Space-time crystal and space-time group

TL;DR

This paper introduces the concept of space-time crystals and their symmetry groups, extending static crystal theory to dynamic systems with intertwined space-time periodicities, and classifies their symmetry groups in 1+1D and 2+1D.

Contribution

It develops the space-time group framework for describing symmetries in dynamic crystals, including new symmetry operations like time-screw rotations and time-glide reflections.

Findings

Classification of 13 space-time groups in 1+1D.

Kramers-type degeneracy from glide time-reversal symmetry.

Spectral degeneracies enforced by non-symmorphic space-time symmetries in 2+1D.

Abstract

Crystal structures and the Bloch theorem play a fundamental role in condensed matter physics. We extend the static crystal to the dynamic "space-time" crystal characterized by the general intertwined space-time periodicities in $D+1$ dimensions, which include both the static crystal and the Floquet crystal as special cases. A new group structure dubbed "space-time" group is constructed to describe the discrete symmetries of space-time crystal. Compared to space and magnetic groups, space-time group is augmented by "time-screw" rotations and "time-glide" reflections involving fractional translations along the time direction. A complete classification of the 13 space-time groups in 1+1D is performed. The Kramers-type degeneracy can arise from the glide time-reversal symmetry without the half-integer spinor structure, which constrains the winding number patterns of spectral dispersions. In 2+1D, non-symmorphic space-time symmetries enforce spectral degeneracies, leading to protected Floquet semi-metal states. Our work provides a general framework for further studying topological properties of the $D+1$ dimensional space-time crystal.

Tables

Table 1. Table 1: The 2D magnetic point groups and their relations to the usual ten 2D point groups. The magnetic groups are denoted by using the international notation, where ′ means the time reversal operation. In the third column, the symmetry generators for each magnetic point group are listed. The symmetry operation m t subscript 𝑚 𝑡 m_{t} stands for the time reversal, R θ subscript 𝑅 𝜃 R_{\theta} represents rotation in the x 𝑥 x - y 𝑦 y plane at the angle of θ 𝜃 \theta , m x subscript 𝑚 𝑥 m_{x} and m y subscript 𝑚 𝑦 m_{y} are the reflections with respect to the x 𝑥 x - and y 𝑦 y -directions, respectively. The symbols of the magnetic group with ′

Point Group	$G_m\left(2\right)$	Generators
$C_1$	$1$
	${11}^{\prime }$	$m_t$
$D_1$	$m$	$m_x$
	$m{1}^{\prime }$	$m_x,m_t$
	$m^{\prime }$	$m_x{m}_t$
$C_2$	$2$	$R_{\pi }$
	${21}^{\prime }$	$R_{\pi },m_t$
	$2^{\prime }$	$R_{\pi }{m}_t$
$D_2$	$mm2$	$m_x,m_y$
	$mm{21}^{\prime }$	$m_x,m_y,m_t$
	$m^{\prime }m{2}^{\prime }$	$m_x{m}_t,m_y$
	$m^{\prime }{m}^{\prime }2$	$m_x{m}_t,m_y{m}_t$
$C_3$	$3$	$R_{2\pi /3}$
	${31}^{\prime }$	$R_{2\pi /3},m_t$
$D_3$	$3m$	$R_{2\pi /3},m_x$
	$3m{1}^{\prime }$	$R_{2\pi /3},m_x,m_t$
	$3m^{\prime }$	$R_{2\pi /3},m_x{m}_t$
$C_4$	$4$	$R_{\pi /2}$
	${41}^{\prime }$	$R_{\pi /2},m_t$
	$4^{\prime }$	$R_{\pi /2}{m}_t$
$D_4$	$4mm$	$R_{\pi /2},m_x$
	$4mm{1}^{\prime }$	$R_{\pi /2},m_x,m_t$
	$4^{\prime }{m}^{\prime }m$	$R_{\pi /2}{m}_t,m_x$
	$4m^{\prime }{m}^{\prime }$	$R_{\pi /2},m_x{m}_t$
$C_6$	$6$	$R_{\pi /3}$
	${61}^{\prime }$	$R_{\pi /3},m_t$
	$6^{\prime }$	$R_{\pi /3}{m}_t$
$D_6$	$6mm$	$R_{\pi /3},m_x$
	$6mm{1}^{\prime }$	$R_{\pi /3},m_x,m_t$
	$6^{\prime }{m}^{\prime }m$	$R_{\pi /3}{m}_t,m_x$
	$6m^{\prime }{m}^{\prime }$	$R_{\pi /3},m_x{m}_t$

Table 2. Table 2: Classification of the 2+1D space-time groups. There are 7 space-time crystal systems, 14 space-time Bravais lattices (The primitive trigonal lattice is the same as the primitive hexagonal lattice). The 31 magnetic point groups are uniquely assigned to the 7 crystal systems, as listed in the third column. In the 4th and 5th columns, the numbers of the ACC and the space-time groups for each Bravais lattice are listed.

