class implies that any G f -symmetric Hamiltonian of the form (5.5) is necessarily gapless when quadratic in the fermions. Theorem 1 then predicts that any G fsymmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum must break spontaneously at least one of the symmetries responsible for the noninteracting spectrum being gapless.  when The indices (6.6c) will thus be shown to form the cyclic group Z 4 with respect to the stacking rule (6.7). The local fermionic Fock space F j of dimension D = 2 is generated by the doublet of Majorana operatorŝ One verifies thatû when expressed in terms of the fermion-number operator n j =ĉ † jĉj ,n jĉ † j |0 = δ j,j n j ĉ † j |0 , n j = 0, 1, j, j ∈ Λ. (6.11c) It is thus diagonalized in the fermion-number basis F Λ = span{|n 1 , · · · , n 2M } (6.12a) in which it is represented by the classical Ising Hamiltonian The subspace F gs in the global Fock space F Λ = F 0 ⊕ F 1 that is spanned by the linearly independent ground states is twofold degenerate. It is spanned by either the ferromagnetic states F gs = span{|0, 0, · · · , 0, 0 , |1, 1, · · · , 1, 1 } ⊂ F 0 (6.13a) when λ > 0 or the antiferromagnetic states F gs = span{|1, 0, · · · , 1, 0 , |0, 1, · · · , 0, 1 } ⊂ F M mod 2 (6.13b) when λ < 0. Because we made sure that the lattice is bipartite (|Λ| = N = 2M ), F gs is homogeneous 75 . All ground states are separated from all excited states by the gap |2λ|. The action of "time reversal" in the fermion representation is that of a "particle-hole" transformation under whichn j → (1 −n j ) and σ j → −σ j . The action of parity is trivial (the identity) in the fermion representation. Reversal of time is broken spontaneously at zero temperature and in the thermodynamic limit in the sense that applying either a uniform or staggered magnetic field that couples to the Ising spins through a Zeeman coupling, taking the thermodynamic limit, and switching off the Zeeman coupling selects one of the two degenerate ground states when λ > 0 and λ < 0, respectively.
The complexity of the Hamiltonian of the form (5.5) that is of quartic order and of range r = 2 increases dramatically. For any cluster made of the three repeat unit cells j − 1, j, and j + 1, one can construct three groups of five monomials. The first group is made of . =χ j−1,1χj−1,2χj,1χj,2 , (6.14a) with fourteen real-valued couplings. If we set the twelve couplings λ 3 = · · · λ 14 = 0 to zero, we obtain the classical Ising model in the same fermion-number basis of the Fock space that delivered the classical Ising Hamiltonian (6.12b). Any perturbation with one of the couplings λ 3 , · · · , λ 14 breaks the fermion-number conservation down to the conservation of the fermion-parity number. Any gapped phase in the 14-dimensional coupling space is predicted by Theorem 1 to be either degenerate or break spontaneously time-reversal or translation symmetry. This prediction can be verified explicitly for the classical Ising model (6.16) with nearest-and next-nearest-neighbor interactions. The local fermionic Fock space F j of dimension D = 4 is generated by the quartet of Majorana operators 76 χ j ≡ χ j,1 · · ·χ j,4 T , j = 1, · · · , 2M. (6.17) One verifies that It is thus diagonalized in the fermion-number basis in which it is represented by two Ising chains (labeled 1 and 3) coupled through their rungs only (but not along the chains), The subspace F gs in the global Fock space F Λ = F 0 ⊕ F 1 that is spanned by the linearly independent ground states is 2 2M -fold degenerate. It is spanned by either F gs = span{| 0 0 , · · · , 0 0 , · · · , | 1 1 , · · · , 1 1 } ⊂ F 0 (6.22a) when λ > 0 or the antiferromagnetic states F gs = span{| 0 1 , · · · , 0 1 , · · · , | 1 0 , · · · , 1 0 } ⊂ F 0 (6.22b) when λ < 0. All ground states are separated from all excited states by the gap |2λ|. The action of "time reversal" in the fermion representation is that of a "particle-hole" transformation under whichn j,2b−1 → (1 −n j,2b−1 ) and σ j,2b−1 → −σ j,2b−1 . The action of parity is trivial (the identity) in the fermion representation.
