Spectrum–Complexity Duality in Many-Body Quantum Systems: How Isospectral Transformations, Operator Growth, Quantum Revivals, and Thermalization Structure Reveal a Candidate Framework for Dynamics Beyond the Energy Eigenvalue
A central assumption of equilibrium quantum statistical mechanics is that the energy spectrum of a Hamiltonian largely determines the dynamical character of the system it governs. Recent preprints across quant-ph and cond-mat.stat-mech collectively challenge this assumption from several independent angles, motivating what we term a *spectrum–complexity duality*: the observation that two systems may share an identical or structurally equivalent spectrum while exhibiting radically different operator dynamics, thermalization pathways, entanglement growth, and relaxation timescales. This is a heuristic reading across multiple findings, not a single unified derivation; we state this explicitly at the outset. The synthesis draws on six corpus sources: (i) an exact isospectral pair of fermion chains in which a nonlocal unitary maps a quadratic to a strongly interacting model, revealing that operator complexity — measured by Lanczos growth — is invisible to the spectrum [corpus:arxiv:2606.05294]; (ii) self-similar quantum revivals driven by quasiperiodic spiral kicks, where a dynamical attractor organizes recurrences at hierarchically nested times independently of eigenvalue statistics [corpus:arxiv:2606.05288]; (iii) a rigorous lower bound on entropy production per oscillation in autonomous stochastic oscillators, showing that a mode-uniformity factor — not eigenvalue location alone — controls the thermodynamic cost of coherence [corpus:arxiv:2606.05498]; (iv) adiabatic ramp dynamics across the ETH–MBL crossover in a disordered XXZ chain, where localized dynamical behavior survives under slow driving even as spectral statistics shift [corpus:arxiv:2606.04926]; (v) thermalization via Gaussian quantum cellular automata, where a quantum Riemann–Lebesgue lemma governs infinite-temperature relaxation independently of the spectral gap [corpus:arxiv:2606.05542]; and (vi) asymmetry dynamics in open quantum systems exhibiting a quantum Mpemba effect, where symmetry restoration rate is governed by mode structure rather than energy distance from equilibrium [corpus:arxiv:2606.07188]. Together these findings suggest that dynamical complexity, thermalization, and coherence are properties of the *eigenvector and mode structure* of a quantum system, not merely its eigenvalues. A concrete falsification path: constructing a family of Hamiltonians with identical spectra but systematically varied eigenvector localization, then measuring Krylov complexity growth and thermalization time as functions of eigenvector spread alone. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.04926, 2606.05288, 2606.05294, 2606.05498, 2606.05542, 2606.06107, 2606.06365, 2606.06455, 2606.06863, 2606.07121, 2606.07188, 2606.07206, 2606.07339, 2606.07377, 2606.07472
A central assumption of equilibrium quantum statistical mechanics is that the energy spectrum of a Hamiltonian largely determines the dynamical character of the system it governs. Recent preprints across quant-ph and cond-mat.stat-mech collectively challenge this assumption from several independent angles, motivating what we term a spectrum–complexity duality: the observation that two systems may share an identical or structurally equivalent spectrum while exhibiting radically different operator dynamics, thermalization pathways, entanglement growth, and relaxation timescales. This is a heuristic reading across multiple findings, not a single unified derivation; we state this explicitly at the outset. The synthesis draws on six corpus sources: (i) an exact isospectral pair of fermion chains in which a nonlocal unitary maps a quadratic to a strongly interacting model, revealing that operator complexity — measured by Lanczos growth — is invisible to the spectrum [corpus:arxiv:2606.05294]; (ii) self-similar quantum revivals driven by quasiperiodic spiral kicks, where a dynamical attractor organizes recurrences at hierarchically nested times independently of eigenvalue statistics [corpus:arxiv:2606.05288]; (iii) a rigorous lower bound on entropy production per oscillation in autonomous stochastic oscillators, showing that a mode-uniformity factor — not eigenvalue location alone — controls the thermodynamic cost of coherence [corpus:arxiv:2606.05498]; (iv) adiabatic ramp dynamics across the ETH–MBL crossover in a disordered XXZ chain, where localized dynamical behavior survives under slow driving even as spectral statistics shift [corpus:arxiv:2606.04926]; (v) thermalization via Gaussian quantum cellular automata, where a quantum Riemann–Lebesgue lemma governs infinite-temperature relaxation independently of the spectral gap [corpus:arxiv:2606.05542]; and (vi) asymmetry dynamics in open quantum systems exhibiting a quantum Mpemba effect, where symmetry restoration rate is governed by mode structure rather than energy distance from equilibrium [corpus:arxiv:2606.07188]. Together these findings suggest that dynamical complexity, thermalization, and coherence are properties of the eigenvector and mode structure of a quantum system, not merely its eigenvalues. A concrete falsification path: constructing a family of Hamiltonians with identical spectra but systematically varied eigenvector localization, then measuring Krylov complexity growth and thermalization time as functions of eigenvector spread alone. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.04926, 2606.05288, 2606.05294, 2606.05498, 2606.05542, 2606.06107, 2606.06365, 2606.06455, 2606.06863, 2606.07121, 2606.07188, 2606.07206, 2606.07339, 2606.07377, 2606.07472
