Floquet Supersymmetry

We show that time-reflection symmetry in periodically driven (Floquet) quantum systems enables an inherently nonequilibrium phenomenon structurally similar to quantum-mechanical supersymmetry. In particular, we find Floquet analogs of the Witten index that place lower bounds on the degeneracies of states with quasienergies 0 and $\pi$. Moreover, we show that in some cases time-reflection symmetry can also interchange fermions and bosons, leading to fermion-boson pairs with opposite quasienergy. We provide a simple class of disordered, interacting, and ergodic Floquet models with an exponentially large number of states at quasienergies 0 and $\pi$, which are robust as long as the time-reflection symmetry is preserved. Floquet supersymmetry manifests itself in the evolution of certain local observables as a period-doubling effect with dramatic finite-size scaling, providing a clear signature for experiments.

Figure 1

Statistics of quasienergy levels outside of the degenerate subspaces in the model (7), as measured by the parameter $r$ defined in Ref. [60]. All data points are averaged over disorder realizations and quasienergy (see main text). Gray dashed lines at $r=0.386$ and 0.527 indicate the expected values for the Poisson and Wigner-Dyson distributions, respectively.

Figure 2

A representative time series of the magnetization $M_X$ starting from an initial state with all spins polarized in the $X=+1$ direction, for a single disorder realization in the interacting version of the model (7) with $L=8$ and $g=1$ [see Eqs. (11) and (12)]. Period-two oscillations around the expected infinite-temperature value $⟨M_X⟩=0$ are clearly visible at late times. (Inset) Power spectrum $⟨I_X(\omega )⟩$ for the same $L$ and $g$, averaged over 20 000 disorder realizations. The dominant coherent structure in the power spectrum is the peak at $\omega =\pi$, which results from the period-two oscillations.

Figure 3

Magnitude of the peak in $⟨I_X(\omega )⟩$ at $\omega =\pi$ as a function of system size, with an exponential fit (gray, dashed line) and the estimate (13) (orange) plotted for reference. The model used to generate the data is that used in Fig. 2. Data are averaged over 40 000, 20 000, 10 000, and 5000 disorder realizations for $L=6$, 8, 10, and 12, respectively.