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 off-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.
I consider a gas of ``free'' anyons with statistical parameter \ensuremath{\delta}, with hard cores, on a two-dimensional square lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on the same lattice coupled to a Chern-Simons gauge theory with coupling \ensuremath{\theta}=1/2\ensuremath{\delta}. At the semiclassical level, the system is found to be equivalent to a gas of fermions, with the same density, in an average effective magnetic field \ensuremath{\rho}/\ensuremath{\theta}. I consider the case in which an integer number of the Landau bands of the saddle-point problem are completely filled. If \ensuremath{\delta}=\ensuremath{\pi}/m and the density \ensuremath{\rho}=r/q, with m, r, and q integers, the system is a superfluid, provided that q is larger than twice the largest common factor of m and r. If q is even and the system is half filled, the state may be either a superfluid or a quantum Hall phase. For all other values of \ensuremath{\rho} and \ensuremath{\delta}, compatible with integer filling of the Landau bands, the system is in a quantum Hall phase. The dynamical stability of the superfluid state is ensured by the topological invariance of the quantized Hall conductance of the fermion problem. I find a close analogy between anyon superconductivity and the Schwinger mechanism. The effective Lagrangian for the low-energy modes coupled to the electromagnetic field is derived. The energies of fermion and flux states are logarithmically divergent, but finite for the anyon state. The system has flux quantization, a zero-temperature Hall effect with a quantized Hall conductance, Meissner effect, charged vortices, screening with induced magnetic fields for static charges, and different masses for the longitudinal and transverse components of the electromagnetic field.
We discuss the BCS-like descriptions of the finite temperature s -wave superconductivity in the two-component anyon gas, and show that in the content of the BCS theory the long range gauge force can not be renormalized at the finite temperature T ≠ 0, and this singularity is inherent and can not be overcome by adding other interactions of any form. We attribute this to the fake vortex excitations, and show that an elementary excitation over the supercondutive s -wave ground state in the two-component anyon gas is always in company with the excitation of a fake vortex. The finite temperature superconductivity in the two-component anyon gas is similar with the two-dimensional superfluid. There are two KT phase transitions.
We show that a non-interacting gas of charge-
e/3 anyons with exchange statistics
θ=−π/3 can superconduct through a hidden weak-pairing mechanism. Such an anyon gas arises naturally in doped fractional Chern insulators at filling
1/3 or
2/3, where projective lattice translations enforce three degenerate anyon pockets. Exploiting this three-pocket structure, we develop a flux-attachment construction in which the average statistical flux vanishes, thereby mapping the problem to three species of composite fermions (CFs) in zero effective magnetic field. We show that the anyon statistics itself, encoded in statistical gauge field fluctuations, supplies the pairing glue and drives the CFs into a
p−ip paired state, which corresponds to a
f−if physical superconductor. The CF strong-pairing phase is adiabatically connected to Laughlin's picture of anyon superconductivity, where charge-
e/3 anyons bind into charge-
2e/3 molecules, which then lead to superconductivity. By contrast, the more natural weak-pairing phase of CFs realizes a distinct superconducting phase - its edge is characterized by a chiral central charge
c−=−1/2, in contrast to the prediction of integer
c− for the anyon superconductor based on Laughlin's picture, thereby resolving the discrepancy between previous theories and recent numerical results. Our theory provides a natural framework for understanding superconductivity near fractional Chern insulators, as observed in recent experiments. Finally, we discuss extensions of our theory that predict new chiral superconductors adjacent to FCIs at other fillings.
We have studied the ground-state (GS) properties of fractional-statistics particles (anyons) by exact numerical solution for small systems. We have seen in our results (depending on the choice of parameters for the Hamiltonian) the fractional-statistics analogs of a number of well-known ground states for conventional particles, including particularly the quantum Hall state (which we relate to the observed ‘hierarchy’ states), and a superconducting GS, the latter being formed (for half-statistics particles or semions) by pairing. We suggest possible experimental signatures for distinguishing semion from fermion superconductors. We also directly test the ‘mean-field theory’ of statistics against our exact results, and find it to treat GS properties well.