We study a pair of exactly solvable, isospectral fermion chains, one strongly interacting and one quadratic, that nevertheless display remarkably different phase structures and operator dynamics. A nonlocal nonlinear unitary transformation maps one onto the other while preserving the entire many-body spectrum and converting local fermion operators into extended many-body strings. Thus, operators that evolve within a closed linear subspace in the quadratic model become interacting operators that generate increasingly higher-body terms and exhibit asymptotic Lanczos growth bn∝n. Despite their identical spectra, the two models realize distinct phases and sharply different notions of operator complexity. Our results demonstrate that free many-body spectra and interacting operator dynamics are fundamentally compatible.
Under the framework of unified time scale, boundary time geometry, Matrix Universe THE--MATRIX, and Quantum Cellular Automaton (QCA) Universe, we construct a structural unified theory for ``Quantum Chaos and Eigenstate Thermalization''. The Bohigas--Giannoni--Schmit (BGS) conjecture suggests that spectral statistics of quantum chaotic systems follow Random Matrix Theory (RMT), while the Eigenstate Thermalization Hypothesis (ETH) explains thermalization of isolated quantum systems from the perspective of matrix elements of local observables. Based on the unified time scale mother formula equation \kappa(\omega) =\varphi'(\omega){\pi} =\rho_{rel}(\omega) =1{2\pi}tr Q(\omega), equation this paper interprets quantum chaos and thermalization as: ergodicity of scattering phase flow in the Matrix Universe and operator mixing in the QCA Universe; and further regards spectral rigidity and RMT universality as statistical inevitable results of unified time scale density under maximum entropy principle. Specifically: 1. In the Matrix Universe, defining the ``scattering phase flow'' dynamical system on the energy shell, we prove that when the scattering matrix S(\omega) exhibits chaotic behavior (positive Lyapunov exponent or mixing), the spectral correlation function of the unified time scale \kappa(\omega) asymptotically approaches the RMT (GOE/GUE) kernel function equation R_2(\omega_1,\omega_2) \simeq 1-\Bigl(\sin[\pi\bar\kappa(\omega_1-\omega_2)]{\pi\bar\kappa(\omega_1-\omega_2)}\Bigr)^2, equation thereby providing a scattering--spectral unified proof for the BGS conjecture. 2. In the QCA Universe, considering the operator algebra \mathcal A_{qloc} of local cell observables and the Heisenberg evolution generated by the unitary update U, we prove that for generic chaotic QCA (satisfying algebraic mixing condition), local operators spread to global operators under time evolution, and their matrix elements in energy eigenstates satisfy the ETH ansatz equation \langle E_a|\mathcal O|E_b\rangle \simeq \mathcal O(E)\delta_{ab} +e^{-S(E)/2}f_{\mathcal O}(E,\Delta E)R_{ab}, equation where R_{ab} is a random variable, establishing the microscopic mechanism of thermalization in the discrete universe. 3. Investigating the growth of entanglement entropy S_{ent}(t) in QCA time evolution, we find that under unified time scale control, the entanglement growth rate is bounded by the Lieb--Robinson velocity and related to the ``boundary time area law''. For chaotic systems, linear growth S_{ent}(t)\propto t is consistent with the entropy production rate of the scattering matrix. 4. Under unified time scale constraints, writing the spectral form factor (SFF) K(t) as the Fourier transform of the two-point correlation of \kappa(\omega), we demonstrate the ``slope--dip--plateau'' structure of SFF in the Matrix--QCA universe, interpreting the linear ramp region as a direct manifestation of the spectral rigidity of the unified time scale. 5. The appendix systematically organizes: standard definitions of quantum chaos, RMT spectral statistics and SFF; operator spreading and out-of-time-order correlator (OTOC) description in QCA; rigorous formulation of ETH; and the maximum entropy derivation of unified time scale spectral correlations. Results indicate a purely structural unified picture: quantum chaos and eigenstate thermalization can be viewed as the statistical behavior of scattering phase flow in the Matrix Universe and operator mixing in the QCA Universe under the unified time scale, thereby answering the structural version of ``why quantum chaos and thermalization occur'' within the unified universe mother structure.
