Long-Lived Interacting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems

We show how a large family of interacting nonequilibrium phases of matter can arise from the presence of multiple time-translation symmetries, which occur by quasiperiodically driving an isolated, quantum many-body system with two or more incommensurate frequencies. These phases are fundamentally different from those realizable in time-independent or periodically driven (Floquet) settings. Focusing on high-frequency drives with smooth time dependence, we rigorously establish general conditions for which these phases are stable in a parametrically long-lived “preheating” regime. We develop a formalism to analyze the effect of the multiple time-translation symmetries on the dynamics of the system, which we use to classify and construct explicit examples of the emergent phases. In particular, we discuss time quasicrystals which spontaneously break the time-translation symmetries, as well as time-translation symmetry-protected topological phases.

Popular Summary

Matter at equilibrium organizes itself into a variety of phases with sharply distinct physical properties, be it solids, liquids, magnets, or Bose-Einstein condensates. The concept of symmetries is an organizing principle in delineating these phases: A solid has a regular structure of atoms, while a liquid does not. Recently, phases were discovered in matter subjected to an external time-periodic drive. These phases are characterized by an unusual dynamical symmetry and are fundamentally far from equilibrium. We establish the existence of a large family of nonequilibrium phases in a different setting: matter subjected to a quasiperiodic drive made from two or more incommensurate frequencies. Such drives are temporal analogs of spatial quasicrystals. We develop a mathematical formalism to describe these systems and show the existence of multiple dynamical symmetries underlying the new phases.

Driven systems of matter often heat quickly, destroying all signatures of the nonequilibrium phases. We find a general class of quasiperiodic drives where we rigorously show phase stability for a long time—a so-called prethermal regime. In this regime, we explicitly construct phases—discrete time quasicrystals—that break the multiple dynamical symmetries. Additionally, we construct and classify topological phases that build upon the presence of the multiple dynamical symmetries.

Our results point toward the richness of the landscape of phases in nonequilibrium settings, including those with complex drives. The ideas and techniques we introduce provide a route to explore this exciting and largely uncharted territory.

Figure 1

(a) Cartoon of an ensemble of spins that is quasiperiodically driven. (b) A time-quasiperiodic Hamiltonian $H(t)$ arises from evaluating a Hamiltonian $H(\mathit{\theta })$ defined on the $m$-dimensional torus ${\mathbb{T}}^m\ni \mathit{\theta }$ on the trajectory $\mathit{\theta }(t)=\mathit{\omega }t+{\mathit{\theta }}_0$ (blue arrows). As the frequency vector $\mathit{\omega }$ is a set of rationally independent frequencies, the trajectory never returns to itself but covers the torus uniformly. Shown here is the case $m=2$, with $\tan \alpha ={\omega }_2/{\omega }_1$ irrational.

Figure 2

Prethermalization in quasiperiodically driven systems at high frequencies. Shown is a cartoon of the dynamics of a generic traceless local observable $⟨O(t)⟩$ under time evolution by Eq. (6). There are three regimes. First, a brief transient regime, where the local observable relaxes on a short timescale $t_r\sim 1/J$, where $J$ is the local energy scale of the system. Second, a prethermalization regime, where the system has locally equilibrated to a thermal ensemble of an effective static Hamiltonian $D$, when viewed in the rotating frame defined by $P(t)$. The evolution $⟨O(t)⟩$ in the laboratory frame shows a plateau around the prethermal value (black dashed line), with small time-quasiperiodic oscillations of amplitude of $\sim J/\omega$ (red dashed lines); here, $\omega =|\mathit{\omega }|$ is the norm of driving frequencies. This regime lasts up to the long heating time $t_{*}\sim \exp [C(\omega /J{)}^{1/(m+\epsilon )}]$. Third, a final featureless infinite-temperature state, reached after the system has fully heated. The inset shows a zoom-in on the orange-shaded region in the prethermal plateau.