The complexity of the Hamiltonian of the form (5.5) that is of quartic order and of range r = 1 increases dramatically. For any cluster made of the two repeat unit cells j and j + 1, the generic cluster Hamiltonian h j that is summed over in Hamiltonian (5.5) is the sum over 70 (choose 4 out of 8) monomials of the form χ j,aχj,bχj+1,cχj+1,d , 1 ≤ a < b ≤ 4, 1 ≤ c < d ≤ 4, (6.23) each one weighted by a real-valued coupling. Of these 70 couplings, 69 are independent by translation symmetry. Three of the 69 monomials are compatible with fermionnumber conservation. All other monomials break the fermion-number conservation down to the conservation of the fermion-parity number. If these 66 couplings are set to zero and the remaining three couplings are set to λ, we obtain the classical Ising ladder in the same fermion-number basis of the Fock space that delivered Hamiltonian (6.21b). Any gapped phase in the 66-dimensional coupling space is predicted by Theorem 1 to be either degenerate or break spontaneously timereversal or translation symmetry. This prediction can be verified explicitly for the classical Ising ladder (6.24) for which the subspace F gs in the global Fock space F Λ = F 0 ⊕ F 1 that is spanned by the linearly independent ground states is 2-fold degenerate. It is spanned by either when λ > 0 or the antiferromagnetic states when λ < 0. Reversal of time is broken spontaneously at zero temperature and in the thermodynamic limit in the sense that applying either a uniform or staggered (within the repeat unit cell) magnetic field that couples to the Ising spins through a Zeeman coupling, taking the thermodynamic limit, and switching off the Zeeman coupling selects one of the two degenerate ground states when λ > 0 and λ < 0, respectively. The local fermionic Fock space F j of dimension D = 8 is generated by the sextet of Majorana operatorŝ χ j ≡ χ j,1 · · ·χ j,6 T , j = 1, · · · , 2M. (6.26) One verifies that u j (t) . . = −iχ j,2χj,4χj,6 K, (6.27a) u j (p) . . = iχ j,1χj,2χj,3χj,4χj,5χj,6 , (6.27b) realizes the projective algebra (6.6) with with λ a 1 a 2 a 3 a 4 a real-valued coupling. Each monomial j,a i , 1 ≤ a 1 < a 2 < a 3 < a 4 ≤ 6, (6.29) is a Hermitian operator with the eigenvalues ±1 as it squares to the identity. Any two such monomials on the right-hand side of Eq. (6.29) either commute or anticommute. In the fermion-number basis (6.2), the 3 monomials are the only ones that are compatible with conservation of the fermion-number. All remaining 12 monomials break the conservation of the fermion-number in the basis (6.2) down to that of the fermion-parity number. Any gapped phase in the 15-dimensional coupling space is predicted by Theorem 1 to be either degenerate or break spontaneously time-reversal or translation symmetry. This prediction can be verified explicitly by summing with the same weight the three monomials defined in Eq. (6.30). One obtains three Ising chains coupled through their rungs only. The ground state is then 2 2M -fold degenerate. This macroscopic degeneracy becomes 2-fold by increasing the range from r = 0 to r = 1 and considering the three Ising chains (labeled 1, 3, 5) with the Hamiltonian [in the fermion-number basis (6. with λ ∈ R, say. The subspace F gs in the global Fock space F Λ = F 0 ⊕ F 1 that is spanned by the linearly independent ground states is either   , · · · , 1 0 1 ⊂ F 0 (6.32b) when λ < 0. Reversal of time is broken spontaneously at zero temperature and in the thermodynamic limit in the sense that applying either a uniform or staggered (within the repeat unit cell) magnetic field that couples to the Ising spins through a Zeeman coupling, taking the thermodynamic limit, and switching off the Zeeman coupling selects one of the two degenerate ground states when λ > 0 and λ < 0, respectively. The local fermionic Fock space F j of dimension D = 16 is generated by the octuplet of Majorana operatorŝ χ j ≡ χ j,1 · · ·χ j, 8 T , j = 1, · · · , 2M. (6.33) One verifies that One verifies that the Majorana octupet (6.33) is even under conjugation byû j (t) and odd odd under conjugation byû j (p). Time-reversal symmetry forbids any Hermitian quadratic form for the quartet (6.33). Theorem 1 is inoperative. It is possible to find examples of both nondegenerate and degenerate gapped Hamiltonians that are translation invariant and G f invariant.
To prove this claim, it suffices to consider a generic Hamiltonian of the form (5.5) that is of quartic order and of range r = 0. It is given by are the only ones that are compatible with conservation of the fermion-number in the fermion-number basis (6.2). All remaining 64 monomials break the conservation of this fermion-number down to conservation of the fermion-parity number. Among these, the 16 monomials generated by expanding χ j,1 +χ j,2 χ j,3 +χ j,4 χ j,5 +χ j,6 χ j,7 +χ j,8 (6.38) are special because they form the basis to represent all 16 terms of the form . =ĉ j,1ĉj,3ĉj,5ĉ † j,7 , · · · , A j|0111 . . =ĉ † j,1ĉj,3ĉj,5ĉj,7 , A j|1111 .
The counterpart to this mechanism to lift the two-fold degeneracy of a 2-rung Ising Hamiltonian is not available in Sec. (VI A 3) because of fermion-parity conservation [to exchange the ferromagnetic ground states, one would need the odd-parity perturbationĉ † j,1ĉ † j,3ĉ † j,5 + H.c]. The same is true in Sec. (VI A 2). Lifting the two-fold degeneracy of a 1-rung Ising Hamiltonian with the help of the perturbationĉ † j,1ĉ † j,3 + H.c is not possible because timereversal symmetry prohibits any local quadratic term.
By identifying the set 1, 0), (1, 0, 0), (0, 1, 0), (0, 0, 0)} (6.43a) with the four distinct group cohomology classes (6.6) and by defining a group operation using the stacking rules (6.7), we have justified the identification g → (1, 1, 0), where g is the generator of the cyclic group The lattice is Λ = {1, · · · , N } with N = 2M an even integer and the global fermionic Fock space F Λ is of dimension 2 n M with n the number of local Majorana flavors. By choosing the cardinality |Λ| = 2M to be even, we make sure that the lattice is bipartite. This allows to treat the two values of the indicator µ . . = n mod 2 in parallel. The symmetry group G f . . = G×Z F 2 is a split group. As usual, Z F 2 is generated by p. We choose G . . = Z 2 × Z 2 . The Abelian group G has hence two generators g 1 and g 2 that commute pairwise, while each of them squares to the identity. We shall only consider the case when the local number of Majorana flavors n = 2m is an even positive integer. The indicator µ then takes the value µ = 0.
