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Journal of Physics Conference Series

A 4-pole approach to the Hubbard model within the Composite Operator Method

Adolfo Avella, Ferdinando Mancini, Gerardo Sica

Published Dec 14, 2012• 1 citation
Publisher

Abstract

The Hubbard model is studied within the Composite Operator Method framework in a 4-pole approximation. The operatorial basis is chosen according to both the hierarchy of equations of motion and the exact solution of the model reduced to the minimal cluster (2 sites) where all Hamiltonian terms are still active. Such a recipe amounts to include into the basis not only the two Hubbard operators, which usually take care of the scale of energy related to the on-site Coulomb repulsion U , but also two other non-local operators describing the Hubbard operators dressed by charge, spin and pair fluctuations on the nearest-neighboring sites and addressing the dynamically generated scale of energy related to the exchange J = 4 t 2 / U . The calculation framework is outlined and a very first comparison to numerical simulations, showing very good qualitative and quantitative agreement, is reported. As reference, the results in the 2-pole approximation are also reported and the differences discussed.

DOI
10.1088/1742-6596/391/1/012151

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The Hubbard model is studied within the Composite Operator Method framework in a 4-pole approximation. The operatorial basis is chosen according to both the hierarchy of equations of motion and the exact solution of the model reduced to the minimal cluster (2 sites) where all Hamiltonian terms are still active. Such a recipe amounts to include into the basis not only the two Hubbard operators, which usually take care of the scale of energy related to the on-site Coulomb repulsion U , but also two other non-local operators describing the Hubbard operators dressed by charge, spin and pair fluctuations on the nearest-neighboring sites and addressing the dynamically generated scale of energy related to the exchange J = 4 t 2 / U . The calculation framework is outlined and a very first comparison to numerical simulations, showing very good qualitative and quantitative agreement, is reported. As reference, the results in the 2-pole approximation are also reported and the differences discussed.
The electronic states of the two-dimensional Hubbard model are investigated by means of a 4-pole approximation within the composite operator method. In addition to the conventional Hubbard operators, we consider other two operators which come from the hierarchy of the equations of motion and carry information regarding nearest-neighbor spin and charge configurations. By means of this operatorial basis, we can study the physics related to the energy scale of J=4t2∕UJ=4{t}^{2}∕UJ=4t2∕U in addition to the one of UUU. Present results show relevant physical features, well beyond those previously obtained by means of a 2-pole approximation, such as a four-band structure with shadow bands and a quasiparticle peak at the Fermi level. The Fermi level stays pinned to the band flatness located at $(\ensuremath{\pi},0)$-point within a wide range of hole-doping $(0\ensuremath{\leqslant}\ensuremath{\delta}\ensuremath{\leqslant}0.15)$. A comprehensive analysis of double occupancy, internal energy, specific heat and entropy features have also been performed. All reported results are in excellent agreement with the data of numerical simulations.
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.
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.
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We study the two-dimensional three-band Hubbard model by means of a four-pole approximation within the Composite Operator Method framework. The model has been solved by considering as basic composite field a four-component spinor field, which includes the p field, the two Hubbard operators for the d field, and a composite operator describing the p field dressed by the spin excitations of the d field. This solution correctly reproduces many results of numerical simulations. In this manuscript, we investigate the relationship between p -and d - populations and the band structure of the model, by varying the on-site potential, the charge-transfer gap, the doping and the hopping integral between the p orbitals.
A well-established method to deal with highly correlated systems is based on the expansion of the Green's function in terms of spectral moments. In the context of the Composite Operator Method one approximation is proposed: a set of n composite fields is assumed as fundamental basis and the dynamics is considered up to the order s. The resulting Green's function has a sn-pole structure. The truncation of the hierarchy of the equations of motion is made at the s-th order and the first s-1 equations are treated exactly. A theorem, which rules the conservation of the spectral moments, is presented. The procedure is applied to the Hubbard model and a recurrence relation for the calculation of its electronic spectral moments is derived.