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arXiv:cond-mat/9609024 · Physical review. B, Condensed matter

Estimation of properties of low-lying excited states of Hubbard models: A multiconfigurational symmetrized projector quantum Monte Carlo approach

Bhargavi Srinivasan, S. Ramasesha, H. R. Krishnamurthy

Published Sep 15, 1997• 4 citations
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Abstract

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 $4n$ and $4n+2$ 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.

DOI
10.1103/physrevb.56.6542

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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×\else\texttimes\fi24\ifmmode\times\else\texttimes\fi{}24\ifmmode×\else\texttimes\fi2 ladder and the 3\ifmmode×\else\texttimes\fi43\ifmmode\times\else\texttimes\fi{}43\ifmmode×\else\texttimes\fi4 lattice with periodic boundary conditions. The 3\ifmmode×\else\texttimes\fi43\ifmmode\times\else\texttimes\fi{}43\ifmmode×\else\texttimes\fi4 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×\else\texttimes\fi24\ifmmode\times\else\texttimes\fi{}24\ifmmode×\else\texttimes\fi2 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.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.
Fixed-node Green's-function Monte Carlo calculations have been performed for very large 16\ifmmode\times\else\texttimes\fi{}16 two-dimensional Hubbard lattices, large interaction strengths U=10,20,U=10,20,U=10,20, and 404040, and many $(15\ensuremath{\sim}20)$ densities between empty and half-filling. The nodes were fixed by a simple Slater-Gutzwiller trial wave function. For each value of U we obtained a sequence of ground-state energies which is consistent with the possibility of a phase separation close to half-filling, with a hole density in the hole-rich phase which is a decreasing function of UUU. The energies suffer, however, from a fixed-node bias: more accurate nodes are needed to confirm this picture. Our extensive numerical results and their test against size, shell, shape, and boundary-condition effects also suggest that phase separation is quite a delicate issue, on which simulations based on smaller lattices than considered here are unlikely to give reliable predictions.
We describe and discuss a recently proposed quantum Monte Carlo algorithm to compute the ground-state properties of various systems of interacting fermions. In this method, the ground state is projected from an initial wave function by a branching random walk in an overcomplete basis of Slater determinants. By constraining the determinants according to a trial wave function |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉 is exact. We illustrate the method by describing in detail its implementation for the two-dimensional one-band Hubbard model. We show results for lattice sizes up to 16\ifmmode\times\else\texttimes\fi{}16 and for various electron fillings and interaction strengths. With simple single-determinant wave functions as |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, the method yields accurate (often to within a few percent) estimates of the ground-state energy as well as correlation functions, such as those for electron pairing. We conclude by discussing possible extensions of the algorithm.
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.