The present state of the gauge theory of high-temperature superconductivity in strongly-correlated two-dimensional spin systems is reviewed. Basic ideas on the statistics of elementary excitations in spatially two-dimensional systems are presented and are used in an analysis of the structure of the energy spectrum and the form of the wave function of a set of anyon quasiparticles in both the long-wave continuous and lattice limits. The continuous and lattice theories are used to classify the phase states, and the hierarchy of phase transitions is described in terms of the topological quantum field theory. Thermodynamic and electrodynamic properties of anyon systems are described, and the experimental consequences of the Chern–Simons theory of high-Tc superconductivity are discussed.
The present state of the gauge theory of high-temperature superconductivity in strongly-correlated two-dimensional spin systems is reviewed. Basic ideas on the statistics of elementary excitations in spatially two-dimensional systems are presented and are used in an analysis of the structure of the energy spectrum and the form of the wave function of a set of anyon quasiparticles in both the long-wave continuous and lattice limits. The continuous and lattice theories are used to classify the phase states, and the hierarchy of phase transitions is described in terms of the topological quantum field theory. Thermodynamic and electrodynamic properties of anyon systems are described, and the experimental consequences of the Chern–Simons theory of high-Tc superconductivity are discussed.
The principal purpose of this topical review is to single out some of the universal features of high-temperature (high-Tc) and related strongly-correlated systems, which can be compared with experiment in a quantitative manner. The description starts with the concept of exchange-interaction-mediated (real-space) pairing, combined with strong correlations among narrow band electrons, for which the reference state is that of the Mott–Hubbard insulator. The physical discussion of concrete properties relies on variational approach, starting from generalized renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and its subsequent generalization, i.e., a systematic Diagrammatic Expansion of the Variational (Gutzwiller-type) Wave Function (DE-GWF). The solution leads to the two energy scales, one involving unconventional quasiparticles close to the Fermi energy, and the other reflecting the fully correlated state, involving electrons deeper below the Fermi surface. Those two regimes are separated by a kink in the dispersion relation, which is observed in photoemission. As a result, one obtains both the doping dependent properties in the correlated state, and effective renormalized quasiparticles. The reviewed ground-state characteristics for high-Tc systems encompass high-Tc superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state. The calculated dynamic properties are: the universal Fermi velocity, the Fermi wave-vector, the effective mass enhancement, the pseudogap; and the d-wave gap magnitude. We discuss that, within the variational approach, the minimal realistic model is represented by the so-called t-J-U Hamiltonian. Inadequacy of the t-J and Hubbard models, particularly of their RMFT versions, is discussed explicitly. For heavy fermion systems, modeled by the Anderson lattice model and with the DE-GWF approach, we discuss the phase diagram, encompassing superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. The superconducting state is then a two-d-wave gap system. If the orbital degeneracy of f-electrons is included, the coexistent ferromagnetic- (spin–triplet) superconducting phases appear and match those observed for UGe2 in a semiquantitative manner. Finally, in the second part, we generalize our approach to the collective spin and charge fluctuations in high-Tc systems, starting from variational approach, defining the saddle-point state, and combining it with 1/N expansion. The present scheme differs essentially from that starting from the saddle-point Hartree–Fock approximation, and incorporating the fluctuations in the random phase approximation (RPA). The spectrum of collective spin and charge excitations is determined for the Hubbard and t-J-U models, and subsequently compared quantitatively with recent experiments. The Appendices provide formal details to make this review self-contained.
Abstract This paper considers modern ideas about the topological nature of superconductivity and the Mott–Hubbard state in strongly correlated systems. These systems are described in terms of spin and charge separation, i.e., deconfinement of spinons and holons in a gauge field. The role of orbital degrees of freedom is discussed. Analogies with topological insulators, quantum Hall effect, and Landau quantization are considered. The importance of the Chern–Simons term for the correct description of the Hubbard splitting is demonstrated.
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.
The discovery of high-Tc superconductivity in cuprates in 1986 moved strongly correlated systems from exotic worlds interesting only for pure theorists to the focus of solid-state research. In recent decades, the majority of hot topics in condensed matter physics (high-Tc superconductivity, colossal magnetoresistance, multiferroicity, ferromagnetism in diluted magnetic semiconductors, etc.) have been related to strongly correlated transition metal compounds. The highly successful electronic structure calculations based on density functional theory lose their predictive power when applied to such compounds. It is necessary to go beyond the mean field approximation and use the many-body theory. The methods and models that were developed for the description of strongly correlated systems are reviewed together with the examples of response function calculations that are needed for the interpretation of experimental information (inelastic neutron scattering, optical conductivity, resonant inelastic X-ray scattering, electron energy loss spectroscopy, angle-resolved photoemission, electron spin resonance, and magnetic and magnetoelectric properties). The peculiarities of (quasi-) 0-, 1-, 2-, and 3- dimensional systems are discussed.
Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the any- onic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts “à la Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e., the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Hal- dane’s statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former. Finally, on the purely combinatorial side, the minimal difference partition problem — partition of integers with minimal difference constraints on their parts — can also be mapped on an abstract exclusion statistics model with a constant one-body density of states, which is neither the LLL-anyon model nor the Calogero model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.