Space-Time Crystal and Space-Time Group

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 a “space-time” group is constructed to describe the discrete symmetries of a space-time crystal. Compared to space and magnetic groups, the 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 one-plus-one dimensions ($1+1\mathrm{D}$) 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 $2+1\mathrm{D}$, nonsymmorphic space-time symmetries enforce spectral degeneracies, leading to protected Floquet semimetal states. We provide a general framework for further studying topological properties of the ($D+1$)-dimensional space-time crystal.

Figure 1

Folding the band dispersions of the $1+1\mathrm{D}$ space-time crystal into the first rhombic MEBZ in the weak lattice limit. The momentum-energy reciprocal lattice vectors of nonzero $V_B$ values are represented by dashed lines. The low-energy part of the free dispersion curve evolves to closed loops. (a) Two loops with the winding numbers ${\mathbf{w}}_{\mathbf{r}}=(1,0)$ (the red curves) and ${\mathbf{w}}_{\mathbf{b}}=(0,1)$ (the blue curves). (b) An extra nonzero $V_G$ value connects two loops in (a), forming a new one with $\mathbf{w}={\mathbf{w}}_{\mathbf{r}}+{\mathbf{w}}_{\mathbf{b}}$.

Figure 2

The classification of 13 space-time groups in $1+1\mathrm{D}$ and the associated crystal configurations. The solid oval marks the twofold space-time axis, and the parallelogram represents the twofold axis without reflection symmetries. The thick solid and dashed lines represent reflection and glide-reflection axes, respectively. Configurations of the triangles and the diamond denote the local symmetries under reflections. (a) The oblique lattices with and without twofold axes. Their basis vectors are generally space-time mixed. (b) Primitive and (c) centered orthorhombic lattices. According to their reflection and glide reflection symmetries, the crystal configurations are classified into eight groups in (b), and three groups in (c).

Figure 3

(a) The Floquet-Bloch band spectrum with the space-time lattice potential possessing the glide time-reversal symmetry $g_t$. When applied to the states with ${\kappa }_x=\pi /\lambda$, $g_t$ becomes a Kramers symmetry protecting the double degeneracy. (b) Lifting the Kramers degeneracy by adding a glide time-reversal symmetry breaking term.

Figure 4

(a) The $2+1\mathrm{D}$ space-time lattice structure of the Hamiltonian equation (7). The bond directions are indicated as ${\overrightarrow{e}}_{1,3}=\pm \frac{1}{2}(\widehat{x}+\widehat{y})$ and ${\overrightarrow{e}}_{2,4}=\mp \frac{1}{2}(\widehat{x}-\widehat{y})$. (b) The time-dependent hopping pattern rotates 90° every one quarter period. The bonding strengths $w_{e_i}(t)$ of the $R$, $B$, $G$, and $Y$ bonds equal 0.2, 3, $-3.2$, and 0.5, respectively. (c) The momentum Brillouin zone with high symmetry points $\mathrm{\Gamma }=(0,0)$, $M=(\pm \pi ,\pm \pi )$, and $X=(0,\pm \pi )$ and $(\pm \pi ,0)$. (d) The dispersions along the cuts from $\mathrm{\Gamma }$ to $X$ to $M$ to $\mathrm{\Gamma }$. Twofold degeneracies appear at $X$ and $M$.