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arXiv:cond-mat/9909352 · Physical review. B, Condensed matter

Strong-coupling theory for the Hubbard model

Ansgar Dorneich, M. G. Zacher, C. Gröber, R. Eder

Published May 15, 2000• 25 citations
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Abstract

We reanalyze the Hubbard-I approximation by showing that it is equivalent to an effective Hamiltonian describing Fermionic charge fluctuations, which can be solved by Bogoliubov transformation. As the most important correction in the limit of large U and weak spin correlations we augment this Hamiltonian by further effective particles, which describe composite objects of a Fermionic charge fluctuation and a spin-, density-, or $\ensuremath{\eta}$ excitation. The scheme is valid for positive and negative U. We present results for the single particle Green's function for the two-dimensional Hubbard model with and without ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$ terms, and compare to quantum Monte-Carlo results for the paramagnetic phase. The overall agreement is significantly improved over the conventional Hubbard-I or two-pole approximation.

DOI
10.1103/physrevb.61.12816

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We reanalyze the Hubbard-I approximation by showing that it is equivalent to an effective Hamiltonian describing Fermionic charge fluctuations, which can be solved by Bogoliubov transformation. As the most important correction in the limit of large U and weak spin correlations we augment this Hamiltonian by further effective particles, which describe composite objects of a Fermionic charge fluctuation and a spin-, density-, or \ensuremathη\ensuremath{\eta}\ensuremathη excitation. The scheme is valid for positive and negative U. We present results for the single particle Green's function for the two-dimensional Hubbard model with and without t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ and t\ensuremath′′{t}^{\ensuremath{''}}t\ensuremath′′ terms, and compare to quantum Monte-Carlo results for the paramagnetic phase. The overall agreement is significantly improved over the conventional Hubbard-I or two-pole approximation.
The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.
We investigate the charge- and spin-dynamical structure factors for the two-dimensional one-band Hubbard model in the strong coupling regime within the dynamical cluster approximation (DCA) that is explicitly extended to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitation. In the spirit of the DCA, the effective vertex is calculated by using the quantum Monte Carlo methods on a finite cluster. On the basis of a comparison to high temperature auxiliary field quantum Monte Carlo data, we show that near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin, and single-particle excitations.
The charge and spin dynamics of the two-dimensional Hubbard model in the paramagnetic phase is first studied by means of the two-pole approximation within the framework of the composite operator method. The fully self-consistent scheme requires: no decoupling, the fulfillment of both Pauli principle and hydrodynamic constraints, the simultaneous solution of fermionic and bosonic sectors, and a very rich momentum dependence of the response functions. The temperature and momentum dependencies, as well as the dependency on the Coulomb repulsion strength and the filling, of the calculated charge and spin susceptibilities and correlation functions are in very good agreement with the numerical calculations present in the literature.
We present a novel solution of the 2D Hubbard model in the framework of the Composite Operator Method within a four-pole approximation. We adopt a basis of four fields: the two Hubbard operators plus two fields describing the Hubbard transitions dressed by nearest-neighbor spin fluctuations. We include these nonlocal operators because spin fluctuations play a dominant role in strongly correlated electronic systems with respect to other types of nonlocal charge, pair and double-occupancy fluctuations. The approximate solution performs very well once compared with advanced (semi-) numerical methods from the weak-to the strong-coupling regime, being by far less computational-resource demanding. We adopt this solution to study the single-particle properties of the model in the strong coupling regime, where the effects of strong short-range magnetic correlations are more relevant and could be responsible for anomalous features. In particular, we will focus on the characterization of the Fermi surface and of its evolution with doping.
We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.
The cluster slave-spin method is introduced to systematically investigate the solutions of the Hubbard model including the symmetry-broken phases. In this method, the electron operator is factorized into a fermionic spinon describing the physical spin and a slave-spin describing the charge fluctuations. Following the U(1)U(1)U(1) formalism derived by Yu and Si [Phys. Rev. B 86, 085104 (2012)], it is shown that the self-consistent equations to explore various symmetry-broken density wave states can be constructed in general with a cluster of multiple slave-spin sites. We employ this method to study the antiferromagnetic (AFM) state in the single band Hubbard model with the two- and four-site clusters of slave spins. While the Hubbard gap, the charge gap due to the doubly occupied states, scales with the Hubbard interaction UUU as expected, the AFM gap $\mathrm{\ensuremath{\Delta}}$, the gap in the spinon dispersion in the AFM state, exhibits a crossover from the weak- to strong-coupling behaviors as UUU increases. Our cluster slave-spin method reproduces not only the traditional mean-field behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}U$ in the weak-coupling limit, but also the behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{t}^{2}/U$ predicted by the superexchange mechanism in the strong-coupling limit. In addition, the holon-doublon correlator as functions of UUU and doping xxx is also computed, which exhibits a strong tendency toward the holon-doublon binding in the strong coupling regime. We further show that the quasiparticle weight obtained by the cluster slave-spin method is in a good agreement with the generalized Gutzwiller approximation in both AFM and paramagnetic states, and the results can be improved beyond the generalized Gutzwiller approximation as the cluster is enlarged from a single site to four sites. Our results demonstrate that the cluster slave-spin method can be a powerful tool to systematically investigate the strongly correlated system.