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Moment formation and pairing in the 2D Hubbard model

Zizhe Cai

Published Jan 1, 2015• 0 citations
Publisher

Abstract

The two-dimensional (2D) Hubbard model is solved in a new way. The zero-frequency components of the Hubbard interaction are separated out and solved to infinite order to avoid the traditional difficulties associated with perturbation theory in the strong-coupling limit. Specifically the zero-frequency spin interaction is decoupled by the introduction of Stratonovich-Hubbard fields, while the zero-frequency charge interaction is treated by "slave" Hartree-Fock. This approach leads to the formation of large fluctuating spin moments, and as a consequence the one-electron spectral density exhibits a pseudogap. A Metropolis Monte Carlo method is used to perform the configuration averaging over Stratonovich-Hubbard fields. Within single Monte Carlo configurations the non-zero frequency components of the Hubbard interaction are treated by a high-order perturbation method, which incorporates the exchange of dynamical particle-hole fluctuations. Since a single configuration has neither translational symmetry nor spin rotational symmetry, these fluctuations do not have a well-defined crystal momentum or total spin. When the dynamical particle-hole fluctuations are included in the calculation, localized electron pairing results at elevated temperatures. Superconductivity presumably onsets when these localized pairs develop coherence, but this is not studied in the present work. The nature of the pairing interaction and the resulting pair eigenfunctions is studied.

DOI
10.25549/usctheses-m748

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The two-dimensional (2D) Hubbard model is solved in a new way. The zero-frequency components of the Hubbard interaction are separated out and solved to infinite order to avoid the traditional difficulties associated with perturbation theory in the strong-coupling limit. Specifically the zero-frequency spin interaction is decoupled by the introduction of Stratonovich-Hubbard fields, while the zero-frequency charge interaction is treated by "slave" Hartree-Fock. This approach leads to the formation of large fluctuating spin moments, and as a consequence the one-electron spectral density exhibits a pseudogap. A Metropolis Monte Carlo method is used to perform the configuration averaging over Stratonovich-Hubbard fields. Within single Monte Carlo configurations the non-zero frequency components of the Hubbard interaction are treated by a high-order perturbation method, which incorporates the exchange of dynamical particle-hole fluctuations. Since a single configuration has neither translational symmetry nor spin rotational symmetry, these fluctuations do not have a well-defined crystal momentum or total spin. When the dynamical particle-hole fluctuations are included in the calculation, localized electron pairing results at elevated temperatures. Superconductivity presumably onsets when these localized pairs develop coherence, but this is not studied in the present work. The nature of the pairing interaction and the resulting pair eigenfunctions is studied.
We describe a new microscopic approach for analyzing interacting electron systems with local moments or, in principle, any local order parameter. We specialize attention to the doped Mott insulator phase of the Hubbard model, where standard weak-coupling perturbation methods fail. A rotationally invariant Stratonovich-Hubbard field is introduced to decouple the static spin components of the interaction at each site. Charge degrees of freedom are then treated by ``slave'' Hartree-Fock in the presence of a spatially varying random spin field. The effective action reduces to a classical Heisenberg model at half filling, insuring that the system has (i) finite-range order at T>0 with an exponentially diverging correlation length and (ii) a one-electron Mott-Hubbard gap in the presence of disorder. Away from half-filling properties are determined by strongly non-Gaussian fluctuations in the amplitude and orientation of the local spin fields. The saddle point equations of the theory at zero temperature reduce to inhomogeneous Hartree-Fock, so that disordering of domain walls at finite temperature is in principle included. We present preliminary small system results for the intermediate and large interaction regimes obtained by Monte Carlo simulation of random spin field configurations.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature T∗,{T}^{*},T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around T∗{T}^{*}T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
The interactions between holes in the Hubbard model, in the low density, intermediate- to strong-coupling limit, are investigated by systematically improving mean field calculations. The configuration-interaction basis set is constructed by applying to local unrestricted Hartree-Fock configurations all lattice translations and rotations. It is shown that this technique reproduces, correctly, the properties of the Heisenberg model, in the limit of large U. Upon doping, dressed spin polarons in neighboring sites have an increased kinetic energy and an enhanced hopping rate. Both effects are of the order of the hopping integral and lead to an effective attraction at intermediate couplings. The numerical results also show that when more than two holes are added to the system, they do not tend to cluster, but rather hole pairs remain far apart. Hole-hole correlations are also calculated and shown to be in qualitative agreement with exact calculations for $4\ifmmode\times\else\texttimes\fi{}4$ clusters. In particular our results indicate that for intermediate coupling the hole-hole correlation attains a maximum when the holes are in the same sublattice at a distance of 2\sqrt{2}2​ times the lattice spacing, in agreement with exact results and the $t\ensuremath{-}J$ model. The method is also used to derive known properties of the quasiparticle band structure of isolated spin polarons.
The ground state of the two-dimensional (2D) Hubbard model is investigated by adopting improved wave functions that take into account intersite electron correlation beyond the Gutzwiller ansatz. The ground-state energy is lowered considerably, giving the best estimate of the ground-state energy for the 2D Hubbard model. There is a crossover from weakly to strongly correlated regions as the on-site Coulomb interaction UUU increases. The antiferromagnetic correlation induced by UUU is reduced for hole doping when UUU is large, being greater than the bandwidth, thus increasing the kinetic energy gain. The spin and charge fluctuations are induced in the strongly correlated region. These antiferromagnetic and kinetic charge fluctuations induce electron pairings, which would result in high-temperature superconductivity.
We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order U2{U}^{2}U2 contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.