热态的量子与经典熵复杂性：随机矩阵和多体模型中的相干性、退相干以及遍历到局域化转变
Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models
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中文总结 AI 辅助
研究热平均是否保留本征态复杂性特征，通过比较三个无序模型遍历到局域化转变中熵复杂性，发现热平均抑制但未消除本征态复杂性特征，且熵复杂性能分离热态和本征态物理。
中文摘要 AI 辅助
热平均是否保留本征态复杂性的特征？我们研究了三个无序模型从遍历到局域化转变过程中的熵复杂性\(C = S_1 - S_2\)（香农熵减去二阶雷尼熵），这三个模型分别是罗森茨维格 - 波特（RP）系综、幂律带状随机矩阵（PLBRM）系综和随机场海森堡链。我们通过迹（\(C_{tr}\)）和对角（\(C_{diag}\)）复杂性，比较了波函数层面的量\(C_{eig}\)与指针基退相前后热（吉布斯）态的复杂性。答案大多是否定的：热平均强烈抑制但并未消除本征态复杂性特征。\(C_{eig}\)在RP中以及在结构独立的PLBRM模型中在匹配分形维数时出现明显的中间阶段最大值；高统计量扫描在热对角复杂性中解析出该峰值的微弱（约10%）但可重现的热影子。此特征局限于两个随机矩阵模型；在海森堡链中，本征态复杂性反而在多体局域化转变处达到峰值。第二个真正的热特征，即在遍历相内部的一个边缘，没有本征态对应物，并且在PLBRM中随系统大小而消退。这两个特征都由尺度不变的转变指标跟踪：迹和对角转变温度的对数比，在局域化时消失，以及相干相对熵。因此，熵复杂性清晰地分离了热态和本征态物理：一个尖锐的波函数层面特征，在不相关的随机矩阵构造中可重现，在热可观测量上仅留下微弱的、结构上不同的印记。
英文摘要
Does thermal averaging preserve signatures of eigenstate complexity? We study the entropic complexity C = S_1 - S_2 (Shannon minus second-order Renyi entropy) across the ergodic-to-localized crossover of three disordered models: the Rosenzweig-Porter (RP) ensemble, the power-law banded random matrix (PLBRM) ensemble, and the random-field Heisenberg chain. We compare a wavefunction-level quantity C_eig to the complexity of the thermal (Gibbs) state before and after pointer-basis dephasing, via the trace (C_tr) and diagonal (C_diag) complexities. The answer is mostly no: thermal averaging strongly suppresses, but does not eliminate, eigenstate-complexity signatures. C_eig develops a pronounced mid-phase maximum in RP and, at matching fractal dimension, in the structurally independent PLBRM model; a high-statistics scan resolves a weak (about 10%) but reproducible thermal shadow of this peak in the thermal diagonal complexity. This feature is confined to the two random-matrix models; in the Heisenberg chain the eigenstate complexity instead peaks at the many-body localization transition. A second, genuinely thermal feature, an edge just inside the ergodic phase, has no eigenstate counterpart and, in PLBRM, recedes with system size. Both features are tracked by scale-invariant crossover indicators: the log-ratio of the trace and diagonal crossover temperatures, which vanishes upon localization, and the relative entropy of coherence. Entropic complexity thus cleanly separates thermal-state and eigenstate physics: a sharp wavefunction-level feature, reproducible across unrelated random-matrix constructions, leaves only a faint, structurally distinct imprint on thermal observables.
