A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction emerges simply by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces the large $U$ superexchange. Also derived, for $U=\pm\infty$, is the Hamiltonian to study Nagaoka ferromagnetism. In this representation, the infinite-$U$ Hubbard problem becomes elegant and easier to handle. Interestingly, the ferromagnetism in Hubbard model is found to be related to the gauge invariance of the spinless fermions. Generalization of this representation for the multicomponent fermions, a new representation for bosons, the notion of a `soft-core' fermion, and some interesting unitary transformations are introduced and discussed in the appendices.
By using variational Monte Carlo and auxiliary-field quantum Monte Carlo methods, we perform an accurate finite-size scaling of the $s$-wave superconducting order parameter and the pairing correlations for the negative-$U$ Hubbard model at zero temperature in the square lattice. We show that the twist-averaged boundary conditions (TABCs) are extremely important to control finite-size effects and to achieve smooth and accurate extrapolations to the thermodynamic limit. We also show that TABCs is much more efficient in the grand-canonical ensemble rather than in the standard canonical ensemble w…
By means of a new canonical transformation for the one-band Hubbard model at half filling we show the existence of Cooper pairs formed by strongly interacting quasiparticles.
The $t-J$ and Heisenberg models are truncated expansions of a canonically transformed Hubbard model coinciding with it at $U\to \infty$. We show that a modified canonical transformation applied to the Hubbard model leads to alternative models of the same form, but whose convergence properties with respect to the expansion are more favourable, resulting in a good description of the half-filled ground state even at $0<U\leq1$. We investigate the transformed Hamiltonian and observables for metallic and insulating variational wave-functions.
We have recently developed a canonical transformation of the Hubbard and related models, valid for systems of arbitrary size and for the full plane; this is particularly suited to study hole pairing. In this work we show that exact diagonalization results of the one band Hubbard model for small clusters with periodic boundary conditions agree well with the analytical ones obtained by means of our canonical transformation. In the presence of a pairing instability, the analytic approach allows us to identify the Cooper pairs. They are W=0 pairs, that is, singlet two-hole eigenstates of the Hubba…
We prove that the t-J-U model can be deduced from the Hubbard model at a large but finite U by a canonical transformation. We argue that the system may have a metal-insulator transition at a critical on-site Coulomb interaction whose value, however, is smaller than that in previous calculations in which the kinetic energy has a double counting. In a mean field theory and a special choice of the parameters, we show that the metallic state may be equivalent to the gossamer superconducting state proposed by Laughlin recently.
We calculate the exact analytical coefficients of the $β$ expansion of the grand-canonical partition function of the unidimensional Hubbard model up to order $β^5$, using an alternative method, based on properties of the Grassmann algebra. The results derived are non-perturbative and no restrictions on the set of parameters that characterize the model are required. By applying this method we obtain analytical results for the thermodynamical quantities, in the high-temeprature limit, for arbitrary density of electrons in the unidimensional chain.
We propose a canonical tranformation approach to the effective interaction $W_{eff}$ between two holes, based on the three-band Hubbard model but ready to include extra interactions as well. An effective two-body Hamiltonian can in principle be obtained including any kind of virtual intermediate states. We derive the closed-form analytic expression of the effective interaction including 4-body virtual states, describing the exchange of an electron-hole pair to all orders. The resulting integral equation, valid for the full plane, leads to a Cooper-like instability of the Fermi liquid. The two-…
A novel canonical transformation is offered as the mean for studying properties of a system of strongly correlated electrons. As an example of the utility of the transformation, it is used to demonstrate the existence of a quantum phase transition in a Hubbard model on a square lattice. An Appendix presents two cases with a negative result.
A microscopic theory for electronic spectrum of the CuO2 plane within an effective p-d Hubbard model is proposed. Dyson equation for the one-electron Green function in terms of the Hubbard operators is derived which is solved self-consistently for the self-energy evaluated in the noncrossing approximation. Electron scattering on spin fluctuations induced by kinematical interaction is described by a model dynamical spin susceptibility. Doping and temperature dependence of electron dispersions, spectral functions, the Fermi surface and the coupling constant are studied.
