Inhomogeneous field theory inside the arctic circle

TL;DR

This paper analyzes a fermionic toy-model exhibiting the arctic circle phenomenon, revealing inhomogeneous Dirac field behavior in curved space and extending techniques to other lattice models, providing explicit correlation expressions.

Contribution

It introduces a novel inhomogeneous Dirac field framework for the arctic circle phenomenon and extends solution techniques to multiple lattice models.

Findings

Explicit correlations in the critical region are derived.

The inhomogeneous Dirac theory effectively describes phase separation.

Connections to height models and Gaussian free fields are established.

Abstract

Motivated by quantum quenches in spin chains, a one-dimensional toy-model of fermionic particles evolving in imaginary-time from a domain-wall initial state is solved. The main interest of this toy-model is that it exhibits the arctic circle phenomenon, namely a spatial phase separation between a critically fluctuating region and a frozen region. Large-scale correlations inside the critical region are expressed in terms of correlators in a (euclidean) two-dimensional massless Dirac field theory. It is observed that this theory is inhomogenous: the metric is position-dependent, so it is in fact a Dirac field theory in curved space. The technique used to solve the toy-model is then extended to deal with the transfer matrices of other models: dimers on the honeycomb and square lattice, and the six-vertex model at the free fermion point ($\Delta=0$). In all cases, explicit expressions are given for the long-range correlations in the critical region, as well as for the underlying Dirac action. Although the setup developed here is heavily based on fermionic observables, the results can be translated into the language of height configurations and of the gaussian free field, via bosonization. Correlations close to the phase boundary and the generic appearance of Airy processes in all these models are also briefly revisited in the appendix.