Crystal System	Bravais Lattice	MP Group	ACC	$ST(2,1)$
Triclinic	Primitive	$1,2^{\prime }$	$2$	$2$
T-Monoclinic	Primitive	${11}^{\prime },2,{21}^{\prime }$	$3$	$8$
	Centered		$3$	$5$
R-Monoclinic	Primitive	$m,m^{\prime },m^{\prime }m{2}^{\prime }$	$3$	$8$
	Centered		$3$	$5$
Orthorhombic	Primitive	\pbox20cm$mm2,m^{\prime }{m}^{\prime }2$ $mm{21}^{\prime },m{1}^{\prime }$	$4$	$68$
	T-Base-Centered		$4$	$15$
	R-Base-Centered		$5$	$22$
	Face-Centered		$4$	$7$
	Body-Centered		$4$	$15$
Tetragonal	Primitive	\pbox20cm$4,{41}^{\prime },4^{\prime }$ $4mm,4mm{1}^{\prime }$ $4^{\prime }{m}^{\prime }m,4m^{\prime }{m}^{\prime }$	$8$	$49$
	Body-Centered		$8$	$19$
Trigonal	Primitive	\pbox20cm$3,6^{\prime },3m$ $3m^{\prime },6^{\prime }{m}^{\prime }m$	$8$	$18$
	Rhombohedral		$5$	$7$
Hexagonal	Primitive	\pbox3cm$6,{61}^{\prime },{31}^{\prime }$
$6mm,6m^{\prime }{m}^{\prime }$
$6mm{1}^{\prime },3m{1}^{\prime }$	$8$	$27$

Equations

$$V(\mathbf{r},t)=V(\mathbf{r}+{\mathbf{u}}^i,t+{\tau }^i),i=1,2,...,D+1,$$

$${\psi }_{\kappa ,m}(\mathbf{r},t)$$

$$\sum _{B^{\prime }}\{[{\epsilon }_0(\mathbf{k}+\mathbf{G})-\Omega ]{\delta }_{B,B^{\prime }}+V_{B-B^{\prime }}\}{c}_{m,B^{\prime }}={\omega }_m{c}_{m,B},$$

$$\Gamma (\mathbf{r},t)=(R\mathbf{r}+\mathbf{u},st+\tau ),$$

$$M_{g_t}^2{\psi }_{\kappa }={\psi }_{\kappa }(x-\lambda ,t)=e^{-i{\kappa }_x\lambda }{\psi }_{\kappa }=-{\psi }_{\kappa },$$

$$g_1{g}_2=T{g}_2{g}_1,$$

$$\displaystyle H(t)=-\sum_{\vec{r}\in A,~{}~{}\vec{r}+\frac{a}{2}\vec{e}_{i}\in B}\big{\{}w_{\vec{e}_{i}}(t)c^{\dagger}_{\vec{r}}d_{\vec{r}+\frac{a}{2}\vec{e}_{i}}+h.c.,\big{\}},$$

$$1\to M\to ST\to G_m\to 1,$$

$$c(1)=0,c(g_1{g}_2)=c(g_1)+g_{1}c(g_2).$$

$$(c(g_1),g_1)(c(g_2),g_2)=(c(g_1{g}_2),g_1{g}_2).$$

$$c\cdot d(g)=c(g)+d(g),$$

$$c_u(g)=gu-u,$$

$$H^1(G_m,T_{D+1}/M)=Z^1(G_m,T_{D+1}/M)/B^1(G_m,T_{D+1}/M),$$

$$g=T_{\mathbf{r}}(\mathbf{u})T_t(\tau )R{m}_t^s$$

$$g_1{g}_2$$

$$\tilde{R}$$

$$\tilde{\mathbf{u}}$$

$$\tilde{t}$$

$$M_{g_1}{M}_{g_2}=e^{i\mathbf{k}\cdot \tilde{\mathbf{u}}-i\omega \tilde{t}}{M}_{g_2}{M}_{g_1}.$$

