Stefan Mashkevich

The third virial coefficient of anyons revisited

Abstract We use the method of solving the three-anyon problem developed in our earlier publication to evaluate numerically the third virial coefficient of free anyons. In order to improve precision, we explicitly correct for truncation effects. The present calculation is about three orders of magnitude more precise than the previous Monte Carlo calculation and indicates the presence of a term a sin4 πν with a very small coefficient a ⋍ −1.65 × 10 −5 .

MAGNETIC MOMENT and PERTURBATION THEORY with SINGULAR MAGNETIC FIELDS

The spectrum of a charged particle coupled to Aharonov-Bohm/anyon gauge fields displays a nonanalytic behavior in the coupling constant. Within perturbation theory, this gives rise to certain singularities which can be handled by adding a repulsive contact term to the Hamiltonian. We discuss the case of smeared flux tubes with an arbitrary profile and show that the contact term can be interpreted as the coupling of a magnetic moment spinlike degree of freedom to the magnetic field inside the flux tube. We also clarify the ansatz for the redefinition of the wave function.

The nature of the three-anyon wave functions

Abstract We show how to separate variables in the problem of three anyons with a harmonic oscillator potential. The anyonic symmetry conditions from cyclic permutations are separable in our coordinates. The condition from two-particle transpositions are not separable, but can be expressed as reflection symmetry conditions on the wave function and its normal derivative on the boundary of a circle. We solve this one-dimensional problem numerically by discretization.We solve, by separation of variables, the problem of three anyons with a harmonic oscillator potential. The anyonic symmetry conditions from cyclic permutations are separable in our coordinates. The conditions from two-particle transpositions are not separable, but can be expressed as reflection symmetry conditions on the wave function and its normal derivative on the boundary of a circle. Thus the problem becomes one-dimensional. We solve this problem numerically by discretization. N -point discretization with very small N is often a good first approximation, on the other hand convergence as N → ∞ is sometimes very slow.

On the thermodynamics of multispecies anyons

Abstract We address the problem of multispecies anyons, i.e. particles of different species whose wave functions are subject to anyonlike conditions. The cluster and virial coefficients are considered. Special attention is paid to the case of anyons in the lowest Landau level of a strong magnetic field, when it is possible (i) to prove microscopically the equation of state, in particular in terms of Aharonov-Bohm charge-flux composite systems, and (ii) to formulate the problem in terms of single-state statistical distributions.

Quantum mechanics of a particle with two magnetic impurities

Abstract A two-dimensional quantum mechanical system consisting of a particle coupled to two magnetic impurities of different strengths, in a harmonic potential, is considered. Topological boundary conditions at impurity locations imply that the wave functions are linear combinations of two-dimensional harmonics. A number of low-lying states are computed numerically, and the qualitative features of the spectrum are analyzed.

Finite-size anyons and perturbation theory.

We address the problem of finite-size anyons, i.e., composites of charges and finite radius magnetic flux tubes. Making perturbative calculations in this problem meets certain difficulties reminiscent of those in the problem of pointlike anyons. We show how to circumvent these difficulties for anyons of arbitrary spin. The case of spin textonehalf{} is special because it allows for a direct application of perturbation theory, while, for any other spin, a redefinition of the wave function is necessary. We apply the perturbative algorithm to the

Statistical mechanics and thermodynamics for multispecies exclusion statistics

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ANYON TRAJECTORIES AND THE SYSTEMATICS OF THE THREE-ANYON SPECTRUM

-body problem, derive the first-order equation of state, and discuss some examples.

The lowest Landau level anyon equation of state in the anti-screening regime

Abstract Statistical mechanics and thermodynamics for ideal fractional exclusion statistics with mutual statistical interactions is studied systematically. We discuss properties of the single-state partition functions and derive the general form of the cluster expansion. Assuming a certain scaling of the single-particle partition functions, relevant to systems of non-interacting particles with various dispersion laws, both in a box and in an external harmonic potential, we derive a unified form of the virial expansion. For the case of a symmetric statistics matrix at a constant density of states, the thermodynamics is analyzed completely. We solve the microscopic problem of multispecies anyons in the lowest Landau level for arbitrary values of particle charges and masses (but the same sign of charges). Based on this, we derive the equation of state which has the form implied by exclusion statistics, with the statistics matrix coinciding with the exchange statistics matrix of anyons. Relation to one-dimensional integrable models is discussed.

Exact multiplicities in the three-anyon spectrum

We develop the concept of trajectories in anyon spectra, i.e., the continuous dependence of energy levels on the kinetic angular momentum. It provides a more economical and unified description, since each trajectory contains an infinite number of points corresponding to the same statistics. For a system of non-interacting anyons in a harmonic potential, each trajectory consists of two infinite straight line segments, in general connected by a nonlinear piece. We give the systematics of the three-anyon trajectories. The trajectories in general cross each other at the bosonic/fermionic points. We use the (semi-empirical) rule that all such crossings are true crossings, i.e. the order of the trajectories with respect to energy is opposite to the left and to the right of a crossing.