Liouvillian skin effects in two-dimensional electron systems at finite temperatures

TL;DR

The paper demonstrates that a two-dimensional electron system with Rashba spin-orbit coupling and an in-plane magnetic field can host both $\\mathbb{Z}$ and $\\mathbb{Z}_2$ Liouvillian skin effects under GKSL dynamics, with dissipation driving toward Gibbs equilibrium. By mapping the density matrix to a doubled Hilbert space and applying a mean-field approximation, the authors obtain a quadratic Liouvillian whose topology is diagnosed via a $k_y$-resolved winding number and a $\\mathbb{Z}_2$ invariant under transposed time-reversal symmetry. They show that skin effects emerge below the SOC/Zeeman energy scale and are suppressed at higher temperatures, with the $\\mathbb{Z}$ skin producing boundary charge accumulation under quenches and exhibiting scale-free localization; the relaxation time becomes system-size independent in the thermodynamic limit due to the linear growth of the localization length with system size. The work highlights a concrete solid-state platform for non-Hermitian skin physics in electronic systems, revealing how temperature and boundary conditions modulate dynamical behavior and suggesting experimental routes to observe Liouvillian boundary localization and its dynamical consequences.

Abstract

Liouvillian skin effects, manifested as the localization of Liouvillian eigenstates around the boundary, are distinctive features of non-Hermitian systems and are particularly notable for their impact on system dynamics. Despite their significance, Liouvillian skin effects have not been sufficiently explored in electron systems. In this work, we demonstrate that a two-dimensional electron system on a substrate exhibits $\mathbb{Z}$ and $\mathbb{Z}_2$ Liouvillian skin effects due to the interplay among energy dissipations, spin-orbit coupling, and a transverse magnetic field. In addition, our analysis of the temperature dependence reveals that these Liouvillian skin effects become pronounced below the energy scale of band splitting induced by the spin-orbit coupling and the magnetic field. While our $\mathbb{Z}$ Liouvillian skin effect leads to charge accumulation under quench dynamics, its relaxation time is independent of the system size, in contrast to that of previously reported Liouvillian skin effects. This difference is attributed to the scale-free behavior of the localization length, which is analogous to non-Hermitian critical skin effects.

Figures (10)

• Figure 1: (a) Schematic illustration of the system: a two-dimensional electron system on a substrate under an in-plane magnetic field, with the energy exchange (dissipation) between the system and the substrate. (b) Energy eigenvalues of the Hamiltonian [Eq. \ref{['eq: Hamiltonian']}] at $k_y=\pi$. The red and blue curves represent $\varepsilon_{\vb*{k}1}$ and $\varepsilon_{\vb*{k}2}$, respectively. The band splitting is induced by the Rashba SOC. The data are obtained for $t_{\mathrm{h}} = 1$, $\mu = 1$, $\alpha = 0.5$, $H_x = H_y = 0$, and $k_y = \pi$.
• Figure 2: (a)-(d): Eigenvalues of Liouvillian under OBC or PBC in the $x$ direction. Panel (a) [(b)] displays eigenvalues for ($\alpha$,$H_y$)=(0,1) [($\alpha$,$H_y$)=(0.5, 0)] and $k_y=3\pi/5$, indicating the absence of the Liouvillian skin effect. Panel (c) [(d)] displays eigenvalues for ($\alpha$,$H_y$)=(0.5,1) and $k_y=3\pi/5$ [($\alpha$,$H_y$)=(0.5,$10^{-6}$) Hy0Note and $k_y=\pi$]. (e) [(f)]: The weight of the right eigenstates under OBC in the $x$ direction for the same parameter set as panel (c) [(d)]. Here, the weight is defined as $\abs{|\rho^{\mathrm{R}}_{n,j}\rangle\rangle} \equiv \sum_{\sigma = \uparrow,\downarrow;\, \tau = \mathrm{K,B}} \abs{|\rho^{\mathrm{R}}_{n,j\sigma\tau}\rangle\rangle}$. The extreme sensitivity of eigenvalues and the localization of eigenstates displayed in panels (c) and (e) [(d) and (f)] are attributed to the nontrivial value of the winding number $w=1$ [$\mathbb{Z}_2$ invariant $\nu=1$]. The data are obtained for $t_{\mathrm{h}}=1$, $\mu=1$,$H_x = 0$, $\gamma=1$, $\gamma_{\mathrm{d}}=0.4$, $T=0.5$, and $L=60$.
• Figure 3: Temperature dependence of the Liouvillian spectrum and localization of the eigenstates. (a) and (c): Eigenvalues of the Liouvillian under PBC for $T = 0.04$, $0.4$, and $1.2$ in (a), and for $T = 0.01$, $0.1$, and $0.3$ in (c). Panel (a) [(c)] is obtained for ($\alpha$, $H_y$, $k_y$) = (0.2, 0.4, $5\pi/6$) [(0.1, $10^{-6}$, $\pi$)] ParametersNote. (b) [(d)]: The weight of the right eigenstates under OBC in the $x$ direction obtained for the same parameters as those of panel (a) [(c)]. All data are obtained for $t_{\mathrm{h}} = 1$, $\mu = 1$, $H_x = 0$, $\gamma = 1$, $\gamma_{\mathrm{d}} = 0.4$, and $L = 60$.
• Figure 4: System-size dependence of the Liouvillian skin effect. (a): Eigenvalues of the Liouvillian under OBC for $L = 30$ (red), 60 (orange), and 120 (gray), and under PBC for $L=200$ (blue). (b): Average localization length $\bar{\xi}$ as a function of the system size $L$ in the $x$ direction. The quantity $\bar{\xi}$ is defined as the average of the localization lengths over all eigenstates. (c) and (d): Spatial profiles of the right eigenstates under OBC for $L = 100$ and $L = 200$, respectively. The temperature is set to $T = 0.2$ in panels (a), (c), and (d), and $T = 0.3$ in panel (b). Other parameters are fixed as $t_{\mathrm{h}} = 1$, $\mu = 1$, $\alpha = 0.7$, $H_x = 0$, $H_y = 1$, $\gamma = 1$, $\gamma_{\mathrm{d}} = 0.4$, $k_y = 3\pi/4$.
• Figure 5: Dynamical accumulation of electrons near the system boundary. (a): Time evolution of the deviation in electron number $\Delta n_j$ from its initial value, for $t=0$ (circle), $0.5$ (square), $1$ (triangle), $3$ (diamond), and $6$ (star), with $H_y = 0.1$. (b): Magnified version of the right-edge region in panel (a). (c): Electron accumulation near the left edge for opposite magnetic field $H_y = -0.1$. To ensure that the bulk particle density matches between the initial and steady states, the initial chemical potential was slightly adjusted to $\mu = 0.9998945$, while the time evolution was computed using $\mu = 1$. All data are obtained for $t_{\mathrm{h}} = 1$, $\alpha=0.5$, $H_x = 0$, $\gamma = 1$, $\gamma_{\mathrm{d}} = 1$, $k_y = 3\pi/4$, $T=0.5$, and $L = 50$.