Projective Spacetime Symmetry of Spacetime Crystals

Wigner's seminal work on the Poincar\'e group revealed one of the fundamental principles of quantum theory: symmetry groups are projectively represented. The condensed-matter counterparts of the Poincar\'e group could be the spacetime groups of periodically driven crystals or spacetime crystals featuring spacetime periodicity. In this study, we establish the general theory of projective spacetime symmetry algebras of spacetime crystals, and reveal their intrinsic connections to gauge structures. As important applications, we exhaustively classify (1,1)D projective symmetry algebras and systematically construct spacetime lattice models for them all. Additionally, we present three consequences of projective spacetime symmetry that surpass ordinary theory: the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless Floquet crystals, and symmetry-enforced crossings in the Hamiltonian spectral flows. Our work provides both theoretical and experimental foundations to explore novel physics protected by projective spacetime symmetry of spacetime crystals.


Introduction
The Poincaré group as the symmetry group of relativistic spacetime is fundamental in high energy physics. In analyzing the representation of the Poincaré group by quantum matter, E. Wigner's revealed one of the most fundamental principles of quantum theory: symmetry groups are projectively represented in his seminal work [1].
On the other hand, the broad range of applications for the theory may be seen from the fact that PRSTGs can be induced from gauge fluxes over spacetime lattices. spacetime crystals are mainly experimented by artificial crystals, most of which feature engineerable gauge fields. Additionally, gauge structures can emerge from the periodic evolution of spacetime crystals [18]. As a result, PRSTGs ubiquitously exist in spacetime crystals, and the theory of PRSTGs is expected to generate widespread interests, given the variety of artificial crystals available, including photonic and acoustic crystals, cold atoms in optical lattices, periodic mechanical systems and electriccircuit [19][20][21][22][23][24][25][26][27][28][29]. Notably, the projective representations of space groups for static crystals with gauge structures have recently attracted intensive attention in the field of topological physics [30][31][32][33][34][35][36][37][38].
This work establishes the general theory of projective representations of PRSTGs and approaches it from several perspectives. The first step is to develop a method for classifying all projective representations of a given spacetime group. We then apply this method to exhaustively classify all (1, 1)D spacetime groups, and characterize each class of projective representations using projective symmetry algebras (PSAs). The PSAs are particularly useful for practical applications. The classifications and corresponding PSAs are tabulated in Tab. 1. Based on these PSAs, we develop systematical methods to construct a canonical spacetime-lattice model for each (1, 1)D spacetime group. By appropriately varying the flux configuration of the canonical model, it is possible to realize all PSAs of the spacetime group. It is noteworthy that recently the classification of PSAs for static crystals with time-reversal invariance has been made in Ref. [33]. Besides different symmetry settings and physical systems, the classification of PSAs for spacetime crystals bears an important technical difference compared with the classification in Ref. [33], i.e., all spacetime symmetries reversing the time direction are anti-unitary.
Our research provides a fundamental framework for investigating the unique physics of spacetime crystals that may emerge from gauge-field enriched spacetime symmetry. Our findings include several noteworthy results, such as the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless spacetime crystals, and symmetryenforced crossings in the Hamiltonian spectral flows. All these discoveries surpass the conventional theory of spacetime symmetry. Hence, upon the theoretical foundation established here, we expect a multitude of novel physics resulting from projective spacetime symmetry to be explored by the community.

arXiv:2310.09577v1 [cond-mat.mes-hall] 14 Oct 2023
Projective spacetime symmetry algebras Let us start with recalling some basics of spacetime groups. Since the time translation operator L T is contained in any spacetime group, each (d, 1)D spacetime group is isomorphic to a (d + 1)D space group. However, not every (d + 1)D space group can be interpreted as a (d, 1)D spacetime group [13]. This is because symmetry operations cannot mix spatial and temporal dimensions, i.e., time reversal, time-glide reflection and time-screw rotation operations are allowed, but spacetime rotations more than twofold are forbidden. Hence, only a subset of (d + 1)D space groups can be interpreted as spacetime groups of H(t). For (1, 1)D time crystals, all possible spacetime groups are listed in Tab. 1, using notation adapted from 2D space groups.
Consider a spacetime group G st . Under a projective representation, the multiplication rules of the symmetry operators S are modified with additional phase factors. If S 0 S 1 = S 2 , then S 0 S 1 = Ω(S 0 , S 1 )S 2 , where S i are symmetry elements in G st . Ω is called the multiplier of the projective representation and is valued U (1). Linear and anti-linear operators always satisfy the associativity, i.e., (S 1 S 2 )S 3 = S 1 (S 2 S 3 ) for three arbitrary operators S 1 , S 2 and S 3 . Hence, the multiplier Ω satisfies the equation, (1) Here, c S is an operator which depends on S. c S := 1 if S is unitary, c S := K if S is antiunitary, where K is the complex conjugation operator. For spacetime groups, S is antiunitary if it reverses time. Meanwhile, one may modify the phase of each S by χ(S)S with χ(S) ∈ U (1). This transforms the multiplier to an equivalent one, i.e., Ω(S 1 , S 2 ) ∼ Ω(S 1 , S 2 ) c S1 [χ(S 2 )]χ(S 1 )/χ(S 1 S 2 ). (2) The equivalence classes of multipliers form an Abelian group H 2,c (G st , U (1)), called the twisted second cohomology group of G st [39]. It is clear that all projective representations of G st are classified by H 2,c (G st , U (1)). For (1, 1)D spacetime crystals, we work out all the classifications as listed in the second column of Tab. 1. Notably, because of the anti-unitarity of time reversal, the (d, 1)D spacetime groups have distinct projective representations compared with the corresponding (d + 1)D space groups.
Under projective representations, the original group algebraic relations of symmetry operators are modified by Ω, resulting in the modified algebras called the PSAs [30-32, 34, 35, 37]. These algebras are classified by H 2,c (G st , U (1)), which gives us the number of algebraic classes. However, for physical applications, it is essential to exhibit the explicit forms of these algebras. To achieve this in (1, 1) dimensions, we carefully select a set of generators (in the third column of Tab. 1) for each G st , such that each PSA corresponds to a modification of the corresponding multiplication relations by a set of independent phase factors (in the fourth column of Tab. 1).
The derivation for all these PSAs can be found in the Supplementary Information (SI).
For example, P m x can be generated by unit translations L x , L T and spatial reflection M x , which satisfy the following relations: [ However, phase factors γ 1 , γ 2 can be set to be 1 by redefin- Moreover, phase factors η 1 , η 2 can only take values in L T γ 2 , leading to η 2 1 = η 2 2 = 1. All the possible values of phase factors, i.e., η 1 , η 2 = ±1 correspond to the Z 2 2 classification of the projective symmetry algebras of P m x .
It is significant to note that all phase factors in Tab. 1 are invariant under multiplying each symmetry operator by an arbitrary phase factor χ. Therefore, they are solely determined by the equivalence class. Conversely, for each G st , the free phase factors in Tab. 1 are complete, i.e., sufficient to specify equivalence classes of H 2,c (G st , U (1)). Thus, for each G st we call these phase factors a complete set of cohomology invariants.
Nevertheless, two elements in H 2,c (G st , U (1)) may correspond to isomorphic algebras. The number of independent algebras for each spacetime group is given in the last column of Tab. 1. Let P m x be an example.