Figure 3

Twisted TTS and action of linear transformation $R$ on $\mathit{\theta }$ space for a specific $m=2$ example. (a) The lattice $\mathcal{L}$ of points (black) is defined by the translation vectors ${\mathit{\tau }}_1=2\pi (1,0)$ and ${\mathit{\tau }}_2=2\pi (0,1)$. The original driving Hamiltonian $H(\mathit{\theta })$, Eq. (15), has periodicity on the standard torus (grey shaded area). The interaction Hamiltonian $H_{\mathrm{int}}(\mathit{\theta })$ generally has different periodicity. We show an example where the periodicity is on a sheared torus (red shaded area), given by the translation vectors ${\mathit{\tau }}_1^{\prime }=2\pi (2,1)$ and ${\mathit{\tau }}_2^{\prime }=2\pi (1,2)$, which defines a lattice ${\mathcal{L}}^{\prime }$ (red points). (b) Invertible linear transformation $R$ mapping $\mathit{\theta }\mapsto R\mathit{\theta }$ so that $({\mathit{\tau }}_1^{\prime },{\mathit{\tau }}_2^{\prime })\mapsto (R{\mathit{\tau }}_1^{\prime },R{\mathit{\tau }}_2^{\prime })=({\mathit{\tau }}_1,{\mathit{\tau }}_2)$. The Hamiltonian $H_{\mathrm{int}}^{\prime }(\mathit{\theta })≔H_{\mathrm{int}}(R^{-1}\mathit{\theta })$ has periodicity on the standard torus (red shaded area) and additionally has $g_{\mathit{\tau }}$ twisted TTSs $H_{\mathrm{int}}^{\prime }(\mathit{\theta }+R\mathit{\tau })=g_{\mathit{\tau }}{H}_{\mathrm{int}}^{\prime }(\mathit{\theta })g_{\mathit{\tau }}^{†}$, where $\mathit{\tau }\in 2\pi {\mathbb{Z}}^2$.

Figure 4

${\mathbb{Z}}_2$ DTQC of Sec. 4b: Driving function (a,c) and time-quasicrystal response (b,d). The spontaneous symmetry breaking of time-translation symmetries as seen in the power spectrum (e) of a local observable. (a) Driving function $f(\mathit{\theta })$ in extended $\mathit{\theta }$ space, with $\mathrm{\Delta }(\theta )$ in Eq. (31) replaced by ${\mathrm{\Delta }}_N(\theta )$ in Eq. (35) with $N=20$. The function is symmetric under translations $\mathit{\tau }\in \mathcal{L}=2\pi {\mathbb{Z}}^2$ (black points). Grey boxes represent the unit cell that is the standard torus. The blue arrow is the single time trajectory $\mathit{\theta }=\mathit{\omega }t$. (b) In contrast, the observable $s(\mathit{\theta })=\mathrm{Tr}[{\sigma }_i^z\rho (\mathit{\theta })]$ is symmetric under translations by ${\mathit{\tau }}^{\prime }=2\pi (1,\pm 1)\mathbb{Z}$, which defines a symmetry-breaking sublattice ${\mathcal{L}}_{\mathrm{SSB}}={\mathcal{L}}^{\prime }$ (red points). The grey boxes represent the unit cell, which is enlarged and rotated by 45° with respect to the driving Hamiltonian’s unit cell. (c) Driving profile $f(t)=f(\mathit{\omega }t)$ as a smooth function in the time domain. Times are measured from the relaxation time $t_r\sim J^{-1}$. (d) Discrete time quasicrystal response $s(t)=s(\mathit{\omega }t)$ in the time domain. (e) Power spectra $\mathcal{P}(\mathrm{\Omega })$ of both the driving function $f(t)$ (red) and the observable $s(t)$ (blue). The power spectrum of $f(t)$ exhibits peaks at the base frequencies $\mathrm{\Omega }=\mathit{n}·\mathit{\omega }$, with dominant peaks at $\mathrm{\Omega }={\omega }_1$ ($A_1$) and $\mathrm{\Omega }={\omega }_2$ ($A_2$). The power spectrum of $s(t)$ has peaks instead at frequencies shifted from the base frequencies by fractional values, reflecting the spontaneous breaking of the time-translation symmetries of the driving Hamiltonian. In particular, the dominant peaks are at $\mathrm{\Omega }=({\omega }_2-{\omega }_1)/2$ ($B_1$), $\mathrm{\Omega }={\omega }_2-{\omega }_1$ ($B_2$), and $\mathrm{\Omega }=({\omega }_1+{\omega }_2)/2$ ($B_3$). The inset shows the power spectra over a wider range of frequencies.

