Random matrices theory elucidates the nonequilibrium critical phenomena

The earlier times of the evolution of a magnetic system contain more information than we can imagine. Capturing correlation matrices built from different time evolutions of a simple testbed spin system, as the spin-1/2 and spin-1 Ising models, we analyzed the density of eigenvalues for different temperatures of the so called Wishart matrices. We observe a transition in the shape of the distribution that presents a gap of eigenvalues for temperatures lower than the critical temperature, or in its roundness, with a continuous migration to the Marchenko-Pastur law in the paramagnetic phase. We consider the analysis a promising method to be applied in other spin systems, with or without defined Hamiltonian, to characterize phase transitions. Our approach differs from the alternatives in literature since it uses the concept of magnetization matrix, not the spatial matrix of single spins.

Automated summary

This study analyzes the density of eigenvalues of so-called Wishart matrices at different temperatures and reveals a transition in the shape of the distribution below the critical temperature, characterized by a gap of eigenvalues or changes in roundness, consistent with the Marchenko-Pastur law in the paramagnetic phase. This analysis method shows promise in characterizing phase transitions in other spin systems.