Composite boson theory of Hall crystals and their transitions to Wigner crystals

TL;DR

This paper develops a composite boson theoretical framework to analyze phase transitions among Hall liquids, Hall crystals, and Wigner crystals in two-dimensional electron systems under magnetic fields, revealing both first-order and continuous transitions.

Contribution

It introduces a unified composite boson model mapping these states onto superconductor, Mott insulator, and supersolid phases, and describes the nature of their phase transitions including the role of roton softening.

Findings

First order transition from Hall liquid to Hall crystal with a soft roton.

Continuous transition from Hall crystal to Wigner crystal at lower roton mass.

Preference for honeycomb lattice Hall crystals in fractional quantum Hall regimes.

Abstract

We consider the crystallization of a two-dimensional electron system in a perpendicular magnetic field using composite boson theory. There are three possible states to consider: the Hall liquid, the Wigner crystal, and the Hall crystal (a state with both broken translation symmetry and a quantized Hall response). Within composite boson theory, these states map onto a superconductor, a Mott insulator, and a supersolid of composite bosons respectively. We show that when a $\nu = 1$ Hall liquid has a sufficiently soft roton, there is a first order transition to a triangular lattice Hall crystal. If we continue to decrease the roton mass, there is a continuous transition from the Hall crystal to a Wigner crystal. {When the Hall crystal exhibits the integer quantum Hall effect,} this transition {is} described by a free Dirac fermion and, at the critical point, the coupling to the phonons of the crystal is irrelevant, {in the {renormalization group} sense}. We extend this analysis to fractional $\nu = 1/m$ Hall liquids. There, due to kinetic frustration arising from flux attachment, honeycomb lattice Hall crystals are preferred over triangular ones at intermediate interaction strength.