In the framework of Unified Time Scale, Matrix Universe THE\text{-MATRIX}, and Quantum Discrete Cellular Automaton Universe U_{qca}, we construct a systematic theory for Quantum Chaos and the Eigenstate Thermalization Hypothesis (ETH). The Unified Time Scale Mother Formula equation \kappa(\omega) =\varphi'(\omega)/\pi =\rho_{rel}(\omega) =(2\pi)^{-1}trQ(\omega) equation unifies the scattering semi-phase derivative, relative density of states, and Wigner--Smith group delay trace into a single time scale density, thereby providing a "Universe Time Mother Ruler" in the Matrix Universe U_{mat} independent of specific Hamiltonian choices. The Universe as a QCA object equation U_{qca}=(\Lambda,\mathcal H_{cell},\mathcal A_{qloc},\alpha,\omega_0) equation describes discrete time evolution via local unitary automorphisms \alpha on a countable lattice, and reconstructs relativistic quantum field theory and geometric structure in the continuum limit. Based on this, we propose an axiomatic system for "Postulated Chaos QCA" and prove the following main results: 1. On any finite region \Omega\subset\Lambda, the restricted finite-dimensional unitary operator U_\Omega satisfies Discrete Time ETH for its quasi-energy spectrum eigenstates with respect to any local operator O_X (support X\subset\Omega): for the vast majority of eigenstates \psi_n within an energy window, equation \langle\psi_n|O_X|\psi_n\rangle =\langle O_X\rangle_{micro}(\varepsilon_n) +\mathcal O(e^{-c|\Omega|}), equation and the squared average of off-diagonal matrix elements decays exponentially with volume, achieving eigenstate thermalization for local observables. 2. Under the axioms of "Postulated Chaos QCA" (finite propagation radius, translation symmetry, local gates generating high-order unitary designs, no extra extensive conserved quantities), the quasi-energy level statistics of the finite region U_\Omega converge after unfolding to the Wigner--Dyson distribution of the CUE random matrix class. Its spectral form factor exhibits a typical "ramp--plateau" structure, constituting a standard quantum chaos diagnosis at the QCA level. 3. Relating QCA--ETH to the Unified Time Scale: Under the unified time scale \tau, the "thermalization time scale" and the growth rate of local entropy density are jointly controlled by the energy shell average of the scale density \kappa(\omega) and the QCA light cone structure. We prove a "Unified Time--ETH--Entropy Growth Theorem": for any family of finite-density local initial states, the entropy density increases monotonically with \tau and approaches the microcanonical entropy density in the long-time limit. 4. On the global QCA universe object, embedding the above local results into the causal network and unified time scale mother structure shows that the "thermal time arrow" and "macroscopic irreversibility" on the cosmic scale can be understood as structural invariants in the unified Matrix--QCA Universe, rather than additionally introduced dynamical postulates. The appendix provides: rigorous definitions and equivalent forms of discrete time ETH; exact mapping between local random circuits and QCA; proof details and constant estimates for the QCA--ETH Theorem and CUE level statistics; and a 1D Postulated Chaos QCA model with specific predictions for numerical verification.
