Classical Prethermal Phases of Matter

Systems subject to a high-frequency drive can spend an exponentially long time in a prethermal regime, in which novel phases of matter with no equilibrium counterpart can be realized. Because of the notorious computational challenges of quantum many-body systems, numerical investigations in this direction have remained limited to one spatial dimension, in which long-range interactions have been proven a necessity. Here, we show that prethermal nonequilibrium phases of matter are not restricted to the quantum domain. Studying the Hamiltonian dynamics of a large three-dimensional lattice of classical spins, we provide the first numerical proof of prethermal phases of matter in a system with short-range interactions. Concretely, we find higher-order as well as fractional discrete time crystals breaking the time-translational symmetry of the drive with unexpectedly large integer as well as fractional periods. Our work paves the way toward the exploration of novel prethermal phenomena by means of classical Hamiltonian dynamics with virtually no limitations on the system’s geometry or size, and thus with direct implications for experiments.

Figure 1

Prethermal discrete time crystals. (a) The system consists of $N=50^3$ classical spins in a three-dimensional lattice with nearest-neighbor interactions. (b) Relevant timescales of a 4-DTC (for $g=0.25$). The prethermalization timescale ${\tau }_{pth}\sim 1/\lambda$ is set by the Lyapunov exponent $\lambda$. Thermalization to an infinite-temperature state occurs at a much later time ${\tau }_{th}\sim e^{c\omega }$ exponential in the drive frequency $\omega$. Here, $\mathrm{\Delta }=0.01$. (c) By varying the parameter $g$, one can access different higher-order and fractional $n$-DTCs, each corresponding to a plateau at frequency $1/n$ in the system’s spectral response. This is shown by means of the Fourier transform $|\tilde{m}|$ of the magnetization for $n=2$, 3, 4 and $20/7$.

Figure 2

Phenomenology of a prethermal 4-DTC. (a) The average energy $H_T$ reaches its infinite-temperature value 0 after an exponentially long time, the signature of prethermalization. (b) A period 4-tupling is observed in the stroboscopic series of the magnetization $m$ over the whole prethermal regime. (c) The decorrelator initially grows exponentially, $d\sim e^{\lambda t}$, signaling sensitivity to the initial conditions and chaos. After a timescale ${\tau }_{pth}\sim 1/\lambda$, it plateaus at a value $\sim 60\%d_{\infty }$, before ultimately reaching its infinite-temperature value $d_{\infty }=\sqrt{2}$ after a time ${\tau }_{th}\sim e^{c\omega }$. (d) Spin components $S_{\mathit{r}}^{x,z}$ over a two-dimensional cut of the system and at representative times $t$ multiples of $4T$, together with the difference $S_{\mathit{r}}^{x,z}-S_{\mathit{r}}^{x,z\prime }$ between the two initially close copies of the system. At time $t={10}^{2}T<{\tau }_{pth}$, the spins are predominantly polarized along $z$ ($m>0$) while having disordered $x$ components, and the difference between the two copies is still small (mostly white). At time ${\tau }_{pth}\ll t={10}^{4}T\ll {\tau }_{th}$ the system has prethermalized: while both copies of the system are still polarized along $z$, their $x$ and $y$ components have decorrelated. At time $t=4\times {10}^{5}T\sim {\tau }_{th}$, the $z$ polarization is progressively destroyed by the proliferation of domain walls, en route towards the ultimate heat death with completely random spin orientations, shown for $t=5\times {10}^6\gg {\tau }_{th}$. Here, $\omega =2.86$, $g=0.255$, and $\mathrm{\Delta }=0.01$.

Figure 3

Frequency dependence of a prethermal discrete time crystal. The decorrelator initially grows exponentially, $d\sim e^{\lambda t}$. The prethermalization time ${\tau }_{pth}\sim 1/\lambda$ is associated with the time at which $d$ crosses the 10% of its infinite-temperature value $d_{\infty }$, and weakly depends on the frequency $\omega$. By contrast, the prethermal plateau $d\approx 60\%d_{\infty }$ extends for a time that grows exponentially with the frequency. As a reference for the thermalization time ${\tau }_{th}$, we take the time at which the decorrelator $d$ crosses $90\%d_{\infty }$. Note, the axes have linear and logarithmic parts. The insets show the magnetization $m$ and the energy $H_1$ [the first part of $H(t)$ in Eq. (1)] at stroboscopic times $t=4nT$. Here, $g=0.25$ and $\mathrm{\Delta }={10}^{-16}$.