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

Anyons on a torus: Braid group, Aharonov-Bohm period, and numerical study

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

Published May 1, 1991• 21 citations
Publisher

Abstract

We present a careful construction of anyons on a torus starting with the braid-group analysis. The rules of Wen, Dagotto, and Fradkin for putting anyons on a torus are reproduced with some minor improvements. The existence of noncontractible loops leads to braid-group representations characterized not only by anyon statistics \ensuremath{\theta} but also by the magnetic fluxes ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ threading through the holes of the torus. The three parameters are tangled with each other. We explore the symmetries of the torus to separate the effects of ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ from those of \ensuremath{\theta}. It is shown that the anyon system always has a smaller period \ensuremath{\theta}/\ensuremath{\pi} in ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ than the natural period 1. We perform several numerical calculations to investigate the spectral flow and consistency of the method and find interesting features in the spectral flow, which are relevant in understanding the fractional quantum Hall effect.

DOI
10.1103/physrevb.43.10761

Related papers

Braid group and anyons on a cylinder

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

In this paper we present a careful reexamination of anyons on a cylinder (or annulus), starting from the braid-group analysis. Proper attention is paid to the topological features arising from the existence of noncontractible loops. The rule for putting anyons on a square lattice has to be modified when the periodic boundary condition is imposed on one direction. In contrast to the annulus, one extra restriction is needed for the cylinder geometry to recover its symmetry between the two edges. We have performed some finite-system calculations. The consistency of our results has been checked by the agreement from two sets of seemingly different rules and also by that with free fermions. We have also calculated the spectral flow of the excited states with varying statistics or flux and have seen a lot of level crossings, which seem to be a general feature of anyon systems. The mean-field treatment is found to be good until level crossing occurs, and to be better if one starts with the hard-core boson rather than the fermion.

Physical review. B, Condensed matter•Feb 1, 1991•33 citations

Anyons on a torus: Braid group Aharonov-Bohm period and numerical study

Yasuhiro Hatsugai, Mahito Kohmoto, WU YONG-SHI

•May 1, 1991•0 citations

Anyons on a torus

Xiao-Gang Wen, Elbio R. Dagotto, Eduardo Fradkin

We show that the states of an anyon system on a torus are not completely determined by the positions of the anyons. There are q states for each fixed anyon configuration if the statistics of the anyons is given by \ensuremath{\theta}=p\ensuremath{\pi}/q. We explicitly construct the lattice Hamiltonian for the anyon system on the torus. The Hamiltonian is shown, both analytically and numerically, to respect the translation symmetries and the rotation symmetries. The flux of the anyon system is found to be quantized in units of 2\ensuremath{\pi}/q, without any shift. We also write down the effective Hamiltonian for the holons (with \ensuremath{\theta}=\ensuremath{\pi}/2) in the chiral spin state.

Physical review. B, Condensed matter•Oct 1, 1990•47 citations

BRAID GROUPS, ANYONS AND GAUGE INVARIANCE

Yong-Shi Wu

The structure of braid groups on topologically nontrivial surfaces is reviewed. The physical meaning of the braid relations and their implications on quantum mechanical properties of anyonic quasiparticles are discussed. These include not only the exotic statistics of anyons but also the Aharonov-Bohm effect for the anyons on a surface with holes. Several results on the novel properties of anyons or their states, which were previously derived by microscopic considerations, are reproduced by this seemingly kinematic and topology-dependent braid group analysis. It is suggested that in the thermodynamic limit, the global excitations in a system on a surface of nontrivial topology do not interfere with properties of local anyonic quasiparticles.

International Journal of Modern Physics B•Jun 1, 1991•7 citations

Braid Groups, Anyons and Gauge Invariance

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

The structure of braid groups on topologically nontrivial surfaces is reviewed. The physical meaning of the braid relations and their implications on quantum mechanical properties of anyonic quasiparticles are discussed. These include not only the restrictions on statistics of anyons but also the Aharonov-Bohm effect for the anyons on a surface with holes. Several results on the novel properties of anyons of their states, which were previously derived by microscopic considerations, are reproduced by this seemingly kinematic but topology dependent braid group analysis. It is suggested than in the thermodynamic limit, the global excitations in a system on a surface of nontrivial topology do not interfere with properties of local anyonic quasiparticles.

