Efficient evaluation of high-order moments and cumulants in tensor network states

TL;DR

This paper introduces an efficient numerical scheme for calculating higher-order moments and cumulants in tensor network states, aiding in phase transition analysis and critical point detection in spin systems.

Contribution

The paper presents a novel, computationally efficient method for extracting moments and cumulants from tensor network states, applicable to both finite and infinite systems, with demonstrated applications in phase transition studies.

Findings

Accurate estimation of critical points in 1D models using cumulants.

Method reduces computational cost for phase transition detection.

Promising results for extending to 2D systems.

Abstract

We present a numerical scheme for efficiently extracting the higher-order moments and cumulants of various operators on spin systems represented as tensor product states, for both finite and infinite systems, and present several applications for such quantities. For example, the second cumulant of the energy of a state, $\langle \Delta H^2 \rangle$, gives a straightforward method to check the convergence of numerical ground-state approximation algorithms. Additionally, we discuss the use of moments and cumulants in the study of phase transitions. Of particular interest is the application of our method to calculate the so-called Binder's cumulant, which we use to detect critical points and study the critical exponent of the correlation length with only small finite numerical calculations. We apply these methods to study the behavior of a family of one-dimensional models (the transverse Ising model, the spin-1 Ising model, and the spin-1 Ising model in a crystal field), as well as the two-dimensional Ising model on a square lattice. Our results show that in one dimension, cumulant-based methods can produce precise estimates of the critical points at a low computational cost, and show promise for two-dimensional systems as well.