Academic literature on the topic 'Fock lattice'

In this paper, we study phase separation in the two-dimensional single-band Hubbard model with the unrestricted Hartree-Fock(UHF) method and the restricted Hartree-Fock (RHF) method. We perform the calculation for square lattices and rectangle lattices. It is observed that the stripe phase exists and it depends on three aspects: geometry of the lattice, Coulomb interaction U and band filling n. To gain more physical insights, we consider the Hubbard model with spin dependent hoppings: t↑ and t↓, and study the effect of varying [Formula: see text] on the phase separation.

AbstractThis paper introduces and analyzes the new grid-based tensor approach to approximate solutions of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree–Fock equation over a spatial{L_{1}\times L_{2}\times L_{3}}lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large{L_{1}\times L_{2}\times L_{3}}lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic{L\times 1\times 1}lattice chain in a 3D rectangular “tube” withLup to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of largeL.

The magnetic phase diagrams of the two-dimensional Hubbard model forisotropic and anisotropic triangular lattices are constructed within theHartree--Fock and slave boson approximations. The triangular latticespecific non-collinear and spiral magnetic states, as well as phaseseparation between them, are shown to be realized in a wide range of modelparameters along with collinear magnetic states (stripe antiferromagneticand ferromagnetic). Phase transitions of the first and second order arefound, and the boundaries of the phase separation regions are determined. A comparison of the two approximations, Hartree--Fock and slave boson, showsthat electronic correlations suppress magnetic states, the region ofparamagnetism being expand, for values U/t>~=5. At the same time, when theFermi level is near the van Hove singularity, electron correlations do notchange the diagrams qualitatively, which is consistent with the previouslyobtained result for square and cubic lattices. The results are comparedwith the data available in the literature for other methods and approaches. Keywords: Hubbard model, phase separation, spiral magnetic order, triangular lattice, metal-insulator transition\

Abstract We revisit the unrestricted Hartree Fock study on the evolution of the ground state of the Hubbard model on the triangular lattice with hole doping. At half-filling, it is known that the ground state of the Hubbard model on triangular lattice develops a 120 degree coplanar order at half-filling in the strong interaction limit, i.e., in the spin 1/2 anti-ferromagnetic Heisenberg model on the triangular lattice. The ground state property in the doped case is still in controversy even though extensive studies were performed in the past. Within Hartree Fock theory, we find that the 120 degree order persists from zero doping to about 0.3 hole doping. At 1/3 hole doping, a three-sublattice collinear order emerges in which the doped hole is concentrated on one of the three sublattices with antiferromagnetic Neel order on the remaining two sublattices, which forms a honeycomb lattice. Between the 120 degree order and 1/3 doping region, a phase separation occurs in which the 120 degree order coexists with the collinear anti-ferromagnetic order in different regions of the system. The collinear phase extends from 1/3 doping to about 0.41 doping, beyond which the ground state is paramagnetic with uniform electron density. The phase diagram from Hartree Fock could provide guidance for the future study of the doped Hubbard model on triangular lattice with more sophisticated many-body approaches.

The magnetic phase diagrams of the two-dimensional Hubbard model for isotropic and anisotropic triangular lattices are constructed within the Hartree-Fock and slave boson approximations. The triangular lattice specific non-collinear and spiral magnetic states, as well as phase separation between them, are shown to be realized in a wide range of model parameters along with collinear magnetic states (stripe antiferromagnetic and ferromagnetic). Phase transitions of the first and second order are found, and the boundaries of the phase separation regions are determined. A comparison of the two approximations, Hartree-Fock and slave boson, shows that electronic correlations suppress magnetic states, the region of paramagnetism being expand, for values U/t>5. At the same time, when the Fermi level is near the van Hove singularity, electron correlations do not change the diagrams qualitatively, which is consistent with the previously obtained result for square and cubic lattices. The results are compared with the data available in the literature for other methods and approaches.

We study the quantum phase transition from the superfluid to the Mott insulator state in two- and three-dimensional Bose–Einstein condensate with optical lattices using Bose–Hubbard Hamiltonian within the generalized Hatree–Fock–Bogoliubov approximation. The behavior of the depletion and the anomalous fraction has been investigated in the Mott insulator phase. We found that at T = 0, these quantities become significant in two and three dimensions. It is shown also that the dimensionality of the lattice enhances the anomalous density.

