Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit

In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U>0 to U=∞ is also shown. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model.     

Fourth-order perturbation theory for the half-filled Hubbard model in infinite dimensions

We calculate the zero-temperature self-energy to fourth-order perturbation theory in the Hubbard interaction U for the half-filled Hubbard model in infinite dimensions. For the Bethe lattice with bare bandwidth W, we compare our perturbative results for the self-energy, the single-particle density of states, and the momentum distribution to those from approximate analytical and numerical studies of the model. Results for the density of states from perturbation theory at U/W = 0.4 agree very well with those from the Dynamical Mean-Field Theory treated with the Fixed-Energy Exact Diagonalization and with the Dynamical Density-Matrix Renormalization Group. In contrast, our results reveal the limited resolution of the Numerical Renormalization Group approach in treating the Hubbard bands. The momentum distributions from all approximate studies of the model are very similar in the regime where perturbation theory is applicable, . Iterated Perturbation Theory overestimates the quasiparticle weight above such moderate interaction strengths.Received: 9 September 2003, Published online: 30 January 2004PACS: 71.10.Fd Lattice fermion models (Hubbard model, etc.) - 71.27. + a Strongly correlated electron systems; heavy fermions - 71.30. + h Metal-insulator transitions and other electronic transitions     

A new theory for including short range order in the cluster Bethe lattice method: Application to the Hubbard model

A new theory for incorporating short range order (SRO) in the cluster Bethe lattice method (CBLM) for disordered binary alloys is proposed. It is applied to determine the condition for the appearance of moments in the disordered phase of a narrow-band metal for general values of concentration and SRO parameter. The final result (i) yields all the known limiting conditions exactly and (ii) displays in a transparant form the SRO dependence, indicating the merit of the present approach.     

Persistent current and Drude weight for the one-dimensional Hubbard model from current lattice density functional theory

The Bethe ansatz local density approximation (LDA) to lattice density functional theory (LDFT) for the one-dimensional repulsive Hubbard model is extended to current-LDFT (CLDFT). The transport properties of mesoscopic Hubbard rings threaded by a magnetic flux are then systematically investigated by this scheme. In particular we present calculations of ground state energies, persistent currents and Drude weights for both a repulsive homogeneous and a single impurity Hubbard model. Our results for the ground state energies in the metallic phase compare favorably well with those obtained with numerically accurate many-body techniques. Also the dependence of the persistent currents on the Coulomb and the impurity interaction strength, and on the ring size are all well captured by LDA-CLDFT. Our study demonstrates the value of CLDFT in describing the transport properties of one-dimensional correlated electron systems. As its computational overheads are rather modest, we propose this method as a tool for studying problems where both disorder and interaction are present.     

Kinetic antiferromagnetism in the triangular lattice

We show that the motion of a single hole in the infinite-U Hubbard model with frustrated hopping leads to weak metallic antiferromagnetism of kinetic origin. An intimate relationship is demonstrated between the simplest versions of this problem in one and two dimensions, and two of the most subtle many body problems, namely, the Heisenberg Bethe ring in one dimension and the two-dimensional triangular lattice Heisenberg antiferromagnet.     

Magnetic ordering versus lattice distortion in very narrow bands

A simple criterion for antiferromagnetic ordering versus lattice distortion in very narrow band materials is obtained. The model describing these narrow band materials contains the Hubbard Hamiltonian, short ranged interatomic Coulomb and exchange interactions, and the electron-phonon interaction.     

Signature of Mott-insulator transition with ultracold fermions in a one-dimensional optical lattice

Using the exact Bethe ansatz solution of the Hubbard model and Luttinger liquid theory, we investigate the density profiles and collective modes of one-dimensional ultracold fermions confined in an optical lattice with a harmonic trapping potential. We determine a generic phase diagram in terms of a characteristic filling factor and a dimensionless coupling constant. The collective oscillations of the atomic mass density, a technique that is commonly used in experiments, provide a signature of the quantum phase transition from the metallic phase to the Mott-insulator phase. A detailed experimental implementation is proposed.     

