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 revealed one of the most fundamental principles of quantum theory: symmetry groups are projectively represented in his seminal work1.

The condensed-matter analogs of the Poincaré group may be spacetime groups for various periodically driven crystals or spacetime crystals with finite spacetime periodicity2,3,4,5,6,7,8,9,10,11,12,13. These groups are extensions of space groups with finite spacetime-translational symmetry and are commonly used to analyze spacetime crystals13,14,15,16,17. However, the theory of projective representations of spacetime groups (PRSTGs) has yet to be established. Nevertheless, its fundamental importance is evident, as Wigner’s philosophy suggests that every spacetime crystal, regardless of whether or not it has interactions, should projectively represent the spacetime group.

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 crystals18. As a result, PRSTGs ubiquitously exist in spacetime crystals, and the theory of PRSTGs is expected to generate widespread interest, given the variety of artificial crystals available, including photonic and acoustic crystals, cold atoms in optical lattices, periodic mechanical systems and electric-circuit19,20,21,22,23,24,25,26,27,28,29. Notably, the projective representations of space groups for static crystals with gauge structures have recently attracted intensive attention in the field of topological physics30,31,32,33,34,35,36,37,38.

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 Table 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.

Table 1 U(1)-projective symmetry algebras of 13 (1, 1)D spacetime groups.
Full size table

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 results, such as the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless spacetime crystals, and symmetry-enforced 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.

Results

Projective spacetime symmetry algebras

Let us start with recalling some basics of spacetime groups. Since the time-translation operator LT 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 group13. 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 spacetime crystals, all possible spacetime groups are listed in Table 1, using notation adapted from 2D space groups.

Consider a spacetime group Gst. Under a projective representation, the multiplication rules of the symmetry operators S are modified with additional phase factors. If S0S1 = S2, then S0S1 = Ω(S0, S1)S2, where Si are symmetry elements in Gst. 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., (S1S2)S3 = S1(S2S3) for three arbitrary operators S1, S2 and S3. Hence, the multiplier Ω satisfies the equation,

$${{\Omega }}({S}_{1},{S}_{2}){{\Omega }}({S}_{1}{S}_{2},{S}_{3})={{\Omega }}({S}_{1},{S}_{2}{S}_{3}){c}_{{S}^{1}}[{{\Omega }}({S}_{2},{S}_{3})].$$
(1)

Here, cS is an operator that depends on S. cS ≔ 1 if S is unitary, \({c}_{S}:= {{{{{{{\mathcal{K}}}}}}}}\) if S is anti-unitary, where \({{{{{{{\mathcal{K}}}}}}}}\) is the complex conjugation operator. For spacetime groups, S is anti-unitary 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.,

$${{\Omega }}({S}_{1},{S}_{2}) \sim {{\Omega }}({S}_{1},{S}_{2})\,{c}_{{S}_{1}}[\chi ({S}_{2})]\chi ({S}_{1})/\chi ({S}_{1}{S}_{2}).$$
(2)

The equivalence classes of multipliers form an abelian group H2,c(Gst, U(1)), called the twisted second cohomology group of Gst39. It is clear that all projective representations of Gst are classified by H2,c(Gst, U(1)). For (1, 1)D spacetime crystals, we work out all the classifications as listed in the second column of Table 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 PSAs30,31,32,34,35,37. These algebras are classified by H2,c(Gst, 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 Table 1) for each Gst, 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 Table 1). The theoretical foundation for representing PSAs through modified relations of generators is provided in Supplementary Note 1, and the derivation for all PSAs of 13 (1,1)D spacetime groups can be found in Supplementary Note 2.

