Eigenvalue crossing as a phase transition in relaxation dynamics

TL;DR

This paper reveals that eigenvalue crossings in relaxation matrices cause singularities in relaxation dynamics, analogous to phase transitions, demonstrated through minimal and thermodynamic limit models.

Contribution

It identifies eigenvalue crossings as a mechanism for phase transition-like behavior in relaxation dynamics, supported by minimal and large-scale models.

Findings

Eigenvalue crossing causes relaxation trajectory singularities.

Demonstrated in a 4-state system and the 1D Ising model.

Analogous to first-order phase transitions.

Abstract

When a system's parameter is abruptly changed, a relaxation towards the new equilibrium of the system follows. We show that a crossing between the second and third eigenvalues of the relaxation matrix results in a relaxation trajectory singularity, which is analogous to a first-order equilibrium phase transition. We demonstrate this in a minimal 4-state system and in the thermodynamic limit of the 1D Ising model.