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Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model.

Pedro H de Freitas Pimenta1, Daniel A Stariolo2

  • 1Departamento de Física, Universidade Federal Fluminense, Niterói 24210-346, RJ, Brazil.

Entropy (Basel, Switzerland)
|June 28, 2023
PubMed
Summary
This summary is machine-generated.

Finite-size effects in the Sherrington-Kirkpatrick spherical model reveal a slow relaxation regime. This regime, influenced by system size and perturbation strength, is governed by spike random matrix eigenvalues and their gaps.

Keywords:
Langevin dynamicsdisordered systemseigenvalue statisticsnon-equilibrium dynamicsspherical modelspike random matrices

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Random Matrix Theory

Background:

  • The Sherrington-Kirkpatrick (SK) spherical model is a fundamental model in statistical mechanics, often used to study spin glasses.
  • Understanding the impact of perturbations and finite-size effects is crucial for bridging theoretical models with experimental observations.

Purpose of the Study:

  • To numerically analyze the relaxational dynamics of the SK spherical model with an additive non-disordered perturbation for large, finite system sizes (N).
  • To investigate the influence of finite-size effects and perturbation strength on phase transitions and relaxation dynamics.
  • To characterize the statistical properties of spike random matrices and their eigenvalues in different regimes.

Main Methods:

  • Thorough numerical analysis of relaxational dynamics.
  • Investigation of finite-size effects for large N.
  • Characterization of spike random matrix eigenvalues and their gaps.
  • Computation of finite-size scaling for energy relaxation.

Main Results:

  • Finite-size effects introduce a distinct slow relaxation regime, dependent on system size and perturbation strength.
  • The perturbation drives a phase transition from spin glass to ferromagnetic phases in the thermodynamic limit at low temperatures.
  • The statistics of the two largest eigenvalues of spike random matrices and the gap between them are characterized across sub-critical, critical, and super-critical regimes.
  • Power laws governing long-time energy relaxation exhibit exponents dependent on perturbation strength, linked to gap statistics.

Conclusions:

  • Finite-size effects significantly alter the dynamics of the SK spherical model, leading to observable slow relaxation.
  • The study provides detailed characterization of eigenvalue statistics in spike random matrices, offering insights into critical phenomena.
  • Numerical results on gap statistics are presented, aiming to stimulate further analytical research in this area.