For example, Pmx can be generated by unit translations Lx, LT and spatial reflection Mx, which satisfy the following relations: [Mx: LT] = 1, \([{L}_{x}{M}_{x}:{L}_{T}]=1,{M}_{x}^{2}=1\) and \({({L}_{x}{M}_{x})}^{2}=1\). All the PSAs of Pmx can be obtained by modifying these relations with U(1) phase factors. Specifically, [Mx: LT] = η1, [LxMx: LT] = η2, \({{\mathsf{M}}}_{x}^{2}={\gamma }_{1}\) and \({({{\mathsf{L}}}_{x}{{\mathsf{M}}}_{x})}^{2}={\gamma }_{2}\). However, phase factors γ1, γ2 can be set to be 1 by redefining \({{\mathsf{L}}}_{x}\to {{\mathsf{L}}}_{x}^{{\prime} }={\gamma }_{1}^{1/2}{\gamma }_{2}^{-1/2}{{\mathsf{L}}}_{x}\), \({{\mathsf{M}}}_{x}\to {{\mathsf{M}}}_{x}^{{\prime} }={\gamma }_{1}^{-1/2}{{\mathsf{M}}}_{x}\). Moreover, phase factors η1, η2 can only take values in \({{\mathbb{Z}}}_{2}=\{\pm 1\}\), since \({{\mathsf{L}}}_{T}{\gamma }_{1}={{\mathsf{L}}}_{T}{{\mathsf{M}}}_{x}^{2}={\eta }_{1}^{-2}{{\mathsf{M}}}_{x}^{2}{{\mathsf{L}}}_{T}={\eta }_{1}^{-2}{{\mathsf{L}}}_{T}{\gamma }_{1}\) and \({{\mathsf{L}}}_{T}{\gamma }_{2}={{\mathsf{L}}}_{T}{({{\mathsf{L}}}_{x}{{\mathsf{M}}}_{x})}^{2}={\eta }_{2}^{-2}{({{\mathsf{L}}}_{x}{{\mathsf{M}}}_{x})}^{2}{{\mathsf{L}}}_{T}={\eta }_{2}^{-2}{{\mathsf{L}}}_{T}{\gamma }_{2}\), leading to \({\eta }_{1}^{2}={\eta }_{2}^{2}=1\). All the possible values of phase factors, i.e., η1, η2 = ± 1 correspond to the \({{\mathbb{Z}}}_{2}^{2}\) classification of the projective symmetry algebras of Pmx.

It is significant to note that all phase factors in Table 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 Gst, the free phase factors in Table 1 are complete, i.e., sufficient to specify equivalence classes of H2,c(Gst, U(1)). Thus, for each Gst we call these phase factors a complete set of cohomology invariants.

Nevertheless, two elements in H2,c(Gst, U(1)) may correspond to isomorphic algebras. The number of independent algebras for each spacetime group is given in the last column of Table 1. Let Pmx be an example. \({H}^{2,c}(P{m}_{x},U(1))={{\mathbb{Z}}}_{2}^{2}\) consists of four possibilities specified by [Mx: LT] = η1 = ± 1 and [LxMx: LT] = η2 = ± 1. Mx and LxMx, as reflections through parallel lines related by half a translation, are on an equal footing. Hence, (η1, η2) = (−1, 1) and (η1, η2) = (1, −1) correspond to isomorphic algebras, leading to 3 isomorphic classes.

Realization by spacetime gauge fluxes

We proceed to construct spacetime crystal models that can realize the PSAs by gauge fluxes. The general form of the time-periodic Hamiltonian is given by \(H(t)={\sum }_{ij}{h}_{ij}(t){e}^{i{A}_{ij}(t)}{c}_{i}^{{{{\dagger}}} }{c}_{j}+{{{{{{{\rm{H.c.}}}}}}}}\), which can be succinctly written as \(H(t)={\sum }_{ij}{c}_{i}^{{{{\dagger}}} }{H}_{ij}(t){c}_{j}\). Here, the subscripts i, j indicate the positions of lattice sites in space, upon which the spatial symmetries act.

