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arXiv:cond-mat/9606204 · TURKISH JOURNAL OF PHYSICS

Low-Energy Electronic Excitations of the Layered Cuprates and the Hubbard Model

N. BULUT

Published Jan 1, 1996• 5 citations
arXivHTMLPDF

Abstract

We present a review of the Quantum Monte Carlo computation results on the magnetic, charge and single--particle excitations as well as the pairing correlations of the two--dimensional Hubbard model. We are particularly interested in how these quantities are related to each other in the metallic state that forms near half--filling. These results are useful in gaining a better understanding of the low energy excitations of the superconducting cuprates.

DOI
10.55730/1300-0101.2543

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We present a review of the Quantum Monte Carlo computation results on the magnetic, charge and single--particle excitations as well as the pairing correlations of the two--dimensional Hubbard model. We are particularly interested in how these quantities are related to each other in the metallic state that forms near half--filling. These results are useful in gaining a better understanding of the low energy excitations of the superconducting cuprates.
The numerical studies of d x 2 - y 2 -wave pairing in the two-dimensional (2D) and the 2-leg Hubbard models are reviewed. For this purpose, the results obtained from the determinantal Quantum Monte Carlo and the Density-Matrix Renormalization-Group calculations are presented. These are calculations which were motivated by the discovery of the high- T c cuprates. In this review, the emphasis is placed on the microscopic many-body processes which are responsible for the d x 2 - y 2 -wave pairing correlations observed in the 2D and the 2-leg Hubbard models. In order to gain insight into these processes, the results on the effective pairing interaction as well as the magnetic, density and the single-particle excitations will be reviewed. In addition, comparisons will be made with the other numerical approaches to the Hubbard model and the numerical results on the t - J model. The results reviewed here indicate that an effective pairing interaction which is repulsive at ( ~ , ~ ) momentum transfer, and enhanced single-particle spectral weight near the ( ~ ,0) and (0, ~ ) points of the Brillouin zone, create optimum conditions for d x 2 - y 2 -wave pairing. These are two effects which act to enhance the d x 2 - y 2 -wave pairing correlations in the Hubbard model. Finding additional ways is an active research problem.
We present recent theoretical results on superconductivity in correlated-electron systems, especially in the two-dimensional Hubbard model and the three-band d-p model. The mechanism of superconductivity in high-temperature superconductors has been extensively studied on the basis of various electronic models and also electron-phonon models. In this study, we investigate the properties of superconductivity in correlated-electron systems by using numerical methods such as the variational Monte Carlo method and the quantum Monte Carlomethod. The Hubbard model is one of basic models for strongly correlated electron systems, and is regarded as the model of cuprate high temperature superconductors. The d-p model is more realistic model for cuprates. The superconducting condensation energy obtained by adopting the Gutzwiller ansatz is in reasonable agreement with the condensation energy estimated for YBa2Cu3O7. We show the phase diagram of the ground state using this method. We have further investigated the stability of striped and checkerboard states in the under-doped region. Holes doped in a half-filled square lattice lead to an incommensurate spin and charge density wave. The relationship of the hole density x and incommensurability δ, δ~x, is satisfied in the lower doping region, as indicated by the variationalMonte Carlocalculations for the two-dimensional Hubbard model. A checkerboard-like charge-density modulation with a roughly period has also been observed by scanning tunneling microscopy experiments in Bi2212 and Na-CCOC compounds. We have performed a variational Monte Carlo simulation on a two-dimensional t-t′-t″- U Hubbard model with a Bi-2212 type band structure and found that the period checkerboard spin modulation, that is characterized by multi Q vectors, is indeed stabilized. We have further performed an investigation by using a quantumMonte Carlomethod, which is a numerical method that can be used to simulate the behavior of correlated electron systems. We present a new algorithm of the quantum Monte Carlo diagonalization that is a method for the evaluation of expectation value without the negative sign problem. We compute pair correlation functions and show that pair correlation is indeed enhanced with hole doping.
Superconducting correlation of the Hubbard model on a square lattice is reinvestigated by quantum Monte Carlo method. Results from more than one order of magnitude larger Monte Carlo steps than before are presented to discuss the convergence of the data to the ground state. By examining the dependence on the choice of initial trial states as well as on the Trotter number in detail, the superconducting correlation in the ground state is reliably obtained up to 12×12 lattice away from the half-filling and up to 16×16 lattice at the half-filling, which substantially improve former results near the half-filling. Various channels of pairing in the ground state show no system size dependence and remain quite short ranged at any filling including the region close to the half-filling. Numerical data show further evidence that neither extended s - nor d -wave superconducting state is possible in the two-dimensional Hubbard model.
We study the three-band Hubbard model for the copper oxide plane of the high-temperature superconducting cuprates using determinant quantum Monte Carlo and the dynamical cluster approximation (DCA) and provide a comprehensive view of the pairing correlations in this model using these methods. Specifically, we compute the pair-field susceptibility and study its dependence on temperature, doping, interaction strength, and charge-transfer energy. Using the DCA, we also solve the Bethe-Salpeter equation for the two-particle Green's function in the particle-particle channel to determine the transition temperature to the superconducting phase on smaller clusters. Our calculations reproduce many aspects of the cuprate phase diagram and indicate that there is an ``optimal'' value of the charge-transfer energy for the model where Tc{T}_{c}Tc​ is largest. These results have implications for our understanding of superconductivity in both the cuprates and other doped charge-transfer insulators.
Single-particle excitation spectra of the two-dimensional Hubbard model on the square lattice near half filling and at zero temperature are investigated on the basis of the self-consistent projection operator method. The method guarantees a high accuracy of the spectra with high energy and high momentum resolutions. It takes into account long-range intersite correlations as well as the strong on-site correlations. Effects of nonlocal excitations are clarified by comparing the results with those of the single-site approximation. The calculated spectra verify the quantum Monte-Carlo results for finite temperatures. The spectra at the Fermi level yield a hole-like Fermi surface in the underdoped region and an electron-like Fermi surface in the overdoped region. From a numerical analysis of the momentum dependent effective mass and self-energy, it is concluded that a marginal Fermi-liquid like state persists even at finite doping concentrations in the strongly correlated region because a van Hove singularity is pinned to the Fermi surface. It is also found that a kink structure appears in the quasiparticle energy band in the same region. The kink is shown to be caused by a mixing between the quasiparticle band and an excitation band with strong short-range antiferromagnetic correlations. These results suggest an explanation for some of the unusual properties of the normal state in high-TcT_{\rm c}Tc​ cuprates.
We study the three-band Hubbard model for the cuprate superconductors using Quantum Monte Carlo simulations and the fluctuation exchange approximation including a self-consistent many-body renormalization of the single-particle propagator. The energy dispersion of the low-lying one-particle excitations is in very good agreement with spectroscopic measurements for a hole doping concentration δ = 0.25 of the CuO2 planes. At the same doping, we examine the spin susceptibility yielding a non-ordered phase with short-range incommensurate fluctuations near the antiferromagnetic wave vector. In the superconducting regime, the interaction vertex for the pairing correlation function in the extended s-wave channel shows a maximum for δ = 0.20, resembling the dependence of transition temperature Tc on doping.