# Cumulants and Scaling Functions of Infinite Matrix Product States

## TL;DR

This paper introduces a numerical scheme using cumulants, Binder cumulant, and scaling functions to accurately determine critical points and exponents in quantum phase transitions of infinite matrix product states, demonstrated on various models.

## Contribution

The paper presents a novel, efficient method combining cumulants and scaling functions to extract critical exponents from infinite matrix product states in quantum systems.

## Key findings

- Successfully applied to four 1D models showing accurate critical exponents.
- Derived a cumulant exponent relation for consistency checks.
- Extended the approach to a 2D model on an infinite cylinder.

## Abstract

The order parameter cumulants of infinite matrix product ground states are evaluated across a quantum phase transition. A scheme using the Binder cumulant, finite-entanglement scaling and scaling functions to obtain the critical point and exponents of the correlation length and cumulants is presented. Analogous to the scaling relations that relate the exponents of various thermodynamic quantities, a cumulant exponent relation is derived and used to check the consistency and relationship between the cumulant exponents. This scheme gives a numerically economical way of accurately obtaining the critical exponents. Examples of this scheme are shown for four one-dimensional models - the transverse field Ising model, the topological Kondo insulator, the S = 1 Heisenberg chain with single-ion anisotropy and the Bose-Hubbard model. A two-dimensional model is also exemplified in the square lattice transverse field Ising model on an infinite cylinder. These exemplary systems portray a variety of local and string order parameters as well as phase transition classes that can be studied with the scaling functions and infinite matrix product states.