Superconductivity in the t-J model is studied by extending the recently introduced extremely correlated fermi liquid theory. Exact equations for the Greens functions are obtained by generalizing Gor’kov’s equations to include extremely strong local repulsion between electrons of opposite spin. These equations are expanded in a parameter λ representing the fraction of double occupancy, and the lowest order equations are further simplified near Tc, resulting in an approximate integral equation for the superconducting gap. The condition for Tc is studied using a model spectral function embodying a reduced quasiparticle weight Z near half-filling, yielding an approximate analytical formula for Tc. This formula is evaluated using parameters representative of single layer High-Tc systems. In a narrow range of electron densities that is necessarily separated from the Mott–Hubbard insulator at half filling, we find a typical Tc ∼ 102 K.
Superconductivity in the t-J model is studied by extending the recently introduced extremely correlated fermi liquid theory. Exact equations for the Greens functions are obtained by generalizing Gor’kov’s equations to include extremely strong local repulsion between electrons of opposite spin. These equations are expanded in a parameter λ representing the fraction of double occupancy, and the lowest order equations are further simplified near Tc, resulting in an approximate integral equation for the superconducting gap. The condition for Tc is studied using a model spectral function embodying a reduced quasiparticle weight Z near half-filling, yielding an approximate analytical formula for Tc. This formula is evaluated using parameters representative of single layer High-Tc systems. In a narrow range of electron densities that is necessarily separated from the Mott–Hubbard insulator at half filling, we find a typical Tc ∼ 102 K.
We present detailed results from a recent microscopic theory of extremely correlated Fermi liquids, applied to the t-J model in two dimensions, developed recently by Shastry [Phys. Rev. Lett. 107, 056403 (2011); Phys. Rev. B 87, 125124 (2013)]. The second-order theory in the parameter $\ensuremath{\lambda}$, related to the density, is argued to be quantitatively valid in the overdoped regime for $0\ensuremath{\le}n\ensuremath{\lesssim}0.75$, with n denoting the particle density. The calculation involves the self-consistent solution of equations for an auxiliary Fermi liquid Green's function and an adaptive spectral weight. We present numerical results at low as well as high T, at various low to intermediate densities in the normal phase, using a minimal set of band parameters relevant to the cuprate superconductors. We display the momentum space occupation function mk, energy dispersion curves locating the peaks of spectral functions, the optical conductivity, relaxation rates for quasiparticles, and the electronic spectral functions on an absolute scale. The line shapes have an asymmetric shape and a broad background that is also seen in experiments, and our calculations validate approximate recent versions of the theory. The results also display the experimentally noted high-energy kink and provide an in-depth understanding of its origin and dependence on band parameters.
Extreme correlations arise as the limit of strong correlations, when the local interaction constant U goes to infinity. This singular limit transforms canonical fermions to noncanonical Hubbard-type operators with a specific graded Lie algebra replacing the standard anticommutators. We are forced to deal with a fundamentally different and more complex lattice field theory. We study the $t\text{\ensuremath{-}}J$ model, embodying such extreme correlations. We formulate the picture of an extremely correlated electron liquid, generalizing the standard Fermi liquid. This quantum liquid breaks no symmetries, and has specific signatures in various physical properties, such as the Fermi-surface volume and the narrowing of electronic bands by spin- and density-correlation functions. We use Schwinger's source field idea to generate equations for the Green's function for the Hubbard operators. A local (matrix) scale transformation in the time domain to a quasiparticle Green's function is found to be optimal. This transformation allows us to generate vertex functions that are guaranteed to reduce to the bare values for high frequencies, i.e., are ``asymptotically free.'' The quasiparticles are fractionally charged objects, and we find an exact Schwinger-Dyson equation for their Green's function, i.e., the self-energy is given explicitly in terms of the singlet and triplet particle-hole vertex functions. We find a hierarchy of equations for the vertex functions, and further we obtain Ward identities so that systematic approximations are feasible. An expansion in terms of the density of holes measured from the Mott Hubbard insulating state follows from the nature of the theory. A systematic presentation of the formalism is followed by some preliminary explicit calculations. We find a d-wave superconducting instability at low T that formally resembles that found in the resonating valence-bond theory, but with a much reduced Tc.