Symmetry group consists of transformations that leave the system invariant. With gauge degrees of freedom, a symmetry transformation may map the system to a gauge equivalent one, which in general is different from the original system. An elementary example is that under the time translation LT, the Hamiltonian H(t) is mapped to H(t + T), and H(t) and H(t + T) are related by a gauge transformation GT, i.e.,

$$H(t+T)={{\mathsf{G}}}_{T}^{-1}H(t){{\mathsf{G}}}_{T}.$$
(3)

This resembles the spacetime cylinder frequently considered for quantum gauge anomalies40, and moreover, we shall see analogous crossings in the spectral flow diagram later. Consequently, we can define the physical time-translation operator as LT = GTLT with [H(t), LT] = 0. Similarly, for any symmetry S ∈ Gst, the physical symmetry operator is the combination S = GSS with GS the corresponding gauge transformation, and it is S that commutes with the Hamiltonian, i.e.,

$$[H(t),{\mathsf{S}}]=0.$$
(4)

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 S1 and S2 is modified by the multiplier \({{\Omega }}({S}_{1},{S}_{2})={{\mathsf{G}}}_{{S}_{1}}{S}_{1}{{\mathsf{G}}}_{{S}_{2}}{S}_{1}^{-1}{{\mathsf{G}}}_{{S}_{1}{S}_{2}}^{-1}\), which is a U(1)-factor as proved in Supplementary Note 3. It is noteworthy that the equivalence class of Ω(S1, S2) 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 Table 1 for both U(1) and \({{\mathbb{Z}}}_{2}\) gauge fields. In Table 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 Supplementary Note 3.

Fig. 1: Realization of projective algebraic relations.
figure 1

a–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 from the corresponding blue hoppings by a minus sign. The yellow and purple hoppings differ from the corresponding blue hoppings by a phase. Thick and thin hoppings have different hopping strengths. Lx and LT denote space and time primitive translations respectively. Mx and Mt are spatial and time reflections respectively. C = MxMt denotes the spacetime inversion. gt denotes time-glide reflection. In (b), the rhombus means a spacetime inversion center. π flux concentrates on the dark strap at t = 0. In (c), the dark lines mean two time-glide reflection axes corresponding to gt and LTgt, respectively. In (d), The dark line means a spatial reflection axis. In (e), the time reflection is effectively realized by the combination of time reflection and a horizontal twofold rotation. π flux concentrates on the dark strap at t = 0.

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(i) The first class is σ = [L1: L2] for the translation subgroup, where L1 and L2 are primitive spacetime translations. The value of σ corresponds to the flux in a spacetime loop formed by \({{\mathsf{L}}}_{1}{{\mathsf{L}}}_{2}{{\mathsf{L}}}_{1}^{-1}{{\mathsf{L}}}_{2}^{-1}\). Figure 1a shows the case when L1 = Lx, L2 = LT. The general cases of spacetime translations can be found in the Supplementary Note 3.

(ii) The second class concerns cohomology invariants for spacetime inversion, e.g., \(\alpha ={({{\mathsf{M}}}_{x}{{\mathsf{M}}}_{t})}^{2}\). 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., \(\tau ={{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}^{-1}\). The phase factor τ* corresponds to the flux in a loop formed by \({{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}^{-1}\), as illustrated in Fig. 1c.

(iv) The fourth class denoted η modifies the relation between spatial reflections and time translations, e.g., \(\eta ={{\mathsf{M}}}_{x}{{\mathsf{L}}}_{T}{{\mathsf{M}}}_{x}^{-1}{{\mathsf{L}}}_{T}^{-1}\). η = ± 1 corresponds to 0 or π flux in the loops formed by \({{\mathsf{M}}}_{x}{{\mathsf{L}}}_{T}{{\mathsf{M}}}_{x}^{-1}{{\mathsf{L}}}_{T}^{-1}\), as illustrated in Fig. 1d.

(v) The fifth class corresponds to the square of a time reversal, i.e, \(\beta ={{\mathsf{M}}}_{t}^{2}\) with β = ± 1. β = −1 cannot be realized by a 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 Table 1. Here we present the models for P1 and Pgt, respectively, each of which has only one phase factor. The remaining models are available in Supplementary Note 4.