Mott-Hubbard transition in a magnetic field

We study the density of states (DOS) as a function of the interaction U in the half-filled simplified Hubbard model in a magnetic field. This model is considered on the Bethe lattice in the limit of high dimensions. We show that the DOS can be calculated exactly, and that many of its properties have an astonishingly simple form. In particular, the DOS can be investigated explicitly in the limits of weak and strong coupling and near the metal-insulator transition. E.g., we find an explicit result for the critical value U_c, at which the metal-insulator transition occurs, as a function of the magnetization. The relation between the magnetization and the magnetic field is calculated numerically. An important result is that the metal-insulator transition, occurring in the model with B = 0, is continuously connected to the metal-insulator transition in the subspace of single spin flips.     

Bethe lattice spin glass: The effects of a ferromagnetic bias and external fields. I. Bifurcation analysis

We present a rigorous analysis of the ±J Ising spin-glass model on the Bethe lattice with fixed uncorrelated boundary conditions. Phase diagrams are derived as a function of temperature vs. concentration of ferromagnetic bonds and, for a symmetric distribution of bonds, external field vs. temperature. In this part we characterize the bulk ordered phases using bifurcation theory: we prove the existence of a distribution of single-site magnetizations far inside the lattice which is stable with respect to changes in the boundary conditions.     

Slave-boson phase diagram of the two-dimensional extended Hubbard model: Influence of electron-phonon coupling

We present a detailed study of the extended Hubbard-Peierls model on a square lattice using the slave-boson method proposed by Kotliar and Ruckenstein. The emphasis is on the investigation of the ground state phase diagram. To compare the relative stability of several homogenous phases, the effective bosonized action was evaluated by means of a two-sublattice saddlepoint approximation which allows for the symmetry broken states compatible with the underlying bipartite lattice structure. Paying particular attention to the interplay of electron-electron and electron-phonon interaction, we take into account various types of magnetic ordered phases, i.e. para-, ferro-, ferri-, and antiferromagnetic states, as well as charge ordered phases, e.g. a static (, ) Peierls distorted state. Furthermore the approach has been applied to the following special cases: the Hubbard model, the extended Hubbard model, and the Hubbard-Peierls model. A careful numerical solution of the corresponding self-consistency equations enables us to map out the ground-state phase diagrams of the various models at arbitrary band filling over the whole range of interaction strength. In the phase diagram of the Hubbard model we found a large region with ferrimagnetic order away from half-filling. The phase diagram of the halffilled band extended Hubbard model shows a first-order transition from a spin-density-wave to a charge-density-wave state which is displaced from the mean-field lineU=4V towards largerV. At large negativeU andV we obtain a domain with charge separation. The phase compares favorably with earlier quantum Monte-Carlo results. Including the local electron-phonon coupling the charge-density-wave region is considerably enlarged. Away from half-filling the phase diagram becomes more complex: besides the pure magnetic phases we obtain ferri- and paramagnetic states which show additional charge-density order. Aspects of phase separation are discussed. Finally we investigate the variation of the different gap and order parameters along characteristic lines in the parameter space and determine the renormalized quasiparticle bands.     

Ground states of extended Hubbard models in the atomic limit

The possible ground states of an extended Hubbard model in the atomic limit, augmented by an additional nearest neighbour Ising-like interaction and an external magnetic field, are rigorously determined for arbitrary values of the coupling parameters and arbitrary chemical potential. The method used requires only simple convexity arguments and the examination of all possible configurations of small clusters of lattice sites, which may be done by computer. The results are valid for all lattices ofAB type (two interpenetrating sublattices). The type of order found are ferromagnetic, antiferromagnetic, and charge density wave. Perturbation theory suggests that for finite band width there may be a state showing both a charge density wave and ferromagnetic order.     

Ferromagnetism in the Hubbard model: instability of the Nagaoka state on the triangular,honeycomb and kagome lattices

In order to analyse the lattice dependence of ferromagnetism in the two-dimensional Hubbard model we investigate the instability of the fully polarised ferromagnetic ground state (Nagaoka state) on the triangular, honeycomb and kagome lattices. We mainly focus on the local instability, applying single spin flip variational wave functions which include majority spin correlation effects. The question of global instability and phase separation is addressed in the framework of Hartree-Fock theory. We find a strong tendency towards Nagaoka ferromagnetism on the non-bipartite lattices (triangular, kagome) for more than half filling. For the triangular lattice we find the Nagaoka state to be unstable above a critical density of n = 1.887 at U = ∞, thereby significantly improving former variational results. For the kagome lattice the region where ferromagnetism prevails in the phase diagram widely exceeds the flat band regime. Our results even allow the stability of the Nagaoka state in a small region below half filling. In the case of the bipartite honeycomb lattice several disconnected regions are left for a possible Nagaoka ground state.     

