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arXiv:2201.02085 · Physical review. B./Physical review. B

Exponential size scaling of the Liouvillian gap in boundary-dissipated systems with Anderson localization

Bozhen Zhou, Xueliang Wang, Shu Chen

Published Aug 9, 2022• 19 citations
arXivHTMLPDF

Abstract

We carry out a systematical study of the size scaling of the Liouvillian gap in boundary-dissipated one-dimensional quasiperiodic and disorder systems. By treating the boundary-dissipation operators as a perturbation, we derive an analytical expression of the Liouvillian gap, which indicates clearly the Liouvillian gap being proportional to the minimum of boundary densities of eigenstates of the underlying Hamiltonian, and thus give a theoretical explanation why the Liouvillian gap has different size scaling relation in the extended and localized phase. While the Liouvillian gap displays a power-law size scaling ${\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{L}^{\ensuremath{-}3}$ in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation ${\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\kappa}L}$ in the localized phase, where $\ensuremath{\kappa}$ takes the largest Lyapunov exponent of localized eigenstates of the underlying Hamiltonian. By scrutinizing the extended Aubry-Andr\'e-Harper model, we numerically confirm that the Liouvillian gap fulfills the exponential scaling relation and the fitting exponent $\ensuremath{\kappa}$ coincides pretty well with the analytical result of the Lyapunov exponent. The exponential scaling relation is further verified numerically in other one-dimensional quasiperiodic and random disorder models. We also study the relaxation dynamics and show the inverse of the Liouvillian gap giving a reasonable timescale of asymptotic convergence to the steady state.

