The discovery of new materials with properties dominated by strong correlations among electrons has opened the problem of new appropriate calculation schemes.In the case of 1D models the Bethe ansatz provides an exact evaluation of many relevant physical quantities, but not a complete framework.By way of developing new methods appropriate to strongly correlated systems, we study the 1D Hubbard model by means of the Composite Operator Method.We investigate various local quantities.A comparison with exact results and other analytical approaches show a reasonable agreement and determine the applicability range of our approximate scheme.
The discovery of new materials with properties dominated by strong correlations among electrons has opened the problem of new appropriate calculation schemes.In the case of 1D models the Bethe ansatz provides an exact evaluation of many relevant physical quantities, but not a complete framework.By way of developing new methods appropriate to strongly correlated systems, we study the 1D Hubbard model by means of the Composite Operator Method.We investigate various local quantities.A comparison with exact results and other analytical approaches show a reasonable agreement and determine the applicability range of our approximate scheme.
PACS. 71.10.−w – Theories and models of many electron systems. PACS. 71.10.Fd – Lattice fermion models. PACS. 71.15.Th – Other methods. Abstract. – The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases. Introduction. – New materials whose physics is dominated by electron correlations in narrow energy bands are actually a challenge for solid state physicists. The treatment of such systems is not trivial due to the competition between itinerant and localized behaviour of the electrons in these bands. The Hubbard Hamiltonian [1] is regarded as the simplest model
The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any non-zero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases.
We study a class of variational wave functions for strongly correlated systems by expanding the electron operators as composites of spin-1/2 Fermi fields and spinless Fermi fields. The composite particles automatically satisfy the local constraint of no double occupancy and include correlations between opposite-spin particles in a very physical way. We calculate the energy and correlation functions for the one-dimensional U=\ensuremath{\infty} Hubbard model, where a comparison with exact results is made. The method is computationally very tractable and can readily be generalized to higher dimensions.
In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green’s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U- J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated selfconsistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system’s response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the twofield theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential ~U , coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of ~U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries. [edited by Author]
The Hubbard model plays a special role in condensed matter theory as it is considered to be the simplest Hamiltonian model one can write in order to describe anomalous physical properties of some class of real materials. Unfortunately, this model is not exactly solved except in some limits and therefore one should resort to analytical methods, like the Equations of Motion Approach, or to numerical techniques in order to attain a description of its relevant features in the whole range of physical parameters (interaction, filling and temperature). In this paper, the Composite Operator Method, which exploits the above mentioned analytical technique, is presented and systematically applied in order to get information about the behaviour of all relevant properties of the model (local, thermodynamic, single- and two-particle properties) in comparison with many other analytical techniques, the above cited known limits and numerical simulations. Within this approach, the Hubbard model is also shown to be capable of describing some anomalous behaviour of cuprate superconductors.
We present a comparative study of the ground state of the one-dimensional Hubbard model. We first use a new fermion coherent state method in the framework of Fermi liquid theory by introducing a hole operator and considering the interactions of two pairs electrons and holes. We construct the ground state of the Hubbard model as ∣⟩ = [ f +∑'φ c † k 1 σ 1 h † k 2 σ 2 c † k 3 σ 3 h † k 4 σ 4 ∏exp(ρ c † k 1 σ 1 h † k 2 σ 2 )]∣⟩ 0 , where φ and ρ are the coupling constants. Our results are then compared to those of variational methods, density functional theory based on the exact solvable Bethe ansatz solutions, variational Monto–Carlo method (VMC) as well as to the exact result of the infinite system. We find satisfactory agreement between the fermion coherent state scheme and the VMC data, and provide a new picture to deal with the strongly correlated system.