Recently a new phenomenological Hamiltonian was proposed to describe the superconducting cuprates in which correlations and on-site Coulomb repulsion are introduced by partial Gutzwiller projection. This Gossamer Hamiltonian has an exact ground state and differs from the t-J and Hubbard Hamiltonians in possessing a powerful attractive interaction among electrons responsible for Cooper pairing in the d-wave channel. It is a faithful description for a superconductor with strong on-site electronic repulsion. The superconducting tunneling gap remains intact and despite on-site repulsion. Near half-filling the Gossamer superconductor with strong repulsion has suppressed photoemission intensities and superfluid density, is unstable toward an antiferromagnetic insulator and possesses an incipient Mott-Hubbard gap. The Gossamer technique can be applied to metallic ground states thus possibly serving as an apt description of strongly correlated metals. Such a Gossamer metallic phase, just as the Gossamer superconducting one, becomes arbitrarily hard to differentiate from an insulator as one turns the Coulomb correlations up near half-filling. Both the metallic and superconducting states undergo a quantum phase transition to an antiferromagnetic insulator as one increases the on-site Coulomb repulsion. In the Gossamer model we reach the critical point at half-filling by fully projecting out the double occupancy. Such a critical point might be the Anderson Resonating Valence bond state.
We present the detailed formalism of the extremely correlated Fermi liquid theory, developed for treating the physics of the $t\ensuremath{-}J$ model. We start from the exact Schwinger equation of motion for the Green's function for projected electrons, and develop a systematic expansion in a parameter $\ensuremath{\lambda}$, relating to the double occupancy. The resulting Green's function has a canonical part arising from an effective Hamiltonian of the auxiliary electrons, and a caparison part playing the role of a frequency-dependent adaptive spectral weight. This adaptive weight balances the requirement at low $\ensuremath{\omega}$ of the invariance of the Fermi volume, and at high $\ensuremath{\omega}$ of decaying as $\frac{{c}_{0}}{i\ensuremath{\omega}}$, with a correlation-depleted c0<1. The effective Hamiltonian Heff describing the auxiliary fermions is given a natural interpretation with an effective interaction Veff containing both the exchange Jij and the hopping parameters tij. It is made Hermitian by adding suitable terms that ultimately vanish, in the symmetrized theory developed in this paper. Simple but important shift invariances of the $t\ensuremath{-}J$ model are noted with respect to translating its parameters uniformly. These play a crucial role in constraining the form of Veff and also provide checks for further approximations. The auxiliary and physical Green's function satisfy two sum rules, and the Lagrange multipliers for these are identified. A complete set of expressions for the Green's functions to second order in $\ensuremath{\lambda}$ is given, satisfying various invariances. A systematic iterative procedure for higher order approximations is detailed. A superconducting instability of the theory is noted at the simplest level with a high transition temperature.
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.
We present the results for the low-energy properties of the infinite-dimensional $t\text{\ensuremath{-}}J$ model with J=0, using $O({\ensuremath{\lambda}}^{2})$ equations of the extremely correlated Fermi liquid formalism. The parameter $\ensuremath{\lambda}\ensuremath{\in}[0,1]$ is analogous to the inverse spin parameter 1/(2S) in quantum magnets. The present analytical scheme allows us to approach the physically most interesting regime near the Mott insulating state $n\ensuremath{\lesssim}1$. It overcomes the limitation to low densities $n\ensuremath{\lesssim}0.7$ of earlier calculations, by employing a variant of the skeleton graph expansion, and a high-frequency cutoff that is essential for maintaining the known high-T entropy. The resulting quasiparticle weight Z, the low $\ensuremath{\omega},T$ self-energy, and the resistivity are reported. These are quite close at all densities to the exact numerical results of the $U=\ensuremath{\infty}$ Hubbard model, obtained using the dynamical mean field theory. The present calculation offers the advantage of generalizing to finite T rather easily, and allows the visualization of the loss of coherence of Fermi liquid quasiparticles by raising T. The present scheme is generalizable to finite dimensions and a nonvanishing J.