Generators
Cohomology invariants N Tab. 1. U (1)-projective algebras of 13 (1, 1)D spacetime groups. The columns in order list the underlying spacetime groups G st , the second cohomology group H 2,c (G st , U (1)), the selected generators, the modified algebraic relations, and the numbers N of isomorphic classes of algebras. The multiplicative commutator is defined as Similarly, for any symmetry S ∈ G st , the physical symmetry operator is the combination S = G S S with G S the corresponding gauge transformation, and it is S that commutes with the Hamiltonian, i.e., [H(t), S] = 0.
Note that a gauge transformation is a diagonal matrix indexed by the lattice sites. Because of gauge transformations, the multiplication of two arbitrary symmetry operators S 1 and S 2 is modified by the multiplier Ω(S 1 , S 2 ) = G S1 S 1 G S2 S −1 1 G −1 S1S2 , which is a U (1)-factor as proved in the SI. It is noteworthy that the equivalent class of Ω(S 1 , S 2 ) is completely determined by the flux configuration, i.e., two gauge equivalent gauge connection configurations correspond to equivalent multipliers.
We realize all PSAs for the 13 spacetime groups in Tab. 1 for both U (1) and Z 2 gauge fields. In Tab. 1, we have categorized the phase factors into five classes, denoted by σ, α, β, η and τ . Each of these classes can be realized by lattice building blocks with appropriate gauge fluxes, which are illustrated in Fig. 1 and elucidated below. The technical details are provided in the SI.
(i) The first class is σ = [L 1 : L 2 ] for the translation subgroup, where L 1 and L 2 are primitive spacetime translations. The value of σ corresponds to the flux in a Figure 1a shows the case when L 1 = L x , L 2 = L T . The general cases of spacetime translations can be found in the SI.
(ii) The second class concerns cohomology invariants for spacetime inversion, e.g., α = (M x M t ) 2 . α = ±1 corresponds to flux 0 or π in every spacetime inversion invariant loop, as illustrated in Fig. 1b. (iii) The third class is denoted by τ , relating time translations and time glide reflections, e.g., τ = g t L T g −1 t L −1 T . The phase factor τ * corresponds to the flux in a loop formed by g t L T g −1 t L −1 T , as illustrated in Fig. 1c. (iv) The fourth class denoted η modifies the relation between spatial reflections and time translations, e.g., Fig. 1d.
(v) The fifth class corresponds to the square of a time reversal, i.e, β = M 2 t with β = ±1. β = −1 cannot be realized by one-layer model, here we realize it by a two layer-model, where the time reversal is effectively realized by the combination of time reversal and a horizontal twofold rotation, as illustrated in Fig. 1e.
These building blocks form the basis for systematic model constructions for all projective symmetry algebras in Tab. 1. Here we present the models for P 1 and P g t , respectively, each of which has only one phase factor. The remaining models are available in the SI.
Projective P1 symmetry. Our first example is simply P 1 consisting of only spacetime translations. For simplicity, we choose symmetry operators as spatial and time Fig. 1. Realization of projective algebraic relations. a,b,c,d,e illustrate the realization of nontrivial cohomology invariants σ, α, η, τ, β respectively. The dashed lines indicate the evolution of hopping amplitudes with time. The dark areas indicate fluxes, which are realized by tunning hopping amplitudes. The red hoppings differ the corresponding blue hoppings by a minus sign. The yellow and purple hoppings differ the corresponding blue hoppings by a phase.
translators, L x and L T . We construct the lattice model with only the nearest-neighbor hopping amplitudes as shown in Fig. 2a. The momentum-space Hamiltonian is given by where w 1,2 are the two hopping amplitudes within each cell, and J(t) is the hopping amplitude between two neighboring cells.
For simplicity, we choose [L x : L T ] = −1, namely {L x , L T } = 0, which can be realized by inserting odd times of π-flux in each spacetime plaquette formed by L x and L T . This can be implemented by setting w 1,2 (t) = −w 1,2 (t + T ) and J(t) = −J(t + T ). In this setting, the flux in each spacetime unit cell is 3π, and we note that any odd times of π-flux is appropriate and leads to the same projective algebra. Since the gauge condition preserves L x , L x = L x . But translating H(k, t) by L T should be combined with a gauge transformation G T (see Fig.  2a), which in momentum space is represented by where L G 2 denotes a half reciprocal translation, namely, sending k to k + G/2. Then, it is easy to verify {L x , L T } = 0 with L T = G T L T and L x = e ika . The anti-commutativity can also be pictorially observed: In Fig. 2a, the signs ± of G T are exchanged by L x , i.e., L x G T L −1 x = −G T . Projective P g t symmetry. Our second example is P g t with two generators g t and L T . In particular, g t is the glide time-reflection, which combines the time inversion M t with a half translation L x/2 along the spatial lattice, namely g t = L x/2 M t . From Tab. 1, we consider the projective algebra, g t L T g −1 t L T = −1. The model is constructed as in Fig. 2b, and is described by the Hamiltonian, The nontrivial projective algebra can be realized by inserting a π-flux into each loop formed by g t L T g −1 t L T (see Fig. 2b). Particularly, in Eq. (7), this is satisfied by w 1 (t) = −w 3 (−t), w 2 (t) = J(−t) and w 1 (t) = −w 1 (t + T ), w 2 (t) = w 2 (t + T ), w 3 (t) = −w 3 (t + T ) and J(t) = J(t + T ). Now, g t should be accompanied by the gauge transformation G gt (see Fig. 2b). In momentum space, G gt = σ 3 ⊗ σ 3 with σ's the Pauli matrices, and where I k (I t ) is the inversion of k (t), and K the complex conjugation. Moreover, one can check that the gauge transformation G T for L T is equal to G gt , i.e., Physical consequences of projective spacetime symmetry We proceed to discuss the aforementioned three fascinating consequences of the projective spacetime symmetry algebras.
The electric Floquet-Bloch theorem for projective spacetime translational symmetry is the counterpart of the magnetic Bloch theorem for static systems with projective spatial translational symmetry in uniform magnetic fields [12,41,42]. Without loss of generality, let us consider a (1, 1)D spacetime crystal with flux 2πp/q through each spacetime unit cell. Then, the proper time and lattice translation operators L T and L x that commute with the Hamiltonian satisfy the gauge-invariant commutation relation [L x : L T ] = e i2πp/q . We choose two commuting operators L x and (L T ) q , which have the common eigenstates corresponding to the quantum numbers (k, ϵ n ). Note that ϵ n denote the eigenvalues of U (k, qT ) = T exp[−i qT 0 H(k, t)dt], which are known as "quasi-energies", with T indicating the time ordering. The Floquet-Bloch states labeled by (k, ϵ n ) can be written as where u k,n satisfies u k,n (x+a, t) = u k,n (x, t), u k,n (x, t+qT ) = λ(x)u k,n (x, t).
Here, the phase λ(x) = e −2πipx/a arises from the assumption of a uniform electric field, and vanishes at each lattice site. The proof of the electric Floquet-Bloch theorem is given in the Methods. It is noteworthy that ψ k,n , L T ψ k,n , ..., (L T ) q−1 ψ k,n are q-fold degenerate states for the quasi-energy, where (L T ) m ψ k,n has momentum k + 2πmp/qa. This is exemplified by the model (5). If ψ k,n is an eigenstate of U (k, 2T ), then U −1 (k + π/a, T )M −1 ψ k,n is an eigenstate of U (k +π/a, 2T ) with the same eigenvalue. Accordingly, as observed in Fig. 2c, the quasi-energy spectrum is the same at k and k + π/a, and hence has a half period.
(2) Kramers degeneracy protected by projective symmetry. For static systems, it has been shown that projective symmetries can lead to the Kramers degeneracy for spinless systems [30]. Here, we show that this extraordinary phenomenon also occurs in the quasi-energy spectra for time crystals with projective symmetry algebras.
Let us illustrate this by P g t symmetry of (7), while another example for group P 2 can be found in the SI. The symmetry operator (8) constrains the Hamiltonian as M g H(k, t)M −1 g = H * (−k, −t), which leads to the evo- Hence, for U (k, 2T ), we should consider the anti-unitary symmetry operator M g (k)K with (M g (k)K) 2 = −e ika . It is significant to notice that the square of M g (k)K equals −1 at k = 0, which leads to the Kramers degeneracy at k = 0 for the quasi-energy. Likewise, one can show that another time glide symmetry L T g t protects the Kramers degeneracy at k = π/a. The Kramers degeneracies at both k = 0 and π/a can be seen in Fig. 2d. It is noteworthy that the Kramers degeneracy at k = π/a also appears for ordinary P g t group [13] (see Fig. 2e), but that at k = 0 only appears for projective P g t algebra.
(3) Symmetry-enforced crossings of the Hamiltonian spectral flow. We first note that a sublattice symmetry Γ operates on a bipartite lattice exactly as a Z 2 gauge transformation: Each A-sublattice (B-sublattice) site is multiplied by a sign + (−). Let us consider a periodic time evolution restores the crystalline system up to the gauge transformation Γ specified by the sublattice symmetry, i.e., H(k, T ) = ΓH(k, 0)Γ. Since {Γ, H} = 0 for sublattice symmetry, we have H(k, T ) = −H(k, 0). Thus, generally, for a gapped H(k, 0), a continuous H(k, t) with respect to t exchanges valence and conduction bands in a period T , which enforces energy band crossings intermediately. Note that here we consider the instantaneous spectrum of the Hamiltonian H(t) rather than the quasienergy spectrum. Above band crossing is illustrated by a t-dependent dimerized model as a generalization of the Su-Schrieffer-Heeger model in the Methods.

