论文标题
通过安德森定位的边界隔离系统中liouvillian差距的指数尺寸缩放
Exponential size scaling of the Liouvillian gap in boundary-dissipated systems with Anderson localization
论文作者
论文摘要
我们对边界衰减的一维准静脉和疾病系统中Liouvillian差距的尺寸缩放的大小缩放进行了系统的研究。通过将边界隔离的操作员视为扰动，我们得出了liouvillian差距的分析表达，这清楚地表明了与基础汉密尔顿基本特征密度的最低边界密度成正比的liouvillian差距，因此给出了liouvill差异的范围尺寸的规模范围，从而给出了liouvill spractian scaping的范围。 While the Liouvillian gap displays a power-law size scaling $Δ_{g}\propto L^{- 3}$ in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation $Δ_{g}\propto e^{- κL}$ in the localized phase, where $κ$ takes the largest Lyapunov exponent基础哈密顿官的本地特征状态。通过仔细检查扩展的Aubry-André-Harper模型，我们从数值上确认Liouvillian Gap满足了指数缩放的关系，并且拟合指数$κ$与Lyapunov指数的分析结果相吻合。在其他一维准静脉疾病和随机疾病模型中，进一步验证了指数缩放关系。我们还研究了松弛动力学，并显示了liouvillian间隙的倒数，从而使渐近融合到稳态的时间尺度合理。
We carry out a systematical study of the size scaling of 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 $Δ_{g}\propto L^{- 3}$ in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation $Δ_{g}\propto e^{- κL}$ in the localized phase, where $κ$ takes the largest Lyapunov exponent of localized eigenstates of the underlying Hamiltonian. By scrutinizing the extended Aubry-André-Harper model, we numerically confirm that the Liouvillian gap fulfills the exponential scaling relation and the fitting exponent $κ$ coincides pretty well with the analytical result of 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 Liouvillian gap giving a reasonable timescale of asymptotic convergence to the steady state.