Figure 5

${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ DTQC with the model and observables detailed in Appendix pp4-s1. (a) Observable $s(\mathit{\theta })$ in $\mathit{\theta }$ space at leading order in inverse frequency, displaying the symmetry-breaking pattern lattice ${\mathcal{L}}_{\mathrm{SSB}}$ (red points) as well as the original lattice of translations $\mathcal{L}$ (black points). Grey boxes denote the symmetry-breaking unit cell. (b) Discrete time quasicrystal response $s(t)=s(\mathit{\omega }t)$. (c) Power spectra $\mathcal{P}(\mathrm{\Omega })$ of both the driving function (red) and the observable $s(t)$ (blue). Dominant peaks in the driving function correspond to $\mathrm{\Omega }={\omega }_1$ ($A_1$) and $\mathrm{\Omega }={\omega }_2$ ($A_2$). Dominant peaks of the observable $s(t)$ correspond to $\mathrm{\Omega }={\omega }_1/2$ ($B_1$), $\mathrm{\Omega }={\omega }_2/2$ ($B_2$), $\mathrm{\Omega }=3{\omega }_1/2$ ($B_3$). Going beyond leading order in inverse frequency, an observable’s power spectrum can exhibit peaks at the frequencies $\mathrm{\Omega }=\frac{1}{2}\mathit{n}·\mathit{\omega }$ for $\mathit{n}\in {\mathbb{Z}}^2$.

Figure 6

${\mathbb{Z}}_3\times {\mathbb{Z}}_2$ DTQC with the model and observable detailed in Appendix pp4-s2. (a) Observable $s(\mathit{\theta })$ in $\mathit{\theta }$ space at leading order in inverse frequency, displaying the symmetry-breaking pattern lattice ${\mathcal{L}}_{\mathrm{SSB}}$ (red points) as well as the original lattice of translations $\mathcal{L}$ (black points). Grey boxes denote the symmetry-breaking unit cell. (b) Discrete time quasicrystal response $s(t)=s(\mathit{\omega }t)$. (c) Power spectra $\mathcal{P}(\mathrm{\Omega })$ of both the driving function (red) and the observable $s(t)$ (blue). Dominant peaks in the driving function correspond to $\mathrm{\Omega }={\omega }_1$ ($A_1$) and $\mathrm{\Omega }={\omega }_2$ ($A_2$). Dominant peaks of the observable $s(t)$ correspond to $\mathrm{\Omega }={\omega }_1/3$ ($B_1$), $\mathrm{\Omega }=2{\omega }_1/3$ ($B_2$), $\mathrm{\Omega }=({\omega }_1+{\omega }_2)/2$ ($B_3$). Going beyond leading order in inverse frequency, an observable’s power spectrum can exhibit peaks at frequencies $\mathrm{\Omega }=n_1({\omega }_1/3)+n_2[-({\omega }_1/6)+({\omega }_2/2)]$ for $n_1$, $n_2\in \mathbb{Z}$.

Figure 7

(a) A symmetry flux for a symmetry $g$ consists of a line terminating at a point. (b) Terms of the Hamiltonian (shown as an oval) that cross the line get conjugated by the unitary action, restricted to act only one side of the line [denoted by $U_R(g)$].

Figure 8

Lattice of spins used in the construction of a minimal Hamiltonian displaying the quasiperiodic symmetry-protected topological phase introduced in Sec. 5b1. Spins are placed on the links ($A$, black), vertices ($B$, blue), and plaquettes ($C$, red) of a 2D square lattice. The ground state is a superposition of fluctuating domain walls of Ising orders of the $B$ spins (blue hatched region) and $C$ spins (red hatched region). In the topological phase, the intersection of these domain wall (yellow square) binds an Ising charge of $A$.