Anderson-Hubbard Model Research Articles

Theory of strongly correlated electron systems: Hubbard–Anderson models from an exact Hamiltonian, and perturbation theory near the atomic limit within a nonorthogonal basis set

AbstractThe theory of correlated electron systems is formulated in a form that allows the use as a reference point a density functional theory in the local density approximation (LDA DFT) for solids or molecules. The theory is constructed in two steps. As a first step, the total Hamiltonian is transformed into a correlated form. To elucidate the microscopic origin of the parameters of the periodic Hubbard–Anderson model (PHAM) the terms of the full Hamiltonian that have the operator structure of PHAM are separated. It is found that the matrix element of mixing interaction includes ion configuration‐ and number of particles‐dependent contributions from the Coulomb interaction. In a second step the diagram technique (DT) is developed by means of generalization of the Baym–Kadanoff method for correlated systems. The advantages of the method are that: (1) A nonorthogonal basis can be used, in particular the one generated by LDA DFT; (2) the equations for Green's functions (GFs) for the Fermi and Bose types of quasiparticles can be formulated in the form of a closed system of functional equations. The latter allows us to avoid the question of the nonunique decoupling procedure existing in previous versions of the DT and perform the expansion in terms of dressed GFs. Although the expressions for all interactions depend on the overlap matrix, it is shown that the theory is formally equivalent to one with orthogonal states with redefined interactions. When the PHAM is treated from the atomic‐limit side the vertexes are generated by kinematic interactions. The latter arise due to nontrivial commutation relations between X‐operators and come from the mixing, hopping, and overlap of states. The equations for GFs are derived within the nonorthogonal basis set in Hubbard‐I, one‐loop and random‐phase approximations with respect to kinematic interactions. The self‐consistent equation for “Hubbard Us” is derived. The technique developed is general, in particular its “bosonic” part can be used for description of spin systems with arbitrary anisotropy, systems with orbital ordering, or ordering of multipoles. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 113–143, 2003

PHASE LOCKING EFFECT OF PERSISTENT CURRENTS IN MESOSCOPIC ANDERSON–HUBBARD RINGS

The effects of the Anderson impurities as well as weak electron–electron interactions on the persistent current in a mesoscopic ring are studied by using the Anderson–Hubbard model. In this system, the single-particle energy spectrum possesses a flat b and and the s-d hybridization plays an interesting role in the phase dependence of the persistent current. In the weak s-d hybridization, the phases can be locked for different electron numbers. This phase locking effect originates from the existence of the flat band. In addition, it is found from the mean field approach that weak electron–electron interactions lead to the spin splitting of the energy b and and the shift of the electron number for the phase locking effect.

Kondo lattice model with correlated conduction electrons

We investigate a Kondo lattice model with correlated conduction electrons. Within dynamical mean-field theory the model maps onto an impurity model where the host has to be determined self-consistently. This impurity model can be derived from an Anderson-Hubbard model both by equating the low-energy excitations of the impurity and by a canonical transformation. On the level of dynamical mean-field theory this establishes the connection of the two lattice models. The impurity model is studied numerically by an extension of the non-crossing approximation to a two-orbital impurity. We find that with decreasing temperature the conduction electrons first form quasiparticles unaffected by the presence of the lattice of localized spins. Then, reducing the temperature further, the particle-hole symmetric model turns into an insulator. The quasiparticle peak in the one-particle spectral density splits and a gap opens. The size of the gap increases when the correlations of the conduction electrons become stronger. These findings are similar to the behavior of the Anderson-Hubbard model within dynamical mean-field theory and are obtained with much less numerical effort.

Spin interactions in an Anderson-Hubbard model

We study the magnetic properties of a site-disordered Anderson-Hubbard model at half-filling on a simple cubic lattice, via a mapping of its low-frequency transverse spin excitations onto those of an effective underlying Heisenberg model with self-consistently determined exchange couplings. Exact in the strong-coupling limit, the mapping remains accurate over the dominant region of the phase plane where the ground state is a disordered antiferromagnet. The effect of disorder and interaction strength on the resultant exchange couplings is examined in detail, and rationalized microscopically. Frustration is found to occur, even within the antiferromagnetic phase, although the ground state is shown to be stable with respect to zero-point quantum spin fluctuations. To probe finite-temperature magnetic properties, an Onsager reaction-field approach to the effective Heisenberg model in the paramagnetic phase is employed. We focus on the effect of disorder on the Néel temperature and the nature of the thermal transition to the ordered phase.

Magnetic correlations in the two-dimensional Anderson-Hubbard model

The two dimensional Hubbard model in the presence of diagonal and off-diagonal disorder is studied at half filling with a finite temperature quantum Monte Carlo method. Magnetic correlations as well as the electronic compressibility are calculated to determine the behavior of local magnetic moments, the stability of antiferromagnetic long range order (AFLRO), and properties of the disordered phase. The existence of random potentials (diagonal or ``site'' disorder) leads to a suppression of local magnetic moments which eventually destroys AFLRO. Randomness in the hopping elements (off-diagonal disorder), on the other hand, does not significantly reduce the density of local magnetic moments. For this type of disorder, at half-filling, there is no ``sign-problem'' in the simulations as long as the hopping is restricted between neighbor sites on a bipartite lattice. This allows the study of sufficiently large lattices and low temperatures to perform a finite-size scaling analysis. For off-diagonal disorder AFLRO is eventually destroyed when the fluctuations of antiferromagnetic exchange couplings exceed a critical value. The disordered phase close to the transition appears to be incompressible and shows an increase of the uniform susceptibility at low temperatures.

