Anyon Crystallization by Statistics

Abstract

We revisit the long-standing problem of the anyon gas. We study the solvable near-bosonic limit of a large number of anyons $N$ at fixed $N/k$, where $1/k$ characterizes the statistical interaction. In this limit, the ground state is a crystal with order $N/k$ lattice sites. The statistical interactions, mediated by a long-range gauge field, are completely screened within the crystalline phase. The resulting state is a conventional crystal without superfluidity. We characterize this anyon crystal by determining its unit-cell size, ground-state energy, acoustic excitations, angular momentum crossings, and other properties. Our results indicate a quantum phase transition between the crystal at $k>k_c$ and a superfluid at $k<k_c$, with a critical value $k_c\simeq2\text{--}5$. These results imply that an anyon gas with only statistical interactions is not necessarily a superfluid, contrary to common lore. We also present a detailed analysis of the renormalization-group evolution of anyon--anyon interactions, identifying many new potential fixed points and elucidating finite-size effects.