Discussion
To summarize, our work presents a general theory for the projective spacetime symmetry of spacetime crystals. We have achieved a complete classification and presented the PSAs of (1, 1)D spacetime crystals, and all systematically constructed models can be easily realized through the use of engineerable gauge fields in artificial crystals. Although our focus has been on spinless systems, the extension of our theory to spinful systems is a straightforward matter. In such cases, spin-1/2 will contribute a phase of -1 to the squares of spacetime rotations and time reversals. Therefore, to include spin degrees of freedom, we can simply replace α and β in Tab. 1 with α = (−1) 2s α and β = (−1) 2s β, respectively. Fig. 2 Projective P 1 and P g t symmetry model. a Spacetime tight-binding model with projective algebra of P 1. The dashed lines indicate a continuous evolution of hopping amplitudes. The three hopping amplitudes are explicitly shown at t = 0, T, 2T . The signs marked in red at each site at t = T describe the gauge transformation G T needed to restore the initial connections. b Spacetime tight-binding model with projective representation of P g t . Glide axes are plotted as dark horizontal lines. The distribution of hopping amplitudes is explicitly marked at five times, respectively. We observe that the glide time-reflection through t = T /2 is manifestly preserved, while that for t = 0 or T is preserved up to a gauge transformation G gt . The signs for G gt at a given time are marked in red. The arrows in a and b are the primitive lattice vectors correspond to L x and L T . c Quasibands of U (2T ) of the model. The present work provides a comprehensive framework for investigating the fascinating physics of spacetime crystals with projective spacetime symmetry. Through a thorough analysis of the PSAs, we have uncovered three significant physical consequences. Moving forward, an intriguing direction for future research would be to explore spacetime topological phases with projective symmetry. This research avenue is particularly compelling given the systematic classifications of crystalline topological phases that have emerged from the study of static crystal symmetries [43-48].
Finally, it is worth noting that while gauge fluxes on spacetime lattices can be used to realize PSAs, they may not be capable of representing every possible PSA in higher dimensions. Certain PSAs may necessitate intrinsic many-body states, indicating a potential avenue for further exploration.

Methods
Proof of the electric Floquet-Bloch theorem Here, we prove the Floquet-Bloch theorem in a uniform electric field, which we call electric Floquet-Bloch theorem. We only concern the 1+1D case, while the generazation to 3+1D is straightforward.
The Hamiltonian for a periodic driving system in a uniform electric field E x = E can be written as where the potential U (x, t) is periodic, i.e, U (x + a, t) = U (x, t + T ) = U (x, t) and we choose the gauge A 0 = 0, A x (t) = −Et. Due to the electric field, H(x, t) is not periodic at the time direction, i.e., H(t+T ) ̸ = H(t). If we define two translation operators L x , L T by L x f (x, t) = f (x + a, t) and L T f (x, t) = f (x, t + T ), then H(x, t) commutes with L x but not with L T . However, we observe that H(x, t + T ) only differs from H(x, t) by a gauge transformation This gauge transformation can be equivalently defined as where G T = e iχ(x) . So we can define a proper time translation operator L T = G T L T , which commutes with H(x, t), L T H(x, t)L −1 T = H(x, t). One can check the two proper translation operators L T and L x satisfy We assume the electric flux in every unit cell is Φ E = pΦ 0 /q = 2πp/q. To find the constrain of the translation symmetry on the wavefunction, we have to find commutative operators which commute with the Hamiltonian. Here, we can take L x , (L T ) q as two generators, they generate an abelian spacetime translation group G. We define then solutions satisfy H(x, t)ψ(x, t) = 0. All solutions form a solution space V . Because every group element g ∈ G commutes with H(x, t), the solution space V is a representation of G, which can be decomposed by irreducible representations of G. Since G is abelian, its irreducible representations are one dimensional, which are labelled by (k, ϵ), with character that is, We can rewrite it as Plug this ansatz into H(x, t)ψ(x, t) = 0, we can obtain a set of eigenvalues ϵ n (k) and eigenstates u n,k (x, t). Thus we can label a state with by (k, n) and we complete the proof of the electric Floquet-Bloch theorem.
is also a solution. Moreover, ψ ′ (x, t) also has the quasienergy ϵ n (k), but has momentum k + 2πp/qa, this can be seen by Band crossing due to projective symmetry Here, we consider a driven Su-Schrieffer-Heeger (SSH) model where v(t) and w(t) are real. We require H(t) to evolve adiabatically. If this model has projective time translation symmetry H(t + T ) = −H(t), (which can be written as , the instantaneous bands must cross at some time t. This is easy to see: In there must be w(t * ) = v(t * ) at some t * . And we know that SSH model is gapless when |w| = |v|, so there is band crossing at t * for this driven SSH model.

Data availability
The data generated and analyzed during this study are available from the corresponding author upon request.

Code availability
All code used to generate the plotted band structures is available from the corresponding author upon request.  An A-projective representation of group G is a vector space V , with a map ρ : G → GL(V ) which is "almost a homomorphism" in the sense that where function ν : G × G → A is called a multiplier for G. Associativity (g 1 g 2 )g 3 = g 1 (g 2 g 3 ) requires the multiplier ν satisfy the 2-cocycle equation: We denote the set of all 2-cocycles by Z 2 (G, A), which is an abelian group under the multiplication of functions. Two projective representations ρ and ρ ′ related by ρ ′ (g) = χ(g)ρ(g) are considered as equivalent, where χ(g) ∈ A. Correspondingly, their multipliers ν, ν ′ is related by A function f : G × G → A in the form f (g 1 , g 2 ) = χ(g 1 )χ(g 2 )/χ(g 1 g 2 ) or f (g 1 , g 2 ) = χ(g 1 g 2 )/χ(g 1 )χ(g 2 ) is called a 2-coboundary. Two 2-cocycles is equivalent if they differ by a 2-coboundary. We denote the set of all 2-coboudaries by B 2 (G, A), which is also an abelian group under the multiplication of functions. Then, the set of nonequivalent multipliers (2-cocycles) is given by the quotient group which is called the second cohomology group. Thus, H 2 (G, A) classifies nonequivalent A-valued multipliers for G.
In quantum mechanics, time-reversal operations are represented by anit-unitary operators rather than unitary operators. Therefore they have complex conjugate action on the A-valued multiplier ν (in general A = U (1)). Consequently, if G contains time-reversal elements, the 2-cocycle equation should be modified to a twisted 2-cocycle equation: where g is complex conjugate if ρ(g) is anti-unitary, otherwise g is trivial. We denote the set of twisted 2-cocycles as Z 2,c (G, A), where c indicate the complex conjugate action. In a similar manner, the 2-coboundaries now should also be twisted, taking the form χ(g 1 ) g 1 (χ(g 2 ))/χ(g 1 g 2 ) or χ(g 1 g 2 )/χ(g 1 ) g 1 (χ(g 2 )). We denote the set of them by B 2,c (G, A). The nonequivalent multipliers are now classfied by the twisted second cohomology group