Using the strong coupling diagram technique, we study the one-band repulsive Hubbard model on a two-dimensional square lattice in a wide range of chemical potentials $μ$. Infinite sequences of diagrams describing interactions of electrons with spin and charge fluctuations are taken into account. At low temperatures, regions of the negative electron compressibility are found for $μ<0$ and $μ>U$, where $U$ is the Hubbard repulsion. For even lower temperatures, in the neighborhood of these regions, states with the phase separation are revealed.
The momentum, electronic density, spin density, and interaction dependences of the exponents that control the $(k,ω)$-plane singular features of the $σ=\uparrow,\downarrow$ one-electron spectral functions of the 1D Hubbard model at finite magnetic field are studied. The usual half-filling concepts of one-electron lower Hubbard band and upper Hubbard band are defined for all electronic density and spin density values and the whole finite repulsion range in terms of the rotated electrons associated with the model Bethe-ansatz solution. Such rotated electrons are the link of the non-perturbative …
Exact ground states of interacting electrons on the diamond Hubbard chain in a magnetic field are constructed which exhibit a wide range of properties such as flat-band ferromagnetism and correlation induced metallic, half-metallic or insulating behavior. The properties of these ground states can be tuned by changing the magnetic flux, local potentials, or electron density.
A new numerical algorithm for interacting fermion systems to treat the grand-canonical ensemble is proposed and examined by extending the path-integral renormalization group method. To treat the grand-canonical ensemble, the particle-hole transformation is applied to the Hamiltonian and basis states. In the interaction-term projection, the Stratonovich-Hubbard transformation which hybridizes up and down spin electrons is introduced. By using this method, the phase diagram of the two-dimensional Hubbard model with next-nearest-neighbor transfer is accurately determined by treating the filling-c…
The Hubbard model extended by either nearest-neighbour Coulomb correlation and/or nearest neighbour Heisenberg exchange is solved analytically for a triangle and tetrahedron. All eigenvalues and eigenvectors are given as functions of the model parameters in a closed form. The groundstate crossings and degeneracies are discussed both for the canonical and grand-canonical energy levels. The grand canonical potential and the electron occupation of the related cluster gases were calculated for arbitrary values (attractive and repulsive) of the three interaction constants. In the pure Hubbard model…
The influence of next-nearest neighbor electron hopping, $t^{\prime}$, on the polaron and bipolaron formation in a square Hubbard-Holstein model is investigated within a variational approach. The results for electron-phonon and electron-electron correlation functions show that a negative value of $t^{\prime}$ induces a strong anisotropy in the lattice distortions favoring the formation of nearest neighbor intersite bipolaron. The role of $t^{\prime}$, electron-phonon and electron-electron interactions is briefly discussed in view of the formation of charged striped domains.
Using a functional renormalization group method, we obtain the phase diagram of the two-leg ladder system within the Holstein-Hubbard model, which includes both electron-electron and electron-phonon interactions. Our renormalization group technique allows us to analyze the problem for both weak and strong electron-phonon coupling. We show that, in contrast results from conventional weak coupling studies, electron-phonon interactions can dominate electron-electron interactions because of retardation effects.
We study electron correlations in the half-filled Hubbard model on two-dimensional Penrose lattice. Applying the real-space dynamical mean-field theory to large clusters, we discuss how low-temperature properties are affected by the quasiperiodic structure. By calculating the double occupancy and renormalization factor at each site, we clarify the existence of the Mott transition. The spatially-dependent renormalization characteristic of geometrical structure is also addressed.
In cond-mat/0306131 Huang {\em et al.} reported for the first time a numerical evidence for the predominance of small phonon momenta in the electron-phonon coupling induced by electronic correlation. They report also an upturn of the electron-phonon vertex function for small momenta as function of the Hubbard repulsion $U$ for sufficient large $U$. In this comment we compare their Quantum Monte Carlo data with analytical results based on slave-boson techniques for generic $U$. We associate the upturn of the electron-phonon vertex function with the tendency towards a phase separation instabilit…
In the first paper of this series it was found that the $η$-spin 1/2 holons, spin 1/2 spinons, and $c$ pseudoparticles whose occupancy configurations describe the energy eigenstates of the one-dimensional Hubbard model emerge from the electron - rotated-electron unitary transformation. In this second paper we discuss and clarify how the relation of the electrons to the above objects can be used in a program for evaluation of finite-energy few-electron spectral functions. As a first step, here we characterize the dominant holon and spinon microscopic physical processes that originate more than …
By comparing the responses to an in-plane magnetic field near the metal-insulator transition (MIT), we find that the observed MIT in Si MOSFETs can be described by the non-perturbative Mott-Hubbard scenario. Interrelations between independent measurables are uncovered and confirmed by replotting the experimental data. A universal critical energy scale vanishing at the MIT is extracted from the experimental data and the critical exponent found.