Projective P1 symmetry

Our first example is simply P1 consisting of only spacetime translations. For simplicity, we choose symmetry operators as spatial and time translators, Lx and LT. We construct the lattice model with only the nearest-neighbor hopping amplitudes as shown in Fig. 2a. The momentum-space Hamiltonian is given by

$$\begin{array}{r}H(k,t)=\left[\begin{array}{ccc}0&{w}_{1}(t)&J(t){e}^{-ika}\\ {w}_{1}(t)&0&{w}_{2}(t)\\ J(t){e}^{ika}&{w}_{2}(t)&0\end{array}\right],\end{array}$$
(5)

where w1,2 are the two hopping amplitudes within each cell, and J(t) is the hopping amplitude between two neighboring cells.

Fig. 2: Projective P1 and Pgt symmetry model.
figure 2

a Spacetime tight-binding model with projective P1 symmetry. The projective algebraic relation is [Lx: LT] = −1, where Lx and LT are gauge-modified space and time primitive translations, respectively. The dashed lines indicate a continuous evolution of hopping amplitudes. The three hopping amplitudes are explicitly shown at t = 0, T, 2T. The different thicknesses and lengths of hoppings mean different hopping strengths. The red hoppings at t = T differ from the corresponding blue hoppings at t = 0 by a minus sign. Each spacetime plaquette has flux 3π. The signs marked in red at each site at t = T describe the gauge transformation GT needed to restore the initial connections. b Spacetime tight-binding model with projective Pgt symmetry. The projective algebraic relation is \({{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}=-1\), where gt is the gauge-modified time-glide reflection. Glide axes are plotted as dark horizontal lines. The distribution of hopping amplitudes is explicitly marked at five times, respectively. The different thicknesses and lengths of hoppings mean different hopping strengths. A red hopping differs from its blue counterpart with the same thickness and length by a minus sign. Each loop formed by \({{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}\) contains π flux. 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 \({{\mathsf{G}}}_{{g}_{t}}\). The signs for \({{\mathsf{G}}}_{{g}_{t}}\) at a given time are marked in red. c Quasibands of U(2T) for the model in (a). d Quasibands of U(2T) for the model in (b). e Quasibands of Floquet model with ordinary Pgt symmetry. The parameter values for (c), (d) and (e) are given in Supplementary Note 5.

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For simplicity, we choose [Lx: LT] = −1, namely {Lx, LT} = 0, which can be realized by inserting odd times of π-flux in each spacetime plaquette formed by Lx and LT. This can be implemented by setting w1,2(t) = −w1,2(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 Lx, Lx = Lx. But translating H(k, t) by LT should be combined with a gauge transformation GT (see Fig. 2a), which in momentum space is represented by

$${{\mathsf{G}}}_{T}=M{{{{{{{{\mathcal{L}}}}}}}}}_{\frac{G}{2}},\quad M=\left[\begin{array}{ccc}-1&0&0\\ 0&1&0\\ 0&0&-1\end{array}\right],$$
(6)

where \({{{{{{{{\mathcal{L}}}}}}}}}_{\frac{G}{2}}\) denotes a half reciprocal translation, namely, sending k to k + G/2. Then, it is easy to verify {Lx, LT} = 0 with LT = GTLT and Lx = eika. The anti-commutativity can also be pictorially observed: In Fig. 2a, the signs ± of GT are exchanged by Lx, i.e., \({L}_{x}{{\mathsf{G}}}_{T}{L}_{x}^{-1}=-{{\mathsf{G}}}_{T}\).

Projective P g t symmetry

Our second example is Pgt with two generators gt and LT. In particular, gt is the glide time-reflection, which combines the time inversion Mt with a half translation Lx/2 along the spatial lattice, namely gt = Lx/2Mt. From Table 1, we consider the projective algebra, \({{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}=-1\).

