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arXiv:cond-mat/0409691 · Physical Review B

Symmetry-projected variational approach to the one-dimensional Hubbard model

K.W. Schmid, T. Dahm, Jérôme Margueron, H. Müther

Published Aug 12, 2005• 19 citations
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Abstract

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 $N=12$ and $N=14$ 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 $N=12$ lattice is investigated and, finally, an $N=30$ 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.

DOI
10.1103/physrevb.72.085116

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Critiquing Variational Theories of the Anderson-Hubbard Model: Real-Space Self-Consistent Hartree-Fock Solutions

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Abstract. 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 〈Si · Sj〉, 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 disorderinteraction phase diagram such physics may be important. PACS numbers: 71.27.+a,71.23.-k,71.10.-wCritiquing HF for the Anderson-Hubbard Model 2 1.

•Aug 4, 2013•5 citations
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 present a symmetry-projected configuration mixing scheme to describe ground and excited states, with well defined quantum numbers, of the two-dimensional Hubbard model with nearest-neighbor hopping and periodic boundary conditions. Results for the half-filled $2\ifmmode\times\else\texttimes\fi{}4$, $4\ifmmode\times\else\texttimes\fi{}4$, and $6\ifmmode\times\else\texttimes\fi{}6$ lattices as well as doped $4\ifmmode\times\else\texttimes\fi{}4$ systems, compare well with available results, both exact and from other state-of-the-art approximations. We report spectral functions and density of states obtained from a well-controlled ansatz for the $({N}_{e}\ifmmode\pm\else\textpm\fi{}1)$-electron system. Symmetry projected methods have been widely used for the many-body nuclear physics problem but have received little attention in the solid state community. Given their relatively low (mean-field) computational cost and the high quality of results here reported, we believe that they deserve further scrutiny.
We present a multireference configuration mixing scheme for describing ground and excited states, with well-defined spin and space-group symmetry quantum numbers, of the one-dimensional Hubbard model with nearest-neighbor hopping and periodic boundary conditions. Within this scheme, each state is expanded in terms of nonorthogonal and variationally determined symmetry-projected configurations. The results for lattices up to 30 and 50 sites compare well with the exact Lieb-Wu solutions as well as with results from other state-of-the-art approximations. In addition to spin-spin correlation functions in real space and magnetic structure factors, we present results for spectral functions and density of states computed with an Ansatz whose quality can be well controlled by the number of symmetry-projected configurations used to approximate the systems with Ne{N}_{e}Ne​ and ${N}_{e}\ifmmode\pm\else\textpm\fi{}1$ electrons. The intrinsic symmetry-broken determinants resulting from the variational calculations have rich structures in terms of defects that can be regarded as basic units of quantum fluctuations. Given the quality of the results here reported, as well as the parallelization properties of the considered scheme, we believe that symmetry-projection techniques, which have found ample applications in nuclear structure physics, deserve further attention in the study of low-dimensional correlated many-electron systems.
The few determinant (FED) approximation introduced in our previous work [Phys. Rev. B 87, 235129 (2013)] is used to describe the ground state, characterized by well-defined spin and space group symmetry quantum numbers as well as doping fractions Ne/Nsites{N}_{e}/{N}_{\mathrm{sites}}Ne​/Nsites​, of one-dimensional Hubbard lattices with nearest-neighbor hopping and periodic boundary conditions. Within this multireference scheme, each ground state is expanded in a given number of nonorthogonal and variationally determined symmetry-projected configurations. The results obtained for the ground-state and correlation energies of half-filled and doped lattices with 30, 34, and 50 sites compare well with the exact Lieb-Wu solutions as well as with those obtained with other state-of-the-art approximations. The structure of the intrinsic symmetry-broken determinants resulting from the variational procedure is interpreted in terms of solitons whose translational and breathing motions can be regarded as basic units of quantum fluctuations. It is also shown that in the case of doped one-dimensional lattices, a part of such fluctuations can also be interpreted in terms of polarons. In addition to momentum distributions, both spin-spin and density-density correlation functions are studied as functions of doping. The spectral functions and density of states, computed with an ansatz whose quality can be well controlled by the number of symmetry-projected configurations used to approximate the ${N}_{e}\ifmmode\pm\else\textpm\fi{}1$ electron systems, display features beyond a simple quasiparticle distribution, as well as spin-charge separation trends.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
The few-determinant (FED) methodology, introduced in our previous works [R. Rodr\'{\i}guez-Guzm\'an et al., Phys. Rev. B 87, 235129 (2013); Phys. Rev. B 89, 195109 (2014)] for one-dimensional (1D) lattices, is here adapted for the repulsive two-dimensional Hubbard model at half filling and with finite doping fractions. Within this configuration mixing scheme, a given ground state with well-defined spin and space group quantum numbers is expanded in terms of a nonorthogonal symmetry-projected basis determined through chains of variation-after-projection calculations. The results obtained for the ground-state and correlation energies of half-filled and doped $4\ifmmode\times\else\texttimes\fi{}4,\phantom{\rule{0.16em}{0ex}}6\ifmmode\times\else\texttimes\fi{}6,\phantom{\rule{0.16em}{0ex}}8\ifmmode\times\else\texttimes\fi{}8$, and $10\ifmmode\times\else\texttimes\fi{}10$ lattices, as well as momentum distributions and spin-spin correlation functions in small lattices, compare well with those obtained using other state-of-the-art approximations. The structure of the intrinsic determinants resulting from the variational strategy is interpreted in terms of defects that encode information on the basic units of quantum fluctuations in the considered 2D systems. The varying nature of the underlying quantum fluctuations, reflected in a transition to a stripe regime for increasing on-site repulsions, is discussed using the intrinsic determinants belonging to a $16\ifmmode\times\else\texttimes\fi{}4$ lattice with 56 electrons. Such a transition is further illustrated by computing spin-spin and charge-charge correlation functions with the corresponding multireference FED wave functions. In good agreement with previous studies, the analysis of the pairing correlation functions reveals a weak enhancement of the extended sss-wave and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing modes. Given the quality of results here reported together with those previously obtained for 1D lattices and the parallelization properties of the FED scheme, we believe that symmetry projection techniques are very well suited for building ground-state wave functions of correlated electronic systems, regardless of their dimensionality.
Abstract. 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 〈Si · Sj〉, 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 disorderinteraction phase diagram such physics may be important. PACS numbers: 71.27.+a,71.23.-k,71.10.-wCritiquing HF for the Anderson-Hubbard Model 2 1.