The spinless Falicov-Kimball model on the simple cubic lattice is analyzed in the Hubbard-I and dynamical mean field (DMFT) approximations. The Matsubara and real frequency itinerant electron Green's functions, the evolution of the system with doping, and the range of phase separation are found in two approximations. At large values of the on-site Coulomb repulsion both approximations give similar results. The phase separation can be also favorable for a more general model, where heavy electrons have a finite bandwidth. This indicates that the phase separation phenomenon is an inherent feature…
In this paper we describe the pseudoparticles, holons, and spinons whose occupancy configurations describe the energy eigenstates of the one-dimensional (1D) Hubbard model in terms of rotated electrons. Rotated electrons are related to electrons by a mere unitary transformation such that rotated electron double occupation is a good quantum number. Moreover, we introduce the concepts of local pseudoparticle and effective pseudoparticle lattice. Our findings provide useful information about the relation of the exotic pseudoparticles, holons, and spinons that diagonalize the non-perturbative many…
The momentum and energy dependence of the weight distribution in the vicinity of the one-electron spectral-function singular branch lines of the 1D Hubbard model is studied for all values of the electronic density and on-site repulsion $U$. To achieve this goal we use the recently introduced pseudofermion dynamical theory. Our predictions agree quantitatively for the whole momentum and energy bandwidth with the peak dispersions observed by angle-resolved photoelectron spectroscopy in the quasi-1D organic conductor TTF-TCNQ.
We have performed extensive density matrix renormalization group (DMRG) studies of the Hubbard model on a honeycomb ladder. The band structure (with Hubbard U=0) exhibits an unusual quadratic band touching at half filling, which is associated with a quantum Lifshitz transition from a band insulator to a metal. %SAK as a function of a third-neighbor hopping parameter. For one electron per site, non-zero $U$ drives the system into an insulating state in which there is no pair-binding between added electrons; this implies that superconductivity driven directly by the repulsive electron-electron i…
We study the repulsive Hubbard model both analytically and numerically on a family of highly frustrated lattices which have one-electron states localized on isolated trapping cells. We construct and count exact many-electron ground states for a wide range of electron densities and obtain closed-form expressions for the low-temperature thermodynamic quantities. Furthermore, we find that saturated ferromagnetism is obtained only for sufficiently high electron densities and large Hubbard repulsion $U$ while there is no finite average moment in the ground states at lower densities.
We study intersite electron correlations in the half-filled Hubbard model on square lattices with periodic and open boundary conditions by means of a real-space dual fermion approach. By calculating renormalization factors, we clarify that nearest-neighbor intersite correlations already significantly reduce the critical interaction. The Mott transition occurs at U/t~6.4, where $U$ is the interaction strength and t is the hopping integral. This value is consistent with the quantum Monte Carlo results. This shows the importance of short-range intersite correlations, which are taken into account …
The iterative perturbation theory of the dynamical mean field theory is generalized to arbitrary electron occupation in case of multi-orbital Hubbard bands. We present numerical results of doubly degenerate Eg bands in a simple cubic lattice. The spectrum shows the electron ionization and affinity levels of different electron occupations. For sufficiently large Coulomb integral, a gap opens in the spectrum at integer filling of electrons and the system becomes insulator. The present scheme is easy to combine with the LSDA electronic structure theory.
When speaking about molecular electronics, the obvious question which occurs is how does one study it theoretically. The simplest theoretical model suitable for application in molecular electronics is the two dimensional Hubbard model. The aim of the present paper is to introduce this model, and give some examples of the systems which it can describe. After a short mathematically oriented discussion, it will be shown how to calculate the electrical conductivity of a particular planar system: a rectangular lattice with mutually independent conductivities along the two axes,but without using the…
By the example of the Hubbard model we analytically and numerically examine the formating and coexisting of localized electron--electron pairs (doublons) and localized electron--hole pairs (Frenkel--type excitons) . Here we demonstrate that at a variation of the on-site Coulomb repulsion U there occurs a quantum transition from the doublon to the exciton regime that is conditioned by the low-energy effective hybridization of the valence and conductivity subbands. We calculate momentum distribution functions and electronic spectrum functions for different U, which reveal topologically nontrivia…
We perform a detailed study of the phase transitions and mechanisms of electron localization in the extended Hubbard model using the dynamical cluster approximation on a $2\times 2$ cluster. We explore the interplay of charge order and Mott physics. We find that a nearest-neighbor Coulomb interaction $V$ causes "screening" effects close to the Mott phase transition, pushing the phase boundary to larger values of $U$. We also demonstrate the different effects of $V$ on correlations in metallic and insulating regimes and document the different correlation aspects of charge order and Mott states.