Progress of Theoretical Physics Supplement•Jan 1, 1992•8 citations

Hierarchical wave functions and fractional statistics in fractional quantum Hall effect on the torus

Dingping Li

One kind of hierarchical wave functions of Fractional Quantum Hall Effect (FQHE) on the torus are constructed. The multi-component nature of anyon wave functions and the degeneracy of FQHE on the torus are very clear reflected in this kind of wave functions. We also calculate the braid statistics of the quasiparticles in FQHE on the torus and show they fit to the picture of anyons interacting with magnetic field on the torus obtained from braid group analysis

•Jan 1, 1993•9 citations
We present a careful construction of anyons on a torus starting with the braid-group analysis. The rules of Wen, Dagotto, and Fradkin for putting anyons on a torus are reproduced with some minor improvements. The existence of noncontractible loops leads to braid-group representations characterized not only by anyon statistics \ensuremath{\theta} but also by the magnetic fluxes \ensuremathΦx{\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}\ensuremathΦx​ and \ensuremathΦy{\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}\ensuremathΦy​ threading through the holes of the torus. The three parameters are tangled with each other. We explore the symmetries of the torus to separate the effects of \ensuremathΦx{\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}\ensuremathΦx​ and \ensuremathΦy{\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}\ensuremathΦy​ from those of \ensuremath{\theta}. It is shown that the anyon system always has a smaller period \ensuremath{\theta}/\ensuremath{\pi} in \ensuremathΦx{\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}\ensuremathΦx​ and \ensuremathΦy{\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}\ensuremathΦy​ than the natural period 1. We perform several numerical calculations to investigate the spectral flow and consistency of the method and find interesting features in the spectral flow, which are relevant in understanding the fractional quantum Hall effect.
In this paper we present a careful reexamination of anyons on a cylinder (or annulus), starting from the braid-group analysis. Proper attention is paid to the topological features arising from the existence of noncontractible loops. The rule for putting anyons on a square lattice has to be modified when the periodic boundary condition is imposed on one direction. In contrast to the annulus, one extra restriction is needed for the cylinder geometry to recover its symmetry between the two edges. We have performed some finite-system calculations. The consistency of our results has been checked by the agreement from two sets of seemingly different rules and also by that with free fermions. We have also calculated the spectral flow of the excited states with varying statistics or flux and have seen a lot of level crossings, which seem to be a general feature of anyon systems. The mean-field treatment is found to be good until level crossing occurs, and to be better if one starts with the hard-core boson rather than the fermion.
We show that the states of an anyon system on a torus are not completely determined by the positions of the anyons. There are q states for each fixed anyon configuration if the statistics of the anyons is given by \ensuremath{\theta}=p\ensuremath{\pi}/q. We explicitly construct the lattice Hamiltonian for the anyon system on the torus. The Hamiltonian is shown, both analytically and numerically, to respect the translation symmetries and the rotation symmetries. The flux of the anyon system is found to be quantized in units of 2\ensuremath{\pi}/q, without any shift. We also write down the effective Hamiltonian for the holons (with \ensuremath{\theta}=\ensuremath{\pi}/2) in the chiral spin state.
The structure of braid groups on topologically nontrivial surfaces is reviewed. The physical meaning of the braid relations and their implications on quantum mechanical properties of anyonic quasiparticles are discussed. These include not only the exotic statistics of anyons but also the Aharonov-Bohm effect for the anyons on a surface with holes. Several results on the novel properties of anyons or their states, which were previously derived by microscopic considerations, are reproduced by this seemingly kinematic and topology-dependent braid group analysis. It is suggested that in the thermodynamic limit, the global excitations in a system on a surface of nontrivial topology do not interfere with properties of local anyonic quasiparticles.
The structure of braid groups on topologically nontrivial surfaces is reviewed. The physical meaning of the braid relations and their implications on quantum mechanical properties of anyonic quasiparticles are discussed. These include not only the restrictions on statistics of anyons but also the Aharonov-Bohm effect for the anyons on a surface with holes. Several results on the novel properties of anyons of their states, which were previously derived by microscopic considerations, are reproduced by this seemingly kinematic but topology dependent braid group analysis. It is suggested than in the thermodynamic limit, the global excitations in a system on a surface of nontrivial topology do not interfere with properties of local anyonic quasiparticles.