Abstract We study the revival of Hofstadter butterfly due to the competition between disorder and electronic interaction using mean field approximation of unrestricted Hartree Fock method at zero temperature for two dimensional square and honeycomb lattices. Interplay of disorder and electronic correlation to nullify each other is corroborated by the fact that honeycomb lattice needs more strength of electronic correlation owing to its less co-ordination number which enhances the effect of disorder. The extent of revival of the butterfly is better in square lattice than honeycomb lattice due to higher coordination number. The effect of disorder and interaction is also investigated to study entanglement entropy and entanglement spectrum. We find that for honeycomb lattice area law of entanglement entropy is obeyed in all cases but for square lattice there is some departure from area law for larger subsystems. The entanglement spectrum have the reflection symmetry of the original butterfly of the Hofstadter spectrum. The interaction induces a gap in the entanglement spectrum as well conforming the correspondence between physical spectrum and entanglement spectrum. The effect of disorder closes the interaction induced gap in the entanglement spectrum establishing the nullification of interaction due to disorder and vice versa.

Doutoramento em Física
O modelo de Hubbard é um dos modelos mais simples para descrever o movimento e a interacção de electrões em sólidos. Tem sido largamente estudado pelas suas aplicações na descrição de condutores orgânicos e na procura de supercondutividade a cada vez mais altas temperaturas. O objectivo desta tese é contribuir para a melhor compreensão do comportamento do modelo de Hubbard a duas dimensões quando a geometria da rede é alterada, nomeadamente torcendo as condições de fronteira ou introduzindo frustração geométrica. Começa-se por fazer uma extensão do diagrama de fases magnéticas do modelo de Hubbard numa rede quadrada usando a aproximação de campo médio, introduzindo a possibilidade de modulação da densidade de spin, contrastando assim com estudos anteriores. Isto foi conseguido dividindo a rede quadrada em duas sub-redes, podendo as suas densidades de spin ser diferentes. Concluiu-se que, em algumas regiões do diagrama de fases, esta densidade de spin modulada permite ao sistema baixar a sua energia livre. Em segundo lugar, introduz-se uma variação da rede quadrada, a que chamamos rede helicoidal. Estas duas redes são equivalentes no limite termodinâmico, visto que apenas diferem nas condições de fronteira. É apresentado um Hamiltoniano efectivo que descreve as correcções de energia em primeira ordem devidas aos saltos transversais no limite de acoplamento forte (strong-coupling limit). Devido à introdução destes saltos, observa-se uma dinâmica de spins, mesmo no limite de interacção electrónica infinita (ou seja, sem as correcções de Heisenberg). É apresentada uma expressão analítica para a correcção energética no caso de uma lacuna e um spin invertido, bem como representações gráficas das correcções para vários spins invertidos, obtidas numericamente. Em terceiro lugar, apresenta-se uma unificação dos estados localizados de redes quadradas decoradas. Esta unificação é apresentada na forma de "regras de origami", que incluem dobrar e desdobrar estados localizados de Hamiltonianos sem interacções (tight-binding ). Mostra-se que os estados localizados das redes decoradas de Lieb, Mielke e Tasaki podem ser obtidos uns a partir dos outros aplicando estas regras. Seguidamente, dá-se ênfase às redes decoradas da classe de Lieb. Começa-se por estudar a evolução temporal dos seus estados localizados quando um campo magnético é aplicado lentamente e perpendicularmente ao plano da rede. Conclui-se que, em concordância com o teorema adiabático, o estado localizado mantém-se localizado desde que haja uma diferença energética finita entre a sua energia e o resto do espectro do Hamiltoniano. Além disto, mostra-se que a forma como o estado localizado evolui pode ser descrita por um Hamiltoniano mais simples, com apenas três níveis energéticos, cuja solução é análoga a um movimento de precessão clássico. Finalmente, introduz-se a interacção de Hubbard na rede de Lieb e, usando a aproximação de campo médio, obtém-se o diagrama de fases magnéticas desta rede, previamente inexistente na literatura. Conclui-se que, no caso de redes bipartidas com diferente número de átomos em cada sub-rede, a abordagem de campo médio tradicional não reproduz resultados correctos na situação de um electrão por sítio (half filling ). Posto isto, segue-se uma abordagem em campo médio mais complexa (Hartree-Fock generalizada), que permite que as sub-redes tenham diferentes magnetizações e densidades de carga. Com estas modificações, a nova abordagem de campo médio já reproduz correctamente os resultados exactos em half filling, dados pelo teorema de Lieb e pelo teorema da densidade uniforme.
The Hubbard model is one of the simplest models to describe the motion and interaction of electrons in solids. It has been widely studied due to its applications in the description of organic conductors and in the search for high-Tc superconductivity. The aim of this thesis is to contribute for the better understanding of the behavior of the two-dimensional Hubbard model when the geometry of the lattice is changed, namely by twisting the boundary conditions or introducing geometric frustration. We begin by extending the mean-field magnetic phase diagram of the Hubbard model in a square lattice, by adding the possibility of spin density modulation, in contrast with previous studies. This was done by considering a square lattice divided into two sublattices, which were allowed to have different spin densities. We found that, in some regions of the phase diagram, nonuniform spin density throughout the lattice leads to a lower free energy. Secondly, we introduce a variation of the square lattice, which we call the helicoidal lattice. This lattice and the square lattice are equivalent in the thermodynamic limit, as they differ only in the boundary conditions. We present an effective Hamiltonian that describes the first-order energy corrections due to transversal hoppings in the strong-coupling limit, and show that interesting spin dynamics arises, even without the Heisenberg correction, due to hole hoppings in the transversal direction. We present an analytic expression for the energy correction in the case of one hole and one inverted spin. The numerically-obtained corrections for higher number of inverted spins are also shown. Thirdly, we present a unifying picture for localized states of decorated square lattices. This unification is presented in the form of what we call the "origami rules", which include folding and unfolding localized states of tight-binding Hamiltonians. We show that localized states of decorated lattices of the Lieb, Mielke and Tasaki classes can be obtained from each other by applying these rules. We then focus on the decorated lattices of the Lieb class. We begin by studying the time evolution of its localized states when a magnetic field is slowly applied perpendicularly to the plane of the lattice. We find that, as stated by the adiabatic theorem, the localized eigenstate remains localized as long as there is an energy gap between its energy and the rest of the Hamiltonian spectrum. Furthermore, we show that the way that the localized state evolves can be described by a simple three-level toy Hamiltonian, whose solution is analogous to a classical precession motion. Lastly, we introduce the Hubbard interaction in the Lieb lattice and, using the mean-field approximation, obtain the magnetic phase diagram of this lattice, previously absent from the literature. We find that, in the case of bipartite lattices with a different number of atoms on each sublattice, the traditional mean-field approach fails to yield correct results at half-filling. Therefore, we follow a more complex (generalized Hartree-Fock) mean-field approach, which allows the sublattices to have different magnetizations and charge densities. Under these new considerations, the mean-field approach correctly reproduces the exact results at half-filling, given by Lieb’s theorem and the uniform density theorem.

