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Physical review. B, Condensed matter

Statistics and superfluidity

David L. Feder, Catherine Kallin

Published Oct 1, 1995• 0 citations
Publisher

Abstract

Using finite-size scaling, anyons on a triangular lattice at zero temperature with statistics \ensuremath{\nu}=1/3, 2/3, 1/4, and 3/4 are tested for flux quantization. It is found that all anyons, except those with \ensuremath{\nu}=1/3, form a superfluid for some range of densities. These results compare favorably with previous mean-field investigations of lattice anyons, although the \ensuremath{\nu}=1/4 case is not predicted to be a superfluid by the usual fermion-based mean-field theories.

DOI
10.1103/physrevb.52.9197

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Using finite-size scaling, anyons on a triangular lattice at zero temperature with statistics \ensuremath{\nu}=1/3, 2/3, 1/4, and 3/4 are tested for flux quantization. It is found that all anyons, except those with \ensuremath{\nu}=1/3, form a superfluid for some range of densities. These results compare favorably with previous mean-field investigations of lattice anyons, although the \ensuremath{\nu}=1/4 case is not predicted to be a superfluid by the usual fermion-based mean-field theories.
Small systems of particles which obey fractional statistics with statistics parameter \ensuremath{\alpha}=1/2, i.e., semions, on a lattice are studied by numerical diagonalization. The change in ground-state energy due to flux enclosure is calculated for various sizes of square lattices with periodic boundary conditions. Partial use is made of the translational symmetry properties of anyons on lattices with toroidal geometry. Using finite-size scaling, it is argued that the system exhibits true flux quantization, a signature of superfluidity, at rational fillings of the lattice less than 1/2. In agreement with previous work, no evidence of superfluidity is observed at lattice filling \ensuremath{\nu}=1/2. However, in contrast to previous numerical studies, the results for \ensuremath{\nu}>1/2 are noticeably different than those for \ensuremath{\nu}1/2. In particular, the signature of flux quantization generally is weaker for \ensuremath{\nu}>1/2 and, in fact, is absent for some densities. These results are discussed in the context of a mean-field treatment of anyons on a lattice.
I consider a gas of ``free'' anyons with statistical parameter \ensuremath{\delta}, with hard cores, on a two-dimensional square lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on the same lattice coupled to a Chern-Simons gauge theory with coupling \ensuremath{\theta}=1/2\ensuremath{\delta}. At the semiclassical level, the system is found to be equivalent to a gas of fermions, with the same density, in an average effective magnetic field \ensuremath{\rho}/\ensuremath{\theta}. I consider the case in which an integer number of the Landau bands of the saddle-point problem are completely filled. If \ensuremath{\delta}=\ensuremath{\pi}/m and the density \ensuremath{\rho}=r/q, with m, r, and q integers, the system is a superfluid, provided that q is larger than twice the largest common factor of m and r. If q is even and the system is half filled, the state may be either a superfluid or a quantum Hall phase. For all other values of \ensuremath{\rho} and \ensuremath{\delta}, compatible with integer filling of the Landau bands, the system is in a quantum Hall phase. The dynamical stability of the superfluid state is ensured by the topological invariance of the quantized Hall conductance of the fermion problem. I find a close analogy between anyon superconductivity and the Schwinger mechanism. The effective Lagrangian for the low-energy modes coupled to the electromagnetic field is derived. The energies of fermion and flux states are logarithmically divergent, but finite for the anyon state. The system has flux quantization, a zero-temperature Hall effect with a quantized Hall conductance, Meissner effect, charged vortices, screening with induced magnetic fields for static charges, and different masses for the longitudinal and transverse components of the electromagnetic field.
The ground-state properties of particles obeying fractional statistics, on lattices with spherical topology, are studied numerically. The possible existence of superfluidity in semion systems is examined by studies of pairing energies, flux quantization, and the effects of applied magnetic fields. Evidence of superfluidity is observed, in agreement with previous numerical studies on cylindrical topologies. Comparison of the results for these different topologies allows some conclusions to be made about finite-size and boundary effects. In particular, we argue that there is no constant shift in the flux-quantization values.
Recent years have shown an upsurge of interest in the physics of fractional statistics. In this same period, a number of authors have chosen to approach the rather knotty problem of many anyons using exact numerical calculations for small systems. In the following, I will attempt to review and summarize the works known to me which fall under this category. I will discuss both the methods used (including a detailed discussion of the problem of boundary conditions for anyons on multiply-connected spaces), and the results obtained. Finite-size studies have provided interesting and useful results in a number of areas, each of which is addressed below: tests of the mean-field theory of statistics; investigation of possible superfluid or superconducting properties in the ground state; and elucidation of possible macroscopic signatures of broken time-reversal [Formula: see text] symmetry for both one and many planes of anyons.
We study a tight-binding model for anyons on a square lattice using a self-consistent mean-field theory, where the anyon gauge field is approximated by its expectation value. For any anyon density per site ${\mathrm{\ensuremath{\rho}}}_{0}$=p/q and statistics \ensuremath{\nu}=1-1/n, the anyon system has a linear collective mode if and only if q is greater than twice the largest common factor of p and n. The collective mode dispersion, \ensuremath{\omega}(k), is calculated numerically for n=1 (hard-core bosons) and n=2 (semions) at certain densities.
I consider a gas of “free” anyons with statistical paremeter δ on a two dimensional lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on a lattice coupled to a Chern-Simons gauge theory with coupling [Formula: see text]. I show that if [Formula: see text] and the density [Formula: see text], with r and q integers, the system is a superfluid. If q is even and the system is half filled the state may be either a superfluid or a Quantum Hall System depending on the dynamics. Similar conclusions apply for other values of ρ and δ. The dynamical stability of the Fetter-Hanna-Laughlin goldstone mode is insured by the topological invariance of the quantized Hall conductance of the fermion problem. This leads to the conclusion that anyon gases are generally superfluids or quantum Hall systems.