Moment Formalisms applied to a solvable Model with a Quantum Phase Transition. I. Exponential Moment Methods

TL;DR

This paper applies exponential moment methods to analyze the quantum phase transition in the Ising chain in a transverse field, providing exact solutions and insights into the behavior of cumulants, generating functions, and energy gaps across phases.

Contribution

It introduces a moment formalism approach with exact solutions for generating functions and cumulants, revealing phase-dependent properties and convergence behaviors near the critical point.

Findings

Exact solutions for generating functions and cumulants at arbitrary couplings.

Identification of cuts in the complex plane related to phase behavior.

Convergence of the t-expansion up to the critical point in the disordered phase.

Abstract

We examine the Ising chain in a transverse field at zero temperature from the point of view of a family of moment formalisms based upon the cumulant generating function, where we find exact solutions for the generating functions and cumulants at arbitrary couplings and hence for both the ordered and disordered phases of the model. In a t-expansion analysis, the exact Horn-Weinstein function E(t) has cuts along an infinite set of curves in the complex Jt-plane which are confined to the left-hand half-plane Im Jt < -1/4 for the phase containing the trial state (disordered), but are not so for the other phase (ordered). For finite couplings the expansion has a finite radius of convergence. Asymptotic forms for this function exhibit a crossover at the critical point, giving the excited state gap in the ground state sector for the disordered phase, and the first excited state gap in the ordered phase. Convergence of the t-expansion with respect to truncation order is found in the disordered phase right up to the critical point, for both the ground state energy and the excited state gap. However convergence is found in only one of the Connected Moments Expansions (CMX), the CMX-LT, and the ground state energy shows convergence right to the critical point again, although to a limited accuracy.