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 $b_n\propto\sqrt n$. 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.
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.
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 bstract Quantum observables of generic many-body systems exhibit a universal pattern of growth in the Krylov space of operators. This pattern becomes particularly manifest in the Lanczos basis, where the evolution superoperator assumes the tridiagonal form. According to the universal operator growth hypothesis, the nonzero elements of the superoperator, known as Lanczos coefficients, grow asymptotically linearly. We introduce and explore broad families of Lanczos coefficients that are consistent with the universal operator growth and lead to the exactly solvable dynamics. Within these families, the subleading terms of asymptotic expansion of the Lanczos sequence can be controlled and fine-tuned to produce diverse dynamical patterns. For one of the families, the Krylov complexity is computed exactly.
A bstract We investigate minimal two-body Hamiltonians with random interactions that generate spectra resembling those of Gaussian random matrices, a phenomenon we term quadratic quantum chaos . Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Krylov complexity and the late-time decay of higher-order out-of-time-ordered correlators (OTOCs); the latter reveals the emergence of freeness in the sense of free probability. Moreover, the fractal dimension and Stabilizer Rényi entropy of a representative mid-spectrum eigenstate show finite-size deviations yet converge toward Haar-randomness as the system size increases. This convergence, constrained by local interactions, highlights the “weakly chaotic” character of these eigenstates. Owing to their simplicity and bosonic nature, these minimal models may constitute promising and resource-efficient candidates for probing quantum chaos and information scrambling on near-term quantum devices.
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.
We study the Lanczos coefficients in a quadratic model given by an impurity interacting with a multi-mode field of fermions, also known as resonant level model. We analytically derive closed expressions for the Lanczos coefficients of Majorana fermion operators of the impurity for different structures of the coupling to the hybridization band at zero temperature. While the model remains quadratic, we find that the growth of the Lanczos coefficients structurally depends strongly on the chosen coupling. Concretely, we find (i) approximately constant, (ii) exactly constant, (iii) square root-like as well as (iv) linear growth in the same model. We further argue that in fact through suitably chosen couplings, essentially arbitrary Lanczos coefficients can be obtained in this model. These altogether evince the inadequacy of the Lanczos coefficients as a reliable criterion for classifying the integrability or chaoticity of the systems. Eventually, in the wide-band limit, we find exponential decay of autocorrelation functions in all the settings (i)−(iv), which demonstrates the different structures of the Lanczos coefficients not being indicative of different physical behavior.
Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, nonintegrable many-body systems, the so-called Lanczos coefficients associated with an autocorrelation function grow asymptotically linear, with a logarithmic correction in one-dimensional systems. In contrast, the growth is expected to be slower in integrable or free models. In this paper, we numerically test this hypothesis for a variety of exemplary systems, including one-dimensional and two-dimensional Ising models as well as one-dimensional Heisenberg models. While we find the hypothesis to be practically fulfilled for all considered Ising models, the onset of the hypothesized universal behavior could not be observed in the attainable numerical data for the Heisenberg model. The proposed linear bound on operator growth associated with the hypothesis eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We derive and investigate a related geometric bound, and we find that while the bound itself is not sharply achieved for any considered model, the hypothesis is nonetheless fulfilled in most cases.