Projective symmetry algebra and cohomology invariants
The multiplicative relations of a projective representations is called a projective symmetry algebra. It is determined by the multiplicative relations of group elements and the multiplier ν. However, the multiplier is typically complicated and contains much redundancy. To concisely capture the essential information of the projective symmetry algebra, we use projective symmetry algebraic relations of generators and cohomology invariants, which will be introduced below.
A finite generated group G can be presented by generators and relations: where S = {s 1 , s 2 , ...} is the set of generators and R = {r 1 (s), r 2 (s), ...} is the set of relations. In general, the relations have the following form : .. correspond to the same group element if they can be reduced to be equal using relations.
If ρ is a representation of G, its constrain on generators also satisfies the algebraic relations in R, because However, if ρ is a projective representation of G with multiplier ν, its constrain on generators does not satisfy the algebraic relations in R, but with additional phases, for example, The occurance of α i is due to the multiplier ν of the projective representation. If we define a "modified" presentation ..} is the projective symmetry algebraic relations of generators. Then the constrain of ρ on S satisfies the relations in R. In the "modified" presentation, two words correspond to the same group element if they can be reduced to be equal up to a phase using relations. Now, we state the core conclusion of this section: The "modified" presentation G = ⟨S| R⟩ contains the information of the equivalence class of multiplier ν. More concretly, if G is obtained by modifying R to R due to multiplier ν, we can infer the equivalence class of multiplier ν from G, though ν itself cannot be inferred. This conclusion is very useful because it tells us that to capture the information of a projective symmetry algebra (in an equivalent sense), we only need a "modified" presentation G = ⟨S| R⟩, which is much simpler than the multiplier.
To prove this conclusion, we proceed as follows. For a fixed S, we can assign a "standard" word w(g) = s n1 1 s n2 2 s n3 3 . . . to each group element g ∈ G. Then for a given G = ⟨S| R⟩, we can obtain a multiplier ν by where g c = g a g b and the multiplier ν is obtained by using the relations in R. We now prove ν is in the equivalence class of ν. From ρ(g a )ρ(g b ) = ν(g a , g b )ρ(g c ), we have where χ(g i ) is the phase due to decomposing ρ(s 3 ).... Thus we complete the proof.
Although the "modified" presentation is more concise than the multiplier, it may still contain redundancies. Let us consider a "modified" presentation G = ⟨s 1 , s 2 , ...|r 1 (s) = α 1 , r 2 (s) = α 2 , ...⟩. For each generator s, we can do a coboundary transformation s → s ′ = χ(s)s, which changes the relations R to This yields a new "modified" presentation ⟨s 1 , s 2 , ...|r 1 (s) = α ′ 1 , r 2 (s) = α ′ 2 , ...⟩, which corresponds to the same cohomology class of multipliers as G. To reduce this redundancy, we can recombine the relations such that some factors are invariant under coboundary transformations while others can be set to 1. We call a factor that is invariant under coboundary transformations a cohomology invariant. The values of a complete set of cohomology invariants uniquely determine the equivalence class of a multiplier (and hence the projective symmetry algebra).
In the following sections, we will utilize "modified" presentation and cohomology invariants to describe projective symmetry algebras.
B. Projective symmetry algebras of (1,1)D spacetime crystalline groups In this section, we derive all the nonequivalent U (1) and Z 2 projective symmetry algebras for all 13 (1,1)D spacetime crystalline groups. The results are summarized in Table. I and Table. II.  Gst H 2 Lx, LT , Mx  The general method to derive all possible nonequivalent A-projective symmetry algebras for G is as follows. First, we choose a presentation G = ⟨S|R⟩ = ⟨s 1 , s 2 , ...|r 1 (s) = 1, r 2 (s) = 1, ...⟩. Then we modify the raltions R into R = {r 1 (s) = α 1 , r 2 (s) = α 2 , ...} by factors α 1 , α 2 , ... ∈ A. The remaining task is to determine all possible values of factors α i . From Eq. (11), we see that G determines a multiplier ν, which is a function of n ai , n bi , i = 1, 2, ..., and also a fucntion of α 1 , α 2 , .... The multiplier ν must satisfy the cocycle equation for any three group element g a , g b , g c , which leads to a equation involving factors α 1 , α 2 , ... and parameters n ai , n bi , n ci , i = 1, 2, .... This equation should hold for all possible values of n ai , n bi , n ci , i = 1, 2, .... Its solutions give all possible values of α 1 , α 2 , ..., and thus give us all cohomology classes of projective symmetry algebras. However, if some factors α i are not cohomology invariants, some solutions may correspond to the same cohomology class. To avoid this situation, we only use cohomology invariants and fix other factors by coboundary transformations at the outset.
While the above cocycle equation method is general, it can be rather tedious. In this paper, we employ a more streamlined approach to identify all possible values of cohomology invariants. Specifically, we use only a small subset of self-consistency conditions derived from the cocycle equation to constrain the cohomology invariants. Since we do not consider all constraints, some of our solutions may be invalid. So to ensure the accuracy of our solutions, we verify them through other means. For A = Z 2 case, we cross-check our results against the already-known H 2 (G st , Z 2 ) values. If our findings yield the same number of projective symmetry algebras as H 2 (G st , Z 2 ), we can validate our results. In the case of A = U (1), we should cross-check our results against the H 2,c (G st , U (1)) values. However, for a general wallpaper group, H 2,c (G st , U (1)) is not available. Fortunately, in most cases, we still have simple method to ensure the correctness of our results. For groups P 1, P m x , P g x and Cm x , H 2,c (G st , U (1)) = H 2 (G st , U (1)) since they have no anti-unitary operation. So we can cross-check our results with H 2,c (G st , U (1)) (which are known) in these cases. For groups P 2, P m x m t , P g x g t , P m x g t , P m t g x and Cm x m t , we find that their cohomology invariants can be constrained to be Z 2 value, which means their U (1)-projective symmetry algebras are just special cases of Z 2 -projective symmetry algebras. Therefore, once we constrain all cohomology invariants to be Z 2 value, we cannot further constrain them and arrive at the final result. For group P m t , P g t and Cm t , we lack a simple method to verify our results, so we solve the cocyle equation by brute force. After verifying the accuracy of our results for the U (1) cases, we can infer H 2,c (G st , U (1)) from the possible values of the cohomology invariants.
The above description of our method may seem abstract, so we recommend that readers refer to the examples below, as they can be helpful in clarifying the approach. The P 1 spacetime group can be generated by two unit spacetime translations L 1 and L 2 , which are not necessary to be L x and L T . The two generators commute with each other, [L 1 : Two lattice models with P 1 symmetry is shown in Fig. 1, one with ordinary spatial and time translation symmetry, another with spacetime translation symmetry.
Projective symmetry algebras can be obtained by modifying the relation by For Z 2 -projective symmetry algebra, σ takes values in Z 2 = {±1}, and for U (1)-projective symmetry algebra, σ takes values in U (1). It is esay to see that σ is a cohomology invariant. These results are already consistent with that of group cohomology so the value of the factor σ cannot be constrained further. The P 2 spacetime group contains two-fold spacetime rotations besides translations. It can be generated by L 1 , L 2 and spacetime rotation C = M x M t . The generator C reverses the direction of translation L 1 , L 2 , so the presentation of P 2 is A spacetime tight binding model with P 2 symmetry is shown in Fig. 2, where we choose L 1 , L 2 to be L x , L T although they can be space-time mixed translations in general. There are four different conjugacy classes of rotations C s (R), C s (L a C), C s (L b C), C s (L a L b C), whose rotation centers are shown in Fig. 2 in four different colors. We can recombine the relations to present P 2 in terms of the squares of four classes of rotations: Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, α 1 , α 2 , α 3 , α 4 ∈ Z 2 and are cohomology invariants, which is consistent with the result of group cohomology For U (1)-projective symmetry algebras, α 1 , α 2 , α 3 , α 4 are also cohomology invariants. The self-consistency conditions require α 1 , α 2 , α 3 , α 4 ∈ Z 2 , which can be seen by and so on. Here we note we used the anti-unitary property of the operator C = M x M t and so on. Although we only use a few self-consistency conditions to constrain the cohomology invariants, once they are all constrained to be Z 2 , they cannot be further constrained. Because now the U (1)-projective symmetry algebra becomes a Z 2 -projective symmetry algebra and we have shown these values of cohomology invariants are allowed for Z 2 case. From the possible values of cohomology invariants, we can infer the twisted group cohomology to be which is different from the untwisted result H 2 (P 2, U (1)) = U (1).
3. Pmx The P m x spacetime group contains space reflections besides translations. The two unit translations must be L x , L T . The reflection M x reverses L x to L −1 x but leaves L T to be invariant. The presentation can be A spacetime tight binding model with P m x symmetry is shown in Fig. 3. There are two different conjugacy classes of mirror reflections C s (M x ), C S (L x M x ), whose reflection axes are shown in Fig. 3 in different colors. The presentation can be rewritten in terms of squares of the two reflections M x , L x M x and their commutators with L T , Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, η 1 , η 2 , γ 1 , γ 2 ∈ Z 2 and are cohomology invariants. For U (1)-projective symmetry algebras, factors γ 1 , γ 2 can be set to be 1 by redefining Cohomology invariants η 1 , η 2 ∈ Z 2 , which can be seen by These results are consistent with that of the group cohomology The P m t spacetime group contains time reversal besides translations. It can be generated by L x , L T and time reversal operator M t . M t reverses the direction of L T but leaves L x invariant, so the presentation can be A spacetime tight binding model with P m t symmetry is shown in Fig. 4, where we choose nT and nT /2 to be time reversal axes corresponding to two conjugacy classes of time reflections C s (M t ), C s (L T M t ). The presentation can also be rewritten as Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, cohomology invariants σ, η, β 1 , β 2 ∈ Z 2 , which is consistent with the result of the group cohomology For U (1)-projective symmetry algebras, factor η can be trivialized by redefining L x → L ′ x = η 1/2 L x . Cohomology invariants β 1 , β 2 ∈ Z 2 since M t and L T M t are anti-unitary (the proof is similar to Eq. (21)). However, there is no constrain on the cohomology invariant σ, so σ ∈ U (1). The validity of this result was checked by the general method of cocycle equations, which we do not present here. From the possible values of cohomology invariants, we can infer that For Z 2 -projective symmetry algebras, the projective symmetry algebras can also be written in the form where η 1 = η, η 2 = ση ∈ Z 2 . The P m x m t spacetime group contains space reflections and time reversal besides translations. It can be generated by L x , L T , M x , M t . The presentation is given by A spacetime tight binding model with P m x m t symmetry is shown in Fig. 5. There are two classes of spatial mirror axes at x direction and t direction respectively, and their intersections are two-fold rotation centers. The relations can also be expressed in terms of the squares of the four rotations and the four reflections: Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, cohomology invariants α 1 , α 2 , α 3 , α 4 , γ 1 , γ 2 , β 1 , β 2 ∈ Z 2 , which is consistent with the result of group cohomology For U (1)-projective symmetry algebras, γ 1 , γ 2 can be trivialized by redefining M x → M ′ 1 L x . The cohomology invariants α 1 , α 2 , α 3 , α 4 , β 1 , β 2 ∈ Z 2 due to the anti-unitary property of rotation and time reversal operators. Since the U (1)-projective symmetry algebras have been reduced to special cases of Z 2 -projective symmetry algebras, the cohomology invariants cannot be further constrained. We can infer that Besides translations, the P g x spacetime group contains a glide reflection operation g x , which is a reflection at x direction followed by a T /2 translation at t direction. Since L T = g 2 x , we can take L x , g x as generators. g x reverses the direction of L x , so the presentation can be given by A spacetime tight binding model is shown in Fig. 6. There are two classes of glide-reflection axes, which are shown in different colors.
Projective symmetry algebras can be obtained by modifying the relation by For Z 2 -projective symmetry algebras, cohomology invariant τ ∈ Z 2 , while for U (1)-projective symmetry algebras, τ can be trivialized by redefining L x → L ′ x = τ −1/2 L x . These results are consistent with that of the group cohomology Besides translations, the P g t spacetime group contains a glide reflection operation g t , which is a reflection at t direction followed by a half translation at x direction, i.e., g t = L x/2 M t . Since L x = g 2 t , we can take L T , g t as generators. g t reverses the direction of L T , so the presentation can be given by A spacetime tight binding model is shown in Fig. 7. There are two classes of glide-reflection axes, which are shown in different colors.
Projective symmetry algebras can be obtained by modifying the relation by For Z 2 -projective symmetry algebras, cohomology invariant τ ∈ Z 2 , which is consistent with the result of group cohomology For U (1)-projective symmetry algebras, cohomology invariant τ ∈ U (1). One can check this result by the general method of cocycle equation. From this result, we infer that The P g x g t spacetime group contains glide reflections at both x and t directions. Since g 2 x = L T , g 2 t = L x , the generators can be chosen as g x , g t . The combination operations g x g t and g x g −1 t are two-fold rotations. The presentation can be written as A spacetime tight binding model with P g x g t symmetry is shown in Fig. 8. There are two classes two-fold rotation centers corresponding to C s (g x g t ) and C s (g x g −1 t ) respectively. Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, cohomology invariants α 1 , α 2 ∈ Z 2 , which is consistent with the result of group cohomology For U (1)-projective symmetry algebras, cohomology invariants α 1 , α 2 ∈ Z 2 , which is due to the anti-unitary property of g x g t and g x g −1 t . Since the U (1)-projective symmetry algebras have been reduced to Z 2 -projective symmetry algebras, α 1 and α 2 cannot be constrained further. From this result, we can infer that The P m x g t spacetime group contains reflections at x direction and glide-reflections at t direction. It can be generated by L T , M x , g t . The presentation can be given by A spacetime tight binding model with P m x g t symmetry is shown in Fig. 9. There is one class of glide reflection axes, and two classes of time mirror axes, also there are two classes of two-fold rotation centers, corresponding to C s (M x g t ) and C s (L T M x g t ) respectively. We can also rewrite the presentation in terms of the commutator [M x : L T ] and the squares of rotations and reflection, Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, cohomology invariants η, α 1 , α 2 , γ ∈ Z 2 , which is consistent with the result of group cohomology For U (1)-projective symmetry algebras, factor γ can be trivialized by redefining M x → M ′ x = γ −1/2 M x , and cohomology invariants α 1 , α 2 ∈ Z 2 due to the anti-unitary property of M x g t and L T M x g t . Cohomology invariants η ∈ Z 2 , which can be seen by Since the U (1)-projective symmetry algebras have been reduced to special cases of Z 2 -projective symmetry algebras, the cohomology invariants α 1 , α 2 , η cannot be further constrained. We can infer that 10. Pmtgx The P m t g x spacetime group contains reflections at t direction and glide-reflections at x direction. Its generators can be chosen as L x , M t , g x . The presentation can be given by A spacetime tight binding model with P m t g x symmetry is shown in Fig. 10. There is one class of glide-reflection axes and two classes of spatial mirror axes, also there are two classes of two-fold rotation centers, corresponding to C s (M t g x ) and C s (L x M t g x ) respectively. We can also rewrite the presentation in terms of the commutator [M t : L x ] and the squares of rotations and reflection, Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, cohomology invariants η, α 1 , α 2 , β ∈ Z 2 , which is consistent with the result of group cohomology For U (1)-projective symmetry algebras, factor η can be trivialized by redefining L x → L ′ x = η 1/2 L x , and cohomology invariants α 1 , α 2 , β ∈ Z 2 due to the anti-unitary property of M x g t , L T M x g t , M t . Since the U (1)-projective symmetry algebras have been reduced to special cases of Z 2 -projective symmetry algebras, the cohomology invariants α 1 , α 2 , β cannot be further constrained. We can infer that The Cm x spacetime group contains mirror reflection M x , which interchanges the two unit translations L 1 , L 2 . Since the time direction and space direction is inequivalent, L 1 , L 2 must be spacetime mixed. The presentation can be given by A spacetime tight binding model with Cm x symmetry is shown in Fig. 11. There are two classes of mirror axes. Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, η can be trivialized by redefining L 1 → L ′ 1 = η −1 L 1 , and cohomology invariants σ, γ ∈ Z 2 .
For U (1)-projective symmetry algebras, η, γ can be trivialized by redefining Cohomology invariant σ ∈ Z 2 due to the self-consistency condition required by the mirror reflection: These results are consistent with that of the group cohomology (68)