Any local projective representation of G f can be la-  Here, the group cohomology class 0, (0, 0), 0 is interpreted as the trivial representation. Theorem 1 is predictive for any of the remaining 7 group cohomology classes. It is shown in Appendix A 4 that these 8 distinct group cohomology classes form the (stacking) group Z 2 ×Z 2 ×Z 2 , whereby the group composition is defined by the stacking rule (A31e). This (stacking) group is generated by the three group cohomology classes 1, (1, 0), 0 , 1, (0, 1), 0 , and 1, (0, 0), 0 , as we are going to verify explicitly. The local fermionic Fock space F j of dimension D = 4 is generated by the quartet of Majorana operators 79 χ j ≡ χ j,1 · · ·χ j,4 T , j = 1, · · · , 2M. (6.46) One verifies that u j (g 1 ) .
. =χ j,1χj,3 , (6.47b) u j (p) . . =χ j,1χj,2χj,3χj,4 ,  the on-site term iχ j,2χj,4 . The on-site term iχ j,2χj,4 gaps chains 2 and 4. The low-energy sector of the theory is that of two decoupled quantum critical Kitaev chains labeld 1 and 3. The quadratic term iχ j,1χj,3 that would gap chains 1 and 3, thereby delivering a nondegenerate gapped ground state, is odd under conjugation byû j (g 1 ) and thus forbidden by symmetry. The stability of this gapless phase to on-site quadratic perturbations can thus be thought of as a consequence of Theorem 1. Theorem 1 also predicts that any G f -symmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum must break spontaneously at least one of the symmetries responsible for the noninteracting spectrum being gapless. When two copies of this 1, (1, 0), 0 representation are stacked, the local fermionic Fock space F j of dimension D = 16 is generated by the octuplet of Majorana operatorsχ One verifies that u j (g 1 ) .
. =χ j,2χj,3 , (6.52a) . =χ j,1χj,3 , (6.52b) The most general translation-and G f -invariant Hamiltonian of the form (5.5) of order 2, range r = 1, and whose hopping is diagonal with respect to the flavor index is then i.e., four decoupled Kitaev chains that are finetuned to their quantum critical point (4.34) between their symmetry-protected and topologically-inequivalent gapped phases. No on-site quadratic term is allowed by the symmetries. The stability of this gapless phase to on-site quadratic perturbations can thus be thought of as a consequence of Theorem 1. Theorem 1 also predicts that any G f -symmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum must break spontaneously at least one of the symmetries responsible for the noninteracting spectrum being gapless. When two copies of this 1, (0, 0), 0 representation are stacked, the local fermionic Fock space F j of dimension D = 16 is generated by the octuplet of Majorana operators (6.49) with the projective representation u j (g 1 ) .
. =χ j,1χj,5χj,7 , (6.55a) . =χ j,1χj,3χj,5 , (6.55b) with λ a , λ a,b ∈ R for a, b = 1, · · · , 8. By construction, this Hamiltonian is gapless since the quantum critical Kitaev chains 3 and 7 are decoupled from all other Kitaev chains. This gapless phase is stable to any on-site quadratic perturbation since the only on-site quadratic terms iχ j,3χj,a and iχ j,7χj,a that could gap the quantum critical Kitaev chains 3 and 7 are odd under G for any a = 3, 7. The stability of this gapless phase to on-site quadratic perturbations can thus be thought of as a consequence of Theorem 1. Theorem 1 also predicts that any G f -symmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum must break spontaneously at least one of the symmetries responsible for the noninteracting spectrum being gapless. When two copies of this 1, (1, 1), 0 representation are stacked, the local fermionic Fock space F j of dimension D = 256 is generated by 16 Majorana operators. One verifies that u j (g 1 ) .
. =χ j,2χj,3χj,5 , (6.58a) . =χ j,1χj,3χj,5χj,7 , (6.58b) with λ a , λ a,b ∈ R with a, b = 1, · · · , 8. By construction, this Hamiltonian is gapless since the quantum critical Kitaev chains 2 and 6 are decoupled from all other Kitaev chains. This gapless phase is stable to any on-site and quadratic perturbation since the only on-site quadratic terms iχ j,2χj,a and iχ j,6χj,a that could gap the quantum critical Kitaev chains 2 and 6 are odd under G for any a = 2, 6. The stability of this gapless phase to on-site quadratic perturbations can thus be thought of as a consequence of Theorem 1. Theorem 1 also predicts that any G f -symmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum must break spontaneously at least one of the symmetries responsible for the noninteracting spectrum being gapless. When two copies of this 0, (1, 0), 0 representation are stacked, the local fermionic Fock space F j of dimension D = 256 is generated by 16 Majorana operators. One verifies that u j (g 1 ) .
