Single-band Hubbard Hamiltonian Research Articles

A VARIATIONAL THEORY OF QUASI-PARTICLES IN A 3D N x N x N CUBIC LATTICE

The single-band Hubbard Hamiltonian study faces a serious limitation and difficulty as we move away from finite - size lattices to larger N - dimensional lattices. Thus there is the needto develop the means of overcoming the finite - size lattice defects as we pass on to a higher dimension.In this work, a quantitative approximation to the one-band Hubbard model is presented using a variational analytic approach. The goal of this work, therefore, is to explore quantitatively the lowest ground-state energy and the pairing correlations in 3D N x N x N lattices of the Hubbard model. We developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationallythe correlation of two interacting elections on a three-dimensional cubic lattice. We also showed primarily how to derive possible electronic states available for several even and odd3D lattices, although, this work places more emphasis on a 3D 5 x 5 x 5 lattice. The results emerging from our present study compared favourablywith the results of Gutzwillervariational approach (GVA) and correlated variational approach (CVA), at thelarge limit of the Coulomb interaction strength (U/4t). It is revealed in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength.

Collective Excitations of an Imbalanced Fermion Gas in a 1D Optical Lattice

The collective excitations that minimize the Helmholtz free energy of a population-imbalanced mixture of a $^{6}$Li gas loaded in a quasi one-dimensional optical lattice are obtained. These excitations reveal a rotonic branch after solving the Bethe-Salpeter equation under a generalized random phase approximation based on a single-band Hubbard Hamiltonian. The phase diagram describing stability regions of Fulde-Farrell-Larkin-Ovchinnikov and Sarma phases is also analyzed.

Insights on the cuprate high energy anomaly observed in ARPES

Recently, angle-resolved photoemission spectroscopy has been used to highlight an anomalously large band renormalization at high binding energies in cuprate superconductors: the high energy “waterfall” or high energy anomaly (HEA). The anomaly is present for both hole- and electron-doped cuprates as well as the half-filled parent insulators with different energy scales arising on either side of the phase diagram. While photoemission matrix elements clearly play a role in changing the aesthetic appearance of the band dispersion, i.e. creating a “waterfall”-like appearance, they provide an inadequate description for the physics that underlies the strong band renormalization giving rise to the HEA. Model calculations of the single-band Hubbard Hamiltonian showcase the role played by correlations in the formation of the HEA and uncover significant differences in the HEA energy scale for hole- and electron-doped cuprates. In addition, this approach properly captures the transfer of spectral weight accompanying doping in a correlated material and provides a unifying description of the HEA across both sides of the cuprate phase diagram. We find that the anomaly demarcates a transition, or cross-over, from a quasiparticle band at low binding energies near the Fermi level to valence bands at higher binding energy, assumed to be of strong oxygen character.

Effect of strong correlations on the high energy anomaly in hole- and electron-doped high-Tc superconductors

Recently, angle-resolved photoemission spectroscopy (ARPES) has been used to highlight an anomalously large band renormalization at high binding energies in cuprate superconductors: the high energy ‘waterfall’ or high energy anomaly (HEA). This paper demonstrates, using a combination of new ARPES measurements and quantum Monte Carlo simulations, that the HEA is not simply the by-product of matrix element effects, but rather represents a cross-over from a quasi-particle band at low binding energies near the Fermi level to valence bands at higher binding energy, assumed to be of strong oxygen character, in both hole- and electron-doped cuprates. While photoemission matrix elements clearly play a role in changing the aesthetic appearance of the band dispersion, i.e. the ‘waterfall’-like behavior, they provide an inadequate description for the physics that underlies the strong band renormalization giving rise to the HEA. Model calculations of the single-band Hubbard Hamiltonian showcase the role played by correlations in the formation of the HEA and uncover significant differences in the HEA energy scale for hole- and electron-doped cuprates. In addition, this approach properly captures the transfer of spectral weight accompanying both hole and electron doping in a correlated material and provides a unifying description of the HEA across both sides of the cuprate phase diagram.

