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arXiv:hep-th/9703225

Anyonic behavior of quantum group fermionic and bosonic systems

Marcelo R. Ubriaco

Published Mar 31, 1997• 0 citations
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Abstract

We discuss the role that quantum group symmetries, in particular $SU_q(2)$, play in a thermodynamic system at high temperatures. We show that the interactions introduced by the quantum group symmetries, are such that a quantum group gas describe repulsive and attractive behavior in two and three spatial dimensions.

DOI
10.48550/arxiv.hep-th/9703225

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We discuss the role that quantum group symmetries, in particular SUq(2)SU_q(2)SUq​(2), play in a thermodynamic system at high temperatures. We show that the interactions introduced by the quantum group symmetries, are such that a quantum group gas describe repulsive and attractive behavior in two and three spatial dimensions.
We first introduce and discuss the formalism of SUq{\mathrm{SU}}_{\mathrm{q}}SUq​ (N) bosons and fermions and consider the simplest Hamiltonian involving these operators. We then calculate the grand partition function for these models and study the high temperature (low density) case of the corresponding gases for N=2. We show that quantum group gases exhibit anyonic behavior in D=2 and 3 spatial dimensions. In particular, for a SUq{\mathrm{SU}}_{\mathrm{q}}SUq​ (2) boson gas at D=2 the parameter q interpolates within a wider range of attractive and repulsive systems than the anyon statistical parameter.
We consider the simplest SUq(2) invariant fermionic hamiltonian and calculate the low and high temperature behavior for the two distinct cases q> 1 and q < 1. For low temperatures we find that entropy values for the Fermi case are an upper bound for those corresponding to q ̸ = 1. At high temperatures we find that the sign of the second virial coefficient depends on q, and vanishes at q = 1.96. An important consequence of this fact is that the parameter q connects the fermionic and bosonic regions, showing therefore that SUq(2) fermions exhibit fractional statistics in three spatial dimensions.
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We consider the simplest SU q(2) invariant fermionic Hamiltonian and calculate the low and high temperature behavior for the two distinct cases q>1 and q<1. For low temperatures we find that entropy values for the Fermi case are an upper bound for those corresponding to q≠1. At high temperatures we find that the sign of the second virial coefficient depends on q, and vanishes at q=1.96. An important consequence of this fact is that the parameter q connects the fermionic and bosonic regions, showing therefore that SU q(2) fermions exhibit fractional statistics in three spatial dimensions.
Chapter 1. Exact and explicit results are derived for the thermodynamic properties (isochores, isotherms, isobars, response functions, speed of sound) of a quantum gas in dimensions [special characters omitted] ≥ 1 and with fractional exclusion statistics 0 ≤ g ≤ 1 connecting bosons (g = 0) and fermions (g = 1). In [special characters omitted] = 1 the results are equivalent to those of the Calogero-Sutherland model, a gas with long-range two-body interaction. Emphasis is given to the crossover between boson-like and fermion-like features, caused by aspects of the statistical interaction that mimic long-range attraction and short-range repulsion. A phase transition along the isobar occurs at a nonzero temperature in all dimensions. The T-dependence of the speed of sound is in simple relation to isochores and isobars. The effects of soft container walls are accounted for rigorously for the case of a pure power-law potential. Chapter 2. The exact thermodynamics (isochores, isotherms, isobars, response functions, speed of sound) is worked out for a statistically interacting quantum gas in [special characters omitted] dimensions. The results in [special characters omitted] = 1 are those of the thermodynamic Bethe ansatz for the Nonlinear Schrödinger model, a gas with repulsive two-body contact potential. In all dimensions the ideal boson and fermion gases are recovered in the weak-coupling and strong-coupling limits, respectively. For all nonzero couplings ideal fermion gas behavior emerges for [special characters omitted] >> 1 and, in the limit [special characters omitted] → ∞, a phase transition occurs at T > 0. Significant deviations from ideal quantum gas behavior are found for intermediate coupling and finite [special characters omitted]. Chapter 3. Methodology previously developed in the framework of the coordinate Bethe ansatz applied to integrable quantum gas models is employed to calculate some ground-state properties and elementary excitations for quantum gas models in [special characters omitted] = 1 dimensions with statistical interactions that are not equivalent to dynamical interactions. The focus in this comparative study is on modifications of the Calogero-Sutherland and Nonlinear Schrödinger models. Statistical interactions cannot be chosen arbitrarily. They must satisfy certain physicality conditions, which will be discussed in some detail.