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Advances in Condensed Matter Physics

On the Validity of the Dynamical Mean-Field Approximation for Studying the Two-Dimensional Hubbard Model on a Square Lattice

André Neves Ribeiro, Cláudio Andrade Macedo

Published Jan 1, 2010• 1 citation
Publisher

Abstract

The dynamical mean-field approximation (DMFA) becomes exact in the limit of infinite dimensions, and allows results to be obtained in a nonperturbative regime without the limitations normally found with exact diagonalization (ED) and quantum Monte Carlo (QMC) methods. In this paper, we investigate the applicability of the method to lattices with small coordination number in special situations. Specifically we use this approximation to study the two-dimensional (2D) Hubbard model on a square lattice far from half filling. In this situation, we calculate the specific heat and find that when the filling decreases, that is, antiferromagnetic correlations become less important, the agreement between DMFA and QMC results increases. Our results show that the DMFA can be a valuable technique for studying the thermodynamic properties of the Hubbard model also on a square lattice, but within a parameter range in which the antiferromagnetic correlations are not important.

DOI
10.1155/2010/487381

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The dynamical mean-field approximation (DMFA) becomes exact in the limit of infinite dimensions, and allows results to be obtained in a nonperturbative regime without the limitations normally found with exact diagonalization (ED) and quantum Monte Carlo (QMC) methods. In this paper, we investigate the applicability of the method to lattices with small coordination number in special situations. Specifically we use this approximation to study the two-dimensional (2D) Hubbard model on a square lattice far from half filling. In this situation, we calculate the specific heat and find that when the filling decreases, that is, antiferromagnetic correlations become less important, the agreement between DMFA and QMC results increases. Our results show that the DMFA can be a valuable technique for studying the thermodynamic properties of the Hubbard model also on a square lattice, but within a parameter range in which the antiferromagnetic correlations are not important.
The two-dimensional half-filled Hubbard model is studied by a nonperturbative analytic theory of correlator projection. The dynamical mean-field approximation (DMFA) is reproduced at the first-order projection and then improved by systematic inclusion of spatial correlations at higher orders. A geometrical frustration induces a first-order Mott transition surface with a finite-temperature critical end curve and related crossovers. Growth of antiferromagnetic correlations gives single-particle spectra strongly modified from DMFA with shadow bands and flat dispersions observed in high-Tc{T}_{\mathrm{c}}Tc​ cuprates.
We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
We present results for the equation of state of the two-dimensional Hubbard model on an isotropic square lattice as obtained from a controlled and numerically exact large-cluster dynamical mean field simulation. Our results are obtained for large but finite systems and are extrapolated to infinite system size using a known finite-size scaling relation, and are supplemented by reliable error bars accounting for all sources of errors. We establish the importance of examining the decay of spatial spin correlations to determine whether a sufficiently large cluster has been used and with this in mind we present the energy, entropy, double occupancy, and nearest-neighbor spin correlations extrapolated to the thermodynamic limit. We discuss the implications of these calculations on pseudogap physics of the 2D Hubbard model away from half filling, where we find a strong behavioral shift in energy below a temperature T∗{T}^{*}T∗ which becomes more pronounced for larger clusters. Finally, we provide reference calculations and tables for the equation of state for values of doping away from half filling which are of interest to cold-atom experiments.
We present a powerful method for calculating the thermodynamic properties of infinite-dimensional Hubbard-type models using an exact diagonalization of an Anderson model with a finite number of sites. The resolution obtained for Green's functions is far superior to that of quantum Monte Carlo calculations. We apply the method to the half-filled Hubbard model for a discussion of the metal-insulator transition, and to the two-band Hubbard model where we find direct evidence for the existence of a superconducting instability at low temperatures.
We develop a precise and reliable numerical method for the calculation of electronic properties of the two orbital Hubbard model in a square lattice at half filling, based on the Dynamical Mean Field Theory (DMFT). We use the Density Matrix Renormalization Group (DMRG) as the impurity solver for the DMFT's selfconsistent equations to obtain accurate values for the Green's functions on the real axis. This way, reliable densities of states are obtained that do not need to resort to analytical continuation methods as those using quantum Monte Carlo techniques. Large system sizes can be achieved with increasing accuracy.