On Anyon Superconductivity

Abstract

We investigate the statistical mechanics of a gas of fractional statistics particles in 2+1 dimensions. In the case of statistics very close to Fermi statistics (statistical parameter θ=π(1−1/n), for large n), the effect of the statistics is a weak attraction. Building upon earlier RPA calculation of Fetter, Hanna, and Laughlin for the case n=2, we argue that for large n perturbation theory is reliable and exhibits superfluidity (or superconductivity after coupling to electromagnetism). We attempt to describe the order parameter for this superconducting phase in terms of “spontaneous breaking of commutativity of translations” as opposed to the usual pairing order parameters. The vortices of the superconducting anyon gas are charged, and superconducting order parameters of the usual type vanish. We investigate the characteristic P and T violating phenomenology.

Singular or non-Fermi liquids

“…16 It is also worth noting that some of the instabilities are disallowed in the limit of translational invariance. Thus, for example, time-reversal breaking states, such as the "anyon-state" [156,60] cannot be realized because in a translationally invariant problem the current operator cannot be renormalized by the interactions, as we have learnt from Eqs. (18), (33).…”

Non-commutative Chern–Simons for the quantum Hall system and duality

“…Kohn's theorem follows from the fact that the system as a whole must obey the magnetic algebra of the full magnetic field B. (A similar approximation also leads to a spurious zero-field Hall effect in theories of anyon superfluidity [32].) Similarly, the excitation spectrum predicted by this mean field theory is wrong, since it predicts that the excitations of this state are composite fermions instead of particles with fractional charge and fractional statistics.…”

Influence of gauge-field fluctuations on composite fermions near the half-filled state

“…where D −1 µν (q, ω, ∆ω c ) was calculated by several authors [5,6,[22][23][24]. For our purpose, the 2 × 2 matrix form for D −1 µν is sufficient so that µ, ν = 0, 1 and 1 represents the direction that is perpendicular to q [5].…”