We obtain the phase diagram of the half-filled two-dimensional Hubbard model on a square lattice in the presence of Einstein phonons. We find that the interplay between the instantaneous electron-electron repulsion and electron-phonon interaction leads to new phases. In particular, a d$_{x^2-y^2}$-wave superconducting phase emerges when both anisotropic phonons and repulsive Hubbard interaction are present. For large electron-phonon couplings, charge-density-wave and s-wave superconducting regions also appear in the phase diagram, and the widths of these regions are strongly dependent on the p…
We analyze the ground-state properties of strongly-correlated electrons coupled with phonons by means of a generalized Gutzwiller wavefunction which includes phononic degrees of freedom. We study in detail the paramagnetic half-filled Hubbard-Holstein model, where the electron-electron interaction can lead to a Mott transition, and the electron-phonon coupling to a bipolaronic transition. We critically discuss the quality of the proposed wavefunction in describing the various transitions and crossovers that occur as a function of the parameters. Previous variational attempts are recovered as p…
In strongly correlated electron systems the constraint which prohibits the double electron occupation at local sites can be realized by either the infinite Coulomb interaction or the correlated hopping interaction described by the Hubbard operators, but they both render the conventional field theory inapplicable. Relaxing such the constraint leads to a class of correlated hopping models based on the deformed Hubbard operators which smoothly interpolate the locally free and strong coupling limits by a tunable interaction parameter $0\leq λ\leq 1$. Here we propose a coherent state path integral …
We study interplay of electronic correlation, randomness and dimensionality effects in half-filled random Hubbard chains weakly coupled via an interchain one-particle hopping. Based on the two-loop renormalization-group approach, phase diagrams are given in terms of temperature vs. strengths of the intrachain electron-electron umklapp scattering, the random scattering and the interchain one-particle hopping.
In this paper we derive general expressions for few-electron spectral functions of the one-dimensional Hubbard model for values of the excitation energy in the vicinity of the $M^{th}$ upper-Hubbard band lower limit. Here $M=1,2,...$ is the rotated-electron double occupation, which vanishes for the ground state and is a good quantum number for all values of the on-site Coulomb repulsion $U$. Our studies rely on a combination of symmetries of the model with a recent finite-energy holon and spinon description of the quantum problem. We apply our general scheme to the one-electron addition spectr…
We focus our quantitative analysis on the stability of the insulator state in the Hubbard model at a half-filling. Taking into account large-scale fluctuations (with a long relaxation time) of the on-site Coulomb repulsion, we consider the possibility of realizing a stabile state which is characterized by pairing for electrons. The pairing mechanism is as follows: due to fluctuations of on-site repulsion of electrons, holes, as excited states, are formed electron pairs. The bare values of on-site Coulomb repulsion and its fluctuations, for which the states with electron pairing are stable, are…
We investigate the superfluid-insulator transition in the disordered two-dimensional Bose-Hubbard model through quantum Monte Carlo simulations. The Bose-Hubbard model is studied in the presence of site disorder and the quantum critical point between the Bose-glass and superfluid is determined in both the grand canonical ($μ/U=0.375$ close to $ρ=1$) and canonical ensemble ($ρ=1$ and 0.5). Particular attention is paid to disorder averaging and it is shown that an extremely large number of disorder realizations are needed in order to obtain reliable results. Typically we average over more than $…
In this review we recapitulate the basic features of the flat-band spin systems and briefly summarize earlier studies in the field. Main emphasis is made on recent developments which include results for both spin and electron flat-band models. In particular, for flat-band spin systems we highlight field-driven phase transitions for frustrated quantum Heisenberg antiferromagnets at low temperatures, chiral flat-band states, as well as the effect of a slight dispersion of a previously strictly flat band due to nonideal lattice geometry. For electronic systems, we discuss the universal low-temper…
The Hubbard model is exactly solved for two particles with opposite spins on d-dimensional hypercubes. It is shown that the spectrum can be separated into two parts: a trivial (U-independent) part resulting from symmetries of hypercubes and a non-trivial part expressed as a single-impurity problem on a set of finite chains of size $d^\prime+1$ ($d^\prime \le d$). The exact expression for the one-particle Green's function is given. Finally, we discuss the extension of these results to standard hypercubic lattices with periodic boundary conditions.