. = iχ j,2χj,3χj,5χj,9χj,11 , (6.62a) u j (g 2 ) . . = iχ j,1χj,3χj,5χj,7χj,9 , (6.62b) u j (p) . . = λ a iχ j,aχj+1,a + λ 1,7 iχ j,1χj,7 + λ 3,5 iχ j,3χj,5 + λ 2,11 iχ j,2χj,11 + λ 4,6 iχ j,4χj,6 + λ 8,10 iχ j,8χj,10 with λ a , λ a,b ∈ R a, b = 1, · · · , 12. By construction, this Hamiltonian is gapless since the quantum critical Kitaev chains 9 and 12 are decoupled from all other Kitaev chains. This gapless phase is stable to any quadratic onsite perturbation since the only on-site quadratic terms iχ j,9χj,a and iχ j,12χj,a that could gap the quantum critical Kitaev chains 9 and 12 are odd under G for any a = 9, 12. The stability of this gapless phase to on-site quadratic perturbations can thus be thought of as a consequence of Theorem 1. Theorem 1 also predicts that any G f -symmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum must break spontaneously at least one of the symmetries responsible for the noninteracting spectrum being gapless. When two copies of this 0, (1, 1), 0 representation are stacked, the local fermionic Fock space F j of dimension D = 2 12 is generated by 24 Majorana operators. The Z 2 -graded tensor product of the projective representation (6.62) with itself realizes the group cohomology class ([ν], [ρ], µ) = 0, (0, 0), 0 , i.e., the trivial group cohomology class. There exists a bijective map a → a . . = (a + 12) mod 24 such that all on-site terms iχ j,aχj,a with a = 1, · · · , 12 can be shown to be G f symmetric. The ground-state degeneracy of any translationand G f -invariant Hamiltonian of the form (5.5) can be lifted by including the 12 on-site terms iχ j,aχj,a+12 with a = 1, · · · , 12, i.e., Theorem 1 is not predictive. The lattice is Λ = {1, · · · , N } with N = 2M an even integer and the global fermionic Fock space F Λ is of dimension 2 n M with n the number of local Majorana flavors. By choosing the cardinality |Λ| = 2M to be even, we make sure that the lattice is bipartite. This allows to treat the two values of the indicator µ . . = n mod 2 in parallel. We shall only consider the case when n = 2m with m a positive integer and µ = 0. The symmetry group where the identification t 2 = p is made. The upper index T for the cyclic group G ≡ Z T 2 ≡ {e, t} refers to the interpretation of t as reversal of time (see Appendix A 5). As usual, p denotes fermion parity. The symmetry group G f is thus generated by reversal of time t, whereby reversal of time squares to the fermion parity p.
The local antiunitary representationû j (t) of reversal of time generates a projective representation of the group Z F T 4 . According to Appendix A 5, all cohomologically distinct projective representations of Z F T The local fermionic Fock space F j of dimension D = 2 is generated by the doublet of Majorana operatorŝ One verifies that one also verifies that It follows from Eq. (6.69) that the only on-site Hermitian quadratic form iχ j,1χj,2 is odd under reversal of time. Consequently, with λ ∈ R is the most general translation-and G finvariant Hamiltonian of the form (5.5) of order 2, range r = 1, and whose hopping is diagonal with respect to the flavor index. This Hamiltonian describes two Kitaev chains that have been finite-tuned to their quantum critical point (4.34) between their symmetry-protected and topologically-inequivalent gapped phases. The stability of this gapless phase to on-site quadratic perturbations can thus be thought of as a consequence of Theorem 1.
Theorem 1 also predicts that any G f -symmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum must break spontaneously at least one of the symmetries responsible for the noninteracting spectrum being gapless. When two copies of the projective representation (6.67) are stacked, the local fermionic Fock space F j of dimension D = 4 is generated by 4 Majorana operators. The Z 2 -graded tensor product of the projective representation (6.67) with itself realizes the group cohomology class ([ν], [ρ], µ) = 0, 0, 0 , i.e., the trivial group cohomology class. There exists a bijective map a → a . . = (a + 2) mod 4 such that all on-site terms iχ j,aχj,a − iχ j,a+1χj,(a+1) with a = 1, 2 can be shown to be G f symmetric. The ground-state degeneracy of any translation-and G f -invariant Hamiltonian of the form (5.5) can be lifted by increasing the strength of iχ j,aχj,a − iχ j,a+1χj,(a+1) with a = 1, 2, i.e., Theorem 1 is not predictive. The lattice is Λ = {1, · · · , N } with N = 2M an even integer and the global fermionic Fock space F Λ is of dimension 2 (2m+1) M with (2m + 1) the number of local Majorana flavors, i.e., µ = 1. By choosing the cardinality |Λ| = 2M to be even, we make sure that the lattice is bipartite. The symmetry group is G f . . = {p, p 2 } ≡ Z F 2 where p denotes fermion parity. If we reinterpet G f = G × Z F 2 with G = {e} the group with one element, we deduce that the indices [ν] and [ρ] are trivial, i.e., [ν] = [ρ] = 0. The index associated with this group is then (0, 0, 1), for which we illustrate how translation symmetry prevents a nondegenerate gapped ground state in agreement with Theorem 2.
For simplicity, we first choose 2m + 1 = 1 local Majorana flavors. Hamiltonian is the most general translation-and G f -invariant Hamiltonian of the form (5.5) of range r = 1. It is gapless as it describes one Kitaev chain on M sites that has been finite-tuned to its quantum critical point (4.34) between its symmetry-protected and topologically-inequivalent gapped phases.
If we now consider 2m + 1 > 1 local Majorana flavors arranged into the (2m+1)-multipletχ j , the most general translation-and G f -invariant Hamiltonian of the form (5.5) of order 2 and range r = 1 is where the (2m + 1) × (2m + 1)-dimensional matrix M is real-valued and antisymmetric. As M has necessarily a zero eigenvalue, the spectrum of H pbc is gapless. Theorem 2 predicts that any G f -symmetric interaction of the form (5.5) that opens a spectral gap in the noninteracting spectrum of Hamiltonian (6.73) must break spontaneously the symmetries responsible for the noninteracting spectrum being gapless.
