Rigorous solution of a Hubbard model extended by nearest‐neighbour Coulomb and exchange interaction on a triangle and tetrahedron

The Hubbard model extended by both nearest‐neighbour (nn) Coulomb correlation and nearest‐neighbour Heisenberg exchange is solved rigorously for a triangle and tetrahedron. All eigenvalues and eigenvectors are given as functions of the model parameters in a closed analytical form. For fixed electron numbers we found a multitude of level crossings, both in the ground state and in the excited states in dependence on the various model parameters. By coupling an ensemble of clusters to an electron bath we get the cluster gas model or the cluster gas approximation, if an extended array of weak‐interacting clusters is considered. The grand‐canonical potential Ω (μ, T, h) and the electron occupation N (μ, T, h) of the related cluster gases were calculated for arbitrary values (attractive and repulsive) of the three interaction constants. For the cluster gases without the additional interactions we found various steps in N (μ, T = 0, h = 0) higher than one. The reason is the degeneration of ground states differing in their electron occupation by more than one electron. For the triangular cluster gas we have one such degeneration point. For the tetrahedral cluster gas two. As a consequence, we do not find areas with one electron in the μ‐U ground‐state phase diagram of the triangular cluster gas or with one, two and five electrons in the case of the tetrahedral cluster gas. The degeneration point of the triangular cluster gas can not be destroyed by an applied magnetic field. This holds also for the lower degeneration point of the tetrahedral cluster gas. Otherwise, the upper degeneration point breaks down at a critical magnetic field h_c. The dependence of h_c on U shows a maximum for strong on‐site correlation. The influence of nn‐exchange and nn‐Coulomb correlation on the ground‐state phase diagrams is calculated. Whereas antiferromagnetic nn‐exchange breaks the degeneration points of the tetrahedral cluster gas partially only, a repulsive nn‐Coulomb correlation lifts the underlying degeneracies completely. Otherwise both ferromagnetic nn‐exchange and attractive nn‐Coulomb interaction stabilise the degeneration points. The consequences of the cluster gas results for extended cluster arrays are discussed.     

Real‐space multiple‐scattering Hubbard model calculations for d‐ and f‐state materials

Calculations are presented of the electronic structure and X‐ray spectra of materials with correlated d‐ and f‐electron states based on the Hubbard model, a real‐space multiple‐scattering formalism and a rotationally invariant local density approximation. Values of the Hubbard parameter are calculated ab initio using the constrained random‐phase approximation. The combination of the real‐space Green's function with Hubbard model corrections provides an efficient approach to describe localized correlated electron states in these systems, and their effect on core‐level X‐ray spectra. Results are presented for the projected density of states and X‐ray absorption spectra for transition metal‐ and lanthanide‐oxides. Results are found to be in good agreement with experiment.     

Dynamics of Hubbard Nano‐Clusters Following Strong Excitation

The Hubbard model is a prototype for strongly correlated electrons in condensed matter, for molecules and fermions or bosons in optical lattices. While the equilibrium properties of these systems have been studied in detail, the excitation and relaxation dynamics following a perturbation of the system are only poorly explored. Here, we present results for the dynamics of electrons following nonlinear strong excitation that are based on a nonequilibrium Green functions approach. We focus on small systems—“Hubbard nano‐clusters”—that contain just a few particles where, in addition to the correlation effects, finite size effects and spatial inhomegeneity can be studied systematically. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantum coherence and ground-state phase transition in a four-chain Bose–Hubbard model

We study the quantum coherence and ground-state phase transition of a four-chain Bose–Hubbard model with the long-range interaction. In a special four-chain Bose–Hubbard model,i.e., each chain only has one optical potential, four types of the ground-state phases are discovered. The effects of the disorder, the on-site interaction and the long-range interaction on the quantum coherence are studied. For the system without the long-range interaction, the quantum coherence changes from one periodic oscillation to two periodic oscillations as the onsite interaction increases. By considering the long-range interaction, the quantum coherence goes back to one periodic oscillation again. The on-site interaction itself suppresses the quantum coherence, both the on-site interaction and long-range interaction together enhance the quantum coherence with the weak disorder. If the disorder strength is increased beyond a critical value,they start to suppress the quantum coherence. In a regular four-chain Bose–Hubbard model, i.e.,each chain has many optical potentials, the ground-state phase transitions are obtained by using the cluster Gutzwiller mean-field method. Exotic ground-state phases are found, i.e., superfluid phase, integer Mott insulator phase, supersolid phase and loophole insulator phase. The combination of the loophole insulator phase and the supersolid phase expands the lobes with the half-integer filling per site for the small ratio β = t_■/t_⊥.     