Analytical and numerical treatment of the Mott-Hubbard insulator in infinite dimensions

We calculate the density of states in the half-filled Hubbard model on a Bethe lattice with infinite connectivity. Based on our analytical results to second order in t/U, we propose a new Fixed-Energy Exact Diagonalization scheme for the numerical study of the Dynamical Mean-Field Theory. Corroborated by results from the Random Dispersion Approximation, we find that the gap opens at . Moreover, the density of states near the gap increases algebraically as a function of frequency with an exponent in the insulating phase. We critically examine other analytical and numerical approaches and specify their merits and limitations when applied to the Mott-Hubbard insulator.Received: 14 July 2003, Published online: 2 October 2003PACS: 71.10.Fd Lattice fermion models (Hubbard model, etc.) - 71.27.+a Strongly correlated electron systems; heavy fermions - 71.30.+h Metal-insulator transitions and other electronic transitions     

Order parameter quantum fluctuations in a two-dimensional system of mesoscopic Josephson junctions

The boson lattice Hubbard model is used to study the role of quantum fluctuations of the phase and local density of the superfluid component in establishing a global superconducting state for a system of mesoscopic Josephson junctions or grains. The quantum Monte Carlo method is used to calculate the density of the superfluid component and fluctuations in the number of particles at sites of the two-dimensional lattice for various average site occupation numbers n ₀ (i.e., number of Cooper pairs per grain). For a system of strongly interacting bosons, the phase boundary of the ordered superconducting state lies above the corresponding boundary for its quasiclassical limit—the quantum XY-model—and approaches the latter as n ₀ increases. When the boson interaction is weak in the boson Hubbard model (i.e., the quantum fluctuations of the phase are small), the relative fluctuations of the order parameter modulus are significant when n ₀<10, while quantum fluctuations in the phase are significant when n ₀<8; this determines the region of mesoscopic behavior of the system. Comparison of the results of numerical modeling with theoretical calculations show that mean-field theory yields a qualitatively correct estimate of the difference between the phase diagrams of the quantum XY-model and the Hubbard model. For a quantitative estimate of this difference the free energy and thermodynamic averages of the Hubbard model are expanded in powers of 1/n ₀ using the method of functional integration. Zh. éksp. Teor. Fiz. 113, 261–277 (January 1998)     

“Linearized” dynamical mean-field theory for the Mott-Hubbard transition

The Mott-Hubbard metal-insulator transition is studied within a simplified version of the Dynamical Mean-Field Theory (DMFT) in which the coupling between the impurity level and the conduction band is approximated by a single pole at the Fermi energy. In this approach, the DMFT equations are linearized, and the value for the critical Coulomb repulsion can be calculated analytically. For the symmetric single-band Hubbard model at zero temperature, the critical value is found to be given by 6 times the square root of the second moment of the free (U=0) density of states. This result is in good agreement with the numerical value obtained from the Projective Selfconsistent Method and recent Numerical Renormalization Group calculations for the Bethe and the hypercubic lattice in infinite dimensions. The generalization to more complicated lattices is discussed. The “linearized DMFT” yields plausible results for the complete geometry dependence of the critical interaction. Received 6 May 1999 and Received in final form 2 July 1999     

Random walks on the Bethe lattice

We obtain random walk statistics for a nearest-neighbor (Pólya) walk on a Bethe lattice (infinite Cayley tree) of coordination numberz, and show how a random walk problem for a particular inhomogeneous Bethe lattice may be solved exactly. We question the common assertion that the Bethe lattice is an infinite-dimensional system.Supported in part by the U.S. Department of Energy.     

Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model

A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.     

Replica symmetry breaking in the Ising spin glass model on Bethe-like lattices with loop

The Ising spin glass model on Bethe-like lattices (cactus lattices) is studied using replicas in the presence of a magnetic field. Parisi's order parameter function and the de Almeida–Thouless (AT) line are obtained close to the spin glass transition temperature. The results are compared with those for the Bethe lattice to see the effects of loops. The slope of the order parameter function diminishes considerably for both lattices compared with that for the Sherrington–Kirkpatrick (SK) model. The loci of the AT line for the cactus lattices and the Bethe lattice are above and below that for the SK model, respectively.