Physical review. B, Condensed matter

Ground-state energy of a dilute anyon gas

William R. Minor

Published May 15, 1993• 3 citations

Abstract

A calculation of the ground-state energy per particle of two-dimensional (2D) dilute quantum gases is given. The method of calculation is based on the free t-matrix approach and the phase shifts associated with 2D scattering. The continuous interpolation of energy from a Fermi to a Bose gas via the anyon gas is demonstrated both for particles undergoing hard-disk repulsions and for noninteracting gases subject to the statistical interaction alone.

DOI: 10.1103/physrevb.47.12716

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Physical review. B, Condensed matter

Ground-state energy of a dilute anyon gas

William R. Minor

Published May 15, 1993• 3 citations

Publisher

Abstract

DOI: 10.1103/physrevb.47.12716

Related papers

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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 ${r}_{s}$ , respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large ${r}_{s}$ (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 ${r}_{s}$ . 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 ${r}_{s}>1$ . 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 $\ensuremath{\nu}$.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 ${r}_{s}$, respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large ${r}_{s}$ (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 ${r}_{s}$. 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 ${r}_{s}>1$. 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 $\ensuremath{\nu}$.

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{r}_{s}

, respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large

{r}_{s}

(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

{r}_{s}

. 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

{r}_{s}>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 $\ensuremath{\nu}$.

N

fermions interacting with a short-range, positive potential of scattering length

a

. For spin-

1∕2

fermions, this is $E\ensuremath{\sim}{E}^{0}+({\ensuremath{\hbar}}^{2}∕2m)2\ensuremath{\pi}N\ensuremath{\varrho}a$, where

{E}^{0}

is the energy of the noninteracting system and $\ensuremath{\varrho}$ is the density. A similar formula holds in two dimensions (2D), with $\ensuremath{\varrho}a$ replaced by $\ensuremath{\varrho}∕\ensuremath{\mid}\mathrm{ln}(\ensuremath{\varrho}{a}^{2})\ensuremath{\mid}$. Obviously this 2D energy is not the expectation value of a density-independent pseudopotential.

Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density

\rho

ρ in 2 dimensions in the thermodynamic limit

\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}

e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as

\rho a^2 \rightarrow 0

ρ a 2 → 0 . Here

Y= |\log (\rho a^2)|^{-1}

Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.

x