$$M_{g_2}{\psi }_{\kappa }=e^{i\mathbf{k}\cdot {\mathbf{u}}_2-i\omega t_2}{e}^{i\theta }{\psi }_{\kappa },$$

$$M_{g_2}{M}_{g_1}{\psi }_{\kappa }=e^{-i\mathbf{k}\cdot ({\mathbf{u}}_2+\tilde{\mathbf{u}})-i\omega t_2}{e}^{-i\theta }{M}_{g_1}{\psi }_{\kappa }.$$

$$g_1^{\prime }{g}_2=T_{\mathbf{r}}(\mathbf{u})T_t(\tau )g_2{g}_1^{\prime },$$

Full text

The fundamental concept of crystal and the associated band theory based on the Bloch theorem lay the foundation of condensed matter physics. Studies on the crystal symmetry and band structure topology lead to the discoveries of topological insulators, topological superconductors, the Dirac and Weyl semi-metal states Hasan2010 ; Qi2011 ; Chiu2016 . Periodically driving further provides a new route to engineer topological states even in systems originally topologically trivial in the absence of driving, as explored in the irradiated graphene Oka2009 ; Gu2011 , semiconducting quantum wells Lindner2011 , dynamically modulated cold atom optical lattices Jotzu2014 , and photonic systems Rechtsman2013a ; Leykam2016 . The periodicity of the quasi-energy enriches the topological band structures Kitagawa2010 ; Asboth2014 ; Roy2016 , such as the dynamically generated Majorana modes Thakurathi2013a , 1D helical channels Budich2017 and anomalous edge states associated with zero Chern number Rudner2013 ; Titum2016a . Topological classifications for interacting Floquet systems have also been investigated Potter2016 ; Potter2016a ; VonKeyserlingk2016a ; VonKeyserlingk2016b ; Else2016 .

For periodically driven crystals, most studies treat the temporal periodicity separately from the spatial one. In fact, the driven system can exhibit much richer symmetry structures than a simple direct product of spatial and temporal symmetries. In particular, a temporal translation at a fractional period can be combined with the space group symmetries to form novel space-time intertwined symmetries, which, to the best of our knowledge, have not yet been fully explored. For static crystals, the intrinsic connections between the space-group symmetries and physical properties, especially the topological phases, have been extensively studied Fu2011 ; Parameswaran2013 ; Young2015 ; Wang2016a ; Kruthoff2016 ; Watanabe2016 ; Bradlyn2017 ; Bouhon2017 . Therefore, it is expected that the intertwined space-time symmetries could also protect non-trivial properties of the driven system, regardless of microscopic details.

In this article, we propose the concept of “space-time” crystal exhibiting the intertwined space-time symmetries, whose periodicities are characterized by a set of $D+1$ independent basis vectors, generally space-time mixed. The situation of separate spatial and temporal perodicities is a special case and is also included. The full discrete space-time symmetries of space-time crystals form a class of new group structures – dubbed the “space-time” group, which is the generalization of space group by including “time-screw” and “time-glide” operations. A complete classification of the 13 space-time groups in 1+1 D is performed, and their constraints on band structure winding numbers are studied. In 2+1 D, 275 space-time groups are classified. The non-symmorphic space-time symmetry operations, similar to their static space-group counterparts, lead to the protected spectral degeneracies for driven systems, even when the instantaneous spectra are gapped at any given time.

Space-time crystal – We consider the time-dependent Hamiltonian $H=P^{2}/(2m)+V(\mathbf{r},t)$ in the $D+1$ dimensional space-time. $V(\mathbf{r},t)$ exhibits the intertwined discrete space-time translational symmetry as

[TABLE]

where $(\mathbf{u}^{i},\tau^{i})=a^{i}$ is a set of the primitive basis vectors. In general, the space-time primitive unit cell is not a direct product between spatial and temporal domains. There may not even exist spatial translational symmetry at any given time $t$, nor temporal translational symmetry at any spatial location $\mathbf{r}$. Consequently, the frequently used time-evolution operator of one period for the Floquet problem generally does not apply. The reciprocal lattice is spanned by the momentum-energy basis vectors $b^{i}=(\mathbf{G^{i}},\Omega^{i})$ defined through $b^{i}\cdot a^{j}=\sum_{m=1}^{D}G^{i}_{m}u^{j}_{m}-\Omega^{i}\tau^{j}=2\pi\delta^{ij}$. The $D+1$ dimensional momentum-energy Brillouin zone (MEBZ) may also be momentum-energy mixed.

