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arXiv 2501.03076cond-mat.stat-mechquant-ph
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量子点接触几何中的本征态热化假说

The Eigenstate Thermalization Hypothesis in a Quantum Point Contact Geometry

G. C. Levine, B. A. Friedman

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7 pages, 8 figures
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AI总结：

本文研究经少量量子点接触耦合的二维自由费米子系统，发现典型激发态下子系统纠缠熵次广延并在低能小系统极限呈现一维热力学标度，从而揭示受限几何对ETH热化性质的影响。

摘要：

众所周知，自由费米子系统中展现的长程量子纠缠足以使一个小子系统“热化”，因为由组合系统的典型激发本征态计算得到的该子系统约化密度矩阵近似为热态。值得注意的是，不存在任何相互作用的费米子因此被认为满足本征态热化假说（ETH）。当费米子子系统仅通过一个量子点接触（QPC）与量子储库（表现为另一个费米子系统）发生最小耦合时，我们探究了这一假说。在有限能量和基态情形下，考察了由一个或多个量子点连接的两个二维自由费米子系统的纠缠熵。结果表明，当组合系统处于典型激发态时，由少量QPC连接的子系统的纠缠熵是次广延的，其标度与子系统的线性尺寸（$L_A$）成正比。对于足够低的能量（$E$）和小子系统，数值证明纠缠熵$S_A \sim L_A E$，这正是人们对一维系统热力学所预期的结果。在此极限下，我们提出每个额外QPC所携带的熵可利用一维有限尺寸/温度共形标度进行量子化：$ΔS_A = α\log{(1/E)\sinh{(L_AE)}}$。少量QPC情形下的次广延熵应与经典遍历系统和量子混沌系统的预期形成对照；在后两类系统中，受限几何可能影响平衡弛豫时间，但不会影响平衡性质本身，例如广延熵和热容。

英文摘要：

It is known that the long-range quantum entanglement exhibited in free fermion systems is sufficient to "thermalize" a small subsystem in that the subsystem reduced density matrix computed from a typical excited eigenstate of the combined system is approximately thermal. Remarkably, fermions without any interactions are thus thought to satisfy the Eigenstate Thermalization Hypothesis (ETH). We explore this hypothesis when the fermion subsystem is only minimally coupled to a quantum reservoir (in the form of another fermion system) through a quantum point contact (QPC). The entanglement entropy of two 2-d free fermion systems connected by one or more quantum point contacts (QPC) is examined at finite energy and in the ground state. When the combined system is in a typical excited state, it is shown that the entanglement entropy of a subsystem connected by a small number of QPCs is sub-extensive, scaling as the linear size of the subsystem ($L_A$). For sufficiently low energies ($E$) and small subsystems, it is demonstrated numerically that the entanglement entropy $S_A \sim L_A E$, what one would expect for the thermodynamics of a one-dimensional system. In this limit, we suggest that the entropy carried by each additional QPC is quantized using the one-dimensional finite size/temperature conformal scaling: $ΔS_A = α\log{(1/E)\sinh{(L_AE)}}$. The sub-extensive entropy in the case of a small number of QPCs should be contrasted with the expectation for both classical, ergodic systems and quantum chaotic systems wherein a restricted geometry might affect the equilibrium relaxation times, but not the equilibrium properties themselves, such as extensive entropy and heat capacity.

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