Erratic Liouvillian Skin Localization and Subdiffusive Transport

Abstract

Non-Hermitian systems with globally reciprocal couplings -- such as the Hatano-Nelson model with stochastic imaginary gauge fields -- avoid the conventional non-Hermitian skin effect, displaying erratic bulk localization while retaining ballistic transport. An open question is whether similar behavior arises when non-reciprocity originates at the Liouvillian level rather than from an effective non-Hermitian Hamiltonian obtained via post-selection. Here we investigate this scenario in a lattice model with globally reciprocal Liouvillian dynamics and locally asymmetric incoherent hopping, a disordered setting in which Liouvillian-specific effects have remained largely unexplored. While the steady state again shows disorder-dependent, erratic localization without boundary accumulation, we find that global reciprocity in the Liouvillian does not protect transport. Instead, in the regime dominated by incoherent hopping, excitations spread via Sinai-type subdiffusion, dramatically slower than the ordinary diffusion found in symmetric stochastic lattices. Our results reveal a fundamental distinction between globally reciprocal Hamiltonian and Liouvillian dynamics: global reciprocity suppresses the skin effect in both cases, but only in Liouvillian dynamics can it coexist with ultra-slow, disorder-induced subdiffusive transport.

Figures (9)

• Figure 1: Schematic a tight-binding lattice made of $L$ sites with coherent ($J$) and site-dependent asymmetric incoherent($J^R_n, J^L_n$) hopping between adjacent sites. Open boundary conditions are assumed. The incoherent hopping rates are given by $J^L_n=Q \exp(h_n)$ and $J_n^R=Q \exp(-h_n)$, where $h_n$ is the asymmetry parameter. For $h_n=h \neq 0$, the model displays the LSE. When $h_n$ are independent stochastic variables that can assume only the two values $+h$ or $-h$ with the same probability, the system is globally reciprocal and the LSE disappears. Instead, erratic Liouvillian skin localization is observed, associated to sub-diffusive spreading of an initially-localized boson in the lattice bulk.
• Figure 2: Liouvllian skin effect in a tight-binding lattice with uniform asymmetric parameter $h_n=h$. (a) Spectrum (eigenvalues $\lambda_{\alpha}$) of the Liouvillian $\mathcal{L}$ in the single-particle sector under OBC for parameter values $J=0.2$, $Q=1$, $h=0.4$ and lattice size $L=31$. (b) Equilibrium density matrix (plot of $|\rho^e_{n,m}|$ on a pseudo-color map). (c) Distribution $I_{n,m}$ of averaged right eigenvectors of the Liouvillian $\mathcal{L}$. Note the localization of $\rho^e$ and $I_{n,m}$ on the upper left corner, a characteristic signature of the LSE.
• Figure 3: Same as Fig.2, but for $h=0$ (reciprocal and disorder-free model). Note the disappearance of the LSE.
• Figure 4: Same as Fig.2, but in the stochastic lattice where $h_n$ can take only the two values $h_n= \pm h$ with the same probability. Parameter values are as in Fig.2 ($J=0.2$, $Q=1$, $h=0.4$ and $L=31$). The three row in the figures refer to three realizations of the stochastic sequence $\{h_n \}$.
• Figure 5: Excitation spreading in a lattice with uniform asymmetry parameter $h_n=h$. (a,b) Numerically-computed temporal behavior of the excitation center of mass $n_{CM}(t)$ [panel (a)] and second-moment $d^2(t)$ [panel (b)]. (c) Excitation spreading dynamics (plot of $\rho_{n,n}(t)$ on a pseudo-color map). Parameter values are $J=0.2$, $Q=1$ and $h=1$. Lattice size $L=81$.