Liouvillian topology and non-reciprocal dynamics in open Floquet chains

TL;DR

The paper addresses how to extend Floquet non-Hermitian topology from NH Hamiltonians to fully quantum open systems described by Liouvillians. It introduces a microscopic driven 1D chain with Lindblad dynamics and defines a Liouvillian winding invariant, revealing a jump-induced topological phase and signatures of the Liouvillian skin effect in dynamics. The NH skin effect and non-reciprocal transport arise in both transient (PBC) and steady-state localization (OBC), governed by outlier modes in the Liouvillian spectrum. The framework provides a theoretical route to unidirectional transport in dissipative Floquet dynamics and guides experimental realizations by including dephasing and tunable dissipation.

Abstract

Open quantum systems far from thermal equilibrium can exhibit remarkable physical phenomena including topological properties without a direct equilibrium counterpart. Along these lines, in periodically driven dissipative systems within the effective non-Hermitian (NH) Hamiltonian approximation spectral winding numbers have been linked to intriguing nonreciprocal transport properties. Here, going beyond an NH Hamiltonian description, we introduce and study a microscopic lattice model of a driven open quantum system described by a Markovian quantum master equation, which exhibits the mentioned spectral winding within a NH approximation. By encompassing quantum jump processes in the topological analysis, we uncover a distinct \emph{jump-induced} topological phase, which qualitatively corresponds to the richer non-reciprocal transport properties of the fully quantum model. In addition, we find that the NH skin effect, i.e.~the accumulation of a macroscopic number of eigenstates at one end of the system, is already visible in the transient dynamics even for systems with periodic boundary conditions. Our results exemplify the subtle correspondence between NH topological properties and physical manifestations of Liouvillian topological properties in open quantum systems, thus providing a theoretical framework towards understanding unidirectional transport in quantum dissipative Floquet dynamics.

Figures (9)

• Figure 1: Schematic illustration of the dissipative Floquet chain of spin-$\tfrac{1}{2}$ fermions over one Floquet period $T$. Each site $j$ (grey circle) hosts spin-up (red) and spin-down (blue) states. Dotted arrows denote spontaneous decay via $L_j^-(t) = \sqrt{\gamma(t)} \sigma_j^-$, while solid arrows indicate coherent dynamics generated by the Hamiltonians $H_1$ and $H_2$. This serves as the minimal model studied here, alongside more experimentally realistic variants discussed in the text.
• Figure 2: (a) Topological winding number as a function of the parameters $\beta$ and $\gamma$. The parameter sets $(\frac{\beta T}{3}, \gamma) = (\frac{\pi}{8}, 9)$ and $(\frac{3\pi}{8}, 9)$ are highlighted, with their corresponding eigenvalues shown in (d). (b,c) Mean transferred charge $\bar{C}(p)$ after $p=1$ (b) and $p=2$ (c) Floquet cycles, plotted versus $\beta$ and $\gamma$. (d) Spectral distribution of the non-unitary Floquet propagator $F$ in the punctured complex plane; the dashed red circle shows a choice of an imaginary gap $\Gamma$, with respect to which the red spectrum becomes topologically non-trivial. Throughout the manuscript, we set the parameter $\alpha = \frac{3\pi}{2T}$.
• Figure 3: Eigenstate localization of the non-unitary Floquet operator $F$ under open boundary conditions for a chain of length $L=10$. (a) Mean localization $\overline{\langle x \rangle} = \frac{1}{2L}\sum_{a=1}^{2L}\langle \varphi_a| x |\varphi_a\rangle$ of the eigenstates $|\varphi_a\rangle$ as a function of the system parameters $\beta$ and $\gamma$. The parameter sets $(\tfrac{\beta T}{3}, \gamma) = (\tfrac{\pi}{8}, 9)$ and $(\tfrac{3\pi}{8}, 9)$ are highlighted, with their corresponding right eigenstates shown in panels (b) and (c). (b,c) Spatial distributions $\langle \varphi_a| n_j |\varphi_a\rangle$ (black) together with spin-resolved components $\langle \varphi_a|n_{js}|\varphi_a\rangle$ for $s=\uparrow$ (red) and $s=\downarrow$ (blue). Note the presence of a single eigenstate localized at the left edge of the chain in (b).
• Figure 4: Spectral distributions in the punctured complex plane and their dependence on momentum $k$. (a) $\xi_\pm(k)$ of the non-unitary Floquet operator $F(k)$; the dashed red circle shows a choice of an imaginary gap $\Gamma_\mathrm{NH}$, with respect to which the spectrum becomes topologically non-trivial. (b) $\eta_{\pm +}(k, \tilde{q}) = \xi_\pm(\tilde{q}-k)\xi_+^*(\tilde{q})$ from the no-jump Floquet-Liouvillian propagator $\mathcal{F}^{\mathrm{NH}}(k, q) = F(q-k) \otimes F^*(q)$, shown versus $k$ for fixed $\tilde{q}=2\pi\frac{33}{L}$. (c) $\eta_{\pm\pm}(k, q)$ as functions of $k$. (d) $\eta_a(k)$ of the full Floquet–Liouvillian propagator $\mathcal{F}(k)$ for the open Floquet chain; the dashed blue circle shows a choice of an imaginary gap $\Gamma_\mathrm{L}$, with respect to which the spectrum becomes topologically non-trivial. For all plots the parameters chosen are $\alpha = \frac{3\pi}{2T}, \beta = \frac{3}{4}\frac{3\pi}{2T}, \gamma=9$ and $L=200$ unit cells.
• Figure 5: Spectral distributions of the interpolation between the no-jump Floquet-Liouvillian propagator $\mathcal{F}^\mathrm{NH}(k)$ and the fully open Floquet-Liouvillian propagator $\mathcal{F}(k)$. The parameter $\lambda$ is defined by $\mathcal{L}_\lambda(t)[\rho] = -i(H_\mathrm{NH}(t) \rho - \rho H_\mathrm{NH}^\dagger(t)) + \lambda \sum_\kappa \gamma_\kappa(t) L_\kappa \rho L_\kappa^\dagger$. For all plots the parameters chosen are $\alpha = \frac{3\pi}{2T}, \beta = \frac{1}{4}\frac{3\pi}{2T}, \gamma=9$ and $L=200$ unit cells. (d) A non-trivial topology has emerged from the inclusion of quantum jumps; the dashed red circle shows a choice of an imaginary gap $\Gamma$, with respect to which the spectrum becomes topologically non-trivial.