Generalized Floquet-Bloch theorem We generalize the Floquet and Bloch theorems for the time-dependent Schrödinger equation $i\hbar\partial_{t}\psi(\mathbf{r},t)=H(\mathbf{r},t)\psi(\mathbf{r},t)$. Due to the space-time translation symmetry, the lattice momentum-energy vector $\kappa=(\mathbf{k},\omega)$ remains conserved. Only the $\kappa$ vectors inside the first MEBZ are non-equivalent, and those outside are equivalent up to integer reciprocal lattice vectors. The Floquet-Bloch states labeled by $\kappa$ take the form of

[TABLE]

where $m$ marks different states sharing the common $\kappa$. $u_{m}(\mathbf{r},t)$ processes the same space-time periodicity as $H(\mathbf{r},t)$, and is expanded as $u_{m}=\sum_{B}c_{m,B}e^{i(\mathbf{G}\cdot\mathbf{r}-\Omega t)}$ with $B=(\mathbf{G},\Omega)$ taking all the momentum-energy reciprocal lattice vectors. The eigen-frequency $\omega_{m}$ is determined through the eigenvalue problem defined as

[TABLE]

where $\varepsilon_{0}(\mathbf{k})$ is the free dispersion, and $V_{B}$ is the momentum-energy Fourier component of the space-time lattice potential $V(\mathbf{r},t)$. The dispersion based on Eq. 3 is represented by a $D$-dimensional surface in the MEBZ which is a $D$+1 dimensional torus.

Dispersion winding numbers – The band structure of the space-time crystal exhibits novel features different from those of the static crystal. For simplicity, below we use the 1+1 D case for an illustration. The dispersion relation $\omega(k)$ forms closed loops in the 2D toroidal MEBZ, each of which is characterized by a pair of winding numbers $\mathbf{w}=(w_{1},w_{2})$. Compared to the static case in which the band dispersion only winds around the momentum direction, here $\omega(k)$ is typically not single-valued and its winding patterns are much richer. The dispersions in the limit of a weak space-time potential $V(x,t)$ with a rhombic MEBZ are illustrated in Fig. 1 ($a$) and ($b$), with details presented in Supplemental Material (S.M.) Sect. A supp . When folded into the MEBZ, the free dispersion curve $\varepsilon_{0}(k)$ can cross at general points not just on high symmetry ones. A crossing point corresponds to two equivalent momentum-energy points related by a reciprocal vector $G$. When $V_{G}\neq 0$, the crossing is avoided by forming a gap at the magnitude of $2|V_{G}|$. The total number of states at each $\mathbf{k}$ is independent of the strength of $V(x,t)$, hence crossing can only split along the $\omega$-direction and $d\omega/dk$ is always finite. Consequently, trivial loops with the winding numbers $(0,0)$ are forbidden. Generally, the winding directions of the dispersion loops are momentum-energy mixed. Furthermore, different momentum-energy reciprocal lattice vectors can cross each other, leading to composite loops winding around the MEBZ along both directions as shown in Fig. 1 ($b$). Hence, in general all patterns $(w_{1},w_{2})$ are possible except the contractible loops.

Space-time group – To describe the symmetry properties of the $D+1$ dimensional space-time crystals, we propose a new group structure dubbed “space-time” group defined as the discrete subgroup of the direct product of the Euclidean group in $D$ spatial dimensions and that along the time-direction $E_{D}\otimes E_{1}$. Please note that in general the space-time group cannot be factorized as the direct product between discrete spatial and temporal subgroups. It not only includes space and magnetic group transformations in the $D$-spatial dimensions, but also includes operations involving fractional translations along the time-direction. Since space and time are non-equivalent in the Schrödinger equation, space-time rotations are not allowed except the 2-fold case.

