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Physical review. B, Condensed matter

Class of variational singlet wave functions for the Hubbard model away from half filling

Philip Warren Anderson, B. Sriram Shastry, Dionissios T. Hristopulos

Published Nov 1, 1989• 108 citations
Publisher

Abstract

We present a class of variational wave functions for strong-coupling Heisenberg Hubbard models. These are written in the form of three factors---a pair of determinants and a Jastrow function---and are made out of orbitals, a la Hartree-Fock theory, which solve a fictitious one-body problem. The wave functions respect various constraints known from general principles and appear to be potentially useful in understanding the possible behavior of the models in quantitative terms.

DOI
10.1103/physrevb.40.8939

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We present a class of variational wave functions for strong-coupling Heisenberg Hubbard models. These are written in the form of three factors---a pair of determinants and a Jastrow function---and are made out of orbitals, a la Hartree-Fock theory, which solve a fictitious one-body problem. The wave functions respect various constraints known from general principles and appear to be potentially useful in understanding the possible behavior of the models in quantitative terms.
A tractable direct optimization algorithm is developed to optimize orbitals in the Slater determinants (S-dets) in a resonating Hartree-Fock (Res HF) wave function. We reduce the variation space using the orbitals to put the first order energy variation in the steepest descent direction. The orbitals in the next iteration are determined so as to minimize the energy functional including up to the second order variation. This algorithm is applied to the one dimensional Hubbard model of half-filling. The optimized S-dets much deviate from the trial S-dets prepared from the HF calculations. The Res HF ground state generated with a few S-dets explains from 99.9 to 95.0% of the ground state correlation energy in all the correlation regimes. We have spin correlation functions with the correct short and long range behaviors and the lowest triplet and singlet spin excitations with correct dispersions, suggesting that the optimization of orbitals incorporates long range spin fluctuations and their mode-mode couplings.
New variational wave functions are introduced to describe the Hubbard model in strong correlation. Intersite correlation (or Jastrow) factors are taken into account beyond the Gutzwiller wave function (GF). A study is made on the one-dimensional lattice and the two-dimensional square lattice at half-filling as well as less-than-half-filling. Much improvement can be achieved thereby in the description of physical quantities such as the momentum distribution and spin and charge density correlation functions. In addition, the ground state energy is lowered appreciably compared with the GF. In this connection, the Mott-Hubbard-type metal-insulator transition is also re-examined.
: The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian. This work was supported in part by Sonderforschungsbereich 288 of the Deutsche Forschungsgemeinschaft...
We apply a variational method devised for the nuclear many-body problem to the one-dimensional Hubbard model with nearest neighbor hopping and periodic boundary conditions. The test wave function consist for each state out of a single Hartree--Fock determinant mixing all the sites (or momenta) as well as the spin projections of the electrons. Total spin and linear momentum are restored by projection methods before the variation. It is demonstrated that this approach reproduces the results of exact diagonalizations for half-filled N=12N=12N=12 and N=14N=14N=14 lattices not only for the energies and occupation numbers of the ground but also of the lowest excited states rather well. Furthermore, a system of ten electrons in an N=12N=12N=12 lattice is investigated and, finally, an N=30N=30N=30 lattice is studied. In addition to energies and occupation numbers we present the spectral functions computed with the help of the symmetry-projected wave functions as well.
We investigate T=0 properties of the commensurate flux phases, a class of variational wave functions for the t-J model proposed recently by Anderson, Shastry, and Hristopoulos and which generalize the (1/2-flux state of Affleck and Marston away from half-filling. These states can be formally written in the quantum spin liquid form, the singlet bond amplitudes of which break the lattice translational symmetry. The supercell of the electromagnetic gauge (defining the fictitious flux) imposes the actual periodicities of the correlated wave function. Their order of commensurability with the lattice unit length is closely related to the hole density.
An approximate solution scheme, similar to the Gutzwiller approximation, is presented for the Baeriswyl and the Baeriswyl-Gutzwiller variational wave functions. The phase diagram of the one-dimensional Hubbard model as a function of interaction strength and particle density is determined. For the Baeriswyl wave function a metal-insulator transition is found at half filling, where the metallic phase (U<Uc)(U<{U}_{c})(U<Uc​) corresponds to the Hartree-Fock solution, the insulating phase is one with finite double occupations arising from bound excitons. This transition can be viewed as the ``inverse'' of the Brinkman-Rice transition. Close to but away from half filling, the U>UcU>{U}_{c}U>Uc​ phase displays a finite Fermi step, as well as double occupations originating from bound excitons. As the filling is changed away from half-filling bound excitons are suppressed. For the Baeriswyl-Gutzwiller wave function at half filling a metal-insulator transition between the correlated metallic and excitonic insulating state is found. Away from half-filling bound excitons are suppressed quicker than for the Baeriswyl wave function.