Universal Chiral Quasisteady States in Periodically Driven Many-Body Systems

We investigate many-body dynamics in a one-dimensional interacting periodically driven system, based on a partially filled version of Thouless’s topologically quantized adiabatic pump. The corresponding single-particle Floquet bands are chiral, with the Floquet spectrum realizing nontrivial cycles around the quasienergy Brillouin zone. For generic filling, with either bosons or fermions, the system is gapless and is expected to rapidly absorb energy from the driving field. We identify parameter regimes where scattering between Floquet bands of opposite chirality is exponentially suppressed, opening a long time window in which the system prethermalizes to an infinite-temperature state restricted to a single Floquet band. In this quasi-steady state, the time-averaged current takes a universal value determined solely by the density of particles and the topological winding number of the populated Floquet band. This remarkable behavior may be readily studied experimentally in recently developed cold atom systems.

Popular Summary

Quantum effects are typically quite fragile and are easily erased when one tries to manipulate the system. When an ensemble of many interacting particles is subjected to a periodic driving force (such as laser light), for example, the system is expected to absorb energy and heat up rapidly. After a short time, any interesting quantum effects would be lost. Here, we show that the tendency of driven quantum systems to heat up can, in fact, bring about the emergence of a new class of universal quantum phenomena.

Using theoretical models and numerical simulations, we study an example of such a phenomenon in a one-dimensional system that exhibits a persistent motion of particles characterized by a precisely quantized current. Importantly, the quantized value of the current is completely insensitive to any microscopic details of the system, but rather depends only on the density of particles and topological properties of the driving protocol. Quick heating of the low-energy degrees of freedom erases the initial details of the system. The high-energy degrees of freedom, meanwhile, remain cold for a long time, allowing the global, topological properties of the driving force to shine through. This behavior persists for a long time and diminishes only when the high-energy degrees of freedom eventually heat up.

Our work serves as a prototype for a new class of phenomena that can arise in a variety of driven quantum systems. In particular, we predict that these phenomena can be observed in recently developed driven cold-atom systems.

Figure 1

Establishing the universal chiral quasi-steady state in a one-dimensional periodically driven system. (a) Tight binding model [see Eq. (1)]. The dimerized hopping amplitudes and on-site potentials are changed adiabatically as depicted by the magenta curve. The origin $\delta J=\delta V=0$ is a degeneracy point. (b) The Floquet spectrum for $\omega =0.2J_0$ and $\lambda =1$ [see text below Eq. (1)]. The right-moving (left-moving) Floquet band is colored green (yellow). (c) The period-averaged current vs time obtained from a numerical simulation of a system of length $L=16$ unit cells with $N=8$ particles, for two different driving frequencies (see legend). After a few driving periods, the system reaches a quasi-steady state featuring a current $\mathcal{J}\approx \rho /T$, where $\rho =N/L$ is the density of particles. (d) At low frequency ($\omega =0.13J_0$, red), the chiral quasi-steady state persists for thousands of periods without noticeable decay. Interband scattering leads to a decay of the current with a rate that falls exponentially with $1/\omega$ (visible here for $\omega =0.33J_0$, blue). In panels (c) and (d), $\lambda =0.66$, $U=2J_0$.

Figure 2

Two-particle scattering in a partially filled Thouless pump. (a) Intraband scattering rate ${\tilde{\mathrm{\Gamma }}}_{2\mathrm{P}}={\mathrm{\Gamma }}_{2\mathrm{P}}/(J_{0}L)$, evaluated for bosons within Fermi’s golden rule, Eq. (6), with the forward scattering contribution removed. Here, $k_1$ and $k_2$ are the momenta of two incident particles; we take $\lambda =0.66$, $\omega =0.13J_0$, and $U=0.67J_0$. (b) Same as above, for interband scattering in which one particle is scattered from the right- to the left-moving band. (c) Single-particle Floquet states. On the left, the shaded region spans the energies of the instantaneous bands as a function of time. The significant Fourier components $|{\phi }_{1\mathrm{P}}^{(m)}{|}^2=|⟨{\phi }_{1\mathrm{P}}^{(m)}|{\phi }_{1\mathrm{P}}^{(m)}⟩{|}^2$ of a Floquet state $\alpha$ with momentum $k$ fall between the minimum and maximum values of the instantaneous energies $E_{\alpha ,k}(t)$. (d) Fourier components $|{\phi }_{2\mathrm{P}}^{(m)}{|}^2=|⟨{\phi }_{2\mathrm{P}}^{(m)}|{\phi }_{2\mathrm{P}}^{(m)}⟩{|}^2$ of incoming and outgoing two-particle Floquet states. For intraband scattering (bottom), the Fourier components overlap. For interband scattering (top), the overlap is strongly suppressed, leading to a suppression of the interband scattering rate.

