Space-time Floquet operator: Non-reciprocity and fractional topology of space-time crystals

TL;DR

The paper addresses how space–time crystals—systems with intertwined spatial and temporal periodicities—can be analyzed beyond traditional Floquet theory by introducing a space–time Floquet operator that advances dynamics over a reduced time-step $τ_0$. This leads to a space–time band structure on an oblique Brillouin zone, with windings $w(\boldsymbol{g}_0)$ and $w(\boldsymbol{g}_1)$ that govern fractional topological transport via $Q_{τ_0} = w(\boldsymbol{g}_0) + \frac{r_α}{β} w(\boldsymbol{g}_1)$ and $Q_T = β Q_{τ_0}$. The authors connect the space–time spectrum to Floquet bands through $U_k(T) = X_k(τ_0)^β$, derive a fractional Thouless pumping and Floquet–Bloch oscillations, and demonstrate nonreciprocal parametric resonances without the spurious degeneracies of conventional folding. The framework generalizes to higher dimensions and non-Hermitian settings, offering a robust toolkit for predicting long-time transport and resonance phenomena in driven space–time crystals across classical and quantum platforms.

Abstract

We introduce a space-time Floquet operator, a generalization of the conventional Floquet operator, that captures the long-time behavior of space-time crystals - systems where spatial and temporal periodicities are intrinsically intertwined. Unlike the standard Floquet operator, which describes evolution over a full time period, the space-time Floquet operator evolves the system over a fraction of the period, thereby resolving finer details of its dynamics. Its eigenmode spectrum defines a space-time band structure that unfolds conventional Floquet bands to respect the intertwined crystal symmetry in reciprocal wavevector-frequency space. We relate the topology of these space-time bands to quantized transport phenomena, such as Bloch oscillations and adiabatic charge transport, and uncover a fractional version of the latter. We also demonstrate how nonreciprocal parametric resonances are naturally anticipated by our framework. The approach applies broadly to both classical and quantum systems with space-time symmetry, including non-Hermitian crystals.

Figures (8)

• Figure 1: (a) Schematic of a $1$D system with space-time symmetry, $(x,t) \mapsto (x+b,t+\tau)$, where $V(t)$ the onsite potential. PBC ensure that the system is also invariant under $(x,t) \mapsto (x,t+T)$, with $T=\frac{\beta}{\alpha}\tau$, and $(x,t) \mapsto (x+a,t)$, with $a:= \beta b$, for some co-prime integers $\beta$ and $\alpha$ (here, $5$ and $2$ respectively). (b) Primitive vectors $\mathbf{a}_0 = -(r_\alpha b, \tau_0)$ and $\mathbf{a}_1 = (\beta b, 0)$ generate the symmetries of the lattice. The integers $r_\beta$ and $r_\alpha$ (here, $-1$ and $3$ respectively), defined via Eq. \ref{['eq:Bezout']}, determine $\tau_0$ the smallest time-step of numerical integration needed to capture the long-time behavior of the system.
• Figure 2: (a) Schematic of a space-time band structure for a system with $\tau = \frac{\alpha}{\beta}T = \frac{2}{5}T$, two bands, and no degeneracies. The band structure is generated by tiling a column of $\beta = 5$ distinct rectangular plaquettes of height $\frac{2\pi}{\beta \tau_0} = \frac{2\pi}{T}$ and width $\frac{2\pi}{\beta b} = \frac{2\pi}{a}$ (distinguished by colors) at lattice points generated by reciprocal lattice vectors $\mathbf{g}_{0} = \left(0, \frac{2\pi}{\tau_0} \right)$ and $\mathbf{g}_{1} = \left(\frac{2\pi}{\beta b},\frac{2\pi r_\alpha}{\beta \tau_0} \right)$ with $r_\alpha = 3$. (b) Floquet band structure computed by considering the spatial and temporal periodicity of the system independently, with orthogonal reciprocal primitive vectors $\mathbf{g}_{0}^{\textrm{F}} = \left(0, \frac{2\pi}{\beta \tau_0} \right)$ and $\mathbf{g}_{1}^{\textrm{F}} = \left(\frac{2\pi}{\beta b}, 0 \right)$.
• Figure 3: (a) A chain of masses connected by springs modulated in a travelling wave fashion [Eq. \ref{['eq:modulation']} with $\alpha =2, \beta =7$]. (b) The static band structure of the unmodulated chain has two bands and is invariant on translation by the reciprocal lattice vector $\mathbf{g}_{1}' = (\frac{2\pi}{b},0)$ in frequency-wavevector space. (c) Floquet band structure in the limit of infinitesimal modulation, obtained by folding the static band structure along the primitive vectors $\mathbf{g}_{0}^{\textrm{F}} = (0,\frac{2\pi}{T})$ and $\mathbf{g}_{1}^{\textrm{F}} = (\frac{2\pi}{7b},0)$. (d) Real part of the Floquet band structure computed at finite modulation strength $\delta = 0.5$. Resonant modes with doubly-degenerate real part and nonzero imaginary part are indicated in red, signifying parametric amplification of those modes. (e) Space-time band structure at infinitesimal modulation, obtained by further folding the static band structure along $\mathbf{g}_{0}'=(\frac{2\pi\alpha}{\beta b}, \frac{2\pi}{\beta \tau_0})$. (f) Real part of the space-time bands at finite modulation $\delta = 0.5$.
• Figure 4: Three bands from a space-time crystal with $\alpha = 2$ and $\beta = 5$. The primitive vectors (solid arrows) and repeating plaquettes (dashed boxes) of the space-time band structure are shown. The blue band forms a closed loop on the momentum-frequency Brillouin zone with winding numbers $\lparen*\rparen{w(\mathbf{g}_{0}), w(\mathbf{g}_{1})} = (-1, 1)$, whereas the orange band traverses the Brillouin zone twice (solid and dashed branches) before closing on itself and has winding numbers $\lparen*\rparen{w(\mathbf{g}_{0}), w(\mathbf{g}_{1})} = (-1, 1)$. Right: The Floquet band structure, obtained by superimposing the five plaquettes from the left, generates windings of $(-2,1)$ and $(1,2)$ for the blue and orange bands respectively.
• Figure 5: (a) Space-time band structure of tight-binding Hamiltonian in Eq. \ref{['eq:tbham']} with parameters $\beta = 3$, $\alpha =1$, $V_0 = 1.5$, $J = 1$, $T = 25$. The reciprocal lattice vectors as defined in the text are shown. The blue, orange, and green bands have windings $(w\lparen*\rparen{\mathbf{g}_{0}},w\lparen*\rparen{\mathbf{g}_{1}}) = (0,1)$, $(-1,1)$ and $(0,1)$ respectively. Inset: Zoom of green shaded region, showing an avoided crossing with gap $\Delta \omega = 0.0367/\tau_0$ which is ignored when band indices are assigned. Other crossings involve smaller gaps. (b) Net pumped charge $Q(t)$ from numerical integration of Schrödinger's equation on a tight-binding chain with $N=400$. The output of independent simulations initialized with a linear superposition of $N$ states from each band is shown; colors indicate the bands from (a). The crosses show the quantized expectation $Q(n\tau_0) = nQ_{\tau_0}$ at integer multiples of the shortest time period $\tau_0$. Inset shows the pumped charge $\Delta Q(t) = Q(t) - Q(t-\tau_0)$ normalized by the winding prediction.