To be concrete, a general space-time group operation $\Gamma$ on the space-time vector $(\mathbf{r},t)$ is defined as,

[TABLE]

where $R$ is a $D$-dimensional point group operation, $s=\pm 1$ and $s=-1$ indicates time-reversal, and $(\mathbf{u},\tau)=\sum_{i}m_{i}a^{i}$ represents a space-time translation with $m_{i}$ either integers or fractions. If $\tau=0$, $\Gamma$ is reduced to a space group or magnetic group operation according to $s=\pm 1$, respectively. If $\tau\neq 0$, when $(\mathbf{u},\tau$) contains fractions of $a^{i}$, new symmetry operations arise due to the dynamic nature of the crystal potential, including the “time-screw” rotation and “time-glide” reflection, which are a spatial rotation and a reflection followed by a fractional time translation, respectively. The operation of $\Gamma$ on the Hamiltonian is defined as $\Gamma^{-1}H(\mathbf{r},t)\Gamma=H(\Gamma(\mathbf{r},t))$, or, $\Gamma^{-1}H(\mathbf{r},t)\Gamma=H^{*}(\Gamma(\mathbf{r},t))$ for $s=\pm 1$, respectively. Correspondingly, the transformation $M_{\Gamma}$ on the Bloch-Floquet wavefunctions $\psi_{\kappa}(\mathbf{r},t)$ is $M_{\Gamma}\psi_{\kappa}=\psi_{\kappa}(\Gamma^{-1}(\mathbf{r},t))$, or, $\psi_{\kappa}^{*}(\Gamma^{-1}(\mathbf{r},t))$ for $s=\pm 1$, respectively.

Now we present a complete classification of the 1+1 D space-time groups. Due to the non-equivalence between spatial and temporal directions, there are no square and hexagonal space-time crystal systems. The point-group like operations are isomorphic to $D_{2}$, including reflection $m_{x}$, time reversal $m_{t}$, and their combination $m_{x}m_{t}$, i.e., the 2-fold space-time rotation. Consequently, only two space-time crystal systems are allowed – oblique and orthorhombic. There exist two types of glide reflections: the time-glide reflection $g_{x}$, and $g_{t}$ denoted as “glide-time-reversal” is time-reversal followed by a fractional translation along the $x$-direction.

The above 1+1 D space-time symmetries give rise to 13 space-time groups in contrast to the 17 wallpaper space groups characterizing the 2D static crystals. The oblique Bravais lattice is simply monoclinic, while the orthorhombic ones include both the primitive and centered Bravais lattices. The monoclinic lattice gives rise to two different crystal structures with and without the 2-fold space-time axes, whose space-time groups are denoted by $P_{1,2}$, respectively, as shown in Fig. 2 (a). For the primitive orthorhombic lattices, the associated crystal structures can exhibit the point-group symmetries $m_{x}$ and $m_{t}$, and the space-time symmetries $g_{t}$ and $g_{x}$. Their combinations give rise to 8 space-time crystal structures denoted as $Pm_{x}$, $Pm_{t}$, $P2m_{x}m_{t}$, $Pg_{x}$, $Pg_{t}$, $P2g_{x}g_{t}$, $P2m_{x}g_{t}$, $P2g_{x}m_{t}$, respectively, as shown in Fig. 2 (b). Four of them possess the 2-fold space-time axes as indicated by “2” in their symbols. For the centered orthorhombic Bravais lattices, 3 crystal structures exist with space-time groups denoted as $Cm_{x}$, $Cm_{y}$, and $C2m_{x}m_{t}$, respectively, as shown in Fig. 2 (c). They all exhibit glide-reflection symmetries, and the last one possesses the 2-fold space-time axes as well. Two unit cells are plotted for the centered lattices to show the full symmetries explicitly, and their primitive basis vectors are actually space-time mixed.

The classifications of the space-time groups in higher dimensions are generally complicated. A general method is the group cohomology as presented in Sect B of S. M. supp . In particular, the classification of 2+1D space-time group is outlined in Sect C of S. M. supp , whose structures are further enriched by spatial rotations and time-screw rotations. Compared to the 3D static crystals, the cubic crystal systems are not allowed, and two different monoclinic crystal systems appear with the perpendicular axis along the time and spatial directions, respectively. In total, there are 7 crystal systems and 14 Bravais lattices, but 275 space-time groups.

Protection of spectral degeneracy The intertwined space-time symmetries besides translations can protect spectral degeneracies. Below we consider the effects from the Kramers symmetry without spin and the non-symmorphic symmetries for the 1+1 D and 2+1 D space-time crystals, respectively.

