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

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 $ν= 1$ 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 $ν= 1/m$ Hall liquids. There, due to kinetic frustration arising from flux attachment, honeycomb lattice Hall crystals are preferred over triangular ones at intermediate interaction strength.

Figures (9)

• Figure 1: Schematic phase diagram of the composite boson mean-field theory. As a function of the extrapolated roton mass, $\Delta$, there is a first order transition at finite $\Delta_{c,1}>0$ from a Hall liquid to a Hall crystal, followed by a continuous transition to a Wigner crystal at $\Delta_{c,2}<0$. The blue line shows $\rho_Q$, the magnitude of density modulations at finite wavenumber, and the red curve shows $\sigma_H$, the Hall conductance.
• Figure 2: Functional form of the real space interaction. At long distances the interaction is the $1/|\bm{r}|$ Coulomb interaction, $V_C$. As short distances there is a discontinuity due to the phenomenological interaction $V_{R}$.
• Figure 3: The dispersion of density fluctuations, $\omega(|\bm{p}|)$ for $V_C = 50$, and $V_R = 8$ (top), $V_R = 8$ (middle), and $V_R\approx 22.05$ (bottom) where the roton gap closes. $\omega(Q) = \sqrt{\Delta}$ is indicated in the middle plot.
• Figure 4: Top) The density profile in Eq. \ref{['eq:crystalAnsatzDensity']} for a triangular crystal with $\rho_Q > 0$. Bottom) the current profile of the crystal, showing a circulating pattern of currents.
• Figure 5: The difference in energy density between the different crystal states and the liquid state, as a function of the roton mass $\Delta$. The triangular crystal always has the lowest energy.