After stacking two copies of the projective representation (0, 0, 1), it is possible to define a local fermionic Fock space with the gapless Hamiltonian The on-site mass term iχ T j,1χj,2 is compatible with parity conservation. When added to the right-hand side so as to preserve translation symmetry, it selects a nondegenerate gapped ground state. There is no contradiction with Theorem 2 since the the projective representation under stacking rules is the trivial one (0, 0, 0).
As was suggested just after Eq. (6.72), we can always do the reinterpretation i χ o,lχe,l +χ o,lχe,l+1 , (6.75a) . =χ 2l , (6.75b) according to which the enlarged repeat unit cell is labeled by the odd sites Λ A of Λ and there are two flavors (even and odd) of Majorana per enlarged repeat unit cell. It is then tempting to ask if one could use Theorem 1 for some group G to understand the spectrum of Eq. (6.75), in which case the need for Theorem 2 would be superfluous at best or contradictory at worst. However, there is no conflict betweeen Theorem 2 and Theorem 1, as Theorem 1 cannot be applied to understand the spectrum of Eq. (6.75). To see this, we observe that the translation by one original repeat unit cell that is presumed by Theorem 2 is represented in Eq. (6.75) bŷ χ o,l →χ e,l ,χ e,l →χ o,l+1 , l = 1, · · · , M. (6.76) As this transformation is not internal to the enlarged repeat unit cell, it cannot be interpreted as the group G of on-site symmetries needed to establish Theorem 1.
We observe that is nothing but the 2-cocyle condition (3.11b) obeyed by φ.
Example n = 1: The coboundary operator δ 1 is defined by One verifies the important identity We observe that Eq. (3.14) implies that φ is the image of ξ under the coboundary operator δ 1 . Using the coboundary operator, we define two sets and The cochains in Z n (G, M ) are called n-cocyles. The cochains in B n (G, M ) are called n-coboundaries. The importance of the coboundaries is that the identity (A7) generalizes to The nth cohomology group is defined as the quotient of the n-cocyles by the n-coboundaries, i.e., From now on, we omit the superscript n in δ n for convenience. it should be understood that the map δ acting on a cochain φ maps it to a cochain of one higher degree. The nth cohomology group H n (G, M ) is an additive Abelian group. We denote its elements by [φ] ∈ H n (G, M ), i.e., the equivalence class of the n-cocycle φ. Finally, we define the following operation on the cochains. Given two cochains φ ∈ C n (G, N ) and θ ∈ C m (G, M ), we produce the cochain (φ ∪ θ) ∈ C n+m (G, N × M ) through (φ ∪ θ)(g 1 , · · · , g n , g n+1 , · · · , g m ) .
Having introduced the basics of group cohomology, next we shall compute the cohomology groups [φ] ∈ H 2 G f , U (1) c for some specific finite Abelian groups G encountered in Sec. VI and whose projective representations are defined by Eqs. (3.10) and (3.11).

Classification of projective representations of G f
It was described in Sec. III A, how a global symmetry group G f for a fermionic quantum system naturally contains the fermion-number parity symmetry group Z F 2 in its center, i.e., it is a central extension of a group G by Z F 2 . Such group extension are classified by prescribing an element [γ] ∈ H 2 G, Z F 2 , such that we may think of G f as the set of tuples (g, h) ∈ G × Z F 2 with composition rule as in Eq. (3.3e). From this perspective there is an implicit choice of trivialization τ : G f → Z F 2 and projection b : Importantly, τ is related to the extension class γ that defines the group extension via the relation is the pullback of γ via b. It was proved in Ref. 60 that equivalence classes of two-cocycles of [φ] ∈ H 2 G f , U (1) are classified by tuples (ν, ρ) ∈ C 2 G f , U (1) × H 1 G, Z 2 that satisfy the following cocycle conditions The proof follows in three steps which we sketch out here. We refer the reader to Ref. 60 for details: 1. First, given a cocycle φ ∈ Z 2 (G f , U (1)), one can define ρ ∈ Z 1 G, Z 2 via eq. (3.27). The fact that ρ is a cocycle follows from that fact that φ is a cocycle.

The split group
Below we describe how to "measure" these indices as well as the product or monoidal structure that these indices satisfy.
Proof. Any cochain ν belonging to the equivalence class [ν] is defined by the substitution G = Z T 2 in Eq. (3.10) and must satisfy the cocyle and coboundary conditions in (3.11b) and (3.13), respectively. If one chooses Equation (A18) is nothing but the statement that the representation of time reversal should square to either the identity or minus the identity. These two possibilities are not connected by a coboundary. Hence, they correspond to different second cohomology classes. To see this, assume they were connected by a coboundary, i.e., they satisfy the equivalence condition (3.13). On the one hand, choosing g = t and h = t in Eq. (3.13) implies that if ν (t, t) = π and ν(t, t) = 0. However, on the other hand, choosing g = t and h = e in Eq.
since ν(g, e) = ν(e, g) = 0 for all g. Equations (A19) and (A20) contradict each other. This contradiction implies that one cannot consistently define a gauge transformation ϕ that interpolates between ν such that ν(t, t) = π to ν such that ν(t, t) = 0. We denote the cases ν(t, t) = π, 0 with the equivalence classes [ν] = 1, 0, respectively. Proof. For the second index [ρ] ∈ H 1 (Z T 2 , Z 2 ), two 1-cochains ρ and ρ are equivalent if and only if they are 1-cocycles that differ by a coboundary of a zero-cochain. But, by definition, a zero-cochain is the identity and as such has vanishing coboundary. Hence, the coset H 1 (Z T 2 , Z 2 ) is just the set of all distinct 1-cocycles. By definition, a 1-cocyle ρ must obey [recall Eq. (A6)] Choosing g = t and h = p delivers where we used the fact that ρ(p) = 0 by the definition (3.27). The value of ρ(t) in Z 2 indicates whether the projective representation of reversal of time commutes or anticommutes with the projective representation of the fermion parity operator p. Equation (A21b) states that fermion parity of the quantum representation of t is equal to the fermion parity of the quantum representation of t · p. We assign the indices [ρ] = 0, 1 to the values ρ(t) = 0, 1, respectively.
