Columnar Order and Ashkin-Teller Criticality in Mixtures of Hard Squares and Dimers

We show that critical exponents of the transition to columnar order in a mixture of 2 x 1 dimers and 2 x 2 hard squares on the square lattice 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.