DOI
10.1103/physrevb.106.064203

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We carry out a systematical study of the size scaling of the Liouvillian gap in boundary-dissipated one-dimensional quasiperiodic and disorder systems. By treating the boundary-dissipation operators as a perturbation, we derive an analytical expression of the Liouvillian gap, which indicates clearly the Liouvillian gap being proportional to the minimum of boundary densities of eigenstates of the underlying Hamiltonian, and thus give a theoretical explanation why the Liouvillian gap has different size scaling relation in the extended and localized phase. While the Liouvillian gap displays a power-law size scaling \ensuremathΔg\ensuremath∝L\ensuremath−3{\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{L}^{\ensuremath{-}3}\ensuremathΔg​\ensuremath∝L\ensuremath−3 in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation \ensuremathΔg\ensuremath∝e\ensuremath−\ensuremathκL{\mathrm{\ensuremath{\Delta}}}_{g}\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\kappa}L}\ensuremathΔg​\ensuremath∝e\ensuremath−\ensuremathκL in the localized phase, where \ensuremathκ\ensuremath{\kappa}\ensuremathκ takes the largest Lyapunov exponent of localized eigenstates of the underlying Hamiltonian. By scrutinizing the extended Aubry-Andr'e-Harper model, we numerically confirm that the Liouvillian gap fulfills the exponential scaling relation and the fitting exponent \ensuremathκ\ensuremath{\kappa}\ensuremathκ coincides pretty well with the analytical result of the Lyapunov exponent. The exponential scaling relation is further verified numerically in other one-dimensional quasiperiodic and random disorder models. We also study the relaxation dynamics and show the inverse of the Liouvillian gap giving a reasonable timescale of asymptotic convergence to the steady state.
We investigate the dynamics of the Su-Schrieffer-Heeger model with boundary dissipations described by Lindblad master equations and unravel distinct dynamical features in the topologically different phases of the underlying Hamiltonian. By examining the long-time damping dynamics, we uncover a dynamical duality phenomenon between the weak and strong dissipation region, which exists only in the topologically non-trivial phase, linked to the structure of the Liouvillian spectra,particularly the stripe closest to the steady state. When dissipation is confined to a single boundary, the dynamical duality phenomenon still exists. Under this condition, the Liouvillian gap fulfills an exponential size scaling relation in the topologically non-trivial phase and a power-law size scaling relation in the topologically trivial phase. Within the topologically non-trivial region, we identify the existence of boundary-localized dark states in the thermodynamical limit, which is responsible for the exponential size decay of Liouvillian gap.
Quasiperiodic systems host exotic transport regimes that are distinct from those found in periodic or disordered lattices. In this work, we study quantum transport in the Aubry-André-Harper lattice in a two-terminal setup coupled to zero-temperature reservoirs, where the conductance is evaluated via the nonequilibrium Green's function method. In the extended phase, we uncover a universal subdiffusive transport when the bath chemical potential aligns with the band edges. Specifically, the typical conductance displays a scaling of Gtyp∼L−2\mathcal{G}_{\text{typ}}\sim L^{-2}Gtyp​∼L−2 with system size LLL. We attribute this behavior to the emergence of an exceptional point (Jordan normal form) in the transfer matrix in the thermodynamic limit. In the localized phase, the conductance shows exponential decay governed by the Lyapunov exponent. Intriguingly, in the critical phase, we identify pronounced log-periodic oscillations of the conductance as a function of system size, arising from the discrete scale invariance inherent to the singular-continuous spectrum. We further extend our analysis to the generalized Aubry-André-Harper model and provide numerical evidence suggesting that the exact mobility edge resides within a finite spectral gap. This results in a counter-intuitive exponential suppression of conductance precisely at the mobility edge. Our work highlights the distinct transport behaviors in quasiperiodic systems and elucidates how they are rigorously dictated by the underlying local spectral structure.
It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap $\mathrm{\ensuremath{\Delta}}$, but also on the localization length $\ensuremath{\xi}$ of the eigenmodes. In particular, we show that the longest relaxation time $\ensuremath{\tau}$ that is maximized over initial states and local observables is given by $\ensuremath{\tau}\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}1}(1+L/\ensuremath{\xi})$ with LLL being the system size. This implies that the longest relaxation time can diverge for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ without gap closing.
Abstract We study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We establish that the global spectral features, the spectral gap, and the steady-state properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular quantity. Within each regime, we determine the scaling exponents with the dissipation strength and system size. We find that, for two or more dissipation channels, the spectral gap increases with the system size. The spectral distribution of the steady state is Poissonian at low dissipation strength and conforms to that of a random matrix once the dissipation is sufficiently strong. Our results can help to understand the long-time dynamics and steady-state properties of generic dissipative systems.
We study relaxation spectra of a quadratic spinless--fermion helical chain with an Aubry--Andre--type quasiperiodic potential and a single N--th neighbor (helical) hopping. Dissipation and pumping are introduced via local linear Lindblad jump operators and treated exactly using the third--quantization / Majorana covariance formalism. Focusing on periodic boundary conditions (to avoid edge artefacts) we compute the Liouvillian rapidities and their smallest nonzero real part (the rapidity gap) for several spatial dissipation patterns: uniform (all), single--site (one--site) and two--site (two--site) placement, plus pairwise gain/loss on helical partner sites. We show that uniform dissipation yields large, weakly lambda--dependent gaps, while sparse local dissipation produces gaps that shrink rapidly as the quasiperiodic potential lambda induces localization. Increasing t_N enhances relaxation by improving mode overlap with dissipative channels. Finite--size scaling, rapidity level statistics (Poisson vs Wigner--Dyson), and spatial profiles of slow modes provide a consistent picture linking Liouvillian spectral structure to transport and localization. Our results highlight Liouvillian rapidities as compact, experimentally relevant diagnostics of relaxation and sensitivity in engineered open quantum lattices.
We study localization and many-body localization transition in one-dimensional systems in the presence of deterministic quasiperiodic potential. We use single-particle excitations obtained through single-particle Green's function in real space to characterize the localization to delocalization transition. A single parameter scaling analysis of the ratio of the typical to average value of the local density of states (LDOS) of single-particle excitations shows that the critical exponent with which the correlation length $\ensuremath{\xi}$ diverges at the transition point $\ensuremath{\xi}\ensuremath{\sim}|h\ensuremath{-}{h}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, coming from the localized side, satisfies the inequality $\ensuremath{\nu}\ensuremath{\ge}1$ for the noninteracting Aubry-Andre (AA) model. For the interacting system with AA potential, we study single-particle excitations produced in highly excited many-body eigenstates across the MBL transition and found that the critical exponent obtained from finite-size scaling of the ratio of the typical to average value of the LDOS satisfies $\ensuremath{\nu}\ensuremath{\ge}1$ here as well. This analysis of the local density of states shows that the localization and MBL transition in systems with quasiperiodic potential belong to a different universality class than the localization and MBL transition in systems with random disorder where $\ensuremath{\nu}\ensuremath{\ge}2$. In complete contrast to this, finite-size scaling of the level spacing ratio is known to support the same universality class for MBL transitions in systems with quasiperiodic as well as random disorder potentials. For the interacting systems with quasiperiodic potentials, though finite-size scaling of the level spacing ratio shows a transition at hclsr{h}_{c}^{\mathrm{lsr}}hclsr​ which is close to the transition point obtained from LDOS within numerical precision, the critical exponent obtained from finite-size scaling of level spacing ratio is $\ensuremath{\nu}\ensuremath{\sim}0.54$ in close similarity to the MBL systems with random disorder.