Consider a 1+1 D space-time crystal whose unit cell is a direct product of spatial and temporal periods $\lambda$ and $T$, respectively. We assume the system is invariant under the glide time-reversal operation $g_{t}(x,t)=(x+\frac{1}{2}\lambda,-t)$, whose operation on the Hamiltonian is defined as $g_{t}^{-1}Hg_{t}=H^{*}(g_{t}(x,t))$. The corresponding transformation $M_{g_{t}}$ on the Bloch-Floquet wavefunction $\psi_{\kappa}(x,t)$ of Eq. 2 is anti-unitary defined as $M_{g_{t}}\psi_{\kappa}=\psi^{*}_{\kappa}(g_{t}^{-1}(x,t))$. This glide time-reversal operation leaves the line of $\kappa_{x}=\pi/\lambda$ in the MEBZ invariant. $M_{g_{t}}$ becomes a Kramers symmetry for states with $\kappa_{x}=\pi/\lambda$,

[TABLE]

without involving the half-integer spinor structure. It protects the double degeneracy of the momentum-energy quantum numbers of $\psi_{\kappa}$ and $M_{g_{t}}\psi_{\kappa}$. Hence the crossing at $\kappa_{x}=\pi/\lambda$ cannot be avoided and the dispersion winding numbers along the momentum direction must be even.

As a concrete example, we study a crystal potential with the above spatial and temporal periodicities, V(x,t)=V_{0}\big{(}\sin\frac{2\pi}{T}t\cos\frac{2\pi}{\lambda}x+\cos\frac{2\pi}{T}t\big{)}. Except the glide time-reversal symmetry, it does not possess other symmetries. Its Bloch-Floquet spectrum is calculated based on Eq. 3, and a representative dispersion loop is plotted in the MEBZ shown in Fig. 3 (a). The crossing at $\kappa_{x}=\pi/\lambda$ is protected by the glide time-reversal symmetry giving rise to a pair of Kramers doublet. As a result, the winding number of this loop is $\mathbf{w}=(w_{x},w_{t})=(2,0)$. If a glide time-reversal breaking term $\delta V=V_{0}^{\prime}\cos(\frac{2\pi}{\lambda}x)$ is added into the crystal potential, the crossing is avoided as shown in Fig. 3 (b). Consequently, the dispersion splits into two loops, both of which exhibit the winding number $(1,0)$. Similarly, out of the 8 primitive orthorhombic space-time crystals, 3 of them, $Pg_{t}$, $P2g_{x}g_{t}$, and $P2g_{t}m_{x}$, enforce this non-spinor type Kramers degeneracy, while the other 5 generally does not protect such a degeneracy.

Next we present a 2+1 D Floquet semi-metal state, whose spectral degeneracies are protected by non-symmorphic space-time group operations. Consider that the space-time little group of the momentum $\mathbf{k}$ contains two non-symmorphic space-time group operations $g_{1,2}$, both of which do not flip the time direction, hence, they are represented by unitary operators. If they satisfy

[TABLE]

where $T$ is a translation of integer lattice vectors. As shown in Sect. D in S. M. supp , $T$ can only be a spatial translation without involving the time denoted as $T(\mathbf{u})$. Assume $\mathbf{k}\cdot\mathbf{u}=2\pi p/q$ with $p$ and $q$ coprime, we find that the Bloch-Floquet wavefunctions exhibit a $q$-fold degeneracy at the momentum-energy vector $\kappa=(\mathbf{k},\omega)$ proved as follows. Since $g_{1}$ belongs to the little group, $\psi_{\kappa}(\mathbf{r},t)$ can be chosen to satisfy $M_{g_{1}}\psi_{\kappa,1}=\mu\psi_{\kappa,1}$, then $\psi_{\kappa},M_{g_{2}}\psi_{\kappa},M_{g_{2}}^{2}\psi_{\kappa},....,M_{g_{2}}^{q-1}\psi_{\kappa}$ are the common Bloch-Floquet eigenstates sharing the same $\kappa$ but exhibiting a set of different eigenvalues of $g_{1}$ as $\eta,\mu\eta,\mu\eta^{2},...,\mu\eta^{q-1}$ with $\eta=e^{i\pi p/q}$. Then they are orthogonal to each other forming a $q$-fold degeneracy. Compared to the case of non-symmorphic space group protected degeneracy Parameswaran2013 ; Young2015 ; Watanabe2016 , here $g_{1,2}$ are space-time operations for a dynamic space-time crystal. For the case that one or both of $g_{1,2}$ flip the time direction, the situation is more involved due to involving anti-unitary operations. Protected degeneracies are still possible as presented in Sect. D in S. M. supp .