On the (anisotropic) uniform metallic ground states of fermions interacting through arbitrary two-body potentials in d dimensions

We demonstrate that the skeleton of the Fermi surface pertaining to a uniform metallic ground state (corresponding to fermions with spin index σ) is determined by the Hartree–Fock contribution to the dynamic self-energy . That is to say, in order for , it is necessary (but for anisotropic ground states in general not sufficient) that the following equation be satisfied: where stands for the underlying non-interacting energy dispersion and ϵF for the exact interacting Fermi energy. The Fermi surface consists of the set of k points which in addition to satisfying the above equation fulfil where1 The set of k points which satisfy the first of the above two equations but fail to satisfy the second constitute the pseudogap region of the putative Fermi surface of the interacting system. We consider the behaviour of the ground-state momentum distribution function for k in the vicinity of and show that, whereas for the uniform metallic ground states of the conventional single-band Hubbard Hamiltonian, described in terms of an on-site interaction, and (here and denote vectors infinitesimally close to , located respectively inside and outside the underlying Fermi sea), for interactions of non-zero range these inequalities can be violated (without thereby contravening the stability condition ). This aspect is borne out by the pertaining to the normal states of for instance liquid 3He (corresponding to a range of applied pressure) as determined by means of quantum Monte Carlo calculations. We further demonstrate that for Fermi-liquid metallic states of fermions interacting through interaction potentials of non-zero range (e.g. the Coulomb potential), the zero-temperature limit of does not need to be equal to ½ for ; this in strict contrast with the uniform Fermi-liquid metallic states of the single-band Hubbard Hamiltonian (if such states at all exist). This aspect should be taken into account while analysing the deduced from the angle-resolved photoemission data concerning real materials. We discuss, in the light of the findings of the present work, the growing experimental evidence with regard to the ‘frustration’ of the kinetic energy of the charge carriers in the normal states of the copper-oxide-based high-temperature superconducting compounds.

The non-Fermi-liquid nature of the metallic states of the Hubbard Hamiltonian

It has long since been argued that the metallic states of the single-band Hubbard Hamiltonian Ĥ in two spatial dimensions (i.e. for d = 2) should be non-Fermi liquid, a possibility that would lead to the understanding of the observed anomalous behaviour of the doped copper-oxide-based superconducting compounds in their normal metallic states. Here we present a formalism which enables us to express, for arbitrary d, the behaviour of the momentum-distribution function nσ(k) pertaining to uniform metallic ground states of Ĥ close to S F; σ (the Fermi surface of the fermions with spin index σ, σ = ↑, ↓) in terms of a small number of constant parameters which are bound to satisfy certain inequalities implied by the requirement of the stability of the ground state of the system. These inequalities restrict the range of variation in nσ(k) for k infinitesimally inside and outside the Fermi sea pertaining to fermions with spin index σ and consequently the range of variation in the zero-temperature limit of nσ(k) for k on S F; σ On the basis of some available accurate numerical results for nσ(k) pertaining to the Hubbard and the t-J Hamiltonian, we conclude that, at least in the strong-coupling regime, the metallic ground states of Ĥ for d = 2 cannot be Fermi-liquid nor can they in general be purely Luttinger or marginal Fermi liquids. We further rigorously identify the pseudogap phenomenon, or 'truncation' of the Fermi surface, clearly observed in the normal states of underdoped copper-oxide-based superconductors, as corresponding to a line of resonance energies (i.e. these energies strictly do not relate to quasiparticles) located below the Fermi energy, with a concomitant suppression to zero of the jump in nσ(k) over the 'truncated' parts of the Fermi surface. Our analyses make explicit the singular significance of the non-interacting energy dispersion ε k underlying Ĥ in determining the low-energy spectral and transport properties of the metallic ground states of Ĥ.

Spin-wave spectrum of a two-dimensional itinerant electron system: Analytic results for the incommensurate spiral phase in the strong-coupling limit

We study the zero-temperature spin fluctuations of a two-dimensional itinerant-electron system with an incommensurate magnetic ground state described by a single-band Hubbard Hamiltonian. We introduce the (broken-symmetry) magnetic phase at the mean-field (Hartree-Fock) level through a \emph{spiral spin configuration} with characteristic wave vector $\gmathbf{Q}$ different in general from the antiferromagnetic wave vector $\gmathbf{Q_{AF}}$, and consider spin fluctuations over and above it within the electronic random-phase (RPA) approximation. We obtain a \emph{closed} system of equations for the generalized wave vector and frequency dependent susceptibilities, which are equivalent to the ones reported recently by Brenig. We obtain, in addition, analytic results for the spin-wave dispersion relation in the strong-coupling limit of the Hubbard Hamiltonian and find that at finite doping the spin-wave dispersion relation has a \emph{hybrid form} between that associated with the (localized) Heisenberg model and that associated with the (long-range) RKKY exchange interaction. We also find an instability of the spin-wave spectrum in a finite region about the center of the Brillouin zone, which signals a physical instability toward a different spin- or, possibly, charge-ordered phase, as, for example, the stripe structures observed in the high-Tc materials. We expect, however, on physical grounds that for wave vectors external to this region the spin-wave spectrum that we have determined should survive consideration of more sophisticated mean-field solutions.

