Second Virial Coefficient of Anyons without Hard Core

We calculate the second virial coefficient of anyons whose wave function does not vanish at coincidence points. This kind of anyons appear naturally when one generalizes the hard-core boundary condition according to self-adjoint extension method in quantum mechanics, and also when anyons are treated field theoretically by applying renormalization procedure to nonrelativistic ChernSimons field theory. For the anyons which do not satisfy hard-core boundary condition, it is argued that the other scale-invariant limit is more relevant in high-temperature limit where virial expansion is useful. Furthermore, the cusp existing at the bosonic point for hard-core anyons disappears in general; instead it is shown that a new cusp is generated at the fermionic point. A physical explanation is given. PACS number: 11.10.Kk, 05.30.-d, 71.10.Pm ∗[email protected] 1 In this letter we would like to calculate the second virial coefficient of a system of anyons [1] obtained by generalizing the usual hard-core boundary condition at coincidence points according to the self-adjoint extension method [2], and discuss its interesting physical properties. To motivate our calculation, let us briefly summarize the present status. Anyons are particles exhibiting fractional statistics characterized by a statistical parameter α which interpolates between bosonic and fermionic cases. Quantum mechanically, they can be regarded as flux-charge composites and their interaction is essentially AharonovBohm type [3]. From the field theoretic point of view, they can be obtained if the Schrödinger field is coupled to a Chern-Simons gauge field [4,5]. The connection of these two descriptions is now well established and an interesting subject here is the relation between the method of self-adjoint extension in quantum mechanics and the regularization-renormalization procedure in field theory where |φ|4-type contact interaction is necessarily introduced [6–9]. Recently, an all-order analysis has been carried out to establish the precise correspondence and the role of contact term has been emphasized in ref. [9]. In the literature, regularity of wave functions is often assumed and it leads to the hardcore boundary condition [1]. Although this hard-core case is conceptually simple and easier to treat, there is no a priori reason to assume such a boundary condition in general; a real system under study should determine the relevant boundary condition eventually. If we relax the regularity requirement, allowing wave functions to diverge at coincidence points according to the method of self-adjoint extension, we get a one-parameter family of boundary conditions which in general introduces a scale in the theory. (The corresponding anyons were called “colliding anyons” [10].) The scale invariance of the theory is restored only in the limit that the scale parameter goes to either zero or infinity. Field theoretically, when the strength of the |φ|4-interaction is equal to the critical value λ = ±2π|α| m ≡ ±λ0, the field theory becomes ultraviolet finite and scale invariant [9]. These critical values have some more interesting properties. The repulsive case λ = λ0 corresponds to the usual hard-core boundary condition, while in the attractive case λ = −λ0, it is known that the classical theory admits static soliton solutions satisfying a self-dual equation [5]. Also, for λ → −λ0, 2 after performing renormalization, one can relate the renormalized coupling λren to a nonzero finite value of the self-adjoint extension parameter in a specific way [11,9]. Since the free anyon gas has not been exactly solved, one usually resorts to the virial expansion to study thermodynamic properties in high-temperature low-density limit. For example, the second virial coefficient was calculated for anyons with hard core in ref. [12–15]. Here an interesting fact is that, according to recent analyses [11,9], λren = λ0 (hard-core case) is an infrared fixed point, while λren = −λ0 is an ultraviolet fixed point, with λren flowing from λ0 to −λ0 as the renormalization scale increases. This implies that the virial expansion for usual anyons with hard core is not much relevant for the case of colliding anyons in hightemperature limit where the virial expansion is useful. Therefore it may be interesting to calculate the second virial coefficient for anyons with an arbitrary boundary condition and study the behavior as the scale changes. Indeed, we will see that the result of hard-core case does not represent a typical behavior of the other cases. For example, the cusp existing at bosonic point α = 0 in hard-core case is a very special feature of scale-invariant limit which does not occur in general; instead a new cusp is generated at the fermionic point |α| = 1 except for the hard core case. We will give a physical interpretation of this result later. Here we should mention earlier works [16,17] which have a partial overlap with this paper. They calculated the second virial coefficient for the case that the system has a bound state. However they did not pay much attention to physical properties for general boundary conditions except at the fixed point, λren = −λ0. Also, we believe that our method is simpler and physically more transparent. Let us start with the Hamiltonian of a system of free anyons with mass m and statistical parameter α in bosonic description