Mean-field hopping approach of disordered interacting bosons

We study the interplay between disorder ( W) and interaction ( U) in a boson Anderson-Hubbard model. Using a mean-field hopping ( t) approach, we calculate the free energy and we obtain the metalinsulator phase diagram for a system in which the number of bosons is equal to the number of sites. At zero temperature, we study the critical value of hopping ( t c ) as a function of W U . For t > t c , the ground state gives the metallic phase ( M). For t < t c , two different insulating phases are distinguished: (I) for small disorder ( W U < 0.5 ), the ground state is a Mott insulator ( H); (II) for large disorder ( W U > 0.5 ), the ground state shows a Bose glass insulator ( B). Starting from the H, and increasing W we obtain first an insulatorto-metal transition, and then a second transition into the B phase. At finite temperature, we study the critical temperature ( T c ) vs U for different values of W, in units of t. We obtain two different transitions when U is increased: starting from the insulating phase, we find first an insulator-to-metal transition, and then a second transition into the insulating phase. Also, at low temperatures, the phase diagram shows a metallic reentrant behavior: the temperature-induced insulator-to-metal-to-insulator transitions.

Interaction effect in the Kondo energy of the periodic Anderson-Hubbard model.

We extend the periodic Anderson model by switching on a Hubbard $U_d$ for the conduction electrons. The nearly integral valent (Kondo) limit of the Anderson--Hubbard model is studied with the Gutzwiller variational method. The new formula for the Kondo energy contains the $U_d$-dependent chemical potential of the Hubbard subsystem in the exponent, and the correlation-induced band narrowing in the prefactor. Both effects tend to suppress the Kondo scale, which can be understood to result from the blocking of hybridization (this behaviour is the opposite of that found for Kondo--Hubbard models). At half-filling, we find a Brinkman--Rice-type transition which leads from a small-gap Kondo insulator to a Mott insulator.

Disorder-induced fluctuations in the magnetic properties of an Anderson-Hubbard model.

We present a microscopic description of the inhomogeneous magnetic response of a disordered, interacting system, with local susceptibilities determined via a random-phase-approximation-type approach about stable, inhomogeneous mean-field ground states. A careful treatment of the role of disorder is vital in describing: the phase boundary to local moment formation; site-differential inhomogeneity in the distribution of local susceptibilities in a local moment regime, and its marked disorder-induced enhancement; and, on the length scales probed, large disorder-induced fluctuations in the total magnetic susceptibility.

Anderson-Hubbard model in infinite dimensions.

We present a detailed, quantitative study of the competition between interaction- and disorder-induced effects in electronic systems. For this the Hubbard model with diagonal disorder (Anderson-Hubbard model) is investigated analytically and numerically in the limit of infinite spatial dimensions, i.e., within a dynamical mean-field theory, at half-filling. Numerical results are obtained for three different disorder distributions by employing quantum Monte Carlo techniques, which provide an explicit finite-temperature solution of the model in this limit. The magnetic phase diagram is constructed from the zeros of the inverse averaged staggered susceptibility. We find that at low enough temperatures and sufficiently strong interaction there always exists a phase with antiferromagnetic long-range order. A strong coupling anomaly, i.e., an increase of the N\'eel temperature for increasing disorder, is discovered. An explicit explanation is given, which shows that in the case of diagonal disorder this is a generic effect. The existence of metal-insulator transitions is studied by evaluating the averaged compressibility both in the paramagnetic and antiferromagnetic phases. A rich transition scenario, involving metal-insulator and magnetic transitions, is found and its dependence on the choice of the disorder distribution is discussed.

Approximations to the two-hole ground state of the Hubbard-Anderson model: a numerical test

Several resonating-valence-bond-type states are being considered as an approximation of the two-hole ground state of the two-dimensional Hubbard-Anderson model. These states have been carefully constructed by Traa and Caspers with such algebraic properties, as to optimise different contributions of the Hubbard-Anderson hamiltonian. In this paper, the different contributions to their energies are calculated for lattices with sizes from 8 × 8 up to 16 × 16 and periodic boundary conditions, using a variational Monte-Carlo method. We show which state is lowest in energy and, more important, why this is so. In accordance with the optimal state from this tested set, we propose a bound state. It will be shown that this state is indeed the most stable state.

Phase diagram of the two-dimensional Anderson-Hubbard model.

We study the ground state of the two-dimensional Anderson-Hubbard model using a quantum real space renormalization group method. We obtain the phase diagram near half filling. The system is always insulating with disorder. At half filling, the system undergoes a transition from a gapless (Anderson) insulator to an incompressible (Mott-Hubbard) insulator as the interaction [ital U] reaches a critical value [ital U][sub [ital c]]. Away from half filling, the insulating phase is always gapless and is found to be controlled by the Anderson fixed point at half filling. This result is similar to that obtained in the corresponding one-dimensional system and suggests strongly the importance of the electron-electron correlation in this gapless insulating phase.