Noncollinear cluster magnetism in the framework of the Hubbard model

Noncollinear magnetic states in clusters are studied by using the single-band Hubbard Hamiltonian. The unrestricted Hartree-Fock (UHF) approximation is considered without imposing any symmetry constraints neither to the size or orientation of the local magnetic moments $<\vec S_l>$ nor to the local charge densities $<n_l>$. A variety of qualitatively different selfconsistent solutions is obtained as a function of cluster size, structure, number of valence electrons $\nu$ and Coulomb interaction strength $U/t$. This includes inhomogeneous density distributions, paramagnetic solutions, magnetic solutions with collinear moments and noncollinear spin arrangements that show complex antiferromagnetic and ferromagnetic-like orders. The environment dependence of the magnetic properties is analyzed giving emphasis to the effects of antiferromagnetic frustrations in compact structures close to half-band filling. Electron correlation effects are quantified by comparing UHF and exact results for the local magnetic moments, total spin, spin-correlation functions and structural stability of 13-atom clusters. Goals and limitations of the present noncollinear approach are discussed.

Magnetic properties of Hubbard clusters: Noncollinear spins in N-atom clusters

The ground-state magnetic properties of clusters with N-atomic sites are studied by solving the single-band Hubbard Hamiltonian in the unrestricted Hartree–Fock approximation. Three-dimensional arrangements of spins are considered in the solution. Results for stability and magnetic order, as a function of the Coulomb interaction strength U/t, and number of electrons ν, are presented for all the clusters with N=3, 4, and for the set of N=13 fcc-, hcp-, icosahedral-, and bcc-like clusters. In most cases, for band fillings (ν) close to half-band |ν/N−1|⩽0.2−0.4, it is found that the antiferromagneticlike noncollinear spin solutions yield the smallest energy for U/t larger than a critical value (0&amp;lt;(U/t)c⩽6).

Magnetism and structure of small clusters: An exact treatment of electron correlations.

The electronic, magnetic, and structural properties of small clusters are studied in the framework of the single-band Hubbard Hamiltonian. Results for various ground-state and excited-state many-body properties are presented, which were calculated exactly by means of Lanczos's numerical diagonalization method. A full geometry optimization is performed for N\ensuremath{\le}8 atoms by considering all possible nonequivalent cluster structures with fixed nearest-neighbor bond lengths. The most stable structure and the corresponding total spin S are obtained rigorously as a function of the Coulomb interaction strength U/t and number of electrons \ensuremath{\nu}. The resulting interplay between electron correlations, magnetism, and cluster structure is analyzed and the main trends as a function of N, U/t, and \ensuremath{\nu} are derived. The stability of cluster ferromagnetism is studied from two complementary points of view. First, for N\ensuremath{\le}8 and \ensuremath{\nu}=N+1, we determine exactly the stability of the ferromagnetic ground state with respect to electronic excitations and structural changes. It is shown that in small clusters the structural changes can be as important to the temperature dependence of the magnetization as the purely electronic excitations. Second, we determine the stability of the saturated ferromagnetic state with respect to single spin flips as a function of the band filling \ensuremath{\nu}/N. In this case a few selected larger clusters (7\ensuremath{\le}N\ensuremath{\le}43) in the strongly correlated limit (U/t\ensuremath{\rightarrow}+\ensuremath{\infty}) are considered. It is shown that the \ensuremath{\nu}/N dependence of the spin-flip energy \ensuremath{\Delta}${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{sf}}$ shows interesting electronic-shell-like oscillations, which reflect the characteristics of the single-particle energy-level structure and its dependence on the symmetry and size of the cluster. Finally, we conclude by discussing some of the limitations of the model together with relevant extensions. \textcopyright{} 1996 The American Physical Society.

Electron correlations in narrow band alloys: Hubbard subbands and cohesive energies

The effect of strong electron-electron interactions (described by the single-band Hubbard Hamiltonian) on the electronic spectrum and cohesive energies of substitutionally disordered alloys has been studied. The authors use the Hubbard III solution to treat the correlations and coherent-potential approximation for configurational averaging. The use of the Lorentzian form of the host metal density of states enables them to obtain analytical expressions for various quantities of interest. Questions related to the problem of the calculations of cohesive energies in disordered systems with correlations are discussed and partially resolved.