Renormalized quasiparticles in antiferromagnetic states of the Hubbard model

We analyze the properties of the quasiparticle excitations of metallic antiferromagnetic states in a strongly correlated electron system. The study is based on dynamical mean field theory (DMFT) for the infinite dimensional Hubbard model with antiferromagnetic symmetry breaking. Self-consistent solutions of the DMFT equations are calculated using the numerical renormalization group (NRG). The low energy behavior in these results is then analyzed in terms of renormalized quasiparticles. The parameters for these quasiparticles are calculated directly from the NRG derived self-energy, and also from the low energy fixed point of the effective impurity model. From these the quasiparticle weight and the effective mass are deduced. We show that the main low energy features of the k-resolved spectral density can be understood in terms of the quasiparticle picture. We also find that Luttinger's theorem is satisfied for the total electron number in the doped antiferromagnetic state.     

Density-matrix renormalization study of the Hubbard model [4]on a Bethe lattice

The half-filled Hubbard model on the Bethe lattice with coordination number z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy per site E, average local magnetization , its fluctuations and various spin correlation functions are determined as a function of the Coulomb interaction strength U/t. The local magnetic moments increase monotonically with increasing Coulomb repulsion U/t showing antiferromagnetic order between nearest neighbors []. At large U/t, is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins [ for ]. shows a maximum for U/t=2.4-2.9 that results from the interplay between the usual increase of with increasing U/t and the formation of important permanent moments at large U/t. While NN sites show antiferromagnetic spin correlations that increase with increasing Coulomb repulsion, the next NN sites are very weakly correlated over the whole range of U/t. The DMRG results are discussed and compared with tight-binding calculations for U=0, independent DMRG studies for the Heisenberg model and simple first-order perturbation estimates. Received 8 February 1999 and Received in final form 14 June 1999     

Superconductivity as a Kosterlitz–Thouless transition in the two-dimensional Hubbard model

We investigate a superconducting Kosterlitz–Thouless transition in the two-dimensional (2D) Hubbard model using auxiliary quantum Monte Carlo method for the ground state. The pair susceptibility is computed for both the attractive and repulsive Hubbard model. The numerical results show that the s-wave pair susceptibility scales as χ  L² for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ  L² also holds for the d-wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t′.     

A first order superfluid-Mott insulator transition for a Bose–Hubbard model in an emergent lattice

The combination of strong correlation and emergent lattice can be achieved when quantum gases are confined in a superradiant Fabry–Perot cavity. In addition to the discoveries of exotic phases, such as density wave ordered Mott insulator and superfluid, a surprising kink structure is found in the slope of the cavity strength as a function of the pumping strength. In this article, we show that the appearance of such a kink is a manifestation of a liquid–vapour-like transition between two superfluids with different densities. The slopes in the immediate neighborhood of the kink become divergent at the liquid–vapour critical points and display a critical scaling law with a critical exponent 1 in the quantum critical region.     

Strong-coupling perturbation theory of the Hubbard model

The strong-coupling perturbation theory of the Hubbard model is presented and carried out to order (t/U)⁵ for the one-particle Green function in arbitrary dimension. The spectral weight is expressed as a Jacobi continued fraction and compared with new Monte-Carlo data of the one-dimensional, half-filled Hubbard model. Different regimes (insulator, conductor and short-range antiferromagnet) are identified in the temperature-hopping integral (T,t) plane. This work completes a first paper on the subject (Phys. Rev. Lett. 80, 5389 (1998)) by providing details on diagrammatic rules and higher-order results. In addition, the non half-filled case, infinite resummations of diagrams and the double occupancy are discussed. Various tests of the method are also presented. Received 25 October 1999     

Superconductivity and antiferromagnetism in the phase diagram of the frustrated Hubbard model within a variational study

The interplay between antiferromagnetism (AF) and superconductivity (SC) in cuprates is studied for the two-dimensional Hubbard model with a diagonal transfer t′, using a variational Monte Carlo method. Optimizing an improved function for strongly correlated values of U/t, we construct phase diagrams in the δ (doping rate)-t′/t space. It is found that the stable state is sensitive to the value of model parameters: For the extremely large values of U/t, a coexisting state is realized for t′/t ? −0.15, whose range of doping rate extends as t′/t increases. In contrast, for t′/t = −0.3, AF and SC states are mutually exclusive, and a coexisting state does not appear. As U/t decreases, the area of pure AF extends, and that of coexisting state shrinks. As a result, the coexisting state disappears for t′/t = −0.15 and U/t = 12, probable values for hole-doped cuprates. Compared with the t–J model, the Hubbard model has richer phases.     