Cmt
The Cm t spacetime group contains mirror reflection M t , which interchanges two space-time translations L 1 , L 2 . Since the time direction and space direction is inequivalent, L 1 , L 2 must be spacetime mixed. The presentation can be given by A spacetime tight binding model with Cm t symmetry is shown in Fig. 12. There are two classes of time reversal axes.
Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, η can be trivialized by redefining L 1 → L ′ 1 = ηL 1 , and cohomology invariants σ, β ∈ Z 2 . This result is consistent with that of the group cohomology For U (1)-projective symmetry algebras, η can also be trivialized by redefining L 1 → L ′ 1 = ηL 1 . Cohomology invariant β ∈ Z 2 due to the anti-unitary property of M t . There is no constrain on the cohomology invariant σ, so σ ∈ U (1). This result can be checked by the general method of cocycle equation. From the possible values of cohomology invariants, we can infer that (72)

Cmxmt
The Cm x m t spacetime group is generated by L 1 , L 2 , M x , M t . L 1 , L 2 must be spacetime mixed. Both M x and M t interchange L 1 and L 2 . The presentation can be given by A spacetime tight binding model with Cm x m t symmetry is shown in Fig. 13. There are two classes of time reversal axes and two classes of spatial reflection axes. There are also three classes of two-fold rotation centers, corresponding We can rewrite the presentation in terms of squares of rotations and reflections, Projective symmetry algebras can be obtained by modifying the relations by For Z 2 -projective symmetry algebras, cohomology invariants α 1 , α 2 , α 3 , γ, β ∈ Z 2 , which is consistent with the result of group cohomology For U (1)-projective symmetry algebras, factor γ can be trivialized by redefining 1 M t and M t . Since the U (1)projective symmetry algebras have been reduced to special cases of Z 2 -projective symmetry algebras, the cohomology invariants α 1 , α 2 , α 2 , β cannot be further constrained. We can infer that H 2,c (Cm x m t , U (1)) = Z 4 2 . (77)

C. Time crystals with gauge structures
In this section, we present a formalism for studying projective symmetries of time crystals with gauge structures. One interesting feature of this formalism is that cohomology invariants can be interpreted in terms of flux, enabling us to realize different projective symmetry algebras by tuning the flux. Different from the magnetic flux which makes crystal symmetries projectively represented, here the flux makes spacetime crystalline symmetries projectively represented is electric flux. While magnetic flux is more commonly known, electric flux may be less familiar to some readers. Therefore, we provide a brief introduction to electric flux before proceeding further.