We employ a 2+1 D tight-binding space-time model as an example to illustrate the above protected degeneracy. A snap shot of the lattice is depicted in Fig. 4 ($a$), which consists of two sublattices: The $A$-type sites are with integer coordinates $(i,j)$, and each $A$-site emits four bonds along $\vec{e}_{i}$ to its four neighboring $B$ sites at $(i\pm\frac{1}{2},j\pm\frac{1}{2})$. The space-time Hamiltonian within the period $T$ is

[TABLE]

where $a$ is the distance between two nearest $A$ sites, and $w_{\vec{e}_{i}}(t)$’s are hopping amplitudes with different strengths. Their time-dependence is illustrated in Fig. 4 ($b$): Within each quarter period, $w_{\vec{e}_{i}}$ does not vary, and their pattern rotates 90∘ after every $T/4$. At each given time, the lattice possesses a simple 2D space group symmetry $p2111$, which only includes two-fold rotations around the $AB$-bond centers without reflection and glide-plane symmetries. For example, the rotation $R_{\pi}$ around $(\frac{a}{4},\frac{a}{4})$ transforms the coordinate $(x,y,t)\to(\frac{a}{2}-x,\frac{a}{2}-y,t)$. In addition, there exist “time-screw” operations, say, an operation $S$ defined as a rotation around an $A$-site $(0,0)$ at 90∘ followed by a time-translation at $T/4$, which transforms $(x,y,t)\to(y,-x,t+\frac{T}{4})$. $R_{\pi}$ and $S$ are generators of the space-time group for the Hamiltonian Eq. 7. Since $S$ is a time-screw rotation, this space-time group is non-symmorphic. It is isomorphic to the 3D space-group $I4_{1}$, while its 2D space subgroup $p2111$ is symmorphic. We have checked that, for a static Hamiltonian taking any of the bond configuration in Fig. 4 ($b$), the energy spectra are fully gapped. However, the non-symmorphic space-time group gives rise to spectral degeneracies. Its momentum Brillouin zone is depicted in Fig. 4 ($c$). The space-time little group of the $M$-point $(\pi,\pi)$ contains both $R$ and $S$ satisfying $RS=T(a\hat{y})SR=-SR$. Similarly, the $X$-point $(\pi,0)$ is invariant under both $R$ and $S^{2}$ satisfying $RS^{2}=T(a\hat{x}+a\hat{y})S^{2}R=-S^{2}R$. Hence, the Floquet eigen-energies are doubly degenerate at $M$ and $X$-points as shown in Fig. 4 ($d$), showing a semi-metal structure.

In conclusion, we have studied a novel class of $D+1$ dimensional dynamic crystal structures exhibiting the general space-time periodicities. Their MEBZs are $D+1$ dimensional torus and are typically momentum-energy entangled. The band dispersions exhibit non-trivial windings around the MEBZs. The space-time crystal structures are classified by space-time group, which extend space group for static crystals by incorporating time-screw rotations and time-glide reflections. In 1+1D, a complete classification of the 13 space-time groups is performed, and there exist 275 space-time groups in 2+1 D. Space-time symmetries give rise to novel Kramers degeneracy independent of the half-integer spinor structure. The non-symmorphic space-time group operations lead to protected spectral degeneracies for space-time crystals. This work sets up a symmetry framework for exploring novel properties of space-time crystals. It also serves as the starting point for future studies, for example, the dynamical topological phases of matter based on their space-time groups.

Acknowledgments This work is supported by AFOSR FA9550-14-1-0168.

Note added. Upon completing this manuscript, we noticed an interesting and important work by T. Morimoto et. al. Morimoto2017 classifying Floquet topological crystalline insulators with two-fold space-time symmetries.

Appendix A The nearly-free-particle approximation of 1+1 D space-time crystal

In this section, we expand the discussion on the nearly-free-particle band structure of the space-time crystal and explicitly demonstrate the momentum-energy Brillouin zone (MEBZ) folding procedure. We consider a weak space-time lattice potential $V(\mathbf{r},t)$ and work in the framework of the generalized Floquet-Bloch theorem based on Equations (2) and (3) in the main text.

