The electronic $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U\text{\ensuremath{-}}J$ model for the $\mathrm{Cu}{\mathrm{O}}_{2}$ plane in high-${T}_{c}$ superconducting cuprates is elaborated. It incorporates both charge transfer $U$ and exchange correlation $J$ with direct $t$ and diagonal ${t}^{\ensuremath{'}}$ hopping. Making use of the Yang concept of the off diagonal long range order (ODLRO) in the reduced density matrix of the electronic system we study the loss of quantum wave coherence (and thereby superconductivity) due to the Coulomb interaction. To this aim the second-quantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. As a result the electrons emerge as composite particles consisting of of spin-carrying neutral fermions and charged topological bosons in a form of ``flux tubes'' with the quantum phase variable dual to the local electron density. The charge carrying bose field forms a collective variable representing the correlated ``sliding'' motion of the entire electronic system. We found that the spin gap $\ensuremath{\Delta}$ which appears in one-particle spectrum is the result of valence bond pairing of neutral fermions without ODLRO and deduce the solution for $\ensuremath{\Delta}$ with ``$d$-wave'' symmetry. In the bosonic sector we derive the effective quantum rotor model along with microscopic phase stiffnesses setting the energy scale for superconductivity and point out the special role played by the diagonal hopping ${t}^{\ensuremath{'}}$. We explore the relationship of the quantum rotor phase model and the Bose-Hubbard system emphasizing the ring topology of quantum phase variable. Furthermore, we obtain the ground state phase diagrams which facilitate the Mott insulator-superconductor transition scenario. We elucidate the role of the chemical potential (doping) and the huge degeneracy of the effective bosonic system in achieving ODLRO with long-range phase coherence in the presence of large Coulomb repulsion. Finally, we discuss a number of relevant issues pertaining to the the present work which include: theoretical aspects of the hierarchy of the energy scales in cuprates, ``spin-charge separation'' concepts, ``local-pair'' and ``phase-fluctuation'' scenarios as well as experimental findings related to the role of the ``$d$-wave'' symmetry of the spin gap and diagonal hopping ${t}^{\ensuremath{'}}$ in cuprates.
The electronic t\ensuremath-t\ensuremath′\ensuremath-U\ensuremath-J model for the CuO2 plane in high-Tc superconducting cuprates is elaborated. It incorporates both charge transfer U and exchange correlation J with direct t and diagonal t\ensuremath′ hopping. Making use of the Yang concept of the off diagonal long range order (ODLRO) in the reduced density matrix of the electronic system we study the loss of quantum wave coherence (and thereby superconductivity) due to the Coulomb interaction. To this aim the second-quantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. As a result the electrons emerge as composite particles consisting of of spin-carrying neutral fermions and charged topological bosons in a form of flux tubes'' with the quantum phase variable dual to the local electron density. The charge carrying bose field forms a collective variable representing the correlated sliding'' motion of the entire electronic system. We found that the spin gap \ensuremathΔ which appears in one-particle spectrum is the result of valence bond pairing of neutral fermions without ODLRO and deduce the solution for \ensuremathΔ with $d$-wave'' symmetry. In the bosonic sector we derive the effective quantum rotor model along with microscopic phase stiffnesses setting the energy scale for superconductivity and point out the special role played by the diagonal hopping ${t}^{\ensuremath{'}}$. We explore the relationship of the quantum rotor phase model and the Bose-Hubbard system emphasizing the ring topology of quantum phase variable. Furthermore, we obtain the ground state phase diagrams which facilitate the Mott insulator-superconductor transition scenario. We elucidate the role of the chemical potential (doping) and the huge degeneracy of the effective bosonic system in achieving ODLRO with long-range phase coherence in the presence of large Coulomb repulsion. Finally, we discuss a number of relevant issues pertaining to the the present work which include: theoretical aspects of the hierarchy of the energy scales in cuprates, spin-charge separation'' concepts, local-pair'' and phase-fluctuation'' scenarios as well as experimental findings related to the role of the ``d-wave'' symmetry of the spin gap and diagonal hopping t\ensuremath′ in cuprates.
In the generalized-tJ model the effect of a large local Coulomb repulsion is accounted for by restricting the Hilbert space to states with at most one electron per site. In this case the electronic system can be viewed in terms of holes hopping in a lattice of correlated spins, where holes are the carriers doped into the half-filled Mott insulator. To explicitly capture the interplay between the hole dynamics and local spin correlations we derive a formulation of the generalized-tJ model where doped carrier operators are used instead of the original electron operators. This ``doped carrier'' formulation provides a starting point to address doped spin systems, and we use it to develop a fully fermionic, mean-field description of doped Mott insulators. This mean-field approach reveals a mechanism for superconductivity---namely, spinon-dopon mixing---and we apply it to the $t{t}^{\ensuremath{'}}{t}^{\ensuremath{''}}J$ model as of interest to high-temperature superconductors. In particular, we use model parameters borrowed from band calculations and from fitting angle-resolved photoemission spectroscopy data to obtain a mean-field phase diagram that reproduces semiquantitatively that of hole- and electron-doped cuprates. The mean-field approach hereby presented accounts for the local antiferromagnetic and d-wave superconducting correlations which, we show, provide a rational for the role of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$ in strengthening superconductivity as expected by experiments and other theoretical approaches. As we discuss how t, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$ affect the phase diagram, we also comment on possible scenarios to understand the differences between as-grown and oxygen-reduced electron-doped samples.