Electric A-B effect and electric flux
In the seminal paper by Aharonov and Bohm (55), they raised two types of A-B effects: electric A-B effect and magnetic A-B effect. While the latter is well-known, the former gets less attention, which is partly due to the difficulties to observe it. Although the type-II electric A-B effect has been confirmed by experiments (56)(57)(58), the type-I electric A-B effect which requires the particle to experience a vanishing field is still lack of observation.
In general, when an electron circles a spacetime loop C, it acquires an A-B phase e i∆g , where A µ = (−ϕ, ⃗ A). By applying Stokes' theorem, we can write it as where S is an area whose boundary is C and the sign depends on the direction of C. When the loop is in a spatial plane, this formula gives us the well-known magnetic A-B phase. But when the loop is in a spacetime plane, for example the t − x plane, it gives us an electric A-B phase where Φ E is called electric flux.
In this paper, we focus on (1,1)D spacetime lattice systems, where the electric flux is solely induced by E x . Consider a spacetime plaquette with four vertex (x 1 , t 1 ), (x 2 , t 1 ), (x 2 , t 2 ), (x 1 , t 2 ). The electric flux on it is In this paper, we always adopt the gauge ϕ = 0, i.e., E x = −∂A x /∂t. With this gauge, the above flux

Symmetries of spacetime tight binding models with gauge structures
In this paper, we study (1,1)D spacetime tight-binding models, which can be given by the Hamiltonian where the phase A ij (t) can be seen as induced by a guage field, In general, a spacetime symmetry operation R that preserves the lattice (strengths of hoppings) and flux configuration may change the gauge configuration (phase A ij (t)). The transformed gauge configuration is related to the original configuration by a gauge transformation G R , since they correspond to the same flux configuration. Therefore, the proper spacetime symmetry operator should be The action of R on H(t) is given by where R is the complex conjugate if R contains time reversal, R(i), R(t) is defined as (R(i), R(t)) = R(i, t), and G R (i) is the phase of gauge transformation on site i. The Hamiltonian H(t) is invariant under R if the following condition is satisfied: This formula will help us derive the flux interpretation of cohomology invariants in the next section. Now, we prove the claim in main text that Ω(R 1 , factor. First we observe that Ω(R 1 , R 2 ) is a pure gauge transformation, i.e., a diagonal matrix with ith diagonal entry being G R1 (i)G R2 (R −1 1 (i))/G R1R2 (i). It commutes with all possible symmetry-preserving Hamiltonian. If we presume the Hamiltonian is a connected lattice model, Ω(R 1 , R 2 ) will be proportional to the identity matrix, namely [Ω(R 1 , R 2 )] ij = ν(R 1 , R 2 )δ ij with ν(R 1 , R 2 ) ∈ U (1).

Flux interpretation of cohomology invariants
In Table. I and Table. II, there are five types of cohomology invariants: (1) Cohomology invariants of translation subgroups, which we denote by σ.
(3) Cohomology invariants between translations and reflections. We denote the cohomology invariant between L x and M t by η, and the cohomology invariant between L T and M x by η (which can be nontrivial only for Z 2 -projective symmetry algebras).
(4) Cohomology invariants between translations and glide-reflections. We denote the cohomology invariant between L T and g t by τ , and the cohomology invariant between L x and g x by τ (which can be nontrivial only for Z 2 -projective symmetry algebras).
(5) Cohomology invariants of reflections. We denote the cohomology invariants of time reversals by β, and the cohomolgy invariants of spatial reflections by γ (which can be nontrivial only for Z 2 -projective symmetry algebras).
As we will see, cohomology invariants can be interpretated in terms of fluxes.
i. Cohomology invariants of translation subgroups For the translation subgroup, the projective algebraic relation which means that the gauge transformations on sites satisfy First, let us consider the translation group generated by L x and L T with L x L T L −1 A lattice with L x and L T symmetry is shown in Fig. 14, where i = L −1 x (j). There can be some sites between i and j in general, and we define e Aij (t) = e Ai,i+1(t) e Ai+1,i+2(t) · · · e Aj−1,j (t) .
For a general symmetry operator R, using Eq. (88) for every hopping between site i and j, we get Take R = L T , we have The left-hand side is exactly e −iΦ , where Φ is the electric flux in the loop L x L T L −1 x L −1 T , so we have a flux interpretation for the cohomology invariant σ, We then consider a general translation group generated by L 1 , L 2 with L 1 L 2 L −1 1 L 2 = σ, where L 1 , L 2 can be spacetime translations. We assume the lattice model still has a finite period at both time and space directions, meaning that there exist α 1 , β 1 , α 2 , β 2 ∈ Z such that L x = L α1 which indicates the flux Φ ′ contained in every loop formed by L x L T L −1 x L −1 T . One can show that the area circled by L x L T L −1 x L −1 T is exactly α 1 β 2 − α 2 β 1 times the area circled by L 1 L 2 L −1 1 L −1 2 . On the other hand, the flux configuration must satisfies L 1 and L 2 symmetry. Therefore, the flux contained in every loop of the form where sgn(x) is the sign of x. If the direction of the loop ii. Cohomology invariants of rotations For the rotations, the projective algebraic relation C 2 = α requires that so the gauge transformation on sites satisfies Consider a lattice with C symmetry in Fig. 15, where site j = C(i) and there can be some sites between i and j. From Eq. (94), we get the constrain of the phases of hoppings: The left hand side is e −iΦ , where Φ is the flux in the loop (i, t) → (j, t) → (j, Ct) → (i, Ct) → (i, t), so we have the flux interpretation of cohomology invariants of rotations, where Φ ∈ πZ is the flux in every loop which is invariant under C. Note that, since t can be very close to Ct, the loop can be infinitesimal, so the flux must concentrate on the time where the rotation center lies. For a π flux, this indicates the hopping passing through the rotation center changes a minus sign after that time. When t is the time at which the rotation center lies, i.e., t = Ct, we have e iAij (t) e −iAij (Ct) = 1, which seems to indicate the cohomology invariant α must be trivial. However, we can set w ij (t) = 0 at this time, then the phases at this time can be arbitrary chosen, and we can choose e iAij (t) e −iAij (Ct) = α to preserve the projective symmetry.
iii. Cohomology invariants between translations and reflections Consider a lattice with M x and L T symmetry, as shown in Fig. 16(a), where site j = M x (i) and there can be some sites between i and j. Taking R = L T in Eq. (94), we obtain Thus we have a flux interpretation of the cohomology invariant η as where Φ ∈ πZ is the flux in every loop formed by M x L T M −1 x L −1 T . For the cohomology invariants η = [M t , L x ], the gauge transformations on sites satisfy the equation Consider a lattice with M t and L x symmetry, as shown in Fig. 16(b), where site j = L x (i). Takiing R = M t in Eq.
(94), we have If we take all hoppings to be real such that e Aij (t) = e −iAij (t) , then we have where Φ ∈ πZ is the flux in every loop formed by M t L x M −1 t L −1 x . Since the the loop can be infinitesimal, the flux must concentrate on the time reflection axis, and the hoppings that are static under M t should be set to 0 for η = −1 case.

