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Physical Review Letters

Quantum Fluctuations due to Spinons, Polarons, and Stripes in the Two-Dimensional Hubbard Model

Norikazu Tomita, Shuji Watanabe

Published Sep 9, 2009• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.103.116401

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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.
Quantum fluctuations (QFs) in the two-dimensional Hubbard model are visualized by using a resonating Hartree-Fock method. Wave functions of optimized multi-Slater determinants show that QFs in the half-filled system are described by vibration, translation, and rotation of spinon-antispinon pairs, while those motions of polarons constitute dominant QFs in the lightly doped systems. We show that an attractive interaction works between two polarons in the framework of a projected Hartree-Fock approximation. The ground states at about 10% doping have a stripe structure and Q’s due to deviations from a uniform stripe.
Electronic states in the one-dimensional (1D) doped Hubbard model are described by superposition of optimized nonorthogonal Slater determinants (S-dets). Analysis on the S-dets allows us to visualize quantum fluctuations. In the weak and intermediate interaction regimes, quantum fluctuations are described by translation and breathing motions of spin-charge coupled defects called polarons as well as spin-charge decoupled defects called holons and spinons. In the strong-interaction regime, on the other hand, spin and charge fluctuations are mostly separated as spinons and holons, especially in the lightly doped systems $(\ensuremath{\delta}=0.08)$. In the highly doped systems $(\ensuremath{\delta}=0.24)$, polarons also play an important role. Finally, it is shown that in the 1/3-filled systems, the concepts of holon, spinon, and polaron do not work anymore. The electronic structure is qualitatively described by mixtures of two different density waves having triple periodicity to the lattice. The domains of these different density waves make quantum fluctuations.
Electronic structures of the two-dimensional doped Hubbard model are investigated by using the resonating Hartree–Fock method. It is shown that doped holes form polarons in a wide range of doping regions relevant to superconductivity. The momentum distribution of electrons and the spin correlation function indicate that carriers change from polarons to conventional holes as we enhance hole doping. We will visualize quantum fluctuations in these electronic states by analyzing the structures of the Slater determinants generating the resonating Hartree–Fock wave functions.
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.
The internal structure of stripes in the two-dimensional Hubbard model is studied by going beyond the Hartree-Fock approximation. Partially filled stripes with inhomogeneous magnetization and charge along the stripe core, consistent with experimental observations, are stabilized by quantum fluctuations, included through the configuration interaction method. Similar configurations result to be stable for filled stripes, and have a lower energy than the standard paramagnetic core solution. Away from half-filling, solitonlike solutions show up within the Hartree-Fock approximation. The hopping of short regions of the stripes in the transverse direction is comparable to the bare hopping element. The hole-hole correlation shows maxima along the stripe direction, while the integrated value of ${n}_{\stackrel{\ensuremath{\rightarrow}}{k}}$ compares satisfactorily with experimental results.