The concept of topological excitations and the related ground state degeneracy are employed to establish an effective theory of the superconducting state evolving from the Mott insulator for high-Tc cuprates. The theory includes the effects of the relevant energy scales with the emphasis on the Coulomb interaction U governed by the electromagnetic U(1) compact group. The results are obtained for the layered $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}_{\ensuremath{\perp}}\ensuremath{-}U\ensuremath{-}J$ system of strongly correlated electrons relevant for cuprates. Casting the Coulomb interaction in terms of composite-fermions via the gauge flux attachment facility, we show that instanton events in the Matsubara ``imaginary time,'' labeled by topological winding numbers, are essential configurations of the phase field dual to the charge. This provides a nonperturbative concept of the topological quantization and displays the significance of discrete topological sectors in the theory governed by the global characteristics of the phase field. We show that for topologically ordered states these quantum numbers play the role of an order parameter in a way similar to the Landau order parameter for conventionally ordered states. In analogy to the usual phase transition that is characterized by a sudden change of the symmetry, the topological phase transitions are governed by a discontinuous change of the topological numbers signaled by the divergence of the zero-temperature topological susceptibility. This defines a quantum criticality ruled by topologically conserved numbers rather than the Landau principle of the symmetry breaking. We show that in the limit of strong correlations topological charge is linked to the average electronic filling number and the topological susceptibility to the electronic compressibility of the system. We exploit the impact of these nontrivial U(1) instanton phase field configurations for the cuprate phase diagram which displays the ``hidden'' quantum critical point covered by the superconducting lobe in addition to a sharp crossover between a compressible normal ``strange metal'' state and a region characterized by a vanishing compressibility, which marks the Mott insulator. Finally, we argue that the existence of robust quantum numbers explains the stability against small perturbation of the system and attributes to the topological ``quantum protectorate'' as observed in strongly correlated systems.
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.
High temperature superconductivity in copper oxides (cuprates) represents one of the most striking phenomena in condensed matter physics. These materials host superconducting states with the highest known critical temperatures, emerging upon doping antiferromagnetic insulator compounds. The antiferromagnetic (AF) insulating and superconducting (SC) phases defy standard descriptions in terms of band picture and conventional phonon-mediated superconductivity. In fact, these phenomena are genuine signatures of strong electronic correlations which hinder effective single-particle descriptions of the many-electron system. After 40 years since their discovery, it is widely accepted that the Hubbard model on the square lattice provides a minimal description of the low-energy electronic properties of cuprates. The model describes the competition between the electronic kinetic energy, modelled by nearest-neighbours hopping t, and the Coulomb repulsion, modelled by a purely local interaction potential U. At half-filling, this competition results in the so-called "Mott localisation" phenomenon, which distinguishes the metallic phase for U/t « 1 from an insulating phase for U/t » 1. The latter is characterised by the almost complete localisation of electrons due to the high energetic cost of hopping towards occupied sites. Interestingly, this basic competition is intimately related to AF ordering and, upon doping, to the formation of the SC state with characteristic d-wave symmetry. The fact that a purely repulsive interaction U is responsible for the formation of a SC state is remarkable, being superconductivity typically associated with an effective attractive interaction. In fact, a d-wave SC state can also be obtained, in general, by considering an attractive potential between electrons of neighbouring sites. Motivated by this observation, in this thesis, we study a modified version of the Hubbard model, dubbed as the "extended Hubbard model", in which we include a nearest-neighbours attractive potential -V. The general goal of this work is to explore the interplay between the purely local repulsion and the non-local attraction. This is not only motivated by the fact that both interactions are, in principle, responsible for d-wave superconductivity, but also by recent experimental observations which strongly suggest that the term -V needs to be included in order to accurately describe the magnetic spectra of some specific compounds. In this thesis, we study the extended Hubbard model using the Hartree-Fock method. Namely, we approximate the ground state by implementing a variational search in the subspace of the many-body Hilbert space of gaussian states. Despite its simplicity, the Hartree-Fock method provides a powerful framework to study competition among different phases, originating from different symmetry breaking mechanisms. In particular, we study the competition between three different phases: the normal phase, characterised by no symmetry breaking, the AF phase, characterised by a lowering of SU(2) symmetry, and the SC phase, characterised by the breaking of U(1) charge conservation. This thesis work led to a number of original results, which can be summarised by the general observation that the non-local attraction -V acts cooperatively with the local