We study a generalized periodic Anderson model with on-site hybridization between wide- and narrow-band electrons and strong and local coupling with the lattice deformation. Provided that the interaction with the lattice is strong enough, the narrow-band electrons will be turned into small polarons which interact attractively with each other over short distances, leading to the formation of local pairs of narrow-band electrons. This leads to a pinning of the Fermi level which is due to the fact that narrow-band electrons exist only in pair states. By means of a generalized Schrieffer-Wolff transformation we eliminate hybridization and obtain an effective Hamiltonian which describes a contact interaction between local pairs and wide-band electrons as well as the direct hopping of local pairs and interparticle Coulomb interactions. In such a system the two types of mechanisms which can lead to superconductivity have been studied. The first one is due to direct local pair hopping and involves exclusively the narrow-band subsystem giving rise to a superconductivity analogous to superfluidity in $^{4}\mathrm{He}$. The second one is due to a contact interaction between local pairs and pairs of wide-band electrons. This leads to a superconducting state involving both subsystems where the local pairs of the narrow-band subsystem induce Cooper pairing amongst the electrons of the wide-band subsystem. Consequently, the single-particle spectrum of the wide-band electrons opens up a gap around the position of the narrow band of electrons in pair states.We study the phase diagrams and the superconducting properties of two such coupled subsystems involving bosons and fermions as a function of the position of the narrow band with respect to the wide band (or the relative concentration of narrow and wide-band electrons). We also show that the critical temperature ${T}_{c}$ and ${T}_{c}$/${E}_{g}$(0) [${E}_{g}$(0) being the energy gap] follow, in general, a strong and nonmonotonic variation. One of the most striking features of this system is that the pairs of narrow-band electrons exist in the normal phase and condense on approaching ${T}_{c}$ from above. The existence of such a narrow boson band together with a fermion band in the normal phase leads to characteristic behavior in the specific heat over temperature variation being very similar to that observed in recent studies of nonclassical superconductors.
We study a generalized periodic Anderson model with on-site hybridization between wide- and narrow-band electrons and strong and local coupling with the lattice deformation. Provided that the interaction with the lattice is strong enough, the narrow-band electrons will be turned into small polarons which interact attractively with each other over short distances, leading to the formation of local pairs of narrow-band electrons. This leads to a pinning of the Fermi level which is due to the fact that narrow-band electrons exist only in pair states. By means of a generalized Schrieffer-Wolff transformation we eliminate hybridization and obtain an effective Hamiltonian which describes a contact interaction between local pairs and wide-band electrons as well as the direct hopping of local pairs and interparticle Coulomb interactions. In such a system the two types of mechanisms which can lead to superconductivity have been studied. The first one is due to direct local pair hopping and involves exclusively the narrow-band subsystem giving rise to a superconductivity analogous to superfluidity in 4He. The second one is due to a contact interaction between local pairs and pairs of wide-band electrons. This leads to a superconducting state involving both subsystems where the local pairs of the narrow-band subsystem induce Cooper pairing amongst the electrons of the wide-band subsystem. Consequently, the single-particle spectrum of the wide-band electrons opens up a gap around the position of the narrow band of electrons in pair states.We study the phase diagrams and the superconducting properties of two such coupled subsystems involving bosons and fermions as a function of the position of the narrow band with respect to the wide band (or the relative concentration of narrow and wide-band electrons). We also show that the critical temperature Tc and Tc/Eg(0) [Eg(0) being the energy gap] follow, in general, a strong and nonmonotonic variation. One of the most striking features of this system is that the pairs of narrow-band electrons exist in the normal phase and condense on approaching Tc from above. The existence of such a narrow boson band together with a fermion band in the normal phase leads to characteristic behavior in the specific heat over temperature variation being very similar to that observed in recent studies of nonclassical superconductors.
Some ground state properties of a fermion system are investigated. An initial Hamiltonian being a generalization of the lattice Anderson model describes this system as a two-band electron-phonon one where both of electron bands are narrow (one is wider than the other). Two bands are hybridized and electrons from both of them are strongly coupled to the same local phonons. After performing the Lang–Firsov type transformation, a new polaron Hamiltonian is obtained. As a result, there appear three effective polaron-polaron interactions in this Hamiltonian and the bandwidths of both of electron bands are reduced. It is assumed that the reduction of the bandwidth of one of the bands is sufficiently strong to make this band dispersionless. Each of the interactions can lead to superconductivity on its own but only one of them is taken into account at this stage; it is the interaction between electrons from the band with a dispersion. The Bogolyubov’s mean field approach is used in order to find the spectrum of the reduced effective Hamiltonian. The effect of the hybridization on the superconducting gap and the relationship between the critical hybridization and the average number of electrons from the wider band per lattice site are investigated.
