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Journal of Physics Conference Series

Fractional exclusion statistics in systems with localized states

George Alexandru Nemneş, Dragoş-Victor Anghel

Published Feb 8, 2013• 3 citations
Publisher

Abstract

We develop a model based on the fractional exclusion statistics to describe systems with localized states. The local distribution of the energy levels is captured in the formalism by including the positions in the definition of the species. The particle distributions on the energy axis, as well as in the real space are determined for test-case systems with a peak/dip profile in the local density of states.

DOI
10.1088/1742-6596/410/1/012120

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We develop a model based on the fractional exclusion statistics to describe systems with localized states. The local distribution of the energy levels is captured in the formalism by including the positions in the definition of the species. The particle distributions on the energy axis, as well as in the real space are determined for test-case systems with a peak/dip profile in the local density of states.
We develop a model based on the fractional exclusion statistics (FES) applicable to non-homogeneous interacting particle systems. Here the species represent elementary volumes in an (s+1)-dimensional space, formed by the direct product between the s-dimensional space of positions and the quasiparticle energy axis. The model is particularly suitable for systems with localized states. We prove the feasibility of our method by applying it to systems of different degrees of complexities. We first apply the formalism on simpler systems, formed of two sub-systems, and present numerical and analytical thermodynamic calculations, pointing out the quasiparticle population inversion and maxima in the heat capacity, in contrast to systems with only diagonal (direct) FES parameters. Further we investigate larger, non-homogeneous systems with repulsive screened Coulomb interactions, indicating accumulation and depletion effects at the interfaces. Finally, we consider systems with several degrees of disorder, which are prototypical for models with glassy behavior. We find that the disorder produces a spatial segregation of quasiparticles at low energies which significantly affects the heat capacity and the entropy of the system.
I amend the concept of fractional exclusion statistics (FES) by introducing new conditions that make the formalism logically —and therefore physically— consistent. These conditions are imposed on the exclusion statistics parameters, defining how the parameters change if the species of particles in the system are divided into sub-species. Using a simple and intuitive model I deduce the general equations for the change of the exclusion statistics parameters at the redefinition of the particle species in any FES system.
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.
I introduce an ansatz for the exclusion statistics parameters of fractional-exclusion-statistics (FES) systems and I apply it to calculate the statistical distribution of particles from both, bosonic and fermionic perspectives. To check the applicability of the ansatz, I calculate the FES parameters in three well-known models: a Fermi-liquid–type system, a one-dimensional quantum system described in the thermodynamic Bethe ansatz and quasi-particle excitations in a fractional quantum Hall (FQH) system. The FES parameters of the first two models satisfy the ansatz, whereas those of the third model, although close to the form given by the ansatz, represent an exception. In this case I also show that the general properties of the FES parameters, deduced elsewhere ( EPL , 87 (2009) 60009), are satisfied also by the parameters of the FQH liquid.
I discuss Haldane's concept of generalized exclusion statistics (1991 Phys. Rev. Lett. 67 937) and show that it leads to inconsistencies in the calculation of the particle distribution that maximizes the partition function. These inconsistencies appear when mutual exclusion statistics is manifested between different subspecies of particles in the system. In order to eliminate these inconsistencies, I introduce new mutual exclusion statistics parameters, which are proportional to the dimension of the Hilbert subspace on which they act. These new definitions lead to properly defined particle distributions and thermodynamic properties. In another paper ( Preprint 0710.0728), I show that the fractional exclusion statistics manifested in general systems with interaction have these, physically consistent, statistics parameters.