Random matrix theory and critical phenomena in quantum spin chains

TL;DR

This paper analyzes quantum spin chains solvable via free fermions, computes their critical exponents and correlators, and demonstrates the universality of these critical properties across different symmetry classes.

Contribution

It provides exact calculations of critical exponents and correlators for a broad class of quantum spin chains related to classical compact groups, establishing their universality.

Findings

Critical exponents s, ν, z computed for various models.

Ground state correlators exhibit quasi-long-range order.

Universality of critical exponents confirmed across symmetry classes.

Abstract

We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups $U(N)$, $O(N)$ and $Sp(2N)$. In particular we calculate critical exponents $s$, $\nu$ and $z$, corresponding to the energy gap, correlation length and dynamic exponent respectively. We also compute the ground state correlators $\left\langle \sigma^{x}_{i} \sigma^{x}_{i+n} \right\rangle_{g}$, $\left\langle \sigma^{y}_{i} \sigma^{y}_{i+n} \right\rangle_{g}$ and $\left\langle \prod^{n}_{i=1} \sigma^{z}_{i} \right\rangle_{g}$, all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.