The model is constructed as in Fig. 2b, and is described by the Hamiltonian,

$$\begin{array}{r}H(k,t)=\left[\begin{array}{cccc}0&{w}_{1}(t)&0&J(t){e}^{-ika}\\ {w}_{1}(t)&0&{w}_{2}(t)&0\\ 0&{w}_{2}(t)&0&{w}_{3}(t)\\ J(t){e}^{ika}&0&{w}_{3}(t)&0\end{array}\right].\end{array}$$
(7)

The nontrivial projective algebra can be realized by inserting a π-flux into each loop formed by \({g}_{t}{L}_{T}{g}_{t}^{-1}{L}_{T}\) (see Fig. 2b). Particularly, in Eq. (7), this is satisfied by w1(t) = −w3(−t), w2(t) = J(−t) and w1(t) = −w1(t + T), w2(t) = w2(t + T), w3(t) = −w3(t + T) and J(t) = J(t + T). Now, gt should be accompanied by the gauge transformation \({{\mathsf{G}}}_{{g}_{t}}\) (see Fig. 2b). In momentum space, \({{\mathsf{G}}}_{{g}_{t}}={\sigma }_{3}\otimes {\sigma }_{3}\) with σ’s the Pauli matrices, and

$${{\mathsf{g}}}_{t}={{\mathsf{G}}}_{{g}_{t}}{g}_{t}={M}_{g}{{{{{{{{\mathcal{I}}}}}}}}}_{k}{{{{{{{{\mathcal{I}}}}}}}}}_{t}{{{{{{{\mathcal{K}}}}}}}},\,{M}_{g}(k)=\left[\begin{array}{cc}0&{\sigma }_{3}\\ -{e}^{ika}{\sigma }_{3}&0\end{array}\right],$$
(8)

where \({{{{{{{{\mathcal{I}}}}}}}}}_{k}\) (\({{{{{{{{\mathcal{I}}}}}}}}}_{t}\)) is the inversion of k (t), and \({{{{{{{\mathcal{K}}}}}}}}\) the complex conjugation. Moreover, one can check that the gauge transformation GT for LT is equal to \({{\mathsf{G}}}_{{g}_{t}}\), i.e., LT = σ3 ⊗ σ3LT. It is straightforward to verify that \({{\mathsf{g}}}_{t}H(k,t){{\mathsf{g}}}_{t}^{-1}={{\mathsf{L}}}_{T}H(k,t){{\mathsf{L}}}_{T}^{-1}=H(k,t)\), and the projective algebra \({{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}=-1\).

Physical consequences of projective spacetime symmetry

We proceed to discuss the aforementioned three fascinating consequences of the projective spacetime symmetry algebras.

(1) Electric Floquet-Bloch theorem. 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 fields12,41,42. 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 LT and Lx that commute with the Hamiltonian satisfy the gauge-invariant commutation relation [Lx: LT] = ei2πp/q. We choose two commuting operators Lx and \({({{\mathsf{L}}}_{T})}^{q}\), which have the common eigenstates corresponding to the quantum numbers (k, ϵn). Note that ϵn denotes the eigenvalues of \(U(k,qT)={{{{{{{\mathcal{T}}}}}}}}\exp [-i\int\nolimits_{0}^{qT}H(k,t)dt]\), which are known as “quasi-energies”, with \({{{{{{{\mathcal{T}}}}}}}}\) indicating the time ordering. The Floquet-Bloch states labeled by (k, ϵn) can be written as

$${\psi }_{k,n}(x,t)={e}^{i(kx-{\epsilon }_{n}(k)t)}{u}_{k,n}(x,t),$$
(9)

where uk,n satisfies

$${u}_{k,n}(x+a,t)={u}_{k,n}(x,t),\,\,{u}_{k,n}(x,t+qT)=\lambda (x){u}_{k,n}(x,t).$$

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 “Methods”.

It is noteworthy that ψk,n, LTψk,n, . . . , \({({{\mathsf{L}}}_{T})}^{q-1}{\psi }_{k,n}\) are q-fold degenerate states for the quasi-energy, where \({({{\mathsf{L}}}_{T})}^{m}{\psi }_{k,n}\) has momentum k + 2πmp/qa. This is exemplified by the model (5). If ψk,n is an eigenstate of U(k, 2T), then U−1(k + π/a, T)M−1ψk,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 systems30. Here, we show that this extraordinary phenomenon also occurs in the quasi-energy spectra for spacetime crystals with projective symmetry algebras.