Two-particle excitations of the Hubbard Hamiltonian: alloy analogy solution

The spectra of the two-hole and the two-electron excitations are calculated for a single-band Hubbard Hamiltonian. Equations of motion for retarded two-particle double-time Green function are solved using a decoupling based on the alloy analogy. The solution is correct in exactly solvable limiting cases and has also correct behaviour in the regime of the strong electron-electron correlations.

Incommensurate antiferromagnetism within a slave-boson approach to a two-dimensional Hubbard Hamiltonian

We calculate the properties of spiral spin phases at the mean-field level for a single-band Hubbard Hamiltonian with variable hole density, via the Kotliar-Ruckenstein slave-boson approach adapted to a regular twist of the spin quantization axes in a two-dimensional square lattice. Antiferromagnetic and spiral phases are found to coexist in equilibrium over a finite range of hole density.

Doping-induced incommensurate antiferromagnetism in a Mott-Hubbard insulator.

Properties of incommensurate spiral spin phases are calculated at the mean-field level for a single-band Hubbard Hamiltonian with variable hole density, by adapting both the Hartree-Fock decoupling and the Kotliar-Ruckenstein slave-boson approach to a regular twist of the spin quantization axes from site to site in a two-dimensional square lattice. The relative stability of the (1,1) and (1,0) spiral phases, the coexistence of the antiferromagnetic and the spiral phases over a finite range of hole density, and the stiffness of the spirals against fluctuations of their direction and pitch are discussed within the model Hamiltonian over a wide range of hole density and interaction strength.

On the valence band photoemission of highT c superconductors

The valence band photoemission spectrum of highT c -superconductors is discussed based on the half-filled single band Hubbard Hamiltonian with the strong Coulomb interaction. We discuss how to analyze these valence band and deep core level excitation spectra, concerning particularly with which orbital, Cud or Op state, a hole will occupy in the CuO2 plane.

The spectrum of two-particle excitations of the Hubbard Hamiltonian

The spectra of two-hole and two-electron excitations are calculated for a single-band Hubbard Hamiltonian. These spectra are directly connected with the intensity of CVV Auger spectra and with the appearance potential spectra of transition metals with strong electron-electron interactions. The theory is based on a local decoupling procedure applied to the retarded two-particle double-time Green functions in the site representation. It is exact in four important limiting cases: in the non-interacting limit (U to 0), in the atomic limit (band width W to 0), in the limit of low concentration of electrons (n to 0) and holes (n to 2), and it provides a physically reasonable interpolation between these limits. In contrast with previous similar theories, it has proper transformation properties under the electron-hole transformation. The theory is illustrated numerically in a simplified model.

The concentration dependence of the magnetisation of the disordered ferromagnetic alloy FexNi1-x

The magnetisation of the disordered ferromagnetic FexNi1-x alloy is calculated using a single-band Hubbard Hamiltonian within the Hartree-Fock approximation. The occupation number of electrons per site is obtained in a self-consistent way. The calculated magnetisation is compared with experimental results.

Absence of the long-range antiferromagnetic order in doped La2?x Sr x CuO4

Based on a simple argument, it is shown that a small concentration of holes in a single band Hubbard Hamiltonian on a quadratic lattice with moderateU/t ratio is enough to destroy the long range antiferromagnetic order in the ground state. The particular emphasis is made on the role of large amplitude zero-point quantum spin fluctuations inherent in low-dimensional spin system with the small magnitude of spinS=1/2. The relevance to the highT c superconductivity materials La2−x Sr x CuO4 is discussed.

Close Relation between Localized-Electron Magnetism and the Paramagnetic Wave Function of Completely Itinerant Electrons

What is commonly called the Gutzwiller wave function $\ensuremath{\psi}$, an approximate ground state of the single-band Hubbard Hamiltonian, is considered here for $N$-site $N$-electron rings, when the number $D$ of doubly occupied sites is small ($N=6, 10, 14, 18$). For $D\ensuremath{\rightarrow}0$ spin correlations in $\ensuremath{\psi}$ surprisingly close to the exact values at zero bandwidth are found (i.e., those of the antiferromagnetic Heisenberg-model ground state). But the energy is grossly in error. A simple modification of $\ensuremath{\psi}$ reproduces the exact energy with remarkable accuracy.

New Model Hamiltonian for the Metal-Insulator Transition

With use of the symmetry properties of the half-filled single-band Hubbard Hamiltonian, there is derived an effective Hamiltonian on a decimated lattice in which the spin and charge operators occur explicitly. Being a generalization of the Blume-Emery-Griffiths Hamiltonian for ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures, this new statistical mechanical model permits one to give a preliminary discussion of the phase diagram of the correlated electron gas by establishing analogies with their results.