Relaxation Fluctuations of Correlation Functions: Spin and Random Matrix Models

TL;DR

This paper investigates how fluctuations in correlation functions can serve as indicators of quantum chaos, analyzing random matrix models and physical systems to distinguish different phases of quantum behavior.

Contribution

It introduces fluctuation averages and variances of correlation functions as novel diagnostic tools for quantum chaos, extending analysis to the GUE case and physical models like the mixed-field Ising model.

Findings

Correlation fluctuations deviate from RMT in physical systems.

Identified three phases in Rosenzweig-Porter models using fluctuation measures.

Provided an alternative method to characterize quantum chaos.

Abstract

Spectral statistics and correlations are the usual way to study the presence or absence of quantum chaos in quantum systems. We present our investigation on the study of the fluctuation average and variance of certain correlation functions as a diagnostic measure of quantum chaos and to possibly characterize quantum systems based on it. These quantities are related to eigenvector distribution and eigenvector correlation. Using the Random Matrix Theory certain analytical expressions of these quantities, for the Gaussian orthogonal ensemble case, were calculated before. So as a first step, we study these quantities for the Gaussian unitary ensemble case numerically, and deduce certain analytical results for the same. We then carry out our investigations in physical system, such as the mixed-field Ising model. For this model, we find that although the eigenvalue statistics follow the behaviour of corresponding random matrices, the fluctuation average and variance of these correlation functions deviate from the expected random matrix theory behaviour. We then turn our focus on the Rosenzweig-Porter model of the Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble types. By using the fluctuation average and variance of these correlations, we identify the three distinct phases of these models: the ergodic, the fractal, and the localized phases. We provide an alternative way to study and distinguish the three phases and firmly establish the use of these correlation fluctuations as an alternative way to characterize quantum chaos.