Kåre Olaussen

The Third virial coefficient of free anyons

Abstract We use a path integral representation for the partition function of non-interacting anyons confined in a harmonic oscillator potential in order to prove that the third virial coefficient of free anyons is finite, and to calculate it numerically. Our results together with previously known results are consistent with a rapidly converging Fourier series in the statistics angle.

KAZAMA-YANG MONOPOLE - FERMION BOUND STATES. 1. ANALYTIC RESULTS

Abstract We present explicit, approximate, remarkably precise results for the Kazama-Yang monopole-fermion binding energies and wave functions. The results are valid for the states of lowest angular momentum and for the binding energy M − E ⪡ M . They agree very well with the numerically calculated values.

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 .

Resolution of an apparent inconsistency in the electromagnetic Casimir effect

The vacuum expectation value of the electromagnetic energy–momentum tensor between two parallel plates in spacetime dimensions D > 4 is calculated in the axial gauge. While the pressure between the plates agrees with the global Casimir force, the energy density is divergent at the plates and not compatible with the total energy which follows from the force. However, subtracting the divergent self-energies of the plates, the resulting energy is finite and consistent with the force. In analogy with the corresponding scalar case for spacetime dimensions D > 2, the divergent self-energy of a single plate can be related to the lack of conformal invariance of the electromagnetic Lagrangian for dimensions D > 4.

Electromagnetic Casimir energy with extra dimensions

We calculate the energy-momentum tensor due to electromagnetic vacuum fluctuations between two parallel hyperplanes in more than four dimensions, considering both metallic and MIT boundary conditions. Using the axial gauge, the problem can be mapped upon the corresponding problem with a massless, scalar field satisfying, respectively, Dirichlet or Neumann boundary conditions. The pressure between the plates is constant while the energy density is found to diverge at the boundaries when there are extra dimensions. This can be related to the fact that Maxwell theory is then no longer conformally invariant. A similar behavior is known for the scalar field where a constant energy density consistent with the pressure can be obtained by improving the energy-momentum tensor with the Huggins term. This is not possible for the Maxwell field. However, the change in the energy-momentum tensor with distance between boundaries is finite in all cases.

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 interpretation of fermionic zero-modes

Abstract We examine the effect of fermionic zero modes on tunneling amplitudes within some simple quantum mechanical models. It is shown that the fermionic zero modes do not cause a total suppression of the tunneling, although it may be reduced. With a θ term present, due to a non-trivial topology, it is shown that the θ dependence is not eliminated by the zero modes. Instead the non-trivial topology introduces a “twist” in the fermionic coordinates which breaks a symmetry of the hamiltonian.

THE FOURTH VIRIAL COEFFICIENT OF ANYONS

We have computed by a Monte Carlo method the fourth virial coefficient of free anyons, as a function of the statistics angle θ. It can be fitted by a four term Fourier series, in which two coefficients are fixed by the known perturbative results at the boson and fermion points. We compute partition functions by means of path integrals, which we represent diagramatically in such a way that the connected diagrams give the cluster coefficients. This provides a general proof that all cluster and virial coefficients are finite. We give explicit polynomial approximations for all path integral contributions to all cluster coefficients, implying that only the second virial coefficient is statistics dependent, as is the case for two-dimensional exclusion statistics. The assumption leading to these approximations is that the tree diagrams dominate and factorize.

High precision series solutions of differential equations: Ordinary and regular singular points of second order ODEs

A subroutine for a very-high-precision numerical solution of a class of ordinary differential equations is provided. For a given evaluation point and equation parameters the memory requirement scales linearly with precision P, and the number of algebraic operations scales roughly linearly with P when P becomes sufficiently large. We discuss results from extensive tests of the code, and how one, for a given evaluation point and equation parameters, may estimate precision loss and computing time in advance. Program summary Program title: seriesSolveOde1 Catalogue identifier: AEMW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEMW_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 991 No. of bytes in distributed program, including test data, etc.: 488116 Distribution format: tar.gz Programming language: C++ Computer: PC’s or higher performance computers. Operating system: Linux and MacOS RAM: Few to many megabytes (problem dependent). Classification: 2.7, 4.3 External routines: CLN — Class Library for Numbers [1] built with the GNU MP library [2], and GSL — GNU Scientific Library [3] (only for time measurements). Nature of problem: The differential equation (1)−s2(d2dz2+1−ν+−ν−zddz+ν+ν−z2)ψ(z)+1z∑n=0Nvnznψ(z)=0, is solved numerically to very high precision. The evaluation point z and some or all of the equation parameters may be complex numbers; some or all of them may be represented exactly in terms of rational numbers. Solution method: The solution ψ(z), and optionally ψ′(z), is evaluated at the point z by executing the recursion (2)Am+1(z)=s−2(m+1+ν−ν+)(m+1+ν−ν−)∑n=0NVn(z)Am−n(z), (3)ψ(m+1)(z)=ψ(m)(z)+Am+1(z), to sufficiently large m. Here ν is either ν+ or ν−, and Vn(z)=vnzn+1. The recursion is initialized by (4)A−n(z)=δn0zν,for n=0,1,…,N (5)ψ(0)(z)=A0(z). Restrictions: No solution is computed if z=0, or s=0, or if ν=ν− (assuming Reν+≥Reν−) with ν+−ν− an integer, except when ν+−ν−=1 and v0=0 (i.e. when z is an ordinary point for z−ν−ψ(z)). Additional comments: The code of the main algorithm is in the file seriesSolveOde1.cc, which “#include” the file checkForBreakOde1.cc. These routines, and the programs using them, must “#include” the file seriesSolveOde1.cc. Running time: On a Linux PC that is a few years old, at y=10 to an accuracy of P=200 decimal digits, evaluating the ground state wavefunction of the anharmonic oscillator (with the eigenvalue known in advance); (cf. Eq. (6)) takes about 2 ms, and about 40 min at an accuracy of P=100000 decimal digits. References: [1] B. Haible and R.B. Kreckel, CLN — Class Library for Numbers, http://www.ginac.de/CLN/ [2] T. Granlund and collaborators, GMP — The GNU Multiple Precision Arithmetic Library, http://gmplib.org/ [3] M. Galassi et al., GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078., http://www.gnu.org/software/gsl/

USING CONSERVATION LAWS TO SOLVE TODA FIELD THEORIES

We investigate the question of how the knowledge of sufficiently many local conservation laws for a model can be used to solve it. We show that for models where the conservation laws can be written in one-sided forms, like the problem can always be reduced to solving a closed system of ordinary differential equations. We investigate the A1, A2 and B2 Toda field theories in considerable detail from this viewpoint. One of our findings is that there is in each case a transformation group intrinsic to the model. This group is built on a specific real form of the Lie algebra used to label the Toda field theory. It is the group of field transformations which leaves the conserved densities invariant.