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arXiv:1712.06218 · Letters in Mathematical Physics

Fermionic behavior of ideal anyons

Douglas Lundholm, Robert Seiringer

Published May 11, 2018• 16 citations
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Abstract

We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter $$\alpha $$ . The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.

DOI
10.1007/s11005-018-1091-y

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We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter α\alpha α . The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.
We derive an approximate analytic formula for the ground-state energy of the charged anyon gas. Our approach is based on the harmonically confined two-dimensional (2D) Coulomb anyon gas and a regularization procedure for vanishing confinement. To take into account the fractional statistics and Coulomb interaction we introduce a function, which depends on both the statistics and density parameters ($\ensuremath{\nu}$ and rs{r}_{s}rs​, respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large rs{r}_{s}rs​ (the 2D Wigner crystal), and to the Hartree-Fock (HF) energy of the spin-polarized 2D electron gas, and the dense 2D Coulomb Bose gas at very small rs{r}_{s}rs​. The latter is calculated by use of the Bogoliubov approximation. Applied to the boson system $(\ensuremath{\nu}=0)$ our results are very close to recent results from Monte Carlo (MC) calculations. For spin-polarized electron systems $(\ensuremath{\nu}=1)$ our comparison leads to a critical judgment concerning the density range, to which the HF approximation and MC simulations apply. In dependence on $\ensuremath{\nu}$, our analytic formula yields ground-state energies, which monotonously increase from the bosonic to the fermionic side if rs>1{r}_{s}>1rs​>1. For ${r}_{s}\ensuremath{\leqslant}1$ it shows a nonmonotonous behavior indicating a breakdown of the assumed continuous transformation of bosons into fermions by variation of the parameter $\ensuremath{\nu}$.
The ground-state energy and wave function of many anyons in a harmonic oscillator potential is studied using perturbation theory from the fermionic and bosonic ends. As the number of anyons increases to infinity, the ground-state energy scales with the power 3/2. As the statistics parameter changes, there are repeated level crossings between states having different angular momenta. Hence the ground state is only piecewise continuous, and the number of pieces increases with the number of anyons. This is to be contrasted with the problem of interpolating statistics in one dimension where there are no level crossings in the ground state for any number of particles.
We calculate the e/Istatistical potentials between particles in an ideal anyon gas arising from the symmetry property of the wave functions. The statistical potential is known to be purely repulsive for fermions, and to be purely attractive for bosons. In contrast, the statistical potentials for anyons are found to behave differently; the statistical potential for quasibosons is not purely attractive, displaying a minimum at finite interparticle distance. That implies there exists a stable equilibrium interparticle distance between two quasibosons. The statistical potential for quasifermions behaves similarly to that for fermions. Our result supports that the probability amplitude of finding two anyons at the same space point is zero.