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 (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 (nu=0) our results are very close to recent results from Monte Carlo (MC) calculations. For spin-polarized electron systems (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 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 leq 1 it shows a nonmonotonous behavior indicating a breakdown of the assumed continuous transformation of bosons into fermions by variation of the parameter nu .Comment: 18 pages, 3 figures. We have added the new reference and Fig. 3 with its description in the text. The result of Fig. 3 is reflected also in the abstrac

a b s t r a c tRecently, Gomes et al. [1] have visualized the gap formation in nanoscale regions (NRs) above the critical temperature T c in the high-T c superconductor Bi 2 Sr 2 CaCu 2 O 8+d . It has been found that, as the temperature lowers, the NRs expand in the bulk superconducting state consisted of inhomogeneities. The fact that the size of the inhomogeneity [2] is close to the minimal size of the NR [1] leads to a conclusion that the superconducting phase is a result of these overlapped NRs. In the present paper we perform the charge and percolation regime analysis of NRs and show that at the first critical doping x c1 , when the superconductivity starts on, each NR carries the positive electric charge one in units of electron charge, thus we attribute the NR to a single hole boson, and the percolation lines connecting these bosons emerge. At the second critical doping x c2 , when the superconductivity disappears, our analysis demonstrates that the charge of each NR equals two. The origin of x c2 can be understood by introducing additional normal phase hole fermions in NRs, whose concentration appearing above x c1 increases smoothly with the doping and breaks the percolation lines of bosons at x c2 . The last one results in disappearing the bulk bosonic property of the pseudogap (PG) region, which explains the upper bound for existence of vortices in Nernst effect [3]. Since Gomes et al. [1] has demonstrated the absence of NRs at the PG boundary one can conclude that along this boundary, as well as in x c2 , all bosons disappear. As justification of appearance of single bosons, the bosonization of 2D fermions is rigorously proven using the concept of anyons. The linear density dependence of the energy gap between excited fermionic and bosonic ground states describes the Uemura relation for 2D superconductors.

The effect of a weak three-dimensional (3d) isotropic laser speckle disorder on various thermodynamic properties of a dilute Bose gas is considered at zero temperature. First, we summarize the derivation of the autocorrelation function of laser speckles in 1d and 2d following the seminal work of Goodman. The goal of this discussion is to show that a Gaussian approximation of this function, proposed in some recent papers, is inconsistent with the general background of laser speckle theory. In this context we also point out that the concept of a quasi-three dimensional speckle, which appears due to an extension of the autocorrelation function in the longitudinal direction of a transverse 2d speckle, is not applicable for the true 3d speckle, since it requires an additional space dimension. Then we propose a possible experimental realization for an isotropic 3d laser speckle potential and derive its corresponding autocorrelation function. Using a Fourier transform of that function, we calculate both condensate depletion and sound velocity of a Bose-Einstein condensate as disorder ensemble averages of such a weak laser speckle potential within a perturbative solution of the Gross-Pitaevskii equation. By doing so, we reproduce the expression of the normalfluid density obtained earlier within the treatment of Landau. This physically transparent derivation shows that condensate particles, which are scattered by disorder, form a gas of quasiparticles which is responsible for the normalfluid component.

We derive an analytic, albeit approximate, expression for the ground state energy of N Coulomb interacting anyons with fractional statistics ν, 0 ≤ |ν| ≤ 1, confined in a two-dimensional well (with characteristic frequency ω 0 ) and subjected to an external magnetic field (with cyclotron frequency ω c ). We apply a variational principle combined with a regularization procedure which consists of fitting a cut-off parameter to existing exact analytical results in the noninteracting case, and to numerical calculations for electrons in quantum dots in the interacting case. The resulting expression depends upon parameters of the system |ν|, N, ω 0 , r 0 , a B and ω c , where r 0 represents a characteristic unit length and a B the Bohr radius. Validity of the result is critically assessed by comparison with exact, approximate, and numerical results from the literature.