International Journal of Modern Physics B

At this point there is no consensus on which of the proposed theories best explains the properties of the high T_c oxides. In addition to anyon theories, there are variations on the more conventional boson-exchange pairing mechanisms which have been suggested. Analyses of experimental data such as the isotope effect, gap to T_c ratio, and heat capacity discontinuity at T_c put constraints on the applicability of these theories and limit the number of acceptable proposals.

The existence of fractional statistics in nature and the ability of 1/2 statistics to cause superconductivity are now both firmly established. The more difficult question of whether fractional statistics accounts for high–T_c behavior is still unclear. The current status of semionic pairing theory is reviewed, and the case made that the situation is encouraging, notwithstanding the slowness of our progress in developing appropriate mathematical tools. Several key questions are identified and strategies for answering them suggested.

A complete review is given of polar Kerr rotation measurements on samples of superconducting cuprates and related materials. We discuss the details of the experimental technique we have employed to discriminate against linear polarization effects, and we compare our results to those of related experiments, We show that there are possible ways to reconcile all of the experimental data available at this time, and we suggest additional crucial experiments needed to make progress towards a definitive identification of the origin of the effect.

Reflection measurements on a one-domain single crystal of YBa₂Cu₃O_7-δ and transmission measurements on single crystals of Bi₂Sr₂CaCu₂O₈ show a state of spontaneously broken symmetry below certain transition temperatures T_s somewhat above the respective superconduction temperature T_c. In both materials the sign of the effect can be controlled by an external magnetic field. In transmission measurements on films of YBa₂Cu₃O_7-δ near T_s a remanent circular dichroism is observed after applying an external magnetic field. Furthermore, the films show a Curie-Weiss type increase of magnetic circular dichroism near T_s. The magnetic field effects are considered as evidence for a state with broken time reversal symmetry below T_s.

We report the first exact finite-temperature studies of anyons. Our method is an extension to finite T of earlier numerical work with small numbers of anyons on a lattice. We study the spontaneous magnetization M₀(T), since this signature has been identified as a key signature of broken symmetry for anyon models. Our results confirm the two-phase picture suggested by earlier work: we find a low-temperature regime where M₀ is very small or zero, and a high-temperature regime where M₀ is of per particle. In the high-temperature regime we can obtain an excellent estimate of M₀(T) in the thermodynamic limit (which we will call ), since our finite-size results extrapolate smoothly with little scatter. Our values for can then be compared with the results of μSR experiments on high-temperature superconductors, which set an upper experimental bound on the internal fields from such moments. We find that in a bulk material of many planes will almost certainly give a signal well above this threshold if (and only if) the planes are ordered ferromagnetically. In the antiferromagnetic case (which is strongly favored energetically) the signal from is probably undetectable. Finally, we estimate the transition temperature T_c from our finite-size studies, obtaining a value on the order of a few hundred Kelvins.

We present a multiplane model for the anyon problem. Anyons on different planes are coupled via a Coulomb–type interaction. We solve exactly finite clusters and show that the "antiferromagnetic" order of the chirality is favored for both attractive and repulsive interplane coupling. We also discuss a simple model that can be treated analytically and that has the same qualitative behavior as the exact results. Our results apply to the low density limit, in which finite currents exist in each plane. These currents also occur in the high temperature (nonsuperfluid) phase of the anyon system, and so our results should apply also to that regime.

μSR is one of the experimental techniques used to search for manifestations of the broken and symmetries predicted by some theories of highly correlated electron systems. When and are broken, the screening of the positively charged muon by the electron system generates a magnetic field in addition to any intrinsic magnetic field that may be present. We estimate the magnetic field that is induced at a muon site when the muon is screened by the fractional statistics gas. The value depends on the distance between the muon and the anyon plane, and a simple theoretical approach to the calculation of the minimum energy muon sites is outlined and some results for Bi₂Sr₂CaCu₂O₈ and YBa₂Cu₃O₇ are presented.

A study of the Meissner effect and the plasmon spectrum for an interacting anyon system is presented. It is shown that the Meissner effect is not sensitive to, and still persists in, the presence of repulsive Coulomb interaction among anyons. Within the random phase approximation we show that the collective excitation of the system in the long-wavelength limit corresponds to the plasmon mode of a two dimensional interacting electron gas and it does not depend on the statistics associated with the anyons. There is a roton minimum of the collective mode at a finite value of the wavevector. A similar analysis on an array of layered anyon systems is also reported.