iv. Cohomology invariants between translations and glide-reflections
There are two types of cohomology invariants between translations and glide-reflections, τ = g t L T g −1 t L T and τ = g x L x g −1 x L x . For the first type, the gauge transformations on sites satisfy the equation Consider a lattice with L T and g t symmetry, as shown in Fig. 17(a), where site j = g t (i). Taking R = L T in Eq. (94), we obtain The left-hand side is actually e −iΦ , where Φ is the flux in the loop formed by g t L T g −1 t L T acting on site i. So we have the flux interpretation where Φ is the flux in every loop formed by g t L T g −1 t L T . It is interesting to note that which means the flux in a loop formed by L x L T L −1 x L −1 T is two times the flux in a loop formed by g t L T g −1 t L T . For the cohomology invariant τ = g x L x g −1 x L x , the gauge transformations on sites satisfy the equation Consider a lattice with g x and L x symmetry, as shown in Fig. 17 then e iAij (t) e iAjk(t) e iAjk(t+T /2) e iAij (t+T If all hopping amplitudes are real, and the gauge transformations take values in Z 2 , then we have where Φ ∈ πZ is the flux in every loop formed by g x L x g −1 x L x .

v. Cohomology invariants of reflections
There are two types of cohomology invariants of reflections, β = M 2 t and γ = M 2 x . The nontrivial cohomology invariant β = −1 can be realized by spinful systems. However, when there are two classes of time reversal axes corresponding to two cohomology invariants β 1 = M 2 t , β 2 = (L T M t ) 2 respectively, the cases β 1 = −β 2 cannot be realized. These cases occur for group P m t and group P m x m t . To address this issue, we realize β = −1 effectively by a two-layer model as shown in Fig. 18, where M t is replaced by RM t (R is a horizental two-fold spatial rotation). With this design, β 1 = −β 2 cases can be realized effectively by tuning hopping amplitudes.
The nontrivial cohomology invariant γ = −1 only appears in Z 2 -projective symmetry algebras. It cannot be realized in models with real hoppings. We can also realize it by a two-layer model as shown in Fig. 19, where M x is replaced by a horizontal two-fold spatial rotation R effectively. γ = −1 corresponds to π magnetic flux in the loops that are invariant under R.
In the following, we will focus on the realization of cohomology invariants σ, α, η,η, τ,τ , and do not discuss the realization of β and γ anymore.  Projective symmetry algebras of P 1 have one independent cohomology invariant σ = L 1 L 2 L −1 1 L −1 2 . For U (1) case, σ ∈ U (1), and for Z 2 case, σ ∈ Z 2 . According to what we analyzed in last section, we only need to add flux Φ to each unit translation plaquette to realize it, as shown in Fig. 20. The relation between the cohomology invariant and flux is σ = e iΦ .
(120) For spacetime group P 2, there are four independent cohomology invariants α 1 , α 2 , α 3 , α 4 ∈ Z 2 of rotations for both U (1) and Z 2 cases. We can add Φ 1 , Φ 2 , Φ 3 , Φ 4 ∈ πZ into every corresponding loops so that One example is shown in Fig. 21. For U (1)-projective symmetry algebras of spacetime group P m x , there are two independent cohomology invariants η 1 , η 2 ∈ Z 2 . We can realize them by adding Φ 1 , Φ 2 ∈ πZ into every loop formed by M x L T M −1 respectively. One example is shown in Fig. 22. The relations between flux configuration and cohomology invariants are Here we note that the adding of Φ i should not break the spatial reflection symmetry. For Z 2 -projective symmetry algebras of spacetime group P m x , there are four independent cohomology invariants η 1 , η 2 , γ 1 , γ 2 ∈ Z 2 . η 1 , η 2 can be also realized in the same way as U (1) case.  For U (1)-projective symmetry algebras of spacetime group P m t , there are three independent cohomology invariants σ, β 1 , β 2 . σ ∈ U (1) can be realized by adding Φ flux into every loop formed by L x L T L −1 x L −1 T , as shown in Fig. 23. The relation between flux configuration and cohomology invariant σ is For Z 2 -projective symmetry algebras of spacetime group P m t , there are four independent cohomology invariants η 1 , η 2 , β 1 , β 2 ∈ Z 2 . η 1 , η 2 can be realized by adding flux Φ 1 , Φ 2 ∈ πZ into every loop formed by x respectively, as shown in Fig. 24. The relations between flux configuration and cohomology invariants are For U (1)-projective symmetry algebras of spacetime group P m x m t , there are six independent cohomology invariants α 1 , α 2 , α 3 , α 4 , β 1 , β 2 ∈ Z 2 . α 1 , α 2 , α 3 , α 4 can be realized by adding Fig. 25. The relations between flux configuration and cohomology invariants are For Z 2 -projective symmetry algebras of spacetime group P m x m t , there are eight independent cohomology invariants α 1 , α 2 , α 3 , α 4 , β 1 , β 2 , γ 1 , γ 2 ∈ Z 2 . α 1 , α 2 , α 3 , α 4 can be realized in the same way as U (1) case. For U (1)-projective symmetry algebras of spacetime group P g x , there is no nontrivial cohomology invariant. For Z 2 -projective symmetry algebras of spacetime group P g x , there is one independent cohomology invariant τ ∈ Z 2 , which can be realized by adding Φ ∈ πZ flux into everly loop formed by g x L x g −1 x L x , as shown in Fig. 26. We have the relation projective symmetry algebras of P g t have one independent cohomology invariant τ . For U (1) case, τ ∈ U (1), and for Z 2 case, τ ∈ Z 2 . τ can be realized by adding flux Φ into every loop formed by g t L T g −1 t L T , as shown in Fig. 27. The relation between flux configuration and cohomology invariant is For both U (1) and Z 2 projective symmetry algebras of P g x g t , there are two independent cohomology invariants α 1 , α 2 ∈ Z 2 , which can be realized by adding Φ 1 , Φ 2 ∈ πZ flux into every loop which is invariant under rotation g x g t , g x g −1 t respectively. One example is shown in Fig. 28. The relations between flux configuration and cohomology invariants are (128) For U (1)-projective symmetry algebras of spacetime group P m x g t , there are three independent cohomology invariants η, α 1 , α 2 ∈ Z 2 . η can be realized by adding Φ 1 ∈ πZ flux into every loop formed by M x L T M −1 x L −1 T . α 1 , α 2 can be realized by adding Φ 2 , Φ 3 ∈ πZ flux into every loop which is invariant under M x g t and L T M x g t respectively. One example is shown in Fig. 29. The relations between flux configuration and cohomology invariants are For Z 2 -projective symmetry algebras of spacetime group P m x g t , there are four independent cohomology invariants η, α 1 , α 2 , γ ∈ Z 2 . η, α 1 , α 2 can be realized in the same way as U (1) case. For U (1)-projective symmetry algebras of spacetime group P m t g x , there are three independent cohomology invariants α 1 , α 2 , β ∈ Z 2 . α 1 , α 2 can be realized by adding Φ 1 , Φ 2 ∈ πZ flux into every loop which is invariant under M t g x and L x M t g x respectively.
For Z 2 -projective symmetry algebras of spacetime group P m t g x , there are four independent cohomology invariants η, α 1 , α 2 , β ∈ Z 2 . α 1 , α 2 can be realized in the way as U (1) case. η can be realized by adding flux Φ 3 into every loop formed by M t L x M −1 t L −1 x . If η = −1, all hoppings should be real. One example is shown in Fig. 30. The relations between flux configuration and cohomology invariants are For U (1)-projective symmetry algebras of spacetime group Cm x , there are one independent cohomology invariant σ ∈ Z 2 , which can be realized by adding Φ ∈ πZ flux into every loop formed by L 1 L 2 L −1 1 L −1 2 . This can be done by adding 2Φ flux into every loop formed by L x L T L −1 x L −1 T and making the flux configuration satisfy Cm x symmetry. One example is shown in Fig. 31. The relation between flux configuration and cohomology invariant is Here we note that the adding of Φ i should not break the spatial reflection symmetry. For Z 2 -projective symmetry algebras of spacetime group Cm x , there are two independent cohomology invariants σ, γ ∈ Z 2 . σ can also be realized in the same way as U (1) case. For U (1)-projective symmetry algebras of spacetime group Cm t , there are two independent cohomology invariants σ ∈ U (1), β ∈ Z 2 . σ can be realized by adding Φ flux into every loop formed by L 1 L 2 L −1 1 L −1 2 . This can be done by adding 2Φ flux into every loop formed by L x L T L −1 x L −1 T and making the flux configuration satisfy Cm t symmetry. One example is shown in Fig. 32. The relation between flux configuration and cohomology invariant is where the sign is positive if L 1 L 2 L −1 1 L −1 2 has the same direction with L x L T L −1 x L −1 T , otherwise the sign is negative. For Z 2 -projective symmetry algebras of spacetime group Cm t , there are two independent cohomology invariants σ, γ ∈ Z 2 . σ can also be realized in the same way as U (1) case. For U (1)-projective symmetry algebras of spacetime group Cm x m t , there are four independent cohomology invariants α 1 , α 2 , α 3 , β ∈ Z 2 . α 1 , α 2 , α 3 can be realized by adding Φ 1 , Φ 2 , Φ 3 ∈ πZ flux into every loop which is invariant under rotation M x M t , L 1 M x M t and L 1 M x L −1 1 M t respectively. One example is shown in Fig. 33. The relations between flux configuration and cohomology invariants are α i = e iΦi , i = 1, 2, 3.
E. Other details 1. Proof of the electric Floquet-Bloch theorem First, we review the proof of the ordinary Floquet-Bloch theorem. This theorem states that for a periodic driving system H(r, t), where H(r + a i , t) = H(r, t + T ) = H(r, t), the solutions of the time-dependent equation i∂ t ψ(r, t) = H(r, t)ψ(r, t) can be linearly composed by the Floquet-Bloch states ψ k,n (r, t) = e ik·r−iϵn(k)t u k,n (r, t), u k,n (r + a i , t) = u k,n (r, t), u k,n (r, t + T ) = u k,n (r, t). (139) For simplicity, we only review the proof of this theorem for 1+1D, and the generalization to 3+1D is straightforward. In 1+1D, there are two translation operators L x , L T , defined as L x f (x, t) = f (x + a, t), L T f (x, t) = f (x, t + T ), respectively. The spacetime translation group G is generated by L x and L T , and is an abelian group since L x commutes with L T . We define the operator Then, the solutions satisfy H(x, t)ψ(x, t) = 0. All solutions form a solution space V .
Because L x and L T both commute with H(x, t), every group element g ∈ G also commutes with H(x, t). This implies that the solution space V is a representation of G, which can be decomposed into a direct sum of irreducible representations of G. Since G is abelian, its irreducible representations are one dimensional. These irreducible representations are labelled by (k, ϵ), with character χ k,ϵ ((L x ) n (L T ) m ) = e ikna−iϵmT , m, n ∈ Z. The range of (k, ϵ) is k ∈ [0, 2π/a), ϵ ∈ [0, 2π/T ). If a solution ψ(x, t) belongs to an irreducible representation labelled by (k, ϵ), it satisfies L x ψ(x, t) = e ika ψ(x, t), L T ψ(x, t) = e −iϵT ψ(x, t). (141) We can rewrite these conditions as ψ(x, t) = e ikx−iϵt u(x, t), u(x + a, t) = u(x, t), u(x, t + T ) = u(x, t).
(142) Plug this ansatz into H(x, t)ψ(x, t) = 0, we have where H(k, t) = e −ika H(x, t)e ika . By solving this equation, we can obtain a set of eigenvalues ϵ n (k) and corresponding eigenstates u k,n (x, t). So we can label each state with (k, n), i.e., ψ k,n (x, t) = e ikx−iϵn(k)t u k,n (x, t). A general solution is a linear combination of ψ k,n (x, t). Thus, we have complete the proof of the Floquet-Bloch theorem. Now, let us prove the Floquet-Bloch theorem in a uniform electric field, which we call electric Floquet-Bloch theorem. We also only concern the 1+1D case. The Hamiltonian for a periodic driving system in a uniform electric field E x = E can be written as Here we choose the gauge A 0 = 0, A x (t) = −Et. Now, although U (x, t) is periodic, i.e, U (x + a, t) = U (x, t + T ) = U (x, t), H(x, t) is not periodic in time, i.e., H(t + T ) ̸ = H(t). However, we observe that H(x, t + T ) only differs from H(x, t) by a gauge transformation A x → A x + ∂ x χ(x), χ(x) = ET x. This gauge transformation can be equivalently defined as where G T = e iχ(x) . So now we can define a proper time translation operator L T = G T L T which commutes with H(x, t), i.e., L T H(x, t)L −1 T = H(x, t). Moreover, one can verify that the two proper translation operators L T and L x satisfy the following equation: where Φ E = EaT is the electric flux in every spacetime unit cell. We assume it to be a rational, Φ E = pΦ 0 /q = 2πp/q. To find the constrain of the translation symmetry on the wavefunction, we need to find commutative operators that commute with the Hamiltonian. Here, we can take L x , (L T ) q as two generators. They generate an abelian spacetime translation group G. Every group element g ∈ G commutes with H(x, t). Therefore, the solution space V is a representation of G. The one dimensional irreducible representations of G are also labelled by (k, ϵ), but now the range of (k, ϵ) is k ∈ [0, 2π/a), ϵ ∈ [0, 2π/qT ). If a solution ψ(x, t) belongs to a representation labelled by (k, ϵ), it satisfies L x ψ(x, t) = e ika ψ(x, t), which can be rewritten as ψ(x + a, t) = e ika ψ(x, t), ψ(x, t + qT ) = e −iET x e −iϵqT ψ(x, t) = e −i2πpx/a e −iϵqT ψ(x, t).
We can rewrite it as ψ(x, t) = e ikx−iϵt u(x, t), u(x + a, t) = u(x, t), u(x, t + qT ) = e −i2πpx/a u(x, t). (149) Plug this ansatz into H(x, t)ψ(x, t) = 0, we can obtain a set of eigenvalues ϵ n (k) and eigenstates u n,k (x, t). Hence we can label a state with by (k, n) and finally we derive the electric Floquet-Bloch theorem in the main text.

