The Hubbard model extended by nearest‐neighbor Coulomb and exchange interaction on a cubic cluster – rigorous and exact results

The Hubbard model on a cube was revisited and extended by both nearest‐neighbor Coulomb correlation W and nearest‐neighbor Heisenberg exchange J. The complete eigensystem was computed exactly for all electron occupancies and all model parameters ranging from minus infinity to plus infinity. For two electrons on the cluster the eigensystem is given in analytical form. For six electrons and infinite on‐site correlation U we determinded the groundstate and the groundstate energy of the pure Hubbard model analytically. For fixed electron numbers we found a multitude of ground state level crossings depending on the various model parameters. Furthermore the groundstates of the pure Hubbard model in dependence on a magnetic field h coupled to the spins are shown for the complete U‐h plane. The critical magnetic field, where the zero spin groundstate breaks down is given for four and six electrons. Suprisingly we found parameter regions, where the ground state spin does not depend monotonously on J in the extended model. For the cubic cluster gas, i.e. an ensemble of clusters coupled to an electron bath, we calculated the density n (μ, T, h) and the thermodynamical density of states from the grand potential. The ground states and the various spin‐spin correlation functions are studied for both attractive and repulsive values of the three interaction constants. We determined the various anomalous degeneration lines, where n (μ, T = 0, h = 0) shows steps higher than one, since in this parameter regions exotic phenomena as phase separation are to expect in extended models. For the cases where these lines end in triple points, i.e. groundstates of three different occupation numbers are degenerated, we give the related parameter values. Regarding the influence of the nn‐exchange and the nn‐Coulomb correlation onto the anomalous degeneration we find both lifting and inducing of degeneracies depending on the parameter values.     

Thermodynamics of a 4‐site Hubbard model by analytical diagonalization

By use of the conservation laws a four‐site Hubbard model coupled to a particle bath within an external magnetic field in z‐direction was diagonalized. The analytical dependence of both the eigenvalues and the eigenstates on the interaction strength, the chemical potential and magnetic field was calculated. It is demonstrated that the low temperature behaviour is determined by a delicate interplay between many‐particle states differing in electron number and spin if the electron density is away from half‐filling. The grand partition sum is calculated and the specific heat, the susceptibility as well as various correlation functions and spectral functions are given in dependence of the interaction strength, the electron occupation and the applied magnetic field. For both the grand canonical and the canonical ensemble the high‐temperature crossing points of the specific heat are calculated. Whereas in the weak correlation regime the universal value calculated by second order perturbation theory for several Hubbard systems being in the thermodynamic limit is confirmed, these crossing points vanish for intermediate to strong correlation.     

扩展哈伯德模型中原子团簇的结构和热力学性质研究

采用数值精确对角化方法,在加入了近邻电子库仑排斥作用的扩展哈伯德模型中对原子数N=5和N=6的原子团簇进行了研究.首先得到了对应于不同强度的电子在位库仑作用能U、近邻排斥作用能V,以及不同电子填充数的团簇的优化结构和相应的总自旋S.研究结果表明,电子的近邻排斥能V的引入会使团簇向成键数目减少的链状或星状结构改变.然后结合团簇的能级特征,分析了团簇的热容和磁化率等热力学特性,其中热容曲线的峰值位置可由团簇的低能激发给出合理的解释.     

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.     

Quantum phase diagram of the half filled Hubbard model with bond-charge interaction

