One-Dimensional Statistical Mechanics for Identical Particles: The Calogero and Anyon Cases

Veigy, Alain Dasnières de; Ouvry, Stéphane

doi:10.1142/s0217732395000028

Cited by 27 publications

(36 citation statements)

References 16 publications

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“…In the case of interest α in [−1, 0], one has a "fractional" exclusion where one can put more than one particle per quantum state according to the fractional α, but not infinitely many as in the Bose case. The degeneracy (55) originates from the exact N -body spectrum (37). In the case of Haldane statistics as defined in (42), there is no Hamiltonian and no N -body spectrum to begin with.…”

Section: Lll-anyon Thermodynamicsmentioning

confidence: 99%

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Anyons and Lowest Landau Level Anyons

Ouvry

2009

The Spin

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Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the anyonic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts "à la AharonovBohm" with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N -body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N -body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e. the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from 1 arXiv:0712.2174v1 [cond-mat.stat-mech] 13 Dec 2007 anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Haldane's statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former...

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Section: Lll-anyon Thermodynamicsmentioning

confidence: 99%

“…This is not a coincidence. Looking at their harmonic N -body spectrum (37) and (76), one realizes that, up to an irrelevant zero-point energy, the latter is the B → 0 limit of the former. This remains true in the thermodynamic limit ω → 0.…”

Section: Lll-anyon Thermodynamicsmentioning

confidence: 99%

Anyons and Lowest Landau Level Anyons

Ouvry

2009

The Spin

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“…In this limit it is simpler to obtain it using the method presented in this paper rather than obtaining by taking the limit β → 0 in the solution of [20]. The average occupation number at a level i for the exclusion statistics has been studied in [2,3,4,21,22], which also can be obtained from (13) simply through…”

Section: Note Added In Proofmentioning

confidence: 99%

“…Exclusion statistics [1,2,3,4,5,6,7,8]-a generalization of Bose and Fermi statistics-can be defined in the following thermodynamical sense. Let Z(β, z) denote the grand partition function of a quantum gas of particles at inverse temperature β and fugacity z.…”

Section: Introductionmentioning

confidence: 99%

Integer partitions and exclusion statistics: limit shapes and the largest parts of Young diagrams

et al. 2007

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Abstract. We compute the limit shapes of the Young diagrams of the minimal difference p partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young diagram and show that the scaled distribution has a Gumbel form for all p. This Gumbel statistics for the largest part remains unchanged even for general partitions of the form E = P i n i i 1/ν with ν > 0 where n i is the number of times the part i appears.

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“…Therefore, developing statistical mechanics for mutual statistics is of interest both theoretically and phenomenologically. Thus far, some progress has been achieved on the issue [13,21,4], generalizing the results of the better-understood single-species case, where the thermodynamics was studied for a system in a box [6,15,22,23] as well as in a harmonic potential [15,24,25].…”

Section: Introductionmentioning

confidence: 99%

Statistical mechanics and thermodynamics for multispecies exclusion statistics

Isakov

Mashkevich²

1997

Nuclear Physics B

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Statistical mechanics and thermodynamics for ideal fractional exclusion statistics with mutual statistical interactions is studied systematically. We discuss properties of the single-state partition functions and derive the general form of the cluster expansion. Assuming a certain scaling of the single-particle partition functions, relevant to systems of noninteracting particles with various dispersion laws, both in a box and in an external harmonic potential, we derive a unified form of the virial expansion. For the case of a symmetric statistics matrix at a constant density of states, the thermodynamics is analyzed completely. We solve the microscopic problem of multispecies anyons in the lowest Landau level for arbitrary values of particle charges and masses (but the same sign of charges). Based on this, we derive the equation of state which has the form implied by exclusion statistics, with the statistics matrix coinciding with the exchange statistics matrix of anyons. Relation to one-dimensional integrable models is discussed.

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One-Dimensional Statistical Mechanics for Identical Particles: The Calogero and Anyon Cases

Cited by 27 publications

References 16 publications

Anyons and Lowest Landau Level Anyons

Anyons and Lowest Landau Level Anyons

Integer partitions and exclusion statistics: limit shapes and the largest parts of Young diagrams

Statistical mechanics and thermodynamics for multispecies exclusion statistics

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