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Physical review. B, Condensed matter

Anyons on a torus

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

Published Oct 1, 1990• 47 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.42.6110

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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.
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.
We study Gutzwiller-projected variational wavefunctions for charged, spinless holon excitations in chiral spin liquids. We find that these states describe anyons, with a statistical phase ΦsΦ_sΦs​ that is continuously adjustable between 000 and π/2π/2π/2, depending on a variational parameter. The statistical flux attached to each holon is localized to within a lattice constant. By diagonalizing the effective Hamiltonian for charge motion in our variational basis we obtain a two-holon pair state. This two-holon state has large overlap with a two-hole state whose relative wavefunction has ddd-wave symmetry and breaks time-reversal and parity.
The complete set of ground state wave functions for N anyons in an external magnetic field on the torus is found. The cases when the filling factor is less than or equal to one are considered. The single valued description of anyons is employed throughout by coupling bosons to a Chern-Simons field. At the end, the Chern-Simons interaction is removed by a singular gauge transformation as a result of which the wave functions become multi-component in agreement with other studies. It is shown that for statistics of the type θ=πq/p the exactly filled lowest Landau level has a q-fold degeneracy. Statistics of type π/p, which are relevant for the Laughlin theory of the FQHE, are singled out as those with non-degenerate filled ground states.
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.
An adiabatic approach put forward by Greiter and Wilczek interpolates between the integer quantum Hall effects of electrons and composite fermions by varying the statistical flux bound to electrons continuously from zero to an even integer number of flux quanta, such that the intermediate states represent anyons in an external magnetic field with the same ``effective'' integer filling factor. We consider such anyons on a torus, and we construct representative wave functions for their ground as well as excited states. These wave functions involve higher Landau levels in general, but can be explicitly projected into the lowest Landau level for many parameters. We calculate the variational energy gap between the first excited state and the ground state, and we find that it remains open as the statistical phase is varied. Finally, we obtain from these wave functions, both analytically and numerically, various topological quantities, such as the ground-state degeneracy, the Chern number, and the Hall viscosity.
We construct an SU(N)\mathrm{SU}(N)SU(N) toric code model describing the dynamics of SU(N)\mathrm{SU}(N)SU(N) electric and magnetic fluxes on a two-dimensional torus. We show that the model has N2{N}^{2}N2 topologically distinct ground states $|{\ensuremath{\psi}}_{0}{\ensuremath{\rangle}}_{(\mathsf{p},\mathsf{q})}$, which are the loop states characterized by ${Z}_{N}\ensuremath{\bigotimes}{Z}_{N}$ center charges $(\mathsf{p},\mathsf{q}=0,1,2,...,N\ensuremath{-}1)$. We explicitly construct them in terms of coherent superpositions of all possible spin network states on a torus with Wigner coefficients as their amplitudes. All excited quasiparticle states with SU(N)\mathrm{SU}(N)SU(N) electric charges and magnetic fluxes are constructed. We show that the braiding statistics of these SU(N)\mathrm{SU}(N)SU(N) electric or magnetic quasiparticles or non-Abelian anyons are encoded in the Wigner rotation matrices.