Abstract Statistical mechanics provides a framework for describing the physics of large, complex many-body systems using only a few macroscopic parameters to determine the state of the system. For isolated quantum many-body systems, such a description is achieved via the eigenstate thermalization hypothesis (ETH), which links thermalization, ergodicity and quantum chaotic behavior. However, tendency towards thermalization is not observed at finite system sizes and evolution times in a robust many-body localization (MBL) regime found numerically and experimentally in the dynamics of interacting many-body systems at strong disorder. Although the phenomenology of the MBL regime is well-established, the central question remains unanswered: under what conditions does the MBL regime give rise to an MBL phase , in which the thermalization does not occur even in the asymptotic limit of infinite system size and evolution time? This review focuses on recent numerical investigations aiming to clarify the status of the MBL phase, and it establishes the critical open questions about the dynamics of disordered many-body systems. The last decades of research have brought an unprecedented new variety of tools and indicators to study the breakdown of ergodicity, ranging from spectral and wave function measures, matrix elements of observables, through quantities probing unitary quantum dynamics, to transport and quantum information measures. We give a comprehensive overview of these approaches and attempt to provide a unified understanding of their main features. We emphasize general trends towards ergodicity with increasing length and time scales, which exclude naive single-parameter scaling hypothesis, necessitate the use of more refined scaling procedures, and prevent unambiguous extrapolations of numerical results to the asymptotic limit. Providing a concise description of numerical methods for studying ETH and MBL, we explore various approaches to tackle the question of the MBL phase. Persistent finite size drifts towards ergodicity consistently emerge in quantities derived from eigenvalues and eigenvectors of disordered many-body systems. The drifts are related to continuous inching towards ergodicity and non-vanishing transport observed in the dynamics of many-body systems, even at strong disorder. These phenomena impede the understanding of microscopic processes at the ETH-MBL crossover. Nevertheless, the abrupt slowdown of dynamics with increasing disorder strength provides premises suggesting the proximity of the MBL phase. This review concludes that the questions about thermalization and its failure in disordered many-body systems remain a captivating area open for further explorations.
Open quantum systems — those coupled to external baths, reservoirs, or measurement apparatuses — sit at the intersection of thermodynamics, information theory, and non-equilibrium dynamics. A candidate structural pattern emerges from recent preprints across quant-ph, math-ph, and cond-mat.stat-mech: the thermodynamic behavior of open quantum systems is more richly constrained than secular (Lindblad) approximations suggest, and this additional structure manifests simultaneously in (i) the steady-state coherences and entropy production of nonsecular master equations, (ii) the noise signatures accessible near Lindbladian exceptional points, (iii) multi-time quantum correlations that survive thermalization and carry pre-collapse information, (iv) the engineering of bosonic mode symmetries that trade error-protection for logical controllability, and (v) the conserved-charge counting that demarcates quantum integrability from chaos. This synthesis is a *heuristic reading*, not a derivation: the five threads share vocabulary (coherence, entropy, symmetry, information) but do not yet share a unified formal structure. The unifying hypothesis is that *steady-state thermodynamic observables systematically undercount the information retained in open quantum dynamics*, and that the deficit is recoverable through higher-order temporal and spectral probes. Falsification path: if current-noise spectra near a Lindbladian exceptional point show no anomalous time-delay structure beyond what secular theory predicts, and if multi-time correlators in evaporating black-hole analogs reduce exactly to single-time thermal distributions, the hypothesis collapses. Sources span quant-ph and cond-mat.stat-mech preprints from June 2026; all are v1 preprints, not peer-reviewed. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.09790v1, 2606.09805v1, 2606.10141v1, 2606.10744v1, 2606.11010v1, 2606.11392v1, 2606.11991v1, 2606.12161v1, 2606.12173v1, 2606.12310v1, 2606.12313v1, 2606.13251v1, 2606.13377v1, 2606.13383v1, 2606.13404v1, 2606.13453v1, 2606.13504v1
A bstract We explore the relationship between complexity and duality in quantum systems, focusing on how local and non-local operators evolve under time evolution. We find that non-local operators, which are dual to local operators under specific mappings, exhibit behaviour that mimics the growth of their local counterparts, particularly when considering state complexity. For the open transverse Ising model this leads to a neat organisation of the operator dynamics on either side of the duality, both consistent with growth expected in a quadratic fermion model like the Kitaev chain. When examining periodic chains, however, the mapping of boundary terms provides access to multiple branches of highly complex operators. These give rise to much larger saturation values of complexity for parity-mixing operators and are in contrast to what one would expect for a quadratic Hamiltonian. Our results shed light on the intricate relationship between non-locality, complexity growth, and duality in quantum systems.