# Universality of Heisenberg-Ising chain in external fields

## TL;DR

This paper demonstrates that the quantum phase transition in a Heisenberg-Ising chain under various external fields universally falls into the 1D transverse-field Ising model class, supported by analytical and numerical evidence, and relates to recent experiments.

## Contribution

It establishes the universality of 1DTFIM in a broader class of Heisenberg-Ising chains with external fields, extending the understanding of quantum phase transitions in real materials.

## Key findings

- Quantum phase transition exhibits 1DTFIM universality.
- Critical exponents and central charge match 1DTFIM predictions.
- Universal critical surface identified across parameter space.

## Abstract

Motivated by the recent surge of transverse-field experiments on quasi-one-dimensional antiferromagnets Sr(Ba)Co$_2$V$_2$O$_8$, we investigate the quantum phase transition in a Heisenberg-Ising chain under a combination of two in-plane inter-perpendicular transverse fields and a four-period longitudinal field, where the in-plane transverse field is either uniform or staggered. We show that the model can be unitary mapped to the one-dimensional transverse-field Ising model (1DTFIM) when the $x$ and $y$ components of the spin interaction and the four-period field are absent. When these two terms are present, following both analytical and numerical efforts, we demonstrate that the system undergoes a second-order quantum phase transition with increasing transverse fields, where the critical exponents as well as the central charge fall into the universality of 1DTFIM. Our results naturally identify the 1DTFIM universality of 1D quantum phase transitions observed in the existed experiments in Sr(Ba)Co$_2$V$_2$O$_8$ with transverse field applied along either [100] or [110] direction. Upon varying the tuning parameters a critical surface with 1DTFIM universality is determined and silhouetted to exhibit the general presence of the universality in a much wider scope of models than conventional understanding. Thus our results provide a broad guiding framework to facilitate the experimental realization of 1DTFIM universality in real materials.