Consider a Floquet-Bloch state $\psi_{\kappa,m}(\mathbf{r},t)$ with the good quantum number $\kappa=(\mathbf{k},\omega_{m})$ and the band index $m$. Its wavefunction component in terms of each reciprocal lattice vector $B=(\mathbf{G},\Omega)$ is denoted as $c_{m,B}$. By construction, if $c_{m,B}$ is a solution, then $c_{m,B-B\prime}$ with $B^{\prime}=(\mathbf{G^{\prime}},\Omega^{\prime})$ corresponds to the solution with an equivalent quasi-momentum and energy $(\mathbf{k}+\mathbf{G}^{\prime},\omega_{m}+\Omega^{\prime})$. Nevertheless, this remains the same state as before. In order to remove this redundancy, $\kappa$ can be constrained in the first momentum-energy Brillouin zone (FMEBZ), i.e., the unit cell in the reciprocal space centered at the origin. Dispersions residing in high order MEBZs can be folded into the FMEBZ. The general process is as follows: First fold the free spectrum $\varepsilon_{0}(\mathbf{k})$ into the FMEBZ. When it crosses, the corresponding $\kappa$’s of the unfolded spectrum are differed by a momentum-energy reciprocal lattice vector $B$. These two plane-wave states are hybridized in the presence of the nonzero Fourier component $V_{B}$, leading to the level repulsion at the crossing point and yielding the Floquet-Bloch wavefunctions. Consequently, the quadratic spectrum breaks into loops winding around in the FBZ.

Fig. 5 demonstrates an example of zone-folding of the dispersion corresponding to the Fig 1. (a) in the main text. Fig. 5 (a) presents an unfolded dispersion in the presence of a weak space-time potential. The gap opening points are marked in pairs on the unfolded dispersion: The dashed arrows linking two points represent the non-vanishing Fourier components $V_{B}$ of the external potential. In Fig. 5 (b), the extended MEBZ representation is used, i.e., the dispersion is duplicated being shifted by all the momentum-energy reciprocal lattice vectors. The FMEBZ boundary is marked by the solid line. The Floquet-Bloch dispersion forming two loops after folded into the FMEBZ as shown in Fig. 5 (c). The parts of the spectrum from $2\rightarrow 3$ and from $3^{\prime}\to 4^{\prime}$ does not participate in forming these loops, and are omitted. We can smoothly vary the loop structure in the FMEBZ such that the winding numbers of each loop become transparent: The red and blue loops have the winding numbers $(0,1)$ and $(1,0)$, respectively. These two loops actually cross, however, they do not open the gap due to the lack of non-zero Fourer component $V_{B}$ with $B$ connecting two momentum-energy vectors at the crossing point as shown in Fig. 5 (a). Otherwise, we will arrive at the situation shown in Fig 1. (b) in the main text, where the two loops merge into one with the winding number $(1,1)$.

The above example demonstrates the connection between the winding numbers and the non-vanishing Fourier components of the external potential. The crossing of bands can be protected from splitting by space-time symmetries of the system. For example, as the example shown in the main text, the glide time-reversal $g_{t}$ protects the spectral double degeneracy.

Appendix B Construction of the space-time group via group cohomology

Each space group is constructed from a static Bravais lattice. Similarly, each space-time group is constructed from a Bravais lattice $M$ constituted of the $D+1$ dimensional space-time mixed discrete translations, and a magnetic point group in $d$ dimensions $G_{m}(D)$ that leaves $M$ invariant. Here “magnetic” refers to the reflection with respect to time, i.e., time-reversal operation. Below we use denote the space-time group by the symbol $ST$.

B.1 The general procedure

The hierarchical classification scheme of the space-time group starts with the crystal systems. Each crystal system is labeled by a set of Bravais lattices $\{M\}$ and these lattices share the same magnetic point group symmetry. Below we use the same symbol $M$ to represent the lattice translation group associated to the Bravais lattice $M$. The magnetic point group symmetry of a crystal is often smaller than the that of the underlying Bravais lattice. As a result, each crystal system can be further divided into different geometry crystal classes (GCC) according to different magnetic point group symmetries. Each GCC can be further classified into different arithmetic crystal classes (ACC) based on a particular Bravais lattice and a particular magnetic point group. It is worth noting that, the same Bravais lattice and magnetic point group can give rise to different ACCs depending on the realizations of magnetic point group operations. For each ACC, based on its unique Bravais lattice and magnetic point group, different space-time groups can be constructed. Such hierarchy is in parallel to the space group classification scheme of static lattices.Prince2004 .