Using the generalized DMFT+Σ approach, we study the influence of disorder on single-particle properties of the normal phase and the superconducting transition temperature in the attractive Hubbard model. A wide range of attractive potentials U is studied, from the weak coupling region, where both the instability of the normal phase and superconductivity are well described by the BCS model, to the strong-coupling region, where the superconducting transition is due to Bose-Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures much higher than the superconducting transition temperature. We study two typical models of the conduction band with semi-elliptic and flat densities of states, respectively appropriate for three-dimensional and two-dimensional systems. For the semi-elliptic density of states, the disorder influence on all single-particle properties (e.g., density of states) is universal for an arbitrary strength of electronic correlations and disorder and is due to only the general disorder widening of the conduction band. In the case of a flat density of states, universality is absent in the general case, but still the disorder influence is mainly due to band widening, and the universal behavior is restored for large enough disorder. Using the combination of DMFT+Σ and Nozieres-Schmitt-Rink approximations, we study the disorder influence on the superconducting transition temperature T c for a range of characteristic values of U and disorder, including the BCS-BEC crossover region and the limit of strong-coupling. Disorder can either suppress T c (in the weak-coupling region) or significantly increase T c (in the strong-coupling region). However, in all cases, the generalized Anderson theorem is valid and all changes of the superconducting critical temperature are essentially due to only the general disorder widening of the conduction band.
In cuprate superconductors, superconductivity is consistently accompanied by antiferromagnetism. This raises the question of a potential causal link between superconductivity and antiferromagnetic mechanisms. In this study, we consider the non-local Coulomb interaction and solve the single-particle Green function exactly. The exact solution shows the existence of strong inter-spin correlations in the absence of superexchange, eliminating the directional degrees of freedom for spin quantization in the Hatsugai-Kohmoto (HK) model. Strong Coulomb interactions force the energy band splitting into infinitely equidistant sub-bands weighted by a Poisson distribution, exhibiting boson properties (incoherent pairs of fermions) and non-Fermi liquids. Attractive pairing interactions caused by electron-phonon coupling open a pairing gap at the Fermi surface. Based on the Pauli exclusion principle, paired particles with the same spin quantization direction completely occupy the state below the gap, resulting in pairing insulators, superconducting instability, and especially antiferromagnetism. A pseudogap occurs when the higher split sub-band overlaps in energy with the lowest sub-band. The critical temperature Tc of superconductivity exhibits a multi-dome structure and peaks around the maximum density of states. The maximum Tc depends on splitting sub-bands and particle occupancy, consistent with experiments. This study concludes that the non-local effect preserves the Hubbard and HK models' characteristics while renormalizing the system to antiferromagnetism in pairing insulators and Pauli paramagnetism in weak semimetals.
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.
Two-band models (one wide band and one narrow band close to the Fermi energy) and local-pair models (periodic Anderson model) of high- T c superconductors give rise to the same equilibrium thermodynamics despite the different physical pictures underlying the models. The low-energy excitations in these models are, however, not of the same character, which leads to a qualitative difference in their non-equilibrium properties. In particular, this is true for the interband optical absorption at threshold. Using the Golden rule for absorption and a mean-field treatment of the superconducting state, we calculate the threshold absorption at zero temperature for three different cases: a two-band model without additional correlation besides the interband interaction between Cooper pairs; a two-band model with attractive correlation (negative- U ) between electrons on the same site in the narrow band; and a local-pair model with the same correlation as in the previous model, but with a pair interaction due to the hybridization (in second order) of the wide band and the local states. In the two latter models (which are closely related), we assume that the local-pair correlation energy is much higher than T c . In the uncorrelated case the absorption has a square-root singularity at threshold, which is due to the diverging density of states of the one-particle excitations. In the negative- U correlated cases no single-particle transitions are possible at threshold, and only direct two-photon two-particle processes are possible. Further, in the local-pair model the hybridization can act together with a photon, giving an additional one-photon two-electron threshold in the absorption.
A new two-band polaronic Hamiltonian is derived from a generalized periodic Anderson model. This is achieved by making use of a new generalized Lang–Firsov transformation on the periodic Anderson model completed with new terms representing the interactions between local phonons and both single electrons and pairs of them with opposite spins residing on the same lattice site. The novelty consists in the appearance of three- and four-fermion terms in the resulting Hamiltonian. The interaction between phonons and pairs of electrons with opposite spins was introduced at first time by Hirsch in order to describe the weakening of the Coulomb repulsion acting between two electrons with opposite spins occupying the same atomic orbital observed in experiments. The three-fermion terms represent the local interactions between a single polarons from one band and a pair of them with opposite spins from the other one. Whether it is attractive or repulsive depends on the signs of couplings of phonons to electron pairs. The four-fermion term stands for the local attraction between pairs of electrons with opposite spins from two bands. In the case of presence of such an interaction, the formation of local fermion quartets is possible. A simplified version of the generalized Hamiltonian is investigated in the end of the paper. The influence of the three- and the four-fermion terms on the properties of the two-band fermion system with two order parameters is examined. Expressions for the order parameters at zero temperatures and finite ones along with the critical temperatures for two-electron subsystems have been derived. This was achieved in the atomic limit by making use of the mean field method.