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Physical review. B, Condensed matter

Quantum Monte Carlo study of quasiparticles in the Hubbard model

Wolfgang von der Linden, I. Morgenstern, Hans A. De Raedt

Published Mar 1, 1990• 22 citations
Publisher

Abstract

We present an improved version of the projector quantum Monte Carlo method, which has recently been proposed by Sorella et al. This scheme allows a very precise computation of the ground-state energy of fermionic models. The ``minus sign'' has been treated without further approximations and does not influence the accuracy. In this paper we focus on the question of the existence of quasiparticles in the Hubbard model in one dimension and resolve the controversy concerning the discontinuity in the momentum distribution function between recent quantum Monte Carlo and weak-coupling renormalization-group techniques.

DOI
10.1103/physrevb.41.4669

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We present an improved version of the projector quantum Monte Carlo method, which has recently been proposed by Sorella et al. This scheme allows a very precise computation of the ground-state energy of fermionic models. The ``minus sign'' has been treated without further approximations and does not influence the accuracy. In this paper we focus on the question of the existence of quasiparticles in the Hubbard model in one dimension and resolve the controversy concerning the discontinuity in the momentum distribution function between recent quantum Monte Carlo and weak-coupling renormalization-group techniques.
The quantum Monte Carlo method for the Hubbard model using the Hubbard-Stratonovich transformation is reexamined. It is observed that the value of the ground state energy for a 3×2 and 4×4 system is estimated correctly by ignoring the minus-sign of the negative sampling weight appearing in the simulation using the projector method. It is also observed that the average of sign is positive and the correlation between the energy and the sign is weak for these systems. The reasons why the ground state energy is estimated correctly is given on a phenomenological base.
We present in detail the recently developed multiconfigurational symmetrized projector quantum Monte Carlo method for excited states of the Hubbard model. We describe the implementation of the Monte Carlo method for a multiconfigurational trial wave function. We give a detailed discussion of issues related to the symmetry of the projection procedure that validates our Monte Carlo procedure for excited states. In this context we discuss various averaging procedures for the Green function and present an analysis of the errors incurred in these procedures. We study the ground-state energy and correlation functions of the one-dimensional Hubbard model at half-filling to confirm these analyses. We then study the energies and correlation functions of excited states of Hubbard chains. Hubbard rings away from half-filling are also studied and the pair binding energies for holes of 4n4n4n and 4n+24n+24n+2 systems are compared with the Bethe ansatz results of Fye, Martins, and Scalettar [Phys. Rev. B 42, 6809 (1990)]. Our study of the two-dimensional Hubbard model includes the $4\ifmmode\times\else\texttimes\fi{}2$ ladder and the $3\ifmmode\times\else\texttimes\fi{}4$ lattice with periodic boundary conditions. The $3\ifmmode\times\else\texttimes\fi{}4$ lattice is nonbipartite and amenable to exact diagonalization studies and is, therefore, a good candidate for checks on the method. We are able to reproduce accurately the energies of ground and excited states, both at and away from half-filling. We study the properties of the $4\ifmmode\times\else\texttimes\fi{}2$ Hubbard ladder with bond alternation as the correlation strength and filling are varied. The method reproduces the correlation functions accurately. We also examine the severity of sign problem for one- and two-dimensional systems.
In this paper we compare numerical results for the ground state of the Hubbard model obtained by Quantum-Monte-Carlo simulations with results from exact and stochastic diagonalizations. We find good agreement for the ground state energy and superconducting correlations for both, the repulsive and attractive Hubbard model. Special emphasis lies on the superconducting correlations in the repulsive Hubbard model, where the small magnitude of the values obtained by Monte-Carlo simulations gives rise to the question, whether these results might be caused by fluctuations or systematic errors of the method. Although we notice that the Quantum-Monte-Carlo method has convergence problems for large interactions, coinciding with a minus sign problem, we confirm the results of the diagonalization techniques for small and moderate interaction strengths. Additionally we investigate the numerical stability and the convergence of the Quantum-Monte-Carlo method in the attractive case, to study the influence of the minus sign problem on convergence. Also here in the absence of a minus sign problem we encounter convergence problems for strong interactions.
Recent computer simulations performed on the one-dimensional Hubbard model with a new algorithm developed by Sorella et al. confirm the absence of a Fermi surface in this model at half filling. However, these same simulations suggest the existence of a Fermi surface at zero temperature away from half filling, contrary to expectations based on renormalization-group analysis. The results found here, using a more standard algorithm and a simple low-temperature extrapolation, appear consistent with the above-mentioned numerical results in the half-filled and quarter-filled band cases. However, it is argued that the discontinuity at the Fermi level found in all these simulations is likely to come from the zero-temperature finite-size dependence of the quasiparticle weight Z, which is also discussed here.