Given two local projective representationsû 1 andû 2 of the group G f = Z T 2 × Z F 2 acting on the Fock spaces F 1 and F 2 , respectively, we now derive the indices associated with the local projective representationû acting on the graded tensor product F = F 1 ⊗ g F 2 of the Fock spaces F 1 and F 2 . The definition (3.33a) implieŝ for the representations of elements t ∈ Z T 2 and p ∈ Z F 2 . In turn, using the relation (3.33c) and the definition (3.27), we find for the 2-cocyle ν and 1-cocyle ρ associated with the representationû. Here, ν 1 and ν 2 are the 2-cocyles, and ρ 1 and ρ 2 are the 1-cocyles associated with the representationsû 1 andû 2 , respectively. Assignments of indices [ν] and [ρ] to the local projective representations of the group Z T 2 × Z F 2 (as shown above) and the identity (A23) imply that the indices of the tensor product representation are related to the indices of the constituent representations via We note that for the group Z T 2 × Z F 2 the cup product [πρ 1 ρ 2 ] in Eq. (3.34) simplifies to One thus finds that different local projective representations of the group Z T 2 × Z F 2 form the cyclic group Z 4 under the stacking rule (A24a).

The nonsplit group
where the upper index T for the cyclic group Z T 2 ≡ {t, t 2 } refers to the interpretation of t as reversal of time. This central extension of time reversal by fermion parity is specified by the map γ obeying the nonsplit condition (3.30) because of γ(t, t) = p (which implies the group composition rule t · t = p). If so, ν is not a cocyle but a cochain with nonvanishing coboundary according to Eq. (3.31) and (A16). On the other hand, ρ is a 1-cocyle. We thus make the same assignments for the values taken by [ρ] as was the case for the split group Z T 2 × Z F 2 , namely [ρ] = 0, 1 if the projective representation of reversal of time has even or odd fermion parity, respectively. Choosing With the help of Eq. (3.11c) and with the choice of the convention γ(t, t) = p ≡ 1 for the nonsplit group Z F T 4 , we conclude that acting on two Fock spaces F 1 and F 2 , respectively, we shall derive the indices associated with the local projective representationû acting on the graded tensor product F = F 1 ⊗ g F 2 of Fock spaces F 1 and F 2 . The definition (3.33a) implieŝ for the representations of elements t, p ∈ Z F T 4 . In turn, using the relation (3.33c) and the definition (3.27), we find that if ν 1 and ν 2 are 2-cochains, and ρ 1 and ρ 2 are two 1-cocyles associated with the representationsû 1 andû 2 , respectively, then the 2-cochain ν and 1-cocyle ρ associated with the representationû are given by respectively. Although this pair of relations are identical to their counterparts in Eq. (A23), the nonsplit nature of the group Z F T 4 carries over to the stacked projective representation. Indeed, inserting twice (A33) on the right-hand side of (A36a) gives There are four cases to consider. When ρ 1 (t), ρ 2 (t) = (0, 0), ρ(t) = 0 and ν(t, t) = 0. When ρ 1 (t), ρ 2 (t) = (0, 1), ρ(t) = 1 and ν(t, t) = −π/2. When ρ 1 (t), ρ 2 (t) = (1, 0), ρ(t) = 1 and ν(t, t) = −π/2. When ρ 1 (t), ρ 2 (t) = (1, 1), ρ(t) = 0 and ν(t, t) = 0. Hence, ν(t, t) and ρ(t) on the left-hand sides of Eqs. (A36a)  We review the construction of fermionic matrix product states (FMPS). We refer the reader to Refs. 58 and 59 and references therein on the topic of matrix product states (MPS). As in bosonic matrix product states (BMPS), FMPS can be expressed as a contraction of objects belonging to a graded tensor product of vector spaces. The need for graded tensor product of vector spaces stems from the underlying fermionic algebra.

Z 2 -graded vector spaces and their Z 2 -graded tensor products
Any fermionic Fock space F can be seen, in the basis that diagonalizes the total fermionic number operator, to be the direct sum over a subspace F 0 with even total fermionic number and a subspace F 1 with odd total fermionic number. This property endows fermionic Fock space with a natural Z 2 -grading.
A Z 2 -graded vector space V admits the direct sum decomposition We shall identify the subscripts 0 and 1 as the elements of the additive group Z 2 . We say that V 0 (V 1 ) has parity 0 (1). Any vector space is Z 2 -graded since the choice V 0 = V and V 1 = ∅ is always possible. Any subspace of V 0 shares its parity 0. Any subspace of V 1 shares its parity 1. A vector |v ∈ V is called homogeneous if it entirely resides in either one of the subspaces V 0 and V 1 . The parity |v| of the homogeneous state |v is either 0 if |v ∈ V 0 or 1 if |v ∈ V 1 . These observation on the Z 2 -grading of a vector space V only become useful when one demands that any operation acting on V preserves the Z 2 -grading. For example, certain operations need to be defined carefully between two Z 2 -graded vector space V and W that preserve their Z 2 structure. One such operation is the Z 2 -graded tensor product. Let V = V 0 ⊕ V 1 and W = W 0 ⊕ W 1 be two Z 2 -graded vector spaces. We define their graded tensor product as the map such that By design, the operation ⊗ g carries the Z 2 -grading of V and W to their Z 2 -graded tensor product. In particular, for any homogeneous vectors |v ∈ V with parity |v| = 0, 1 and |w ∈ W with parity |w| = 0, 1, the graded tensor product |v ⊗ g |w of two homogeneous vectors has the parity |v ⊗ g |w . . = (|v| + |w|) mod 2.
