A system of N interacting bosons or fermions in a two-dimensional harmonic potential (or, equivalently, magnetic field) whose states are projected onto the lowest Landau level is considered. Generic expressions are derived for matrix elements of any
interaction, in the basis of angular momentum eigenstates. For the fermion ground state (N=1 Laughlin state), this makes it possible to exactly calculate its energy all the way up to the mesoscopic regime N ~ 1000. It is also shown that for N = 3 and Coulomb interaction, several rational low-lying values of energy exist, for bosons and fermions alike.
At 6th order in perturbation theory, the random magnetic impurity problem at second order in impurity density narrows down to the evaluation of a single Feynman diagram with maximal impurity line crossing. This diagram can be rewritten as a sum of or
dinary integrals and nested double integrals of products of the modified Bessel functions $K_{ u}$ and $I_{ u}$, with $ u=0,1$. That sum, in turn, is shown to be a linear combination with rational coefficients of $(2^5-1)zeta(5)$, $int_0^{infty} u K_0(u)^6 du$ and $int_0^{infty} u^3 K_0(u)^6 du$. Unlike what happens at lower orders, these two integrals are not linear combinations with rational coefficients of Euler sums, even though they appear in combination with $zeta(5)$. On the other hand, any integral $int_0^{infty} u^{n+1} K_0(u)^p (uK_1(u))^q du$ with weight $p+q=6$ and an even $n$ is shown to be a linear combination with rational coefficients of the above two integrals and 1, a result that can be easily generalized to any weight $p+q=k$. A matrix recurrence relation in $n$ is built for such integrals. The initial conditions are such that the asymptotic behavior is determined by the smallest eigenvalue of the transition matrix.
The lowest-Landau-level anyon model becomes nonperiodic in the statistics parameter when the finite size of the attached flux tubes is taken into account. The finite-size effects cause the inverse proportional relation between the critical filling fa
ctor and the statistics parameter to be nonperiodically continued in the screening regime, where the fluxes are anti-parallel to the external magnetic field -- at critical filling, the external magnetic field is entirely screened by the mean magnetic field associated with the flux tubes. A clustering argument is proposed to select particular values of the statistics parameter. In this way, IQHE and FQHE fillings are obtained in terms of gapped nondegenerate LLL-anyonic wave functions. Jains series are reproduced without the need to populate higher Landau levels. New FQHE series are proposed, like, in particular, the particle-hole complementary series of the Laughlin one. For fast-rotating Bose-Einstein condensates, a corresponding clustering argument yields particular fractional filling series.
