Solvable relaxation in discrete unitary systems: Ruelle-Pollicott resonances and CMV matrices

• URL: http://arxiv.org/abs/2608.28575v1
• Date: Fri, 28 Aug 2026 17:52:00 GMT
• Title: Solvable relaxation in discrete unitary systems: Ruelle-Pollicott resonances and CMV matrices
• Abstract summary: Leading eigenvalues of the truncated propagator, known as Ruelle-Pollicott (RP) resonances, are an elegant way of addressing the dynamics of unitary many-body systems.<n>We study unitary propagators in their canonical form, known in the mathematical literature as the CMV matrices.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Leading eigenvalues of the truncated propagator, known as Ruelle-Pollicott (RP) resonances, are an elegant way of addressing the dynamics of unitary many-body systems. We study unitary propagators in their canonical form, known in the mathematical literature as the CMV matrices, and obtain a number of exact results for RP resonances and the associated norm-diverging eigenvectors. For the simplest CMV class describing a unilateral shift with an impurity, motivated by operator dynamics in dual-unitary circuits, we obtain closed-form results and in particular show that the three independent ways of obtaining RP resonances -- the truncated propagator, analytic continuation of the resolvent, and the rigged Hilbert space approach -- all give the same results. In more realistic CMV matrices, in which shift-like operator dynamics characteristic of chaotic systems is only asymptotic, we rely on the rich theory of orthogonal polynomials on the unit circle and identify two phases. In the first phase, relaxation occurs due to local operators effectively evolving into increasingly nonlocal ones with negligible backflow. Especially interesting is the second phase, which, surprisingly, exhibits faster relaxation because of contributions from the backflow of large operators. Additionally, in the second phase, RP resonances are not equal to the eigenvalues of the truncated propagator, instead, they are ``hidden'' within a ring of ill-conditioned eigenvalues.

Related papers

• Variational Continuation for Double Pendulum Periodic Orbits [28.921557702004176]
  We present a Hessian-based approach to numerically continue periodic orbits in dynamical systems.<n>Unlike previous work relying on hand-derived Jacobians, our method automates the process by leveraging automatic differentiation.<n>Our method is integrator-free, precisely initializes oscillations around unstable fixed points, and efficiently detects orbit family intersections and subharmonic bifurcations.
  arXiv  Detail & Related papers  (2026-09-04T16:42:51Z)
• Exact mobility rings in non-Hermitian quasiperiodically decorated Lieb lattices [5.437499298223565]
  The mobility ring (MR) is fundamental to understanding the Anderson transition in non-Hermitian (NH) disordered systems.<n>Here, we analytically derive the Lyapunov exponents and obtain exact expressions for the MRs.<n>We reveal distinct evolutionary behaviors of the MRs driven by the quasiperiodic potential strength.
  arXiv  Detail & Related papers  (2026-08-17T15:35:21Z)
• On the Diverse Dynamical Behaviors Arising in Deep Linear Transformers [34.73438474762916]
  We study the inference-time behavior of deep linear encoder-only transformers through the lens of interacting particle systems.<n>We show that the dynamics can be reformulated as a generalized Kuramoto-type model with pure second-harmonic coupling.
  arXiv  Detail & Related papers  (2026-07-20T23:43:26Z)
• Quantum simulation of the Liouville equation in classical mechanics with discontinuous potential via Schrödingerization [42.22857110018992]
  We develop quantum simulation algorithms for the Liouville equation of classical mechanics with discontinuous potential.<n>We combine Hamiltonian-preserving schemes by Jin and Wen in Commun. Math. Sci. 3(3), 285-315 (2005) with the Schrdingerization method, which embeds the resulting nonunitary semi-discrete dynamics into a unitary Schrdinger type system.
  arXiv  Detail & Related papers  (2026-06-13T02:49:05Z)
• Beyond Commutativity: Redesigning Trotter Decomposition via Local Symmetry [3.7159810638207813]
  We introduce a new decomposition principle that goes beyond commutativity, grouping Hamiltonian terms into local three-site clusters.<n>We show that three-site generators fall into at most four SU(2)-symmetry classes, each admitting an effective two-qubit SU(4) representation.<n>We demonstrate the proposed method on several physically relevant spin-lattice models, where the reduced cluster structure can even realise the second-order product formula.
  arXiv  Detail & Related papers  (2026-05-15T14:50:29Z)
• Low Rank Structure of the Reduced Transition Matrix [0.13999481573773073]
  influence-matrix formalism provides an alternative route to the classical simulation of quantum dynamics.<n>Recent work has shown that influence matrices carry strong temporal correlations even in maximally chaotic systems.<n>We show that the reduced transition matrix, a suitable combination of influence matrices that directly determines local expectation values, can nevertheless be efficiently approximated.
  arXiv  Detail & Related papers  (2026-05-12T19:13:34Z)
• Interacting chiral fermions on the lattice with matrix product operator norms [37.69303106863453]
  We develop a Hamiltonian formalism for simulating interacting chiral fermions on the lattice. The fermion doubling problem is circumvented by constructing a Fock space endowed with a semi-definite norm. We demonstrate that the scaling limit of the free model recovers the chiral fermion field.
  arXiv  Detail & Related papers  (2024-05-16T17:46:12Z)
• Three perspectives on entropy dynamics in a non-Hermitian two-state system [41.94295877935867]
  entropy dynamics as an indicator of physical behavior in an open two-state system with balanced gain and loss is presented. We distinguish the perspective taken in utilizing the conventional framework of Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint states and a third case based on an isospectral mapping.
  arXiv  Detail & Related papers  (2024-04-04T14:45:28Z)
• Asymptotic Gaussian Fluctuations of Eigenvectors in Spectral Clustering [24.558241146742205]
  It is shown that the signal $+$ noise structure of a general spike random matrix model is transferred to the eigenvectors of the corresponding Gram kernel matrix. This CLT-like result was the last missing piece to precisely predict the classification performance of spectral clustering.
  arXiv  Detail & Related papers  (2024-02-19T17:25:12Z)
• Unconditional Wigner-negative mechanical entanglement with linear-and-quadratic optomechanical interactions [62.997667081978825]
  We propose two schemes for generating Wigner-negative entangled states unconditionally in mechanical resonators. We show analytically that both schemes stabilize a Wigner-negative entangled state that combines the entanglement of a two-mode squeezed vacuum with a cubic nonlinearity. We then perform extensive numerical simulations to test the robustness of Wigner-negative entanglement attained by approximate CPE states stabilized in the presence of thermal decoherence.
  arXiv  Detail & Related papers  (2023-02-07T19:00:08Z)
• Localization via Quasi-Periodic Bulk-Bulk Correspondence [0.0]
  We report on a direct connection between quasi-periodic topology and the Almost Mathieu (Andre-Aubry) metal insulator transition (MIT) By constructing quasi-periodic transfer matrix equations from the limit of rational approximate projected Green's functions, we relate results from $textSL (2,mathbbR)$ co-cycle theory to consequences of rational band theory.
  arXiv  Detail & Related papers  (2021-10-29T18:16:28Z)
• Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions [0.0]
  The eigenstate thermalization hypothesis provides to date the most successful description of thermalization in isolated quantum systems. We study the distribution of matrix elements for a class of operators in dual-unitary quantum circuits.
  arXiv  Detail & Related papers  (2021-03-22T09:46:46Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.