The connection between the Z 2 -graded vector space V = V 0 ⊕V 1 and fermionic Fock spaces F = F 0 ⊕F 1 , is established through the identifications F 0 → V 0 and F 1 → V 1 . However, a fermionic Fock space has more structure than a mere Z 2 -graded vector space. Wave functions in a fermionic Fock space are fully antisymmetric under the permutation of two fermions. This requirement can be implemented as follows on a Z 2 -graded vector space. The exchange of two fermions can be represented by the isomorphism by which the graded tensor product of the homogeneous vectors |v ∈ V and |w ∈ W obeys |v ⊗ g |w → (−1) |v| |w| |w ⊗ g |v .
The map R is called the reordering operation. It is invertible with itself as inverse since R 2 is the identity map. For every Z 2 -graded vector space V , we define the dual Z 2 -graded vector space V * . We denote an element of the dual Z 2 -graded vector space V * by v|, the dual to the vector |v ∈ V . The dual Z 2 -graded vector space V * inherits a Z 2 grading from assigning the parity |v| to the vector v| ∈ V * if |v ∈ V is homogeneous with parity |v|. The contraction C is the map where ψ|φ denotes the scalar product between the pair |ψ , |φ ∈ V . Hence, holds for any pair of orthonormal and homogeneous basis vectors |i , |j ∈ V . The contraction C * is the map C * : V ⊗ g V * → C defined by its action for any pair of orthonormal basis vectors |i , |j ∈ V . It is common practice to use the same symbol C for both C and C * . Any linear operator can be represented in the orthonormal and homogeneous basis {|i } of V by the matrix The linear operator M has a well defined parity if and only if each term |i ⊗ g j| in the summation has the same parity, in which case |M | . . = (|i| + |j|) mod 2.

Fermionic matrix product state (FMPS)
We attach to each integer j = 1, · · · , N three Z 2 -graded vector spaces V j . . = span{|α) | α = 1, · · · , D v,j }, F j . . = span{|ψ σ | σ = 1, · · · , D j }, V * j . . = span{(β| | β = 1, · · · , D v,j }. (B7a) The basis state |α) and (β| of the dual pair V j and V * j of Z 2 -graded vector spaces are virtual (auxiliary) states. They are denoted by rounded kets and bras and are introduced for convenience. Each auxiliary basis state has a well defined parity by assumption. The basis states {|ψ σ } span the physical fermionic Fock space F j . Each physical basis state |ψ σ has a well defined parity by assumption, as follows from working in the fermion-number basis of F j say. The auxiliary Z 2 -graded vector space V j has the dimension D v,j . The physical fermionic Fock space F j has dimension D j . A fermionic matrix product state (FMPS) takes the form and has a well defined parity provided the objects Q(b), Y , A [1], A [2], · · · , A[N ] are defined as follows. For any once the matrices A σ j , labeled as they are by the basis elements of the local Fock space F j and with the matrix elements (A σ j ) α j β j , have been chosen. The contraction C v labeled by the lower index v is understood to be over all virtual indices belonging to the dual pair (V * j , V j ) of auxiliary Z 2 -graded vector spaces, thereby producing the tensor proportional to were chosen to be the identity The integer b = 0, 1 labels the boundary conditions selected by is needed to fix the fermion parity of |Ψ . More precisely, we demand that the parity (B5d) of are both even, while the parity (B5d) of Y ∈ V 1 ⊗ g V * 1 is either even or odd. Consequently, the parity of |Ψ is determined by the parity of Y since A prerequisite to imposing translation symmetry on any FMPS is that all dimensions D v,j and D j are independent of j = 1, · · · , N . Hence, we assume from now on that 3. Even-parity fermionic matrix product sate (FMPS) The FMPS is an even-parity FMPS obeying periodic (b = 0) or antiperiodic (b = 1) boundary conditions if, for any j = 1, · · · , N , By construction, both Q(b) and Y are of even parity. Moreover, |α j | + |σ j | + |β j | mod 2 = 1 implies that We are going to give an alternative representation of this even-parity FMPS under the assumption that the virtual dimension D v obeys the partition D v = M e + M o where M e ≡ M and M o ≡ M are the numbers of even-and odd-parity virtual basis vectors, respectively. Parity evenness of when the physical state is of even parity [as follows from Eq. (B8c)] or block off diagonal when the physical state is of odd parity [as follows from Eq. (B8c)]. All the blocks are here M × M -dimensional. Parity evenness of Q(b) with matrix elements Q(b) α 1 β 1 and Y with matrix elements Y α 1 β 1 implies that Hereby, we introduced the parity matrix P that satisfies Inserting these explicit representations of Q(b) and Y in Eq. (B8a) delivers where we used the shorthand notation |Ψ σ . . = |ψ σ 1 ⊗ g |ψ σ 2 ⊗ g · · · ⊗ g |ψ σ N . The appearence of the matrix P when b = 0 is counterintuitive. It is needed to eliminate from the sum over all physical basis states {|Ψ σ } those physical basis states of odd parity. The state |{A (0) σ j }; b has even parity since where we used condition (B8c) to establish the first equality and the condition |β j | = |α j+1 | that is imposed by the contractions of virtual indices to establish the second equality.

