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arXiv:1610.02053 · Journal of Physics Condensed Matter

Biorthogonal projected energies of a Gutzwiller similarity transformed Hamiltonian

Jacob M. Wahlen-Strothman, Gustavo E. Scuseria

Published Oct 5, 2016• 7 citations
arXivHTMLPDF

Abstract

We present a method incorporating biorthogonal orbital-optimization, symmetry projection, and double-occupancy screening with a non-unitary similarity transformation generated by the Gutzwiller factor [Formula: see text], and apply it to the Hubbard model. Energies are calculated with mean-field computational scaling with high-quality results comparable to coupled cluster singles and doubles. This builds on previous work performing similarity transformations with more general, two-body Jastrow-style correlators. The theory is tested on 2D lattices ranging from small systems into the thermodynamic limit and is compared to available reference data.

DOI
10.1088/0953-8984/28/48/485502

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We present a method incorporating biorthogonal orbital-optimization, symmetry projection, and double-occupancy screening with a non-unitary similarity transformation generated by the Gutzwiller factor [Formula: see text], and apply it to the Hubbard model. Energies are calculated with mean-field computational scaling with high-quality results comparable to coupled cluster singles and doubles. This builds on previous work performing similarity transformations with more general, two-body Jastrow-style correlators. The theory is tested on 2D lattices ranging from small systems into the thermodynamic limit and is compared to available reference data.
We present a class of Lie algebraic similarity transformations generated by exponentials of two-body on-site Hermitian operators whose Hausdorff series can be summed exactly without truncation. The correlators are defined over the entire lattice and include the Gutzwiller factor ${n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}$, and two-site products of density $({n}_{i\ensuremath{\uparrow}}+{n}_{i\ensuremath{\downarrow}})$ and spin $({n}_{i\ensuremath{\uparrow}}\ensuremath{-}{n}_{i\ensuremath{\downarrow}})$ operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with polynomial computational cost. The proposed similarity transformation generates locally weighted orbital transformations of the reference determinant. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one- and two-dimensional repulsive Hubbard model where it yields accurate results for small and medium sized interaction strengths.
Similarity transformation of the Hubbard Hamiltonian using a Gutzwiller correlator leads to a non-Hermitian effective Hamiltonian, which can be expressed exactly in momentum-space representation and contains three-body interactions. We apply this methodology to study the two-dimensional Hubbard model with repulsive interactions near half filling in the intermediate interaction strength regime (U/t=4)(U/t=4)(U/t=4). We show that at optimal or near optimal strength of the Gutzwiller correlator, the similarity-transformed Hamiltonian has extremely compact right eigenvectors, which can be sampled to high accuracy using the full configuration interaction quantum Monte Carlo (FCIQMC) method and its initiator approximation. Near-optimal correlators can be obtained using a simple projective equation, thus obviating the need for a numerical optimization of the correlator. The FCIQMC method, as a projective technique, is well suited for such non-Hermitian problems, and its stochastic nature can handle the three-body interactions exactly without undue increase in computational cost. The highly compact nature of the right eigenvectors means that the initiator approximation in FCIQMC is not severe and that large lattices can be simulated, well beyond the reach of the method applied to the original Hubbard Hamiltonian. Results are provided in lattice sizes up to 50 sites and compared to auxiliary-field QMC. New benchmark results are provided in the off-half-filling regime, with no severe sign problem being encountered. In addition, we show that methodology can be used to calculate excited states of the Hubbard model and lay the groundwork for the calculation of observables other than the energy.
We present a class of correlated wavefunctions generated by exponentials of two-body on-site Hermitian operators that can be evaluated with polynomial computational cost via a Hamiltonian similarity transformation. Wavefunctions of this form have been studied with variational Monte Carlo methods, but we present a formalism to perform non-stochastic calculations. The Hausdorff series generated by these Jastrow factors can be summed exactly without truncation resulting in a set of equations with polynomial computational cost. The correlators include the density-density, collinear spin-spin, spin-density cross terms, and on-site double occupancy operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with only a small set of correlation terms required for accurate calculations in systems with local interactions. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one and two-dimensional repulsive Hubbard model where it yields accurate results for large systems with low computational cost. Symmetry projection methods are included to further improve the reference wavefunction and results under strong correlation without sacrificing good quantum numbers resulting in very accurate energies for small systems and producing a better ground state for the calculation of other properties.
We report a random-sampling method for computing the expectation value of physical quantities based on the Gutzwiler variational wave function. As the first application, we calculated the double occupancy, which is a critical quantity for understanding the correlation effects in many-body systems, for single-band 1D and 2D lattices. We demonstrated that the random-sampling scheme is more efficient than an existing Metropolis Monte-Carlo algorithm. For the 1D Hubbard model with only nearest-neighbour hopping, our results are almost identical to the exact analytic solution. We have also studied systems to which analytic solutions are not available, including the 1D lattices with next-nearest-neighbour hopping and 2D lattices. In addition, constraints on real-space configurations can be easily implemented in the current scheme to further improve the Gutzwiller wave function. As an example, we calculated the double occupancy for 1D Hubbard model by applying the constraint that all double-occupied sites are paired with an empty site. With enhanced correlation between double-occupied and empty sites, the constraint results in much improved ground-state energy for 1D Hubbard model with strong on-site repulsion.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.
Numerical results for ground-state and excited-state properties (energies, double occupancies, and Matsubara-axis self-energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary-field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed-node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock methods. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic-limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods.