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arXiv:2210.12032 · Physical review. B./Physical review. B

Benchmarks and results of the two-band Hubbard model from the Gutzwiller conjugate gradient minimization theory

Zhuo Ye, Feng Zhang, Yongxin Yao, Cai‐Zhuang Wang, Kai‐Ming Ho

Published Sep 22, 2023• 1 citation
arXivHTMLPDF

Abstract

Ground-state properties, such as energies and double occupancies, of a one-dimensional two-band Hubbard model are calculated using a first-principles Gutzwiller conjugate gradient minimization theory. The favorable agreement with the results from the density matrix renormalization group theory demonstrates the accuracy of our method. A rotationally invariant approach is further incorporated into the method to greatly reduce the computational complexity with a speedup of approximately 50 times. Moreover, we investigate the Mott transition between a metal and a Mott insulator by evaluating the charge gap. With greatly reduced computational effort, our method reproduces the phase diagram in reasonable agreement with the density matrix renormalization group theory.

DOI
10.1103/physrevb.108.115145

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Ground-state properties, such as energies and double occupancies, of a one-dimensional two-band Hubbard model are calculated using a first-principles Gutzwiller conjugate gradient minimization theory. The favorable agreement with the results from the density matrix renormalization group theory demonstrates the accuracy of our method. A rotationally invariant approach is further incorporated into the method to greatly reduce the computational complexity with a speedup of approximately 50 times. Moreover, we investigate the Mott transition between a metal and a Mott insulator by evaluating the charge gap. With greatly reduced computational effort, our method reproduces the phase diagram in reasonable agreement with the density matrix renormalization group theory.
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We introduce Gutzwiller conjugate gradient minimization (GCGM) theory, an ab initio quantum many-body theory for computing the ground-state properties of infinite systems. GCGM uses the Gutzwiller wave function but does not use the commonly adopted Gutzwiller approximation (GA), which is a major source of inaccuracy. Instead, the theory uses an approximation that is based on the occupation probability of the on-site configurations, rather than approximations that decouple the site-site correlations as used in the GA. We test the theory in the one-dimensional and two-dimensional Hubbard models at various electron densities and find that GCGM reproduces energies and double occupancies in reasonable agreement with benchmark data at a very small computational cost.
The Gutzwiller conjugate gradient minimisation (GCGM) theory is an ab initio quantum many-body theory for computing the ground-state properties of infinite systems. Previous applications of GCGM provides satisfying accuracy of ground-state energy of Hubbard models. In the current work, we address the problem of whether the obtained wave function is a good approximation for the true ground state by comparing the correlation functions with the benchmark data. Our results confirms the accuracy of the reproduced ground state of the regular Hubbard model, but with some exception for the frustrated Hubbard model.
We present a self-consistent numerical approach to solve the Gutzwiller variational problem for general multiband models with arbitrary on-site interaction. The proposed method generalizes and improves the procedure derived by Deng et al. [Phys. Rev. B 79, 075114 (2009)], overcoming the restriction to density-density interaction without increasing the complexity of the computational algorithm. Our approach drastically reduces the problem of the high-dimensional Gutzwiller minimization by mapping it to a minimization only in the variational density matrix, in the spirit of the Levy and Lieb formulation of density functional theory (DFT). For fixed density the Gutzwiller renormalization matrix is determined as a fixpoint of a proper functional, whose evaluation requires only ground-state calculations of matrices defined in the Gutzwiller variational space. Furthermore, the proposed method is able to account for the symmetries of the variational function in a controlled way, reducing the number of variational parameters. After a detailed description of the method we present calculations for multiband Hubbard models with full (rotationally invariant) Hund's rule on-site interaction. Our analysis shows that the numerical algorithm is very efficient, stable, and easy to implement. For these reasons this method is particularly suitable for first-principles studies (e.g., in combination with DFT) of many complex real materials, where the full intra-atomic interaction is important to obtain correct results.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.
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.