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International Journal of Modern Physics B

NUMERICAL SIMULATION OF THE 1D AND 2D HUBBARD MODELS: FERMI LIQUID BEHAVIOR AND ITS BREAKDOWN

Sandro Sorella, Erio Tosatti, Stefano Baroni A, Roberto Car, Michele Parrinello

Published Oct 1, 1988• 109 citations
Publisher

Abstract

The ground state of the 1D and of the 2D (square lattice) finite-size Hubbard model is investigated for variable filling using a novel quantum simulation method. We have studied up to 256 sites for both 1D and 2D. Away from half filling the 2D antiferromagnetic order is initially destroyed, albeit without any clear sign of a Fermi liquid behaviour. A metallic jump in n(k) appears only very far from half filling. In the 1D case, by contrast, a Fermi liquid-like jump in n(k) is obtained even very close to half filling.

DOI
10.1142/s0217979288000822

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The ground state of the 1D and of the 2D (square lattice) finite-size Hubbard model is investigated for variable filling using a novel quantum simulation method. We have studied up to 256 sites for both 1D and 2D. Away from half filling the 2D antiferromagnetic order is initially destroyed, albeit without any clear sign of a Fermi liquid behaviour. A metallic jump in n(k) appears only very far from half filling. In the 1D case, by contrast, a Fermi liquid-like jump in n(k) is obtained even very close to half filling.
The ground state of the 2D (square lattice) finite-size Hubbard model is studied at half filling with a newly developed Quantum Monte Carlo method. We have studied square cells from 18 to 242 sites for U=3 with high statistical accuracy to perform a reliable size scaling analysis. The results confirm that the system is an antiferromagnetic insulator. The finite size scaling results are discussed in detail, focusing on the behaviour of magnetization, and most particularly of momentum distribution.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.
The applicability of the Hartree-Fock and random phase approximations to models of strongly correlated electrons is discussed. The 2D Hubbard model is analyzed. An antiferromagnetic phase (at half filling) and Fermi liquid behavior (at low electron densities) are found. In addition, the system shows an unconventional regime near half filling. It is characterized by the formation of spin and charge textures. Local magnetic moments and spin waves are also present, and the quasiparticle pole at the Fermi energy vanishes. The charge excitation spectrum is gapless. The overall features resemble those of the 1D Luttinger liquid.
The two-dimensional Hubbard model with a single spin-up electron interacting with a finite density of spin-down electrons is studied by use of the quantum Monte Carlo technique, the standard second-order perturbation theory and a very good variational wave function introduced recently by Edwards. We also present a new conjugate gradient scheme for the evaluation of this variational state. We performed simulations up to 242 sites at U/t=4 reaching the zero-temperature properties with no ``fermion-sign problem'' and found a surprisingly good accuracy of the Edwards state at low density or low doping. The conjugate gradient method was then applied to the system up to 3698 sites and infinite U. Fermi-liquid theory seems to remain stable in two dimensions for all cases studied with the exception of the half-filling case where a Luttinger-like behavior survives in the Hubbard model, yielding a vanishing quasiparticle weight in the thermodynamic limit.
Employing a combination of functional renormalization group calculations and projective determinantal quantum Monte Carlo simulations, we examine the Hubbard model on the square lattice bilayer at half filling. From this combined analysis, we obtain a comprehensive account of the ground-state phase diagram with respect to the extent of the system's metallic and (antiferromagnetically ordered) Mott-insulating as well as band-insulating regions. By means of an unbiased functional renormalization group approach, we exhibit the antiferromagnetic Mott-insulating state as the relevant instability of the free metallic state, induced by any weak finite on-site repulsion. Upon performing a careful analysis of the quantum Monte Carlo data, we resolve the difficulty of identifying this antiferromagnetic ground state for finite interlayer hopping in the weak-coupling regime, where nonmonotonic finite-size corrections are shown to relate to the two-sheeted Fermi surface structure of the metallic phase. On the other hand, quantum Monte Carlo simulations are well suited to identify the transition between the Mott-insulating phase and the band insulator in the intermediate-to-strong coupling regime. Here, we compare our numerical findings to indications for the transition region obtained from the functional renormalization group procedure.
The half-filled Hubbard model on the honeycomb lattice is investigated by numerically exact large-scale quantum Monte Carlo simulations for lattice sizes up to 2592 sites. By performing careful finite-size scaling for the spin-spin correlation functions, calculated with a high degree of accuracy, we find that the ground state is antiferromagnetically long-range ordered at U / t = 4, where U is the on-site Hubbard interaction and t is the nearest neighbor hopping. Our result is in sharp contrast to a recent report [Meng et al. , Nature 464, 847 (2010)], where instead strong evidence of spin liquid behavior is found.