Primary and Secondary Order Parameters in the Fully Frustrated Transverse Field Ising Model on the Square Lattice

TL;DR

This paper investigates the fully frustrated transverse-field Ising model on a square lattice, analyzing primary and secondary order parameters through simulations and field theory, revealing new insights into symmetry and critical behavior.

Contribution

It introduces a novel analysis of primary and secondary order parameters in the model, linking them to effective theories and predicting secondary critical exponents.

Findings

Both order parameters lead to the same phase diagram.

The spin order scales with conventional exponents.

The secondary dimer order's scaling relates to low-energy effective models.

Abstract

Using quantum Monte Carlo simulations and field-theory arguments, we study the fully frustrated (Villain) transverse-field Ising model on the square lattice. We consider a "primary" spin order parameter and a "secondary" dimer order parameter, which both lead to the same phase diagram but detect $Z_8$ and $Z_4$ symmetry, respectively. The spin order scales with conventional exponents, both in the finite temperature critical phase and at the $T = 0$ quantum critical point. The scaling of the dimer order requires more detailed investigations of the applicable low-energy theories; the height model at $T > 0$ and the $O(2)$ model in 2+1 dimensions at $T = 0$. Relating the order parameters to operators in these effective models, we predict the secondary critical exponents and confirm them numerically. The relationships between the primary and secondary order parameters have not been previously discussed in this context and provide insight more broadly for Ising models whose low-energy physics involves dimer degrees of freedom.