Odd-parity fermionic matrix product sate (FMPS)
The FMPS is an odd-parity FMPS obeying periodic (b = 0) or antiperiodic (b = 1) boundary conditions if, for any j = 1, · · · , N , By construction, Q(b) is of even parity while Y is of odd parity. Moreover, |α j | + |σ j | + |β j | mod 2 = 1 implies We note that the only difference between definitions (B8) and (B11) is the choice for Y . In the former case its parity is even, in the latter case its parity is odd. Analogously to the even FMPS case, we define 2M × 2M dimensional matrices A (1) σ j and Y with the matrix elements A (1) σ j α j β j and Y α 1 β 1 . The parity |Y | = 1 implies that where Y 1 and Y 2 are M × M and dimensional matrices, respectively. Imposing translation symmetry requires that We choose which implies where G σ j are M × M dimensional matrices. Inserting these explicit representations of Q(b) and Y in Eq. (B11a) delivers The state |{A where we used condition (B11c) to establish the first equality. For the second equality we used the conditions |α| = |β N | and |β| = |α 1 | where |α|, |β| are the parities of the virtual indices corresponding to matrix elements Y α β , and |β j | = |α j+1 | for j = 2, · · · , N − 1 that is imposed by the contractions of virtual indices to establish the second equality.
We will prove Theorem 1 for one dimensional systems within the FMPS framework. Our proof follows closely that for the bosonic case 83 introduced in Ref. 32. We will show that a parity-even or parity-odd injective FMPS necessarily requires the local projective representationû j of the symmetry group G f to have trivial second cohomology class [φ] ∈ H 2 G f , U (1) c . In other words, when this cohomology class is nontrivial there is no compatible injective FMPS with even or odd parity. The general forms (B10a) and (B13a) as well as the injectivity conditions (1) and (2) are distinct for even and odd parity FMPS. The proofs for the even-and the odd-parity cases are thus treated successively. For conciseness, we are going to supress the symbol ⊗ g when working with the orthonormal and homogeneous basis of the Fock space be a translation-invariant, G f -symmetric, even-parity, and injective FMPS obeying periodic boundary conditions when b = 0 or antiperiodic boundary conditions when b = 1. For any g ∈ G f , the global representation U (g) of g is defined in Eq. (3.18). By assumption, |{A σ j }; b is a nondegenerate gapped ground state of some local fermionic Hamiltonian in one-dimensional space. Hence, for any g ∈ G f , there exists a phase η(g; b) ∈ [0, 2π) such that The action of the transformation U (g) on the right-hand side of Eq. (C2) gives after using N times the resolution of the identity, one for each local Fock space F j . The right-hand side can be written more elegantly with the definition of the g-dependent 2M × 2M matrix where the D × D matrix U(g), whose matrix elements are the complex-valued coefficients weighting the sum over the , acts on the local Fock space F j and we have defined As usual, K denotes complex conjugation. Equation (C4) becomes which is nothing but the FMPS (C2) with A (0) σ j substituted for A (0) σ j (g). Equating the right-hand sides of Eqs. (C3) and (C5c) implies This equation is satisfied by the Ansatz where κ(g) = 0, 1 dictates if the 2M × 2M unitary matrix U (g) commutes or anticommutes with the 2M × 2M parity matrix P defined in Eq. (B9c), since The existence of the 2M × 2M invertible matrix U (g) is guaranteed because of the injectivity of the FMPS. In an injective even-parity FMPS, the matrices A (0) σ span the simple algebra of all 2M ×2M matrices for any > for some nonvanishing integer . Hence, provided N is sufficiently large, the family of matrices {A U (g) = [0, 2π) and some invertible 2M × 2M matrix U (g) that must also obey U (p) = P, P U (g) P = (−1) κ(g) U (g).
Here, the map κ : G f → {0, 1} specifies the algebra between the similarity transformation U (g) corresponding to element g ∈ G f and the fermion parity P . The effect of the factor (−1) κ(g) is nothing but the phase as follows from Eq. (C6b).
Appendix D: Proof of Theorem 1 with twisted boundary conditions for any Abelian group G f whose projective representations are all unitary The lattice is Λ = {1, · · · , N } ≡ Z N with N an integer. The global fermionic Fock space F Λ is of dimension 2 mN with n = 2m an even number of local Majorana flavors. The local F j and global F Λ Fock spaces are generated by the Hermitian Majorana operatorsχ j,a obeying the Clifford alegbra χ j,a ,χ j ,a = 2 δ j,j δ a,a , j, j = 1, · · · , N, a, a = 1, · · · , n = 2m.
The local and global fermion parity operators arê respectively. Any polynomial h j in the Majorana operators that is of finite order, of finite range r (the integer r is the maximum separation between the space labels of the Majorana operators entering h j ) of even parity ( P Λ h j P Λ = h j ), and Hermitian ( h † j = h j ) is a local Hamiltonian. We define the unitary operator T1 by its action j+1,a , if j = 1, · · · , N − 1 and a = 1, · · · , n = 2m, χ 1,a , if j = N and a = 1, · · · , n = 2m.
Proof. We begin with the proof of Eq. (D14). We choose two elements g, h ∈ G f with the local representationsv 1 (g) andv 1 (h), respectively, both of which are unitary.