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arXiv:1712.02002 · International Journal of Modern Physics B

The hierarchical Green’s function approach to the two-dimensional Hubbard model

Yu-Liang Liu

Published Aug 17, 2018• 5 citations
arXivHTMLPDF

Abstract

By introducing multiple-site correlation functions, we propose a hierarchical Green’s function approach, and apply it to study the characteristic properties of a two-dimensional (2D) square lattice Hubbard model by solving the equation of motions of a one-particle Green’s function and related multiple-site correlation functions. Under a cut-off approximation and taking the Fourier representation of multiple-site correlation functions, we obtain an analytical expression of one-particle Green’s function with static correlation functions. Then we calculate the spectral density function of electrons, and obtain that besides two main peaks corresponding to the lower and upper Hubbard bands in the spectral density function, there emerge some novel states between these two main peaks, and the total spectral weight of these emerged states is proportional to the hole doping concentration [Formula: see text]. Meanwhile, there also emerge some collective modes related to possible charge/spin density wave and/or electronic pairing density wave ordering states. This calculation is completely consistent with the spectroscopy observations of the cuprate superconductors in normal states. On the other hand, the appearance of the static correlation functions in the one-particle Green’s function can be used to describe the intertwined orders observed in the normal state of the cuprate superconductors.

DOI
10.1142/s0217979218502582

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By introducing multiple-site correlation functions, we propose a hierarchical Green’s function approach, and apply it to study the characteristic properties of a two-dimensional (2D) square lattice Hubbard model by solving the equation of motions of a one-particle Green’s function and related multiple-site correlation functions. Under a cut-off approximation and taking the Fourier representation of multiple-site correlation functions, we obtain an analytical expression of one-particle Green’s function with static correlation functions. Then we calculate the spectral density function of electrons, and obtain that besides two main peaks corresponding to the lower and upper Hubbard bands in the spectral density function, there emerge some novel states between these two main peaks, and the total spectral weight of these emerged states is proportional to the hole doping concentration [Formula: see text]. Meanwhile, there also emerge some collective modes related to possible charge/spin density wave and/or electronic pairing density wave ordering states. This calculation is completely consistent with the spectroscopy observations of the cuprate superconductors in normal states. On the other hand, the appearance of the static correlation functions in the one-particle Green’s function can be used to describe the intertwined orders observed in the normal state of the cuprate superconductors.
With the hierarchical Green’s function approach, we study a doped Mott insulator described with the Hubbard model by analytically solving the equations of motion of an one-particle Green’s function and related multiple-point correlation functions, and find that the separation of the spin and charge degrees of freedom of the electrons is an intrinsic character of the doped Mott insulator. For enough of large on-site repulsive Coulomb interaction, we show that the spectral weight of the one-particle Green’s function is proportional to the hole doping concentration that is mainly produced by the charge fluctuation of electrons, while the excitation spectrum of the electrons is composed of two parts: One is contributed by the spin fluctuation of the electrons, which is proportional to the hole doping concentration, and another one is coming from the coupling between the charge and spin fluctuations of the electrons that takes the maximum at undoping. All of these low energy/temperature physical properties originate from the strong on-site Coulomb interaction. The present results are consistent with the spectroscopy observations of the cuprate superconductors and the numerical calculations in normal state above the pseudogap regime.
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The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
A progress report on the development of numerical and rigorous solutions of the effective two-band two-dimensional Hubbard model of the high-temperature superconductivity in cuprates [N.M. Plakida et al., PRB 51, 16599 (1995)], within the equation of motion technique of the thermodynamic Green function approach, is given. Four important consequences of the rigorous mean-field solution are worth mentioning: (1) Occurrence, both in the normal and superconducting states, of the spin-charge separation, conjectured by P.W. Anderson on intuitive grounds. (2) Origin of the static d-wave pairing in the hopping singlet conduction. (3) Complete suppression of the s-type superconducting pairing by the crystal symmetry in square two- dimensional lattices and occurrence of a residual s-type term in rectangular lattices, in agreement with the experiment. (4) The energy band hybridization in the superconducting state points to an overall displacement of the energy spectrum toward lower energies, as pointed by very precise optical measurements.
Electronic states of a two-dimensional Hubbard model near half filling is investigated by use of the composite-operator method, which is improved by including two-site correlations in combined use with the resolvent method. The densities of states obtained for the upper and lower Hubbard bands have three-peek structures at half filling and give shapes above a certain temperature similar to the ones obtained in the numerical simulation by Dagotto, Ortolani, and Scalapino [Phys. Rev. B 46, 3183 (1992)], but show narrowing of the bands at low temperature due to the formation of spin-singlet states among the nearest-neighbor electrons. A doped case shows a coherent peak at the Fermi level and the feature of the density of states is very similar to the one from numerical simulation. The one-particle spectra in the (k,k) direction is also presented. The results show that, at half filling, the spectral weight is not evenly distributed between two bands; most of the weight of the lower Hubbard band appears between (0,0) and (\ensuremath{\pi}/2a,\ensuremath{\pi}/2a), while that of the upper Hubbard band lies in the region between (\ensuremath{\pi}/2a,\ensuremath{\pi}/2a) and (\ensuremath{\pi}/a,\ensuremath{\pi}/a), keeping roughly the original band distribution. With hole doping, a flat band develops at the Fermi level with a large Fermi surface and the lower Hubbard band extends through the whole Brillouin zone.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].