# Critical behavior of hard squares in strong confinement

## TL;DR

This paper analytically studies the critical behavior of a confined quasi-one-dimensional system of hard squares, revealing a phase transition with Ising universality class at a specific channel width and infinite pressure.

## Contribution

It introduces a restricted orientation and position approximation to analytically describe the critical transition in confined hard squares, extending previous transfer operator methods.

## Key findings

- Identifies a critical point at H ≈ 1.8284 and infinite pressure.
- Shows the transition belongs to the 1D Ising universality class.
- Predicts similar behavior for unrestricted orientations and positions.

## Abstract

We examine the phase behavior of a quasi-one-dimensional system of hard squares with side-length $\sigma$, where the particles are confined between two parallel walls and only nearest neighbor interactions occur. As in our previous work (PRE, 94, 050603 (2016)), the transfer operator method is used, but here we impose a restricted orientation and position approximation to yield an analytic description of the physical properties. This allows us to study the parallel fluid-like to zigzag solid-like structural transition, where the compressibility and heat capacity peaks sharpen and get higher as $H \rightarrow H_c=2\sqrt{2}-1\approx 1.8284$ and $p \rightarrow p_c= \infty$. Here $H$ is the width of the channel measured in $\sigma$ units and $p$ is the pressure. We have found that this structural change becomes critical at the $(p_c,H_c)$ point. The obtained critical exponents belong to the universality class of the one-dimensional Ising model. We believe this behavior holds for the unrestricted orientational and positional case.