We present analytic formulas for the ground state energy of two-dimensional (2D) anyon gas in strong magnetic field (Landau level filling factor νL ≤ 1). The formulas are obtained by applying harmonic potential regularization for the vanishing confinement to harmonically confined Coulomb anyon gas. In a case of absence of the Coulomb interaction our analytic result provides an exact solution. It contains a contribution of the anyon gauge field characterized by the anyon parameters ν and νL. In a case of presence of the Coulomb interaction we introduce a function depending on the parameters ν, νL and the density parameter rs. The function is determined by fitting the Fano-Ortolani interpolation equation in the fractional quantum Hall effect regime for spin-polarized electrons and by consistence requirement with known results for the ground state energy of the 2D Coulomb Bose gas in strong magnetic field. We show that our formulas are valid not only for fermions (ν=1) but quite generally for anyons (0 ≤ ν ≤ 1).
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, respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large
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. 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. 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 apply the diagram-technique formalism beyond the Hartree-Fock approximation to a two-dimensional nearly ideal electron gas in a weak perpendicular magnetic field. The case of an almost completely filled upper Landau level [for its filling factor ${\ensuremath{\nu}}_{0}$ holds $(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\ll}1]$ with a quantum number ${N}_{0}\ensuremath{\gg}1$ is considered. We uncover two regimes of renormalization by electron-electron interactions. In the first regime, where ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\ll}1,$ these interactions lead to a splitting of the Landau levels. In the second regime, where ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\gg}1,$ apart of the splitting, a renormalization of the bare Zeeman splitting occurs. The intermediate case ${N}_{0}^{1/2}(1\ensuremath{-}{\ensuremath{\nu}}_{0})\ensuremath{\approx}1$ cannot be studied within our approach. The applicability of the Fermi-liquid description is investigated for both regimes.