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arXiv:1408.2567 · Physical Review B

Variational description of the ground state of the repulsive two-dimensional Hubbard model in terms of nonorthogonal symmetry-projected Slater determinants

R. R. Rodríguez-Guzmán, Carlos A. Jiménez-Hoyos, Gustavo E. Scuseria

Published Nov 7, 2014• 15 citations
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Abstract

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 $s$-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.

DOI
10.1103/physrevb.90.195110

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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×\else\texttimes\fi4,6\ifmmode×\else\texttimes\fi6,8\ifmmode×\else\texttimes\fi84\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{}84\ifmmode×\else\texttimes\fi4,6\ifmmode×\else\texttimes\fi6,8\ifmmode×\else\texttimes\fi8, and 10\ifmmode×\else\texttimes\fi1010\ifmmode\times\else\texttimes\fi{}1010\ifmmode×\else\texttimes\fi10 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×\else\texttimes\fi416\ifmmode\times\else\texttimes\fi{}416\ifmmode×\else\texttimes\fi4 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 dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ 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.
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.
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.
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 determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/81/81/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/51/51/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.
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.
Quantum fluctuations (QF's) in the two-dimensional Hubbard model are visualized by superposition of optimized nonorthogonal Slater determinants. In the half-filled system, QF's consist of rotational and translational motions of spinon-antispinon pairs, while in the lightly doped systems (delta=0.96), those motions of polarons form the QF's. It is shown that an attractive interaction works between two polarons, in the framework of a projected Hartree-Fock picture. At about 10% doping, the ground state has a stripe structure and QF's due to deviations from the uniform stripe. The present method gives the ground state energies comparable or in some cases superior to the variational Monte Carlo method with a Gutzwiller projection parameter.