With the use of a self-consistent mean-field theory, where the anyon gauge field is approximated by its expectation value and the Coulomb interaction is calculated in the Hartree approximation, we have calculated the coupled acoustic anyon plasmon modes for an alternating-layered interacting anyon gas structure. In a single-layer system, we obtain a phonon mode for a noninteracting anyon gas. This is different from the result given by the random- phase approximation. When the Coulomb interaction is included, this phonon mode is suppressed. In a multi-layered system, we get an acoustic anyon plasmon mode with its sound velocity renormalized by the Coulomb interaction. The uniform liquid ground state is shown to be stable against solid crystallization.

The ground and the excited states of a two-dimensional anyon system are studied via finite-size numerical calculations and single-mode approximation in a spherical geometry. It is shown that Laughlin's mean field theory gives an accurate description of the ground state properties. We find, consistent with earlier analytic theories, that the collective mode of the system exhibits a linear dispersion in the long wavelength limit, and predict a novel roton minimum at a finite value of the wavenumber q. We also discuss the consequences on the anyon spectrum of having a model repulsive interaction between the particles.

The recently discovered solitons in Chern–Simons/anyon systems me reviewed.

The long distance physics of the Hall fluid and the anyon superfluid may be described by effective field theories constructed out of the Chern-Simons term. These theories are topological in character and may be of deep mathematical significance. Various physical quantities such as the quantum numbers of the quasiparticles are determined by a matrix appearing in the Chern-Simons term.

We review the topological orders in strongly correlated quantum liquids. The characterization of the topological orders through ground state degeneracy, non-Abelian Berry's phases and edge excitations are discussed.

The structure of braid groups on topologically nontrivial surfaces is reviewed. The physical meaning of the braid relations and their implications on quantum mechanical properties of anyonic quasiparticles are discussed. These include not only the exotic statistics of anyons but also the Aharonov-Bohm effect for the anyons on a surface with holes. Several results on the novel properties of anyons or their states, which were previously derived by microscopic considerations, are reproduced by this seemingly kinematic and topology-dependent braid group analysis. It is suggested that in the thermodynamic limit, the global excitations in a system on a surface of nontrivial topology do not interfere with properties of local anyonic quasiparticles.

We seek the analytic eigenstates for an N-identical anyons in an external harmonic oscillator potential.

We construct exact many-body eigenstates of both energy and angular momentum for the N-anyon problem in an external magnetic field. Such states span the full ground-state eigenspace and have a natural representation in terms of the Fubini-Veneziano vertex operators of string theory.

I describe how, starting from fundamental symmetry principles, a relativistic theory of anyons can be formulated.

I review the concept of statistics transmutation in two dimensions and apply it to the understanding of Fractional quantum-Hall effect and anyon superconductivity. A contrast of the electromagnetic properties of an ordinary 2D superconductor, a quantum-Hall liquid and an anyon superconductor is presented. The relevance of the anyon model to copper-oxide superconductors is also discussed.

In the framework of collective field theory, We apply the Chern-Simon field theory treatment to the constraint equation for the lowest Landau level to investigate the generic properties for the quasi-particles of the FQH system. It shows a transparent connection to the Laughlin's wave functions. If we take an average over the wave functional for the constraint equation, the resulted equation can be interpreted as the vortex equation for the fractionally charged quasi-particles. Introducing a generalized ρ (density)-ϑ (phase) transformation, not only the fractional statistics and the hierarchy scheme can be drawn from the constraint equation, it also gives rise an interesting picture that vortices condense as a Halperin like wave fuction on a Laughlin like background condensate of ν=1/m.

Ideal anyons with statistics ν in a strong magnetic field are studied by means of a similarity transformation. The ground state exhibits "integer" quantum Hall effect at filling factor 1/ν with quasiparticle excitations of charge q/ν and statistics -1/ν. Certain electron FQH states can be considered as realization of this, for example, the sequence 2/5, 3/7, … hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum Hall hierarchy.

The ν=1/q quantum Hall states are characterized by a perfect cancellation of the magnetic (Aharonov-Bohm) and the Fermi statistical phases. The lowering of the ground state energy due to this phase cancellation is illustrated by using a partial analogy with the Helium II problem and via a path integral formulation. For general filling fractions ν=p/q, a cluster of p electrons is considered instead of a single electron. There is again a perfect cancellation of the effective Aharonov-Bohm phase and the statistical phase of the clusters provided that q is odd.