Let us illustrate this by Pgt symmetry of (7), while another example for group P2 can be found in Supplementary Note 5. The symmetry operator (8) constrains the Hamiltonian as \({M}_{g}H(k,t){M}_{g}^{-1}={H}^{* }(-k,-t)\), which leads to the evolution operator \(U(k,2T)={{{{{{{\mathcal{T}}}}}}}}\exp [-i\int\nolimits_{0}^{2T}dtH(k,t)]\) satisfying

$${M}_{g}U(k,2T){M}_{g}^{-1}={U}^{-1* }(-k,2T).$$
(10)

Hence, for U(k, 2T), we should consider the anti-unitary symmetry operator \({M}_{g}(k){{{{{{{\mathcal{K}}}}}}}}\) with \({({M}_{g}(k){{{{{{{\mathcal{K}}}}}}}})}^{2}=-{e}^{ika}\). It is significant to notice that the square of \({M}_{g}(k){{{{{{{\mathcal{K}}}}}}}}\) 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 LT gt 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 Pgt group13 (see Fig. 2e), but that at k = 0 only appears for projective Pgt algebra.

(3) Symmetry-enforced crossings of the Hamiltonian spectral flow. We note that a sublattice symmetry Γ operates on a bipartite lattice exactly as a \({{\mathbb{Z}}}_{2}\) 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 quasi-energy spectrum. Above band crossing is illustrated by a t-dependent dimerized model as a generalization of the Su-Schrieffer-Heeger model in “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 Table 1 with α = (−1)2sα and β = (−1)2sβ, respectively.

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 symmetries43,44,45,46,47,48.

To realize spacetime lattice models with projective symmetries, one needs to realize the required time-dependent hopping amplitudes, which may be done by cold atoms, photonic crystals, ultrafast spintronics or other systems49,50,51,52,53,54.

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 the electric Floquet-Bloch theorem. We only concern the 1+1D case, while the generalization to 3+1D is straightforward.

The Hamiltonian for a periodic driving system in a uniform electric field Ex = E can be written as

$$H(x,t)=-\frac{1}{2m}{({\partial }_{x}-i{A}_{x}(t))}^{2}+U(x,t).$$
(11)

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 A0 = 0, Ax(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 Lx, LT by Lxf(x, t) = f(x + a, t) and LTf(x, t) = f(x, t + T), then H(x, t) commutes with Lx but not with LT. However, we observe that H(x, t + T) only differs from H(x, t) by a gauge transformation Ax → Ax + ∂xχ(x), χ(x) = ETx. This gauge transformation can be equivalently defined as

$$H(x,t)\to {G}_{T}H(x,t){G}_{T}^{-1}={e}^{i\chi (x)}H(x,t){e}^{-i\chi (x)},$$
(12)

where GT = eiχ(x). So we can define a proper time-translation operator LT = GTLT, which commutes with H(x, t), \({{\mathsf{L}}}_{T}H(x,t){{\mathsf{L}}}_{T}^{-1}=H(x,t)\). One can check the two proper translation operators LT and Lx satisfy

$${L}_{x}{{\mathsf{L}}}_{T}{L}_{x}^{-1}{{\mathsf{L}}}_{T}^{-1}={e}^{iEaT}={e}^{i{{{\Phi }}}_{E}}.$$
(13)

We assume the electric flux in every unit cell is ΦE = pΦ0/q = 2πp/q.

To find the constraint of the translation symmetry on the wavefunction, we have to find commutative operators that commute with the Hamiltonian. Here, we can take \({L}_{x},{({{\mathsf{L}}}_{T})}^{q}\) as two generators, they generate an abelian spacetime translation group G. We define

$${{{{{{{\mathcal{H}}}}}}}}(x,t)=H(x,t)-i{\partial }_{t},$$
(14)