The Hubbard model, which augments independent-electron band theory with a single parameter to describe electron-electron correlations, is widely regarded to be the `standard model' of condensed matter physics. The model has been remarkably successful at addressing a range of correlation effects in solids, but beyond one dimension its solution is intractable. Much current research aims, therefore, at finding appropriate approximations to the Hubbard model phase diagram. Here we take the new approach of using ab initio electronic structure methods to design a material whose Hamiltonian is that o…
We calculate the one-electron spectral function of the attractive-U Hubbard model in two dimensions. We work in the intermediate coupling and low density regime and evaluate analytically the self-energy. The results are obtained in a framework based on the self-consistent T-matrix approximation. We also calculate the chemical potential of the bound pairs as a function of temperature. On the basis of this calculation we analyze the low-temperature resistivity and specific heat in the normal state of this system. We compare our results with recent beautiful tunneling experiments in the underdope…
Functional renormalization group without functional integrals: implementing Hilbert space projections for strongly correlated electrons via Hubbard X-operators
We discuss how to construct a tight binding model Hamiltonan for the simplest possible solid, composed of hydrogen-like atoms. A single orbital per atom is not sufficient because the on-site electron-electron repulsion mixes in higher energy orbitals. The essential physics is captured by a dynamic Hubbard model with one electronic orbital and an auxiliary spin degree of freedom per site. We point out that this physics can lead to a substantial shift in the position and width of electronic energy bands relative to what is predicted by conventional band structure calculations.
In this paper we employ a gaussian expansion within the finite-$U$ slave-bosons formalism to investigate the momentum structure of the electron-phonon vertex function in the Hubbard model as function of $U$ and $n$. The suppression of large momentum scattering and the onset a small-${\bf q}$ peak structure, parametrized by a cut-off $q_c$, are shown to be essentially ruled by the band narrowing factor $Z_{\rm MF}$ due to the electronic correlation. A phase diagram of $Z_{\rm MF}$ and $q_c$ in the whole $U$-$n$ space is presented. Our results are in more than qualitative agreement with a recent…
The single band, two dimensional Hubbard Hamiltonian has been extensively studied as a model for high temperature superconductivity. While Quantum Monte Carlo simulations within the dynamic cluster approximation are now providing considerable evidence for a d-wave superconducting state at low temperature, such a transition remains well out of reach of finite lattice simulations because of the "sign problem". We show here that a bilayer Hubbard model, in which one layer is electron doped and one layer is hole doped, can be studied to lower temperatures and exhibits an interesting signal of d-wa…
We elaborate the first theoretical realization of two dimensional itinerant topological magnons, based on the quarter filled Haldane-Hubbard model with a nearly-flat electron band. By using the exact diagonalization method with a projection onto this band, we obtain the spin wave excitations over the itinerant ferromagnetic ground state. In the flatband limit, the excitation exhibits similar dispersion to the free electron band with Dirac magnons. The nonflatness of the electron band opens a topological gap at Dirac points and leads to an acoustic magnon band with a nonzero Chern number. We fu…
We present results on the dynamical correlation functions of the particle-hole symmetric Holstein-Hubbard model at zero temperature, calculated using the dynamical mean field theory which is solved by the numerical renormalization group method. We clarify the competing influences of the electron-electron and electron-phonon interactions particularity at the different metal to insulator transitions. The Coulomb repulsion is found to dominate the behaviour in large parts of the metallic regime. By suppressing charge fluctuations, it effectively decouples electrons from phonons. The phonon propag…
The bilayer Hubbard model with electron-hole doping is an ideal platform to study excitonic orders due to suppressed recombination via spatial separation of electrons and holes. However, suffering from the sign problem, previous quantum Monte Carlo studies could not arrive at an unequivocal conclusion regarding the presence of phases with clear signatures of excitonic condensation in bilayer Hubbard models. Here, we develop a determinant quantum Monte Carlo (DQMC) algorithm for the bilayer Hubbard model that is sign-problem-free for equal and opposite doping in the two layers, and study excito…