Thermodynamics of a quantum dissipative charged magneto‐oscillator

Quantum dissipative effect on the thermodynamics of an electron in the combined presence of a parabolic potential and a uniform (and homogeneous) magnetic field, is investigated here. Starting from the microscopic system plus bath model, we explicitly derive the thermodynamic properties using the reduced partition function of the system which is calculated using the imaginary time path integral method. The quantum heat bath we consider here is a structured heat bath whose spectral density corresponds to a structured thermal harmonic noise. All the statistical thermodynamic functions calculated do reconcile with the requirements of the fundamental axioms of physics. In particular, the specific heat and the entropy vanishes as the temperature approaches its absolute zero value, a necessity of the third law of thermodynamics. Moreover the specific heat satisfies classical equipartition theorem at high temperatures. The coefficients of the leading temperature dependent terms of the thermodynamic quantities depend only on the damping constant but not on other parameters of the bath spectral density, which is similar to the analysis based on the Drude bath spectral density. Our study facilitates the physics of small quantum systems, which are always in contact with some environments, at very low temperatures.

     

Remnant Fermi surface in a pseudogap regime of the two-dimensional Hubbard model at finite temperature

A precursor effect on the Fermi surface in the two-dimensional Hubbard model at finite temperatures near the antiferromagnetic instability is studied using three different itinerant approaches: the second order perturbation theory, the paramagnon theory (PT), and the two-particle self-consistent (TPSC) approach. In general, at finite temperature, the Fermi surface of the interacting electron systems is not sharply defined due to the broadening effects of the self-energy. In order to take account of those effects we consider the single-particle spectral function A(, 0) at the Fermi level, to describe the counterpart of the Fermi surface at T = 0. We find that the Fermi surface is destroyed close to the pseudogap regime due to the spin-fluctuation effects in both PT and TPSC approaches. Moreover, the top of the effective valence band is located around = (π/2,π/2) in agreement with earlier investigations on the single-hole motion in the antiferromagnetic background. A crossover behavior from the Fermi-liquid regime to the pseudogap regime is observed in the electron concentration dependence of the spectral function and the self-energy. Received 8 September 2000 and Received in final form 20 December 2000     

Possibility of non‐adiabatic level crossing by DFT study of tautomerism and potential energy surfaces in of 3‐hydroxy‐5‐(pyrimidin‐2‐yl)‐2H‐pyrrol‐2‐one and its tautomer

3‐Hydroxy‐5‐(pyrimidin‐2‐yl)‐2H‐pyrrol‐2‐one (HYPO, T1) and 2‐hydroxy‐5‐(pyrimidine‐2‐yl)‐3H‐pyrrole‐3‐one (HYPO, T2) have designed in this research to study potential energy curves for their dynamic motions and possibility of crossing between levels. Study of tautomerism shows that T1 tautomer is more stable than T2 (about 5.83 kJ/mol). Dynamic study of possible motions show rate constants (highest possible) equal to 8.82 M/s for tautomerism, 1.70 × 10⁹ M/s for relative rotation of ring (rr) and 3.67 × 10⁶ M/s for rotation of OH bond (br). Moreover, variations of orbital populations, NBO charges, hybridations, and acceptor–donor interactions in IRC steps have been investigated to study the possibility of non‐adiabatic crossing between tautomerism and ring rotation potential energy curves. The data showed that in spite of the fact that these two potentials share three common points, these two potential curves cannot have non‐adiabatic crossing because of different symmetries and a large difference between their barrier energies. Copyright © 2010 John Wiley & Sons, Ltd.     