Regarding electrons as anyons of index α_s pierced with -(m+α_s) flux quanta per particle, and letting the mean field of these fluxes cancel the external magnetic field B, we obtain the filling factor ν=1/(m+α_s), where m must be odd. Demanding the resulting system of anyons to exhibit "anyon supercanductivity", we obtain α_s=±(1-q/n) where q is odd, and n>q is relatively prime to q. For q=1 we recover a formula due to Jain, and resolve the mystery why, for a state with ν=n/(2pn±1)<1 he requires use of the statistical correlation of n filled Landau levels. For q=3,5,⋯, we obtain the fractions 4/11, 4/13, 5/13, etc., which are missing from Jain's list. Thus this non-heirarchical approach to the non-1/m fractional quantum Hall effect has the strengths of Jain's composite-fermion approach, but not its (potential) weaknesses.

The hard core boson gas in (2+1)-dimensions in the presence of a Chern-Simons term is considered, we find that the long wave description is given in terms of Coulomb (longitudinal) and transversal electrodynamics. The longitudinal part is described in terms of electric charges which carry vorticity. When binding of electrical charges occurs the transversal part is described by a photon field and corresponds to a superfluid. Using the same language we find that a spinless fermion is described by a massive photon. We use these concepts to understand the relation between the negative-U Hubbard model in the presence of a magnetic field, and vortices which carry electric charge induced by doping in the repulsive Hubbard model.

We show how the topological Chern-Simons term in three dimensions and the θ-term, in four dimensions can be dynamically generated due to quantum fluctuations in Nambu-Jona-Lasinio models. These results give a vivid demonstration of hidden local symmetry at work. The fluctuation-induced Chern-Simons term provides a natural way to incorporate fractional statistics and thus a more direct connection with anyon superconductivity. Possible phenomenological implications of this approach in particle physics are also discussed.

A topological argument similar to that of Leinaas and Myrheim implies that a non-trivial statistical phase factor can also arise from exchanging twice a pair of distinguishable particles in two dimensions. Some general properties of this phase factor are deduced. Wilczek's model for anyons and the Laughlin theory for the quasiparticles in the fractional quantum Hall ground states are examined in light of these properties, and the former is generalized for systems containing many species of anyons. The statistical properties of holons and spinons relative to each other are briefly discussed as an example.

A Maxwell system of differential forms induces a coarse topology on a space time variety. This topology can be refined by certain extremal constraints of domain closure that lead to three topologically distinct, reactive impedances, Z1, Z2, Z3. These non-dissippative impedances are rational fraction ratios of topological invariants. The closure conditions induce physical constraints that permit Z1 to be identified with ordinary type I and II superconductive domains, and Z3 with the fractional quantum Hall effect. Only Z2 admits the time-reversal and parity symmetry breaking associated with Anyon theories.

We study a general class of variational wavefunctions for strongly correlated lattice fermions. The wavefunctions considered here automatically satisfy local constraints of no double occupancy and include correlations between opposite spin particles in a very physical way. We calculate the energy and correlation functions for the one dimensional U=∞ Hubbard model, where a comparison with exact results is made. We briefly report on preliminary results for the t-J model.

A path-integral formalism of the Hubbard model is used to study the one-dimensional large-U case. It is shown that the bare electron (hole) becomes a composite particle of two decoupled excitations, holon and spinon, together with the non-local string fields. Various correlation functions are analytically derived. The results strongly suggest a U*=∞ fixed point of Hubbard model which is distinct from the weak coupling case.

We studied the magnetic properties of the ring of a Josephson junctions in a magnetic field at different capacitance of superconducting regions. We found that the capacitance which is the characteristic of the size of the superconducting region, does not change qualitatively the magnetic properties of the ring of junctions. At half-quantum of the flux through the ring there appears the magnetic phase transition which is characterised by the singularity in magnetization and in sucseptibility. At the transition there is the trapping of the one quantum of the flux by the ring. The interpretation of experiments, describing the inverse kinetic inductance as a function of a flux per plaquette of a lattice of two-dimensional arrays of the Josephson junctions is proposed. The new singularities in this dependency are predicted.