Effective Hamiltonian for the motion of holes in the Hubbard-Anderson model

The motion of (interacting) holes in the Hubbard-Anderson model for high-Tc superconductivity is translated into the motion of (coupled) spinless fermions. The entities responsible for the coupling are localized spin excitations and are described by bosons. The new description shows resemblance with the BCS description of electrons and photons of the ?classical? superconductors.

The two-hole ground state of the Hubbard-Anderson model, approximated by a variational RVB-type wave function

In this paper the Hubbard-Anderson model on a square lattice with two holes is studied. The ground state (GS) is approximated by a variational RVB-type wave function. The holes interact by exchange of a localized spin excitation (SE), which is created or absorbed if a hole moves to a nearest-neighbour site. An SE can move over the sublattice on which it is created. A variational calculation of the GS and the GS-energy is performed for an open-ended 4 × 4 lattice with two holes with the restriction that the SE is neighbouring both holes and does not move over its sublattice. It is found that the two holes prefer a bound state in which their mutual distance is 1 or 2 (with lattice spacing 1).

Interplay between disorder and electron interactions in a d=3 site-disordered Anderson-Hubbard model: A numerical mean-field study.

We consider a numerical mean-field study, at the unrestricted Hartree-Fock level, of a Gaussian site-disordered Anderson-Hubbard model on a simple cubic lattice. The phase diagram at half filling is obtained, including magnetic and metallic-insulating phases, and all relevant phase boundaries are found to occur in a relatively weak-coupling regime. Variation with filling fraction y is also considered, with particular reference to the y-differential disorder-induced enhancement of electron interactions that lead to site-differential local-moment formation

Interplay between disorder and electron interactions: mean-field phase diagram of an Anderson-Hubbard model

To obtain a broad understanding of the combined effects of disorder and electron interactions, the phase diagram of a half-filled Gaussian site-disordered Anderson-Hubbard model is assessed numerically on a simple cubic lattice, at the unrestricted Hartree-Fock mean field level. Metallic/insulating and magnetic phases are considered. Paramagnetic, disordered antiferromagnetic and spin glass magnetic phases are found. Particular attention is given to obtaining a microscopic picture of the interplay between disorder and interaction-induced local moment formation, which underlies the metal-(gapless) insulator transition.

DISORDER AND SUPERCONDUCTIVITY IN CONNECTION WITH HIGH Tc: AN OVERVIEW

There is a growing interest in the interplay between disorder and superconductivity in high T c phenomenon. After a short review of the field we discuss the work done both on the Anderson-Hubbard model, (semianalytically and numerically) and in the model which describes both extended and localized excitations. Enhancement of the superconducting pairing with disorder appears in some of the approaches.

Absence of spin diffusion in most random lattices.

We examine the metallic phase of the positionally disordered Anderson-Hubbard model as a prototype of a strongly correlated disordered metal. We find that the low-temperature thermodynamics is dominated by spin excitations in rare regions, leading to non-Fermi-liquid behavior as T\ensuremath{\rightarrow}0 with a diverging susceptibility, a specific heat rising faster than T, and a vanishing spin diffusion coefficient. We argue that the results should, in principle, apply throughout the metallic phase of random alloys which undergo a transition, as a function of concentration, to an insulating phase with local moments.

Symmetry breaking in the Anderson-Hubbard model

The Anderson-Hubbard (A-H) model with one or two holes and with periodic boundary conditions on a 4 M x 4 N square lattice is considered. On grounds of an intuitive generalization of Marshall's theorem we split the A-H Hamiltonian ( H A−H) into a zeroth order term ( H 0) and a perturbation term ( H'). With H 0 we construct unfrustrated states: the zeroth order approximation of the degenerate ground state (GS). The one-hole system has a four-fold symmetry broken H 0-GS with k = ( π/2, ± π/2), (- π/2, ± π/2). Group theory shows that this symmetry breaking (SB) may be stable if H' is taken into account. For the two-hole system we derive candidates for the H 0-GS with the corresponding good quantum numbers k and total spin S. Here we find no SB or a two-fold SB: again, this result may hold for the complete H A−H. Second order perturbation calculation possibly describes an effective coupling of two holes.

Single-particle hopping probability on a chain with interaction and disorder by the modified Lanczos algorithm.

We consider numerically a chain of ten sites (half filled) described by the Anderson-Hubbard model. The single-particle hopping probability is calculated as a function of disorder and correlation. A qualitative agreement for the logarithm of this hopping probability is found when compared with results of a calculation of the low-frequency conductivity by the Monte Carlo method.

Modified Lanczos simulation of the localized superconductivity for a chain with interaction and disorder.

Using the modified Lanczos algorithm, we calculate numerically for a chain with ten sites and half-filled the local pair-pair correlation function of the Anderson-Hubbard model (with the interaction both repulsive and attractive). The calculation aims to describe the phase with local superconductivity.