Using quantum field theory and bosonization, we determine the quantum phase diagram of the one-dimensional Hubbard model with bond-charge interaction X in addition to the usual Coulomb repulsion U at half-filling, for small values of the interactions. We show that it is essential to take into account formally irrelevant terms of order X  . They generate relevant terms proportional to X²$X^2$ in the flow of the renormalization group (RG). These terms are calculated using operator product expansions. The model shows three phases separated by a charge transition at U=Uc$U=U_c$ and a spin transition at U=Us>Uc$U=U_s>U_c$. For U<Uc$U<U_c$ singlet superconducting correlations dominate, while for U>Us$U>U_s$, the system is in the spin-density wave phase as in the usual Hubbard model. For intermediate values Uc<U<Us$U_c<U<U_s$, the system is in a spontaneously dimerized bond-ordered wave phase, which is absent in the ordinary Hubbard model with X=0$X=0$. We obtain that the charge transition remains at Uc=0$U_c=0$ for X≠0$X\ne 0$. Solving the RG equations for the spin sector, we provide an analytical expression for Us(X)$U_s(X)$. The results, with only one adjustable parameter, are in excellent agreement with numerical ones for X<t/2$X<t/2$ where t is the hopping.     

Role of the attractive intersite interaction in the extended Hubbard model

We consider the extended Hubbard model in the atomic limit on a Bethe lattice with coordination number z. By using the equations of motion formalism, the model is exactly solved for both attractive and repulsive intersite potential V. By focusing on the case of negative V, i.e., attractive intersite interaction, we study the phase diagram at finite temperature and find, for various values of the filling and of the on-site coupling U, a phase transition towards a state with phase separation. We determine the critical temperature as a function of the relevant parameters, U/|V|, n and z and we find a reentrant behavior in the plane (U/|V|, T). Finally, several thermodynamic properties are investigated near criticality.     

最近邻弱交换相互作用对纳米管中Blume-Capel模型热学性质的研究

利用有效场理论研究了纳米管上最近邻弱交换相互作用下spin-1纳米管中Blume-Capel模型的内能、比热和自由能,得到了系统的内能、比热和自由能与最近邻弱交换相互作用和晶场的关系。结果表明：最近邻弱交换相互作用和晶场强度等诸多因素相互竞争,使系统表现出比 = = =1时的BC模型更为复杂的热学性质;系统内能随温度的变化曲线表现出不连续性;比热随温度的变化出现奇异性;高温对自由能的影响更加明显。     

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_⊥.     

Spin‐1 Bose Hubbard Model with Nearest Neighbour Extended Interaction

A spinor () Bose gas is studied in presence of a density‐density interaction through a mean field approach and a perturbation theory for either sign of the spin dependent interaction, namely the antiferromagnetic (AF) and the ferromagnetic cases. In the AF case, the charge density wave (CDW) phase appears to be sandwiched between the Mott insulating (MI) and the supersolid phases for small values of the extended interaction strength. But the CDW phase completely occupies the MI lobe when the extended interaction strength is larger than a certain critical value related to the width of the MI lobes and hence opens up the possibilities of spin singlet and nematic CDW insulating phases. In the ferromagnetic case, the phase diagram shows similar features as that of the AF case and are in complete agreement with a spin‐0 Bose gas. The perturbation expansion calculations nicely corroborate the mean field phase results in both these cases. Further, we extend our calculations in presence of a harmonic confinement and obtained the momentum distribution profile that is related to the absorption spectra in order to distinguish between different phases.     

外壳层最近邻交换相互作用对Blume-Capel模型相变行为的影响

利用有效场理论研究了纳米管上双模随机晶场中混合自旋Blume-Capel模型的相变行为。结果表明,系统相变行为与取值概率、晶场强度比值、晶场参数、温度以及外壳层最近邻交换相互作用密切相关。取值概率、晶场强度比值、晶场参数和外壳层最近邻交换相互作用等诸多因素相互竞争,影响系统的一级和二级相变以及临界点。     

最近邻弱交换相互作用对spin-1纳米管磁化强度的影响

利用有效场理论研究了纳米管上最近邻原子间弱交换相互作用对Blume-Capel模型磁化强度的影响,得到了系统格点的磁化强度与最近邻弱交换相互作用和晶场强度的关系。结果表明：最近邻弱交换相互作用和晶场强度等诸多因素相互竞争,使系统表现出丰富的磁学特性。正晶场促进系统的磁化强度,而负晶场对其有抑制作用;负晶场作用下,磁化强度基态饱和值小于1,且系统发生一级相变;不同位置的弱交换相互作用对系统的磁化强度影响程度不同。     

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.     