Substituent effect investigation of 3‐(2, 4‐dichlorophenyl)‐1‐(4′‐X‐phenyl)‐2‐propen‐1‐one. Part 1. Correlation analysis of 13C NMR chemical shifts

A series of substituted chlorinated chalcones namely, 3‐(2,4‐dichlorophenyl)‐1‐(4′‐X‐phenyl)‐2‐propen‐1‐one, have been synthesized, X being H, NH₂, OMe, Me, F, Cl, CO₂Et, CN, and NO₂. Dual substituent parameter (DSP) models of ¹³C NMR chemical shift (CS) have revealed that π‐polarization concept could be utilized to explain the reverse field effect at CO, the enhanced substituent field effect at CO, C‐2, and C‐5, and the decreased sensitivity of substituent field effect at C‐6. Chlorine atoms dipole direction at the benzylidene ring either enhances or reduces substituent effect depending on how they couple with the substituent dipole at the probe site. The correlation of ¹³C NMR CS of C‐2, C‐5, and C‐6 with σ and σ indicates that chlorine atoms in the benzylidine ring deplete the ring from charges. Both MSP of Hammett and DSP of Taft ¹³C NMR CS models give similar trends of substituent effects at C‐2, C‐5, and C‐6. However, the former fail to give a significant correlation for CO and C‐6 ¹³C NMR CS. MSP of σ_q and DSP of Taft and Reynolds models significantly correlated ¹³C NMR CS of C_β. MSP of σ_q fails to correlate C‐1′ ¹³C NMR CS. Investigation of ¹³C NMR CS of non‐chlorinated chalcones series: 3‐phenyl‐1‐(4′‐X‐phenyl)‐2‐propen‐1‐one has revealed similar trends of substituent effects as in the chlorinated chalcones series for C‐1′, CO, C_α, and C_β. In contrast, the substituent effect of the non‐chlorinated chalcone series at C‐2, C‐5, and C‐6 did not correlate with any substituent constant. Copyright © 2010 John Wiley & Sons, Ltd.     

Phase diagram of the three-dimensional Hubbard model at half filling

We investigate the phase diagram of the three-dimensional Hubbard model at half filling using quantum Monte Carlo (QMC) simulations. The antiferromagnetic Néel temperature is determined from the specific heat maximum in combination with finite-size scaling of the magnetic structure factor. Our results interpolate smoothly between the asymptotic solutions for weak and strong coupling, respectively, in contrast to previous QMC simulations. The location of the metal-insulator transition in the paramagnetic phase above is determined using the electronic compressibility as criterion. Received 11 April 2000 and Revised in final form 29 June 2000     

Phase diagram of the Mott transition in a two-band Hubbard model in infinite dimensions

The Mott metal-insulator transition in the two-band Hubbard model in infinite dimensions is studied by using the linearized dynamical mean-field theory recently developed by Bulla and Potthoff. The phase boundary of the metal-insulator transition is obtained analytically as a function of the on-site Coulomb interaction at the d-orbital, the charge-transfer energy between the d- and p-orbitals and the hopping integrals between p-d, d-d and p-p orbitals. The result is in good agreement with the numerical results obtained from the exact diagonalization method. Received 5 October 2000 and Received in final form 8 December 2000     

Possible phases of the two-dimensional Hubbard model

We present a stability analysis of the 2D t - t' Hubbard model on a square lattice for various values of the next-nearest-neighbor hopping t' and electron concentration. Using the free energy expression, derived by means of the flow equations method, we have performed numerical calculation for the various representations under the point group C _4ν in order to determine at which temperature symmetry broken phases become more favorable than the symmetric phase. A surprisingly large number of phases has been observed. Some of them have an order parameter with many nodes in -space. Commonly discussed types of order found by us are antiferromagnetism, d _{x2 - y2} -wave singlet superconductivity, d-wave Pomeranchuk instability and flux phase. A few instabilities newly observed are a triplet analog of the flux phase, a particle-hole instability of p-type symmetry in the triplet channel which gives rise to a phase of magnetic currents, an s^*-magnetic phase, a g-wave Pomeranchuk instability and the band splitting phase with p-wave character. Other weaker instabilities are found also. A comparison with experiments is made. Received 25 July 2002 / Received in final form 28 November 2002 Published online 14 February 2003 RID="a" ID="a"Current address: Département de physique and Centre de recherche sur les propriétés électroniques de matériaux avancés, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1 e-mail: vaha@physique.usherb.ca     

Anisotropy of the energy gap in the insulating phase of the U-t-t' Hubbard model

We apply a diagrammatic expansion method around the atomic limit () for the U-t-t ' Hubbard model at half filling and finite temperature by means of a continued fraction representation of the one-particle Green's function. From the analysis of the spectral function we find an energy dispersion relation with a modulation of the energy gap in the insulating phase. This anisotropy is compared with experimental ARPES results on insulating cuprates. Received 18 May 2000 and Received in final form 9 August 2000