无序介质中自由费米子微观流的量子涨落和大偏差原理
Quantum Fluctuations and Large Deviation Principle for Microscopic Currents of Free Fermions in Disordered Media
摘要：
本文推广了[N.J.B.]中的大偏差结果。Aza，J.- B. Bru，W. de Siqueira Pedra，A. Ratsimanetrimanana，J. Math. Pures Appl. 125（2019）209]在无序介质中自由晶格费米子的原子尺度上的电导率理论。无序由（i）随机外部势（如著名的安德森模型）和（ii）具有随机复值振幅的最近邻跳跃项来建模。根据实验观察，通过大偏差形式主义，我们以前的论文表明，在这种情况下，微观电流密度在其（经典）宏观值附近的量子不确定性被抑制，相对于施加外部电场的晶格区域的体积呈指数快速。在这里，线性响应电流的量子涨落被证明存在于热力学极限，我们从数学上证明了它们与电流密度相关的大偏差原理的速率函数有关。我们还表明，在一般情况下，它们不会消失（在热力学极限）和宏观电流密度周围的量子不确定性消失指数快速与指数率成正比的电流从其宏观值和电流波动的平方偏差，相对于不断增长的空间（体积）尺度。
英文摘要：
We contribute an extension of large-deviation results obtained in [N.J.B. Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures Appl. 125 (2019) 209] on conductivity theory at atomic scale of free lattice fermions in disordered media. Disorder is modeled by (i) a random external potential, like in the celebrated Anderson model, and (ii) a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric current densities around their (classical) macroscopic value is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Here, the quantum fluctuations of linear response currents are shown to exist in the thermodynamic limit and we mathematically prove that they are related to the rate function of the large deviation principle associated with current densities. We also demonstrate that, in general, they do not vanish (in the thermodynamic limit) and the quantum uncertainty around the macroscopic current density disappears exponentially fast with an exponential rate proportional to the squared deviation of the current from its macroscopic value and the inverse current fluctuation, with respect to growing space (volume) scales.
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