Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Finite-size studies of particles obeying fractional statistics

Weikang Wu, Catherine Kallin, Andy Brass

Published Aug 1, 1990• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.42.2222

Related papers

Flux quantization for semions on a torus

Catherine Kallin

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.

Physical review. B, Condensed matter•Nov 1, 1993•8 citations

ANYONS, THE QUANTUM HALL EFFECT, AND TWO-DIMENSIONAL SUPERCONDUCTIVITY

Geoffrey Canright, S. M. Girvin

We have studied the ground-state (GS) properties of fractional-statistics particles (anyons) by exact numerical solution for small systems. We have seen in our results (depending on the choice of parameters for the Hamiltonian) the fractional-statistics analogs of a number of well-known ground states for conventional particles, including particularly the quantum Hall state (which we relate to the observed ‘hierarchy’ states), and a superconducting GS, the latter being formed (for half-statistics particles or semions) by pairing. We suggest possible experimental signatures for distinguishing semion from fermion superconductors. We also directly test the ‘mean-field theory’ of statistics against our exact results, and find it to treat GS properties well.

International Journal of Modern Physics B•Dec 1, 1989•23 citations

Finite-size studies of semion systems

X. C. Xie, Song He, S. Das Sarma

The ground and the excited states of a two-dimensional anyon system are studied via finite-size numerical calculations in a spherical geometry. It is shown that Laughlin's mean-field theory gives an accurate description of the ground-state properties of the finite, spherical system. We find, consistent with earlier analytic theories, that the collective mode of the system exhibits a linear dispersion in the long-wavelength limit, and predict a novel roton minimum at a finite value of the wave number q. We also discuss the consequences on the anyon spectrum of having a model repulsive interaction between the particles.

Physical Review Letters•Jul 30, 1990•23 citations

Ground state of the two-dimensional charged particles in a strong magnetic field and the fractional quantum Hall effect

D. Yoshioka

Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at $\frac{1}{3}$ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.

Physical review. B, Condensed matter•Jun 15, 1984•165 citations

Statistics and flux in two dimensions

Geoffrey Canright, S. M. Girvin, Andy Brass

We have investigated the interplay of fractional statistics and magnetic field for anyons on a lattice, by exact calculation of ground-state energies and wave functions for small systems. We find that the two types of ``flux'' (magnetic and statistical) are to some extent interchangeable in determining the ground-state properties, and thus that mean-field theories, which replace statistical phase shifts by magnetic fields, may be an appropriate method for treating fractional statistics.

Physical Review Letters•Nov 13, 1989•73 citations

Superfluidity of the lattice anyon gas and topological invariance

Eduardo Fradkin

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.

Physical review. B, Condensed matter•Jul 1, 1990•67 citations
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.
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.
We have studied the ground-state (GS) properties of fractional-statistics particles (anyons) by exact numerical solution for small systems. We have seen in our results (depending on the choice of parameters for the Hamiltonian) the fractional-statistics analogs of a number of well-known ground states for conventional particles, including particularly the quantum Hall state (which we relate to the observed ‘hierarchy’ states), and a superconducting GS, the latter being formed (for half-statistics particles or semions) by pairing. We suggest possible experimental signatures for distinguishing semion from fermion superconductors. We also directly test the ‘mean-field theory’ of statistics against our exact results, and find it to treat GS properties well.
The ground and the excited states of a two-dimensional anyon system are studied via finite-size numerical calculations in a spherical geometry. It is shown that Laughlin's mean-field theory gives an accurate description of the ground-state properties of the finite, spherical system. We find, consistent with earlier analytic theories, that the collective mode of the system exhibits a linear dispersion in the long-wavelength limit, and predict a novel roton minimum at a finite value of the wave number q. We also discuss the consequences on the anyon spectrum of having a model repulsive interaction between the particles.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 13\frac{1}{3}31​ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.
We have investigated the interplay of fractional statistics and magnetic field for anyons on a lattice, by exact calculation of ground-state energies and wave functions for small systems. We find that the two types of ``flux'' (magnetic and statistical) are to some extent interchangeable in determining the ground-state properties, and thus that mean-field theories, which replace statistical phase shifts by magnetic fields, may be an appropriate method for treating fractional statistics.
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.