Utilizamos o modelo de Bose-Hubbard para estudar as estabilidades dinâmica e termodinâmica dos condensados numa rede óptica periódica circular. O nosso principal objetivo foi investigar a existência de condensados metaestáveis no sistema. Deduzimos e resolvemos a equação de Gross-Pitaevskii e, a partir da análise das soluções, foi possível mostrar que o sistema se condensa em estados com momento modular bem definido. Esses estados formam uma base que diagonaliza o termo que descreve o tunelamento atômico no hamiltoniano de Bose-Hubbard. No contexto da teoria de Bogoliubov deduzimos para cada condensado, o hamiltoniano efetivo cuja diagonalização determina o espectro das excitações coletivas do sistema. Identificamos corretamente o modo de energia zero, conseqüência da violação da conservação do número de átomos, e verificamos que este possui momento modular igual ao do condensado. No estudo da estabilidade vimos que todos os condensados com momento modular nos 2º e 3º quadrantes são termodinamicamente instáveis e as respectivas condições de estabilidade dinâmica dependem dos parâmetros de controle do sistema. Por outro lado os condensados com momento modular nos 1º e 4º quadrantes são todos dinamicamente estáveis enquanto que, nesse caso, é a estabilidade termodinâmica que depende dos parâmetros de controle do sistema. Nessa análise verificamos que o condensado com momento modular zero, que corresponde ao mínimo global da energia, é sempre estável. Determinamos exatamente o intervalo nos parâmetros de controle a partir do qual podemos encontrar condensados metaestáveis no sistema. Examinamos como a competição entre as intensidades dos termos de tunelamento e repulsão local afeta a estabilidade dos condensados. Essa competição define dois regimes distintos: Rabi, onde a coerência entre estados localizados nos sítios é mantida, e Fock, onde não há mais essa coerência e a aplicabilidade da aproximação de Bogoliubov é questionável.
We use the Bose-Hubbard model to study the dynamical and thermodynamical stabilities of condensates in a circular periodic optical lattice. Our main goal was to investigate the existence of metastable condensates in the system. We derive and solve the Gross-Pitaevskii equation, and from the analysis of the solutions it was possible to show that the system condenses in states with well-defined modular momentum. These states constitute a basis that diagonalizes the term of the Bose-Hubbard Hamiltonian which describes the dynamics of atomic tunneling. In the framework of Bogoliubov theory we determine, for each condensate, the effective Hamiltonian whose diagonalization give us the collective excitation spectrum of the system. We show that the mode associated to a zero eigenvalue, which is a consequence of the violation of atoms number conservation, has the same modular momentum of the condensate. The condensates with modular momentum in the 2nd and 3rd quadrants are all thermodynamically unstable whereas the dynamical stability depends on the control parameters. On the other hand, the condensates with modular momentum in the 1st and 4th quadrants are all dynamically stable whereas the thermodynamical stability depends on the control parameters. Our analysis shows that the condensate with modular momentum zero, which corresponds to a global minimum of energy, is always stable independently of the control parameters. We determine, exactly, the range on the control parameters where it is possible to detect metastability in the system. We have studied how the competition between the intensities of the tunneling and local interaction terms affects the stability of the condensates. This competition defines two distinct regimes: Rabi, where the coherence between states localized in the sites is achieved, and Fock, where this coherence is not achieved and the validity of Bogoliubov approximation is questionable.