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.     

Exact solution of the ion‐laser interaction in all regimes

We show that in the trapped ion‐laser interaction all the regimes may be considered analytically. We may solve not only for different laser intensities, but also away from resonance and from the Lamb‐Dicke regime. It is found a dispersive Hamiltonian for the high intensity regime, that, being diagonal, its evolution operator may be easily calculated.     

Pseudogap in the Eliashberg approach based on electron‐phonon and electron‐electron‐phonon interaction

The properties of the superconducting and the anomalous normal state were described by using the Eliashberg method. The pairing mechanism was reproduced with the help of the Hamiltonian, which models the electron‐phonon and the electron‐electron‐phonon interaction (EEPh). The set of the Eliashberg equations, which determines the order parameter function (φ), the wave function renormalization factor (Z), and the energy shift function (χ), was derived. It was proven that for the sufficiently large values of the EEPh potential, the doping dependence of the order parameter () has the analogous course to that observed experimentally in cuprates. The energy gap in the electron density of states is induced by Z and χ ‐ the contribution from φ is negligible. The electron density of states possesses the characteristic asymmetric form and the pseudogap is observed above the critical temperature.

     

Unusually flat hole dispersion relation in the two-dimensional Hubbard model and restoration of coherence by addition of pair-hopping processes

The dispersion relation of a doped hole in the half-filled 2D Hubbard model is shown to follow a law around the and points in the Brillouin zone. Upon addition of pair-hopping processes this dispersion relation is unstable towards a law. The above follows from T=0 Quantum Monte-Carlo calculations of the single particle spectral function on lattices. We discuss finite dopings and argue that the added term restores coherence to charge dynamics and drives the system towards a d _{x2 - y2} superconductor. Received 22 March 1999     

Comment on a recent conjectured solution of the three-dimensional Ising model

In a recent paper published in Philosophical Magazine [Z.-D. Zhang, Phil. Mag. 87 (2007) p.5309], the author advances a conjectured solution for various properties of the three-dimensional Ising model. Here, we disprove the conjecture and point out the flaws in the arguments leading to the conjectured expressions.     

Investigation of the multiplet features of SrTiO3 in X‐ray absorption spectra based on configuration interaction calculations

Synchrotron‐based L_2,3‐edge absorption spectra show strong sensitivities to the local electronic structure and chemical environment. However, detailed physical information cannot be extracted easily without computational aids. Here, using the experimental Ti L_2,3‐edges absorption spectrum of SrTiO₃ as a fingerprint and considering full multiplet effects, calculations yield different energy parameters characterizing local ground state properties. The peak splitting and intensity ratios of the L₃ and L₂ set of peaks are carefully analyzed quantitatively, giving rise to a small hybridization energy around 1.2 eV, and the different hybridization energy values reported in the literature are further addressed. Finally, absorption spectra with different linearly polarized photons under various tetragonal crystal fields are investigated, revealing a non‐linear orbital–lattice interaction, and a theoretical guidance for material engineering of SrTiO₃‐based thin films and heterostructures is offered. Detailed analysis of spectrum shifts with different tetragonal crystal fields suggests that the e_g crystal field splitting is a necessary parameter for a thorough analysis of the spectra, even though it is not relevant for the ground state properties.     

最近邻原子间交换相互作用对纳米管上Blume-Capel模型磁热性质的影响

利用有效场理论研究了纳米管上最近邻原子间交换相互作用对Blume-Capel模型格点的磁化强度、内能、比热和自由能的影响,得到了系统格点的磁化强度、内能、比热和自由能与交换相互作用和晶场强度的关系.结果表明:最近邻交换相互作用和晶场强度等诸多因素相互竞争,使系统表现出丰富的磁学特性;正晶场对系统磁化强度具有促进作用;负晶场对系统磁化强度具有抑制作用;负晶场作用下,系统发生一级相变;晶场强度参数和交换相互作用不同时,系统的内能、比热和自由能也呈现出奇异性.