Kramers degeneracy protected by projective P2 symmetry
There can be Kramers degeneracy protected by projective P 2 symmetry. Consider the lattice model with P 2 symmetry in Fig. 34(a), whose Hamiltonian in momentum space is where w 1 (t), w 2 (t), w 3 (t) are the inner cell hoppings and J(t) is the intra cell hopping. All hoppings are T -periodic. The P 2 symmetry require the hoppings satisfy w 1 (t) = w 3 (−t), w 2 (t) = w 2 (−t), J(t) = J(−t). δ is the unit cell distance. The two-fold spacetime rotation operator C in momentum space is represented as where σ i are Pauli martices, I k (I t ) is the inversion of k(t), and K is the complex conjugation. We have C 2 = 1, (L x C) 2 = 1, (L T C) 2 = 1, (L T L x C) 2 = 1. Now consider the nontrivial projective symmetry algebra C 2 = −1, (L x C) 2 = −1, (L T C) 2 = 1, (L T L x C) 2 = 1. This algebra can be realized by introducing a π flux into every loop that is invariant under C or (L x C), as illustrated in Fig. 34(b). Now the new Hamiltonian H(k, t) has the same form of Eq. (151), but with hoppings satisfy w 1 (−t) = −w 3 (t), w 2 (−t) = −w 2 (t), J(−t) = −J(t) and w i (t) = −w i (t + T ), i = 1, 2, 3, J(t) = −J(t + T ). The proper rotation is C = G C C, where the gauge transformation G C in momentum space is represented as G C = σ 1 ⊗ σ 3 . Therefore, C is represented as It can be verified that the new Hamiltonian satisfies CH(k, t)C −1 = H(k, t), and we obtain the desired projective symmetry algebra C 2 = −1, (L x C) 2 = −1, (L T C) 2 = 1, (L T L x C) 2 = 1. From CH(k, t)C −1 = H(k, t), we have where M C = σ 1 ⊗ iσ 2 . This relation implies the Kramers degeneracy of the quasibands at each k. Here is the proof. The quasienergies ϵ n (k) are defined as eigenvalues of the evolution operator U (k, 2T ) = T exp(−i 2T 0 dt ′ H(k, t ′ )), i.e., U (k, 2T )|ψ k,n ⟩ = e −iϵn(k)2T |ψ k,n ⟩. From Eq. (154) we can prove If |ψ⟩ is an eigenstate of U (k, 2T ), then M −1 C K|ψ⟩ is also an eigenstate. When M C M * C = −1, which is the case for M C = σ 1 ⊗ iσ 2 , M −1 C K|ψ⟩ is linear independent with |ψ⟩. So there is a two-fold Kramers degeneracy protected by the projective P 2 symmetry. The quasibands for one set of parameters are shown in Fig. 35 (a), and we plot the quasibands of a model with ordinary P 2 symmetry in Fig. 35 (b) for comparison.

Band crossing due to projective symmetry
Here, we consider a driven Su-Schrieffer-Heeger (SSH) model