This chapter discusses the energetics of point perturbations arising from intrinsic and extrinsic defects, and intentional and unintentional impurities. Point perturbations can be short range or long range. This requires the inclusion of core potential, exchange correlation and Hartree or Hartree-Fock potential. Hubbard energy, which is useful for Hartree calculations in a localized state, is introduced. An approach to calculating the behavior arising in shallow dopants (long range) and deep centers (short range) is presented. The tight binding defect-molecule model is used to explore the appearance of bonding and antibonding states in vacancies, interstitials and substitutional deep centers and some common complexes, such as the DX center, using configuration coordinates to understand the electronic and lattice energy contributions in the defect behavior. Finally, the chapter summarizes other important centers, such as the Pb center and the F center, before reviewing the implications of centers in light interaction and Poole-Frenkel conduction.

The H+–Li+ exchange technique in LiNbO3 is useful for the preparation of optical waveguides because it creates a large increase in the extraordinary refractive index. We have considered the reason for this increase both experimentally and theoretically. The exchange rate x of H+ and Li+ is about 0.7, as measured by nuclear-reaction analysis with a high-energy 15N ion beam. Electron diffraction shows that the space group does not change, and x-ray diffraction shows that the lattice constant hardly changes. Measurements of the infrared-absorption spectrum suggest that H+ is located in the plane where oxygen atoms are packed most closely. Two optimized positions of H+ in the plane have been decided by calculating the total energy of clusters that include hydrogen and oxygen atoms by the coupled Hartree–Fock method. The calculated first-order polarizability of clusters that include hydrogen and oxygen atoms is much larger than that of clusters including lithium and oxygen atoms. As a result of the decomposition contributions of molecule orbitals, this seems to arise from the s orbital, which is the lowest unoccupied orbital in all of the clusters. It suggests that the small atomic radius of H+ is the reason for the refractive-index change.

This paper mainly concentrates on the design and analysis of the annulus with zero thermal expansion coefficient (ZTE) aiming to solve the heat generation and deformation in high speed bearing. First, a fork-like lattice cell inspired by the basic triangular cell is put forward and further applied to construct an annulus. The stretch-dominated lattice cell utilizes the Poisson’s contraction effect to achieve the tailorable thermal expansion coefficient (CTE). The thermal behaviors differences between the continuous interfaces and lattice cells will lead to the internal stress. Thus, the CTE of the annulus consisting of the lattice cell can be tailored to zero even negative values through the offset between the thermal-strain and force-strain. Then a theoretical model is established with some appropriate assumptions to reveal the quantitative relations among the geometrical parameters, material properties and equivalent CTEs thoroughly. The prerequisites for realizing a zero CTE are further derived in terms of material limitations and geometric constraints. Finally, FEA method is implemented to verify and analyze the thermal behaviors of annulus. The proposed annulus design characterized by the CTE tunability, structure efficiency and continuous interfaces is hopefully to be applied in the high speed bearings, adapters between the shaft and collar and fastener screws.

Modern geometric methods open up prospects for improving the shape and structure of parts. Such improvement can pursue the goals of increasing the strength with constant material consumption, or reducing the mass when it is not necessary to increase the strength. The meaning of geometric methods is to create a part shape the stresses arising in the part material under the action of applied loads are distributed most evenly. Such methods include the use of fractal geometry. This article presents the results of a study of a fractal lattice created on the basis of the Sierpinski triangle. Computer simulation in the SolidWorks, as well as strength studies of parts produced using additive technologies, allowed us to confirm a multiple increase in the strength of the fractal lattice with an increase in the number of fractal iterations. One of the most promising areas of application of fractal structures may be aviation technology. In this area, weight reduction is needful, and the complex shape of the parts is realized with the help of expensive production methods. For this reason, a number of experiments were conducted within the framework of the study, the purpose of which was to test the feasibility of using fractal gratings to reduce the weight of aircraft parts, using the example of the fork of the front landing gear of the combat training aircraft Yak-130.