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Register a new userExact diagonalization analysis of the Anderson-Hubbard model and comparison to real-space self-consistent Hartree-Fock solutions
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We have obtained the exact ground state wave functions of the Anderson-Hubbard model for different electron fillings on a 4x4 lattice with periodic boundary conditions - for 1/2 filling such ground states have roughly 166 million states. When compared to the uncorrelated ground states (Hubbard interaction set to zero) we have found strong evidence of the very effective screening of the charge homogeneities due to the Hubbard interaction. We have successfully modelled these local charge densities using a non-interacting model with a static screening of the impurity potentials. In addition, we have compared such wave functions to self-consistent real-space unrestricted Hartree-Fock solutions and have found that these approximate ground state wave functions are remarkably successful at reproducing the local charge densities, and may indicate the role of dipolar backflow in producing a novel metallic state in two dimensions.
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A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by < S_i . S_j >, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorder-interaction phase diagram such physics may be important.
We present an alternative scheme to the widely used method of representing the basis of one-band Hubbard model through the relation $I=I_{uparrow}+2^{M}I_{downarrow}$ given by H. Q. Lin and J. E. Gubernatis [Comput. Phys. 7, 400 (1993)], where $I_{uparrow}$, $I_{downarrow}$ and $I$ are the integer equivalents of binary representations of occupation patterns of spin up, spin down and both spin up and spin down electrons respectively, with $M$ being the number of sites. We compute and store only $I_{uparrow}$ or $I_{downarrow}$ at a time to generate the full Hamiltonian matrix. The non-diagonal part of the Hamiltonian matrix given as ${cal{I}}_{downarrow}otimes{bf{H}_{uparrow}} oplus {bf{H}_{downarrow}}otimes{cal{I}}_{uparrow}$ is generated using a bottom-up approach by computing the small matrices ${bf{H}_{uparrow}}$(spin up hopping Hamiltonian) and ${bf{H}_{downarrow}}$(spin down hopping Hamiltonian) and then forming the tensor product with respective identity matrices ${cal{I}}_{downarrow}$ and ${cal{I}}_{uparrow}$, thereby saving significant computation time and memory. We find that the total CPU time to generate the non-diagonal part of the Hamiltonian matrix using the new one spin configuration basis scheme is reduced by about an order of magnitude as compared to the two spin configuration basis scheme. The present scheme is shown to be inherently parallelizable. Its application to translationally invariant systems, computation of Greens functions and in impurity solver part of DMFT procedure is discussed and its extention to other models is also pointed out.
Partially-projected Gutzwiller variational wavefunctions are used to describe the ground state of disordered interacting systems of fermions. We compare several different variational ground states with the exact ground state for disordered one-dimensional chains, with the goal of determining a minimal set of variational parameters required to accurately describe the spatially-inhomogeneous charge densities and spin correlations. We find that, for weak and intermediate disorder, it is sufficient to include spatial variations of the charge densities in the product state alone, provided that screening of the disorder potential is accounted for. For strong disorder, this prescription is insufficient and it is necessary to include spatially inhomogeneous variational parameters as well.
We investigate an extended version of the periodic Anderson model (the so-called periodic Anderson-Hubbard model) with the aim to understand the role of interaction between conduction electrons in the formation of the heavy-fermion and mixed-valence states. Two methods are used: (i) variational calculation with the Gutzwiller wave function optimizing numerically the ground-state energy and (ii) exact diagonalization of the Hamiltonian for short chains. The f-level occupancy and the renormalization factor of the quasiparticles are calculated as a function of the energy of the f-orbital for a wide range of the interaction parameters. The results obtained by the two methods are in reasonably good agreement for the periodic Anderson model. The agreement is maintained even when the interaction between band electrons, U_d, is taken into account, except for the half-filled case. This discrepancy can be explained by the difference between the physics of the one- and higher dimensional models. We find that this interaction shifts and widens the energy range of the bare f-level, where heavy-fermion behavior can be observed. For large enough U_d this range may lie even above the bare conduction band. The Gutzwiller method indicates a robust transition from Kondo insulator to Mott insulator in the half-filled model, while U_d enhances the quasi-particle mass when the filling is close to half filling.
Twisted bilayer transition metal dichalcogenides have emerged as important model systems for the investigation of correlated electron physics because their interaction strength, carrier concentration, band structure, and inversion symmetry breaking are controllable by device fabrication, twist angle, and most importantly, gate voltage, which can be varied in situ. The low energy physics of some of these materials has been shown to be described by a moire Hubbard model generalized from the usual Hubbard model by the addition of strong, tunable spin orbit coupling and inversion symmetry breaking. In this work, we use a Hartree-Fock approximation to reach a comprehensive understanding of the moire Hubbard model on the mean field level. We determine the magnetic and metal-insulator phase diagrams, and assess the effects of spin orbit coupling, inversion symmetry breaking, and the tunable van Hove singularity. We also consider the spin and orbital effects of applied magnetic fields. This work provides guidance for experiments and sets the stage for beyond mean-field calculations.
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