Anyon quarks and superconductivity

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 .

1991

We obtain a hierarchy of effective Hamiltonians which allow for a unified treatment of the fractional quantum Hall effect and a gas of fractional-statistics particles (anyons) in two dimensions. Anyon superconductivity is the analog of the fractional quantum Hall effect. For a rational statistics parameter a, P/Q with PQ even, Q anyons bind forming a charge-Qe superfluid.

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.

The European Physical Journal B, 2014

In the paper, two-parametric models of fractional statistics are proposed in order to determine the functional form of the distribution function of free anyons. From the expressions of the second and third virial coefficients, an approximate correspondence is shown to hold for three models, namely, the nonextensive Polychronakos statistics and both the incomplete and the nonextensive modifications of the Haldane-Wu statistics. The difference occurs only in the fourth virial coefficient leading to a small correction in the equation of state. For the two generalizations of the Haldane-Wu statistics, the solutions for the statistics parameters g, q exist in the whole domain of the anyonic parameter α ∈ [0; 1], unlike the nonextensive Polychronakos statistics. It is suggested that the search for the expression of the anyonic distribution function should be made within some modifications of the Haldane-Wu statistics. arXiv:1403.3577v1 [cond-mat.stat-mech]

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.

2012

We study a 2+1 dimensional theory of bosons and fermions with an ω ∝ k 2 dispersion relation. The most general interactions consistent with specific symmetries impart fractional statistics to the fermions. Unlike examples involving Chern-Simons gauge theories, our statistical phases derive from the exchange of gapless propagating bosons with marginal interactions. Even though no gap exists, we show that the anyonic statistics are precisely defined. Symmetries combine with the vacuum structure to guarantee the non-renormalization of our anyonic phases. arXiv:1205.6816v1 [hep-th] 30 May 2012 1 Despite the absence of a gap, the anyonic phase is well-defined, as we show in section 3.1

Physical review letters, 2009

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

Physical Review A, 2008

We investigate the ground state of the one-dimensional interacting anyonic system based on the exact Bethe ansatz solution for arbitrary coupling constant (0 ≤ c ≤ ∞) and statistics parameter (0 ≤ κ ≤ π). It is shown that the density of state in quasi-momentum k space and the ground state energy are determined by the renormalized coupling constant c ′ . The effect induced by the statistics parameter κ exhibits in the momentum distribution in two aspects: Besides the effect of renormalized coupling, the anyonic statistics results in the nonsymmetric momentum distribution when the statistics parameter κ deviates from 0 (Bose statistics) and π (Fermi statistics) for any coupling constant c. The momentum distribution evolves from a Bose distribution to a Fermi one as κ varies from 0 to π. The asymmetric momentum distribution comes from the contribution of the imaginary part of the non-diagonal element of reduced density matrix, which is an odd function of κ. The peak at positive momentum will shift to negative momentum if κ is negative.