Two-parametric fractional statistics models for anyons

We formulate quantum statistical mechanics of particles obeying fractional statistics, including mutual statistics, by adopting a state-counting definition. For an ideal gas, the most probable occupation-number distribution interpolates between bosons and fermions, and respects a generalized exclusion principle except for bosons. Anyons in strong magnetic field at low temperatures constitute such a physical system. Applications to the thermodynamic properties of quasiparticle excitations in the Laughlin quantum Hall fluid are discussed.

Physical Review B, 1998

We define one-dimensional particles as non-abelian representations of the symmetric group SN . The exact solution of an XXZ type Hamiltonian built up with such particles is achieved using the coordinate Bethe Ansatz. The Bethe equations show that fractional statistics, effectively, accounts for coupling an external gauge field to an integer statistics' system. Numbers: 71.10.Pm, 71.27.+a, 75.10.Jm Physical behaviour of quantum systems is deeply affected by the statistics of the constituting effective degrees of freedom. Quasi-particles and quasi-holes in condensed matter physics may obey statistics interpolating between fermionic and bosonic behaviour. Examples are the excitations of two-dimensional electron systems exhibiting Fractional Quantum Hall effect [1]. These excitations are called anyons. They have been a subject of intense study also in connection with superconductivity [2] and superfluidity . Fractional statistics of such particles arises from the trajectory-dependence of the particle exchange procedure in the two-dimensional configuration space. This feature makes the concept of anyons purely two-dimensional. The Fock space formulation of anyon operator algebras takes into account these characteristics. The creation and annihilation operators (introduced as Jordan-Wigner transforms of usual fermions on a twodimensional lattice or as unitary representations of the diffeomorphism group of R 2 [5]) obey deformed commutation relations if the exchange involves anyons at different spatial positions (see Appendix). N -anyon-states are abelian representations of the braid group B N [6] (whereas bosons and fermions furnish, respectively, the identical and alternating abelian representations of the symmetric group S N ). These features make anyons different from q-oscillators, the latter providing a realization of Gel'fand-Farlie quantum group, which is a local deformation of the Weyl-Heisenberg (bosons) or Clifford algebra (fermions) . The path dependence implies that the one-particle state is inextricably related with the complete state of the many body configuration. This intrinsic non-locality makes anyon physics very difficult. Even statistical properties of a free anyon gas are only partially established using the virial expansion .

Annals of Physics, 2009

An explicit realization of anyons is provided, using the three-body Calogero model. The fact that in the coupling domain, −1/4 < g < 0, the angular spectrum can have a band structure, leads to the manifestation of the desired phase in the wave function, under the exchange of the paticles. Concurrently, the momentum corresponding to the angular variable is quantized, exactly akin to the relative angular momentum quantization in two dimensional anyonic system.

Low Temperature Physics

We consider two examples of real physical systems approximately described using fractional nonadditive Polychronakos statistics. The values of two statistics parameters are linked to properties of modeled systems using virial expansion. For a two-dimensional Fermi gas with contact interactions, accuracy up to the third virial coefficient is achieved. An approach to model the second virial coefficient of non-Abelian soft-core anyons is analyzed in detail.

Nuclear Physics B, 1985

We study the statistical mechanics of a two-dimensional gas of free anyons -particles which interpolate between Bose-Einstein and Fermi-Dirac character. Thermodynamic quantities are discussed in the low-density regime. In particular, the second virial coefficient is evaluated by two different methods and is found to exhibit a simple, periodic, but nonanalytic behavior as a function of the statistics determining parameter.

Physical Review Letters, 1990

Using notions of supersymmetry we present an exactly soluble model of anyons with both statistical and scalar interactions in 2+1 dimensions. We demonstrate that half-statistics particles with two spin flavors condense into a local singlet state which is both a charge superfluid and a "spin metal ' in the sense that there is charge-pairing oA'-diagonal long-range order with gapless charge excitations but a gap in the collective spin-mode spectrum. The present results shed considerable light on the mean-field theory of fractional statistics.

Physical review letters, 2009

2007

In two-dimensions, the laws of physics even permit the existence of anyons which exhibit fractional statistics ranging continuously from bosonic to fermionic behaviour. They have been responsible for the fractional quantum Hall effect and proposed as candidates for naturally fault-tolerant quantum computation. Despite these remarkable properties, the fractional statistics of anyons has never been observed in nature directly. Here we report the demonstration of fractional statistics of anyons by simulation of the first Kitaev lattice-spin model on a nuclear magnetic resonance system. We encode four-body interactions of the lattice-spin model into two-body interactions of an Ising spin chain in molecules. It can thus efficiently prepare and operate the ground state and excitations of the model Hamiltonian. This quantum system with convenience of manipulation and detection of abelian anyons reveals anyonic statistical properties distinctly. Our experiment with interacted Hamiltonian co...

EPL, 2010

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. Then, to check the applicability of the ansatz, I calculate the FES parameters in three well-known models: in a Fermi liquid type of system, a one-dimensional quantum systems described in the thermodynamic Bethe ansatz and quasiparticle excitations in the fractional quantum Hall (FQH) systems. 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. With this ocasion I also show that the general properties of the FES parameters, deduced elsewhere (EPL 87, 60009, 2009), are satisfied also by the parameters of the FQH liquid.

Physical Review E, 2010

We extend our earlier study about the fractional exclusion statistics to higher dimensions in full physical range and in the non-relativistic and ultra-relativistic limits. Also, two other fractional statistics, namely Gentile and Polychronakos fractional statistics, will be considered and similarities and differences between these statistics will be explored. Thermodynamic geometry suggests that a two dimensional Haldane fractional exclusion gas is more stable than higher dimensional gases. Also, a complete picture of attractive and repulsive statistical interaction of fractional statistics is given. For a special kind of fractional statistics, by considering the singular points of thermodynamic curvature, we find a condensation for a non-pure bosonic system which is similar to the Bose-Einstein condensation and the phase transition temperature will be worked out. PACS number(s): 05.20.-y, 67.10.Fj