repulsion U for the stabilisation of the various phases hosted by the Hubbard model. For the normal phase, we find that the non-local attraction acts as a source of Mott localisation, leading to an effective reduction of the electronic bandwidth. In particular, for values of V larger than a critical value and close to half-filling, we find a strong Mott localisation characterised by the complete flattening of the electronic bands. Interestingly, this result shows an example of Mott localisation that can be described at the level of the Hartree-Fock approximation, whereas Mott localisation in the conventional Hubbard model usually requires more sophisticated approximation schemes. For the AF phase, we restrict our analysis to the half-filled case. We find that the pairing-induced Mott localisation leads to a significant enhancement of the magnetisation, primarily originating from the local repulsion U. Finally, we focus on the superconducting phase and derive self-consistency equations in both singlet and triplet Cooper pairing channels, showing how Cooper pairing can be mediated simultaneously by both the local repulsion and the non-local attraction. Therefore, we numerically investigate the singlet channel and find evidence of "symmetry mixing", namely, the formation of a superconducting gap that is neither symmetric nor anti-symmetric under rotations of π/2 in the plane. We show that symmetry mixing is connected to the emergence of a finite nematic order parameter, that signals the breaking of planar rotational symmetry. This nematic symmetry breaking results in the fact that the renormalisation of bands becomes anisotropic, producing a difference in electronic kinetics in the x and y directions. To summarise, the original results obtained in this thesis show that the non-local pairing provides a different route towards the formation of key strongly correlated phenomena, including Mott localisation and nematic superconductivity, and call for future investigation using more sophisticated methods such as, for example, the Dynamical Mean Field Theory.
Proximity to a Mott insulating phase is likely to be an important physical ingredient of a theory that aims to describe high-temperature superconductivity in the cuprates. Quantum cluster methods are well suited to describe the Mott phase. Hence, as a step toward a quantitative theory of the competition between antiferromagnetism and d-wave superconductivity in the cuprates, we use cellular dynamical mean-field theory to compute zero-temperature properties of the two-dimensional square lattice Hubbard model. The d-wave order parameter is found to scale like the superexchange coupling J for on-site interaction U comparable to or larger than the bandwidth. The order parameter also assumes a dome shape as a function of doping, while, by contrast, the gap in the single-particle density of states decreases monotonically with increasing doping. In the presence of a finite second neighbor hopping ${t}^{\ensuremath{'}}$, the zero-temperature phase diagram displays the electron-hole asymmetric competition between antiferromagnetism and superconductivity that is observed experimentally in the cuprates. Adding realistic third neighbor hopping ${t}^{\ensuremath{''}}$ improves the overall agreement with the experimental phase diagram. Since band parameters can vary depending on the specific cuprate considered, the sensitivity of the theoretical phase diagram to band parameters challenges the commonly held assumption that the doping vs Tc∕Tcmax phase diagram of the cuprates is universal. The calculated angle-resolved photoemission spectrum displays the observed electron-hole asymmetry. The tendency to homogeneous coexistence of the superconducting and antiferromagnetic order parameters is stronger than observed in most experiments but consistent with many theoretical results and with experiments in some layered high-temperature superconductors. Clearly, our calculations reproduce important features of d-wave superconductivity in the cuprates that would otherwise be considered anomalous from the point of view of the standard Bardeen--Cooper--Schrieffer approach. At strong coupling, d-wave superconductivity and antiferromagnetism naturally appear as two equally important competing instabilities of the normal phase of the same underlying Hamiltonian.
An understanding of the anomalous charge dynamics in the high-Tc cuprates is obtained based on a model study of doped Mott insulators. The high-temperature optical conductivity is found to generally have a two-component structure: a Drude-like part followed by a mid-infrared band. The scattering rate associated with the Drude part exhibits a linear-temperature dependence over a wide range of high temperature, while the Drude term gets progressively suppressed below a characteristic energy of magnetic origin as the system enters the pseudogap phase. The high-energy optical conductivity shows a resonancelike feature in an underdoped case and continuously evolves into a $1∕\ensuremath{\omega}$ tail at higher doping, indicating that they share the same physical origin. In particular, such a high-energy component is closely correlated with the $\ensuremath{\omega}$-peak structure of the density-density correlation function at different momenta, in systematic consistency with exact diagonalization results based on the $t\text{\ensuremath{-}}J$ model. The underlying physics is attributed to the high-energy spin-charge separation in the model, in which the ``mode coupling'' responsible for the anomalous charge properties is not between the electrons and some collective mode but rather between charge carriers, holons, and a topological gauge field controlled by spin dynamics, as the consequence of the strong short-range electron-electron Coulomb repulsion in the doped Mott insulator.