零温无序系统 Kubo 公式的推导
Derivation of Kubo's formula for disordered systems at zero temperature
AI总结：
本文通过局部绝热定理控制随机含时 Hamilton 量动力学，证明二维无序系统 Hall 电导的 Kubo 线性响应公式，并以一维 Anderson 模型的全尺度本征向量杂化说明全局谱流失效。
摘要：
本工作论证了二维无序系统 Hall 电导的线性响应公式。该证明依赖于控制与随机含时 Hamilton 量相关的动力学。主要挑战在于：谱局域化和动力学局域化在扰动下本质上不稳定，而且此前在此背景下用于控制动力学的工具——精确谱流——并不存在。我们通过证明一个局部绝热定理来解决此问题：以高概率，与随机系统相关的局域本征态 $ψ$ 的物理演化，仍接近于瞬时 Hamilton 量在 $ψ$ 主体所支撑的区域 $R$ 上的限制所对应的谱流。允许 $R$ 至多随时间对数增长，可确保物理演化与该谱流的偏差很小。为了证实我们关于无序系统中全局谱流失效的论断，我们证明了一维 Anderson 模型在所有尺度上的本征向量杂化。
英文摘要：
This work justifies the linear response formula for the Hall conductance of a two-dimensional disordered system. The proof rests on controlling the dynamics associated with a random time-dependent Hamiltonian. The principal challenge is related to the fact that spectral and dynamical localization are intrinsically unstable under perturbation, and the exact spectral flow - the tool used previously to control the dynamics in this context - does not exist. We resolve this problem by proving a local adiabatic theorem: With high probability, the physical evolution of a localized eigenstate $ψ$ associated with a random system remains close to the spectral flow for a restriction of the instantaneous Hamiltonian to a region $R$ where the bulk of $ψ$ is supported. Allowing $R$ to grow at most logarithmically in time ensures that the deviation of the physical evolution from this spectral flow is small. To substantiate our claim on the failure of the global spectral flow in disordered systems, we prove eigenvector hybridization in a one-dimensional Anderson model at all scales.
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