Bethe ansatz solution for crossover scaling functions of the asymmetric XXZ chain and the Karder--Parisi--Zhang-type growth model

TL;DR

This paper develops a perturbative approach to analyze the crossover from isotropic to anisotropic KPZ scaling in the asymmetric XXZ chain, providing universal crossover functions and applying them to growth models.

Contribution

It introduces a new perturbative method to compute finite size corrections and crossover scaling functions for the asymmetric XXZ chain near the stochastic line.

Findings

Universal crossover scaling functions for the KPZ to EW transition.

Explicit formulas for mass gaps as functions of growth parameters.

Higher order corrections to the crossover scaling functions.

Abstract

A perturbative method is developed to calculate the finite size corrections of the low lying energies of the asymmetric XXZ hamiltonian near the stochastic line. The crossover from isotropic to anisotropic, Kardar-Parisi-Zhang (KPZ) scaling of the mass gaps is determined in terms of universal crossover scaling functions. At the stochastic line, the asymmetric XXZ hamiltonian describes the time evolution of the single-step or body-centered solid-on-solid growth model in one dimension. The mass gaps of the growth model are found as a function of the growth rate and the substrate slope. Higher order corrections to the growth model mass gaps are also calculated to obtain the first terms of the KPZ to Edward-Wilkinson crossover scaling function in the large argument expansion in the zero slope sector.