Nematicity is a well known property of liquid crystals and has been recently discussed in the context of strongly interacting electrons. An electronic nematic phase has been seen by many experiments in certain strongly correlated materials, in particular, in the pseudogap phase generic to many hole-doped cuprate superconductors. Recent measurements in high $T_c$ superconductors has shown even if the lattice is perfectly rotationally symmetric, the ground state can still have strongly nematic local properties. Our study of the two-dimensional Hubbard model provides strong support of the recent …
One of the challenges in many-body physics is determining the effects of phonons on strongly correlated electrons. The difficulty arises from strong correlations at differing energy scales -- for band metals, Migdal-Eliashberg theory accurately determines electron-phonon coupling effects due to the absence of vertex corrections -- but strongly correlated electrons require a more complex description and the standard Migdal-Eliashberg approach does not necessarily apply. In this work, we solve for the atomic limit Green's function of the Holstein-Hubbard model with both time-dependent electron-e…
Density matrix renormalization group methods are used to investigate the quantum phase diagram of a one-dimensional half-filled ionic Hubbard model with bond-charge attraction, which can be mapped from the Su-Schrieffer-Heeger-type electron-phonon coupling at the antiadiabatic limit. A bond order wave (dimerized) phase which separates the band insulator from the Mott insulator always exists as long as electron-phonon coupling is present. This is qualitatively different from that at the adiabatic limit. Our results indicate that electron-electron interaction, ionic potential and quantum phonon …
The finite-temperature phase diagram of the Hubbard model in $d=3$ is obtained from renormalization-group analysis. It exhibits, around half filling, an antiferromagnetic phase and, between 30%--40% electron or hole doping from half filling, a new $τ$ phase in which the electron hopping strength $t$ asymptotically becomes infinite under repeated rescalings. Next to the $τ$ phase, a first-order phase boundary with very narrow phase separation (less than 2% jump in electron density) occurs. At temperatures above the $τ$ phase, an incommensurate spin modulation phase is indicated. In $d=2$, we fi…
We investigate metal-insulator transitions in the Holstein-Hubbard model as a function of the on-site electron-electron interaction U and the electron-phonon coupling g. We use several different numerical methods to calculate the phase diagram, the results of which are in excellent agreement. When the electron-electron interaction U is dominant the transition is to a Mott-insulator; when the electron-phonon interaction dominates, the transition is to a localised bipolaronic state. In the former case, the transition is always found to be second order. This is in contrast to the transition to th…
We explore the quantum phase transitions between two ordered states in the infinite dimensional Hubbard-Holstein model at half filling. Our study is based on the dynamical mean field theory (DMFT) combined with the numerical renormalization group (NRG), which allows us to handle both strong electron-electron and strong electron-phonon interactions. The transition line is characterized by an effective electron-electron interaction. Depending on this effective interaction and the phonon frequency $ω_0$ one finds either a continuous transition or discontinuous transition. Here, the analysis focus…
We construct exact ground states of interacting electrons on triangle and diamond Hubbard chains. The construction requires (i) a rewriting of the Hamiltonian into positive semidefinite form, (ii) the construction of a many-electron ground state of this Hamiltonian, and (iii) the proof of the uniqueness of the ground state. This approach works in any dimension, requires no integrability of the model, and only demands sufficiently many microscopic parameters in the Hamiltonian which have to fulfill certain relations. The scheme is first employed to construct exact ground state for the diamond H…
Quantum Monte Carlo is used to calculate various pairing correlations of the 2D Hubbard model possessing band features experimentally observed in the cuprates. In the hole-doped case, where the Fermi level lies close to the van Hove singularities around $(0,π)$, the d$_{x^2-y^2}$ pairing correlation is selectively enhanced, while in the electron-doped case, where the singularities are far below the Fermi level and the Fermi surface runs through $(\pm π/2,\pm π/2)$, both d$_{x^2-y^2}$ and d$_{xy}$ correlations are enhanced with the latter having a $\sqrt{2}\times \sqrt{2}$ structure. The two pa…
