Columnar order and Ashkin-Teller criticality in mixtures of hard-squares and dimers

TL;DR

This paper demonstrates that the critical behavior of a mixture of dimers and hard-squares on a square lattice follows Ashkin-Teller criticality, with properties depending on mixture composition, settling a debate about the hard-square transition.

Contribution

It provides the first exact evidence that critical exponents in a polydisperse system depend on composition, confirming Ashkin-Teller theory predictions.

Findings

Critical exponents depend on mixture composition.

The transition in hard-square lattice gas is of Ashkin-Teller type.

First example of composition-dependent criticality in polydisperse systems.

Abstract

We show that critical exponents of the transition to columnar order in a {\em mixture} of $2 \times 1$ dimers and $2 \times 2$ hard-squares on the square lattice {\em depends on the composition of the mixture} in exactly the manner predicted by the theory of Ashkin-Teller criticality, including in the hard-square limit. This result settles the question regarding the nature of the transition in the hard-square lattice gas. It also provides the first example of a polydisperse system whose critical properties depend on composition. Our ideas also lead to some interesting predictions for a class of frustrated quantum magnets that exhibit columnar ordering of the bond-energies at low temperature.