Integrable Many-Body Quantum Floquet-Thouless Pumps

We construct an interacting integrable Floquet model featuring quasiparticle excitations with topologically nontrivial chiral dispersion. This model is a fully quantum generalization of an integrable classical cellular automaton. We write down and solve the Bethe equations for the generalized quantum model and show that these take on a particularly simple form that allows for an exact solution: essentially, the quasiparticles behave like interacting hard rods. The generalized thermodynamics and hydrodynamics of this model follow directly, providing an exact description of interacting chiral particles in the thermodynamic limit. Although the model is interacting, its unusually simple structure allows us to construct operators that spread with no butterfly effect; this construction does not seem possible in other interacting integrable systems. This model exemplifies a new class of exactly solvable, interacting quantum systems specific to the Floquet setting.

Figure 1

Chiral quasiparticles in the DFFA model. (a) Dispersion relations showing the (bare) single-particle Floquet quasienergies $ϵ(k)$ for both $+$ and $-$ quasiparticles, for $\lambda =0.3$ (solid lines) and $\lambda =0.65$ (dashed lines). Both bands are topological (chiral) as they wrap around the periodic quasienergy direction and can only exist in a periodically driven system. Note that, for $\lambda =0.65$, the $\pm$ particles can be left (right) moving for some range of momenta. (b) Soliton gas picture. The scattering events in the DFFA model factorize onto simple two-body processes, which semiclassically correspond to a displacement $\mathrm{\Delta }x=\pm 1$ after a collision, independent of the momenta of the quasiparticles.

Figure 2

Numerical results. Quasienergy level statistics as a function of $\lambda$, for several values of $N_{\pm }$ and $K$ at $L=9$. (a) The $r$ ratio shows good agreement with a Poisson distribution (dashed line) for all $\lambda >0$. (b) (Inset) The distribution of $r$ for $\lambda =1.0$ does not show level repulsion, consistent with integrability. Plot of the OTOC $C(t)$ for $L=14$ unit cells with $N_{+}=1$ and $N_{-}=2$, for (c) $\lambda =0$, corresponding to the FFA model, and (d) $\lambda =0.05$, where we see that the OTOC does not “fill in” behind the front except through the dispersion of the perturbed quasiparticle. All data obtained from exact diagonalization.