Perturbative anyon gas

Abstract

The problem of statistical mechanics of an anyon gas is addressed. A perturbative analysis in the anyonic coupling constant α is reviewed and the thermodynamical potential is computed at first and second order. Particular care is taken of non-analyticity in α. A naive perturbative expansion close to Bose statistics would be ill-defined, due to the very singular nature of the anyon interaction when two anyons approach each other. To cure this perturbative obstruction, one explicitly introduces non-perturbative effects in | α | in the anyonic wave function by imposing it to vanish when any two anyons coincide. Thus, one removes the diagonal of the configuration space, which in turn allows for braid group statistics. As a bonus, one is left only with two-body interactions. An adequate second-quantized formalism (field theory at finite temperature) is proposed. Particular emphasis is given to boundary considerations and regularization procedures since all the quantities computed formally diverge as the volume in the thermodynamic limit. At first order in perturbation theory, the results are strikingly simple: only the second virial coefficient close to bosonic statistics is corrected. At second order, however, the complexity of the anyon model appears. One can compute exactly the perturbative correction to each cluster coefficient. However, and contrary to first order, a closed expression for the equation of state seems out of reach. As an illustration we give the perturbative expressions of a 3, a 4, a 5 and a 6, at second order. Finally, using the same formalism, we analyze the equation of state of an anyon gas in a constant magnetic field at first order in perturbation theory.