then solutions satisfy \({{{{{{{\mathcal{H}}}}}}}}(x,t)\psi (x,t)=0\). All solutions form a solution space V. Because every group element g ∈ G commutes with \({{{{{{{\mathcal{H}}}}}}}}(x,t)\), 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 labeled by (k, ϵ), with character \({\chi }_{k,\epsilon }({({L}_{x})}^{n}{({{\mathsf{L}}}_{T}^{q})}^{m})={e}^{ikna-i\epsilon qmT}\), \(m,n\in {\mathbb{Z}}\). The range of (k, ϵ) is k ∈ [0, 2π/a), ϵ ∈ [0, 2π/qT). If a solution ψ(x, t) is in an irreducible representation labeled by (k, ϵ), it satisfies

$$\begin{array}{rlr}&{L}_{x}\psi (x,t)={e}^{ika}\psi (x,t),&\\ &{({{\mathsf{L}}}_{T})}^{q}\psi (x,t)={e}^{-i\epsilon qT}\psi (x,t),\end{array}$$
(15)

that is,

$$\psi (x+a,t)= \, {e}^{ika}\psi (x,t),\\ \psi (x,t+qT)= \, {e}^{-i2\pi px/a}{e}^{-i\epsilon qT}\psi (x,t).$$
(16)

We can rewrite it as

$$\psi (x,t)= \, {e}^{ikx-i\epsilon t}u(x,t), \\ u(x+a,t)= \, u(x,t),\\ u(x,t+qT)= \, {e}^{-i2\pi px/a}u(x,t).$$
(17)

Plug this ansatz into \({{{{{{{\mathcal{H}}}}}}}}(x,t)\psi (x,t)=0\), we can obtain a set of eigenvalues ϵn(k) and eigenstates un,k(x, t). Thus we can label a state with by (k, n) and we complete the proof of the electric Floquet-Bloch theorem.

Since \([{{{{{{{\mathcal{H}}}}}}}}(x,t),{{\mathsf{L}}}_{T}]\) = 0, if ψk,n(x, t) is a solution, \({\psi }^{{\prime} }(x,t)={{\mathsf{L}}}_{T}{\psi }_{k,n}(x,t)={e}^{i\chi (x)}{\psi }_{k,n}(x,t+T)\) is also a solution. Moreover, \({\psi }^{{\prime} }(x,t)\) also has the quasi-energy ϵn(k), but has momentum k + 2πp/qa, this can be seen by

$${({{\mathsf{L}}}_{T})}^{q}{\psi }^{{\prime} }(x,t) = {{\mathsf{L}}}_{T}{({{\mathsf{L}}}_{T})}^{q}{\psi }_{k,n}(x,t)={e}^{-i{\epsilon }_{n}(k)qT}{\psi }^{{\prime} }(x,t), \\ {L}_{x}{\psi }^{{\prime} }(x,t) = {L}_{x}{{\mathsf{L}}}_{T}{\psi }_{k,n}(x,t)={e}^{i{{{\Phi }}}_{E}}{{\mathsf{L}}}_{T}{L}_{x}{\psi }_{k,n}(x,t)\\ ={e}^{i(k+2\pi p/qa)a}{\psi }^{{\prime} }(x,t).$$
(18)

With this reason, \({{\mathsf{L}}}_{T}^{m}{\psi }_{k,n}(x,t),\,m=0,1,2,...,q-1\) are q-fold degenerate Floquet-Bloch states, wherein \({{\mathsf{L}}}_{T}^{m}{\psi }_{k,n}(x,t)\) has momentum k + 2πmp/qa.

Band crossing due to projective symmetry

Here, we consider a driven Su-Schrieffer-Heeger (SSH) model

$$H(t)=v(t)\mathop{\sum}\limits_{i}{c}_{i,A}^{{{{\dagger}}} }{c}_{i,B}+w(t)\mathop{\sum}\limits_{i}{c}_{i,B}^{{{{\dagger}}} }{c}_{i+1,A}+h.c.$$
(19)

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 GH(t + T)G−1 = H(t), where G = diag( + 1, −1, + 1, −1, . . . ) in real space), the instantaneous bands must cross at some time t. This is easy to see: In one period T, since w(t + T) = −w(t), v(t + T) = −v(t), there must be w(t*) = v(t*) at some t*. And we know that the SSH model is gapless when ∣w∣ = ∣v∣, so there is band crossing at t* for this driven SSH model.