On the virial coefficients of nonabelian anyons

We study a system of nonabelian anyons in the lowest Landau level of a strong magnetic field. Using diagrammatic techniques, we prove that the virial coefficients do not depend on the statistics parameter. This is true for all representations of all nonabelian groups for the statistics of the particles and relies solely on the fact that the effective statistical interaction is a traceless operator. E-mail: [email protected] In two spatial dimensions the group relevant to the quantum statistics of particles is the braid group [1, 2], rather than the permutation group. As a result, the possibility for non-standard statistics exists. A well-studied case is (abelian) anyons [3, 4, 5], transforming in a unitary abelian representation of the braid group. Anyons in the lowest Landau level, in particular, are relevant to the quantum Hall effect [6, 7] and constitute realizations of ideal exclusion statistics [8, 9]. A natural generalization is nonabelian anyon statistics, based on nonabelian representations of the braid group. These would be the anyonic analogs of parastatistics [10, 11]. Just as abelian anyons can be thought of as ordinary (bosonic or fermionic) particles interacting through a non-dynamical abelian gauge field, nonabelian anyons can be though of as particles carrying internal degrees of freedom in some irreducible representation R of a nonabelian group SU(n) and interacting through an appropriate non-dynamical nonabelian gauge field. What fixes the statistics, then, is the group SU(n), the representation R and the coupling strength g of the internal degrees of freedom to the gauge field. A field-theoretic approach to achieving such statistics, in analogy to the abelian case, is to couple the particles to a nonabelian gauge field with a Chern-Simons action [12]. Such a particle-field model was proposed by Verlinde [13]. g, then, is essentially the inverse of the coefficient of the Chern-Simons term and, as such, inherits the quantization condition g = 2 n , n integer (1) This condition does not seem to be crucial for the purely first-quantized approach and, at any rate, will not play any role in this paper. It is of interest to derive the thermodynamics and statistical mechanical properties of nonabelian anyons in order to probe the possibility of new physics deriving from the nonabelian nature of the system. In a recent paper, Isakov, Lozano and Ouvry [14] examined these questions for the simplest case of SU(2) anyons in the fundamental (spin-half) representation. They found that the virial coefficients up to the fifth one do not depend on the statistics parameter g. They conjectured that this holds for all the coefficients and posed the question of a possible underlying symmetry that explains this vanishing dependence. The purpose of this note is to give a complete proof of the independence of all virial coefficient of this model on the statistics parameter g, valid for any group and any representation. It is based on a diagrammatic expression of the cluster coefficients which is useful in deriving them in a simple way and reveals their scaling properties with the volume. It will be apparent that the only feature of the statistical interaction which is relevant for this result is that it is a traceless operator in the space of internal degrees of freedom of the particles. We repeat here the main results for the system that will be used in this paper, as presented in [14]. The model consists of N non-interacting spinless particles on the plane with internal degrees of freedom transforming in some finite-dimensional