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Physical review. B, Condensed matter

Microscopic theory of exclusion statistics

Bill Sutherland

Published Aug 15, 1997• 22 citations
Publisher

Abstract

We present a microscopic theory of exclusion statistics, based on an analogy with the exactly solved integrable one-dimensional systems. The resulting theory differs from the usual theory of mutual exclusion statistics by certain dimension-dependent features. As an example, we carry out a calculation of the ground-state properties, low-lying excitations, and thermodynamics of a system of spinless fermions in two dimensions, obeying pure exclusion statistics.

DOI
10.1103/physrevb.56.4422

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After a very brief introduction to the concept of fractional exclusion statistics (FES), introduced by Haldane in Phys. Rev. Lett. 67, 937 (1991), I shall present some recent results regarding the basic properties of this statistics. These properties have been overlooked for almost two decades and because of that, the application of FES to concrete physical systems was, to a certain degree, arbitrary. Essentially, the FES is a generalization of the Pauli Exclusion Principle which allows one to describe systems of interacting particles (bosons or fermions) as systems of ideal particles, but which obey a more peculiar type of statistics. In the second part of the paper I shall present some application of FES by analyzing concrete systems of interacting particles.

•Jan 1, 2011•0 citations
We present a microscopic theory of exclusion statistics, based on an analogy with the exactly solved integrable one-dimensional systems. The resulting theory differs from the usual theory of mutual exclusion statistics by certain dimension-dependent features. As an example, we carry out a calculation of the ground-state properties, low-lying excitations, and thermodynamics of a system of spinless fermions in two dimensions, obeying pure exclusion statistics.
I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called thermodynamically equivalent and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonization technique, of transforming between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In appendix A, I calculate the fluctuation of the ground-state population of a condensed Bose gas in a grand-canonical ensemble and mean field approximation, while in appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single-particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.
We study statistical characterization of the many-body states in exactly solvable models with internal degrees of freedom. The models under consideration include the isotropic and anisotropic Heisenberg spin chains, the Hubbard chain, and a model in higher dimensions which exhibits the Mott metal-insulator transition. It is shown that the ground states of these systems are all described by that of a generalized ideal gas of particles (called exclusons) which have mutual-exclusion statistics, either between different rapidities or between different species. For the Bethe ansatz solvable models, the low-temperature properties are well described by the excluson description if the degeneracies due to string solutions with complex rapidities are taken into account correctly. For the Hubbard chain with strong but finite coupling, charge-spin separation is shown for thermodynamics at low temperatures. Moreover, we present an exactly solvable model in arbitrary dimensions which, in addition to giving a perspective view of spin-charge separation, constitutes an explicit example of mutual-exclusion statistics in more than two dimensions. \textcopyright{} 1996 The American Physical Society.
We study statistical characterization of the many-body states in the exactly solvable model with internal degree of freedom in more than one dimension. The model exhibits the Mott metal-insulator transition. It is shown that the ground state is described by that of a generalized ideal gas of particles (called exclusons) which have mutual exclusion statistics between different species. In addition to giving a perspective view of spin-charge separation, the model constitutes an explicit example of mutual exclusion statistics in more than two dimensions.
We show that fractional exclusion statistics is manifested in general in interacting systems and we discuss the conjecture recently introduced (J. Phys. A: Math. Theor. 40, F1013, 2007), according to which if in a thermodynamic system the mutual exclusion statistics parameters are not zero, then they have to be proportional to the dimension of the Hilbert space on which they act. By using simple, intuitive arguments, but also concrete calculations in interacting systems models, we show that this conjecture is not some abstract consequence of unphysical modeling, but is a natural--and for a long time overlooked--property of fractional exclusion statistics. We show also that the fractional exclusion statistics is the consequence of interaction between the particles of the system and it is due to the change from the description of the system in terms of free-particle energies, to the description in terms of the quasi-particle energies. From this result, the thermodynamic equivalence of systems of the same, constant density of states, but any exclusion statistics follows immediately.
After a very brief introduction to the concept of fractional exclusion statistics (FES), introduced by Haldane in Phys. Rev. Lett. 67, 937 (1991), I shall present some recent results regarding the basic properties of this statistics. These properties have been overlooked for almost two decades and because of that, the application of FES to concrete physical systems was, to a certain degree, arbitrary. Essentially, the FES is a generalization of the Pauli Exclusion Principle which allows one to describe systems of interacting particles (bosons or fermions) as systems of ideal particles, but which obey a more peculiar type of statistics. In the second part of the paper I shall present some application of FES by analyzing concrete systems of interacting particles.