Figure 3

Matrix elements and energy denominators for high-order scattering. (a) In the Born approximation, the $\mathrm{RR}\to \mathrm{RL}$ scattering matrix element is (super) exponentially suppressed by the small overlap of harmonic-space wave functions. (b) At an order $N_{\min }\sim \mathrm{\Delta }/(\delta m\omega )$ in perturbation theory, a nonsuppressed matrix element between initial and final states for the $\mathrm{RR}\to \mathrm{RL}$ process is constructed by moving through a sequence of off-shell intermediate states, shifted away from one another by $\delta \nu \lesssim \delta m$ harmonics. (c) Energy denominators for the virtual states in the process described in (b). Multiplying these energy denominators gives the expression in Eq. (10).

Figure 4

Universality of the quasi-steady state. Time traces of the current are shown for four choices of parameter values for the single-particle part of the Hamiltonian, Eq. (1), showcasing the topologically trivial ($w=0$) and nontrivial ($w=1$) regimes. After a transient of several driving periods, the current saturates to the value $\mathcal{J}\approx w\rho /T$, where $\rho =0.5$ is the density of particles, per unit cell. In all cases, the system is initialized by filling all states with crystal momentum values in the range $0\le k<\pi /a$, in a single Floquet band. The simulation was done with $\omega =0.23J_0$ and $U=2J_0$.

Figure 5

Interband excitation rate ${\mathrm{\Gamma }}_{\text{inter}}$ obtained from exact numerical evolution of a system with eight fermions on 32 sites ($L=16$ unit cells). Left panel: Log-log plot showing dependence of ${\mathrm{\Gamma }}_{\text{inter}}/(J_{0}L)$ on interaction strength $U$, for three fixed values of frequency $\omega$ and for the band-structure parameter $\lambda =0.56$. Trend lines are linear fits to the data, confirming a power-law dependence consistent with Eq. (11). Inset: Power-law exponent as a function of $\omega$, extracted from linear fits to the data. Right panel: Log-linear plot showing the driving-frequency dependence of ${\mathrm{\Gamma }}_{\text{inter}}/(J_{0}L)$, indicating an exponential dependence on $1/\omega$, for the same model parameters.

Figure 6

Effective Zener tunneling problem for the Floquet states in harmonic space. (a) The Floquet problem, Eq. (a1), is equivalent to a tight-binding problem in harmonic ($m$) space, with a linear potential term $m\omega$. The effective bands have a characteristic width $W$ and gap $\mathrm{\Delta }$. The gray regions indicate the “classically allowed” regions for each fixed value of “energy,” as determined by the sum of the band energies and the linear potential. The Floquet wave functions in the two bands (illustrated by the red and blue bars) are strongly localized in the classically allowed regions and decay rapidly beyond these regions. (b) Modulus squared of a representative Floquet wave function as a function of the Fourier index $m$. Sufficiently far from the maximum, the wave function decays as $\exp (-A|m-m_0{|}^{3/2})$, where $A$ and $m_0$ are constants.

Figure 7

(a) Single-particle Floquet spectrum in the vicinity of an avoided crossing between the two counterpropagating bands. In this calculation, $J_0=1.5$, $J_1=1$, $V_1=3$, $U=3$, and $\omega =0.3$. The single-particle resonance is at $k_{\mathrm{res}}=1.2856033\pi$. We use twisted boundary conditions to tune one of the allowed $k$ points in our $L=16$ system near the resonance; the $k$ points nearest to $k_{\mathrm{res}}$, corresponding to twist angles $\delta {\theta }_1$ and $\delta {\theta }_2$, are shown by blue and red dots, respectively. (b) Total occupation number of the left-moving Floquet band, excluding the $k$ point nearest to $k_{\mathrm{res}}$, as a function of time, for three different twist angles, $\delta {\theta }_{1,2,3}$. Two of the twist angles, $\delta {\theta }_1$ and $\delta {\theta }_2$, are such that there is a $k$ point close to $k_{\mathrm{res}}$ (blue and red curves, respectively), while for the third angle, $\delta {\theta }_3$, none of the $k$ points are near $k_{\mathrm{res}}$ (green curve). This is a simulation of $N=8$ particles in an $L=16$ unit cells system. The interaction strength is $U=3$.

Figure 8

Particle distribution in single-particle Floquet states in the right- and left-moving Floquet bands, $N_k^{(R)}$ and $N_k^{(L)}$, as a function of time. In both figures, we indicate only the maximal and minimal occupations, ${\max }_k{N}_k^{(\alpha )}$ and ${\min }_k{N}_k^{(\alpha )}$, by shading the area between them. In both plots, the yellow line indicates the average occupation $(1/L)\sum _k{N}_k^{(\alpha )}$ within the given band. As can be seen from the left figure, the distribution in the right-moving band relaxes to an approximately uniform distribution within a short time scale of a few driving periods. The right figure clearly shows the interband scattering rate from the right- to the left-moving Floquet band. The parameters for this figure were $\omega =0.27J_0$, $\lambda =0.56$, $U=1.67J_0$, $L=16$, and $\rho =0.5$.

Figure 9

Size dependence of the interband scattering rate normalized by the size of the system, $\mathrm{\Gamma }/(J_{0}L)$. As the number of unit cells, $L$, is changed, the density of particles is held fixed at $\rho =0.5$. The parameters used for this figure were $\omega =0.3$, $\lambda =0.56$, $U=2$.