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arXiv:2607.15798

Quantum and classical entropic complexity of the thermal state: coherence, decoherence, and the ergodic-to-localized crossover in random-matrix and many-body models

Imre Varga, Onur Aktan

Published Jul 17, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2607.15798

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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.
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The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-1/21/21/2 XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.
The Rosenzweig--Porter (RP) random matrix ensemble has emerged as a minimal model for the integrability-to-chaos crossover in quantum many-body systems. Its phase diagram features a region with fractal eigenstates, exhibiting intermediate spectral and localization properties between the fully localized and fully delocalized regimes. In this work, we explore several generalizations of the RP model and determine their level statistics at the scale of the Thouless energy ETE_TET​, which characterizes the crossover. Using tools from free probability theory and the replica method, we compute the full counting statistics in the limit of large system size, and show that it takes a simple, universal scaling form around ETE_TET​, shared across all variations of the model. We validate our analytical predictions using exact numerical diagonalization of large samples, and large-deviation algorithms that resolve the full counting statistics down to probabilities as low as 10−4010^{-40}10−40. We also contrast our predictions with measurements on the quantum random energy model, which is the simplest model displaying many-body localization.
The extreme-value statistics of the entanglement spectrum in disordered spin chains possessing a many-body localization transition is examined. It is expected that eigenstates in the metallic or ergodic phase, behave as random states and hence the eigenvalues of the reduced density matrix, commonly referred to as the entanglement spectrum, are expected to follow the eigenvalue statistics of a trace normalized Wishart ensemble. In particular, the density of eigenvalues is supposed to follow the universal Marchenko-Pastur distribution. We find deviations in the tails both for the disordered XXZ with total SzS_zSz​ conserved in the half-filled sector as well as in a model that breaks this conservation. A sensitive measure of deviations is provided by the largest eigenvalue, which in the case of the Wishart ensemble after appropriate shift and scaling follows the universal Tracy-Widom distribution. We show that for the models considered, in the metallic phase, the largest eigenvalue of the reduced density matrix of eigenvector, instead follows the generalized extreme-value statistics bordering on the Fisher-Tipett-Gumbel distribution indicating that the correlations between eigenvalues are much weaker compared to the Wishart ensemble. We show by means of distributions conditional on the high entropy and normalized participation ratio of eigenstates that the conditional entanglement spectrum still follows generalized extreme value distribution. In the deeply localized phase we find heavy tailed distributions and Lévy stable laws in an appropriately scaled function of the largest and second largest eigenvalues. The scaling is motivated by a recently developed perturbation theory of weakly coupled chaotic systems.