Topological origin of quantized transport in non-Hermitian Floquet chains

We show that non-Hermiticity enables topological phases with unidirectional transport in one-dimensional Floquet chains. The topological signatures of these phases are noncontractible loops in the spectrum of the Floquet propagator that are separated by an imaginary gap. Such loops occur exclusively in non-Hermitian Floquet systems. We define the corresponding topological invariant as the winding number of the Floquet propagator relative to the imaginary gap. To relate topology to transport, we introduce the concept of regularized dynamics of non-Hermitian chains. We establish that, under the conditions of regularized dynamics, transport is quantized insofar as the charge transferred over one period equals the topological winding number. We illustrate these theoretical findings with the example of a Floquet chain that features a topological phase transition and acts as a charge pump in the nontrivial topological phase. We finally discuss whether these findings justify the notion that non-Hermitian Floquet chains support topological transport.

Figure 1

Conceptual overview of the topological scenarios for non-Hermitian chains. (a) The dispersion of a static chain (St) with directional nearest-neighbor hopping is an elliptical loop in the complex energy plane. (b) By splitting the hopping spatially, the chain (Sp) supports two loops separated by a real ($G$) and imaginary ($i\mathrm{\Gamma }$) line gap. (c) By splitting the hopping temporally, the Floquet chain (Fl) supports loops that traverse the complex quasienergy zone with its $\varepsilon \mapsto \varepsilon +2\pi$ periodicity.

Figure 2

[(a) and (b)] Spectrum of the (Floquet) propagator for the chains (Sp) and (Fl) from Fig. 1. Only the Floquet chain (Fl) can exhibit noncontractible loops with nonzero winding number. [(c)–(e)] Topological phase transition in the Floquet chain (Fl) with $J=\pi /3$. The transition at $\gamma ={\gamma }_c\approx 0.55$ (d) separates the trivial [$\gamma =0.2$, (c)] from the nontrivial [$\gamma =0.6$, (e)] phase.

Figure 3

[(a) and (b)] Transferred charge $\overline{C}$ in the chains (Sp) and (Fl) as a function of $\gamma$ (with $J=\pi /3,\mathrm{\Delta }=3i$). The curve is normalized with the factor $\mathrm{\Xi }={max}_{\xi }{\left|\xi \right|}^2$. Hermiticity (at $\gamma =0$) enforces $\overline{C}=0$. (c) Probing transport with wave-packet propagation in the chains (Sp) and (Fl) (with $\gamma =0.8,0.09,1.5$ from top to bottom, and $n_p=40$ periods).

Figure 4

[(a) and (b)] In the RD limit the spectrum of the propagator outside (inside) of the imaginary gap $i\mathrm{\Gamma }$ moves to the unit circle (origin). [(c) and (d)] Transferred charge $\overline{C}$, for various hopping $J=\pi /2,\pi /3,\pi /8$, as one approaches the RD limit in the static chain (Sp) (for $\mathrm{Im}\mathrm{\Delta },{\gamma }^{-1}\to \infty$) or the Floquet chain (Fl) (for $\gamma \to \infty$). In this limit, $\overline{C}=W\left(\mathrm{\Gamma }\right)$.

Figure 5

The Floquet chain (Fl) as a charge pump: Shown is a wave packet, centered initially at site $n=0$, after propagation over $n_p=20$ periods. (a) Propagation for $\gamma =-\infty ,-1,-0.25,0,0.25,1,\infty$ from top to bottom (with $J=\pi /3$). (b) Propagation in the RD limit $\gamma \to \infty$, for disorder $\delta =0,0.25,0.5,0.75$ from top to bottom (with $J=\pi /3$). Inset panel (c): Charge $\overline{C}$ transferred per period. (d) Propagation at $J=\pi /2, \gamma =0$, with $\delta$ as in panel (b).

Figure 6

Topological phase transition in the Floquet chain (Fl) with $J=\pi /3$, as already shown in Fig. 2. Panels (a)–(c) show the real part $\mathrm{Re}{\varepsilon }_{1,2}\left(k\right)$ of the quasienergies, which have been omitted in Fig. 2. Panels (d)–(f) show the spectrum of eigenvalues ${\xi }_{1,2}\left(k\right)$. The pink dots in panels (b) and (e) indicate the exceptional point at the transition.