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Communications in Theoretical Physics

Exact Ground State Solution for Anyons with spin-$\frac {1}{2}$

Shun-Qing Shen, Jianming Cai, Ruibao Tao

Published Jan 1, 1992• 0 citations
Publisher

Abstract

We solve the problem of anyons with spin- in an external field and obtain one of exact wavefunctions for its ground state, which is highly degenerated. Using Laughlin's plasma analogy, we find that there exists the algebraic off-diagonal long-range order for the ground state wavefunction in a certain external magnetic field.
DOI
10.1088/0253-6102/17/1/107

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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.
Anyon superconductivity arises from the condensation of mobile anyons rather than from a conventional Cooper instability, yet a systematic wavefunction description remains lacking. We develop a hierarchy-wavefunction construction for superconducting states derived from parent topological orders and identify their off-diagonal long-range order, condensate charge, chiral central charge, and residual topological order through the plasma analogy and topological field theories. We construct Abelian and non-Abelian examples descending from the semion state, a ν=2/3ν=2/3ν=2/3 hierarchy state, the ν=1/3ν=1/3ν=1/3 Laughlin state, ν=1ν=1ν=1 integer quantum Hall state, and Pfaffian state. Remarkably, for the semion case, the superconducting many-semion wavefunction is equivalent to a state of fermionized anyons filling two effective Landau levels, recovering Laughlin's original construction of semion superconductor. Finally, we show that hierarchy wavefunctions emerge naturally in the dilute, long-distance limit of the anyon-Hilbert-space formulation, which applies to ideal Chern bands and moiré bands of twisted bilayer MoTe2_22​. Our results establish a unified wavefunction-level framework linking anyon condensation, superconducting order, and topological field theory.
We show the existence of an exact ground state in certain parameter regimes of one-dimensional half-filled extended Hubbard model with site-off-diagonal interactions. In this ground state, the bond-located spin correlation exhibits a long-range order. In the case when the spin space is SU(2) symmetric, this ground state degenerates with higher spin states including a fully ferromagnetic state. We also discuss the relation between the exact bond-ordered ground state and the critical bond-spin-density-wave phase.