The 1D Hubbard model

A study of the 1D Hubbard model by means of the COM is reported in Refs. [19,20,21,22]. In particular, the following properties have been computed:

1. Thermodynamic properties (i.e., chemical potential, ground state energy, double occupancy, local magnetic moment, internal energy, free energy, entropy, specific heat) in Refs. [19,20,21,22]

2. Single-particle properties (i.e., energy spectra, charge gap, momentum distribution function, density of states) in Refs. [19,20,22]

3. Response functions (i.e., charge and spin correlations and susceptibilities, density distribution in real space) in Refs. [21,22]

The COM results have been compared with the ones obtained by:

• The Bethe Ansatz technique [23,24,25,26,27,28,29]

• Numerical techniques (i.e., Exact diagonalization, Lanczos and quantum Monte Carlo) [30,31,32,33]

• The Gutzwiller Approximation [34]

• The Ladder and the Self-Consistent Ladder Approximations [35]

• The Renormalization Group [36]

The Bethe Ansatz technique does not provide a complete framework to study the 1D Hubbard model since it does allow the evaluation of the response functions only in some limiting cases. According to this, we have used the COM, within a two-pole approximation, to study those properties which cannot be directly extracted from the Bethe Ansatz. Firstly, we have compared our results with the ones obtained by the Bethe Ansatz as well as by other analytical and numerical approaches. At half filling, in spite of considering a paramagnetic solution, we have found a ground state with strong antiferromagnetic correlations [19,20,22] in agreement with the exact solution [23]. The energy spectra, in proximity of half filling, clearly shows features (i.e., the doubling of the Brillouin zone, the reflection of the upper band and the bandwidth of the order $J$), which are due to strong antiferromagnetic correlations. This antiferromagnetic-like state also shows a gap for any value of the coupling [20,22] in agreement with the Bethe Ansatz result [23]. As function of the Coulomb repulsion, the energy and double occupancy [19,20,22] are in almost perfect agreement with the exact ones. Such a good agreement is not reached by other analytical approaches, like the Gutzwiller approximation, the ladder and self-consistent ladder approximations and the renormalization group. We have also calculated the temperature dependence of the specific heat [19,22]. The locations of the spin- and charge- excitation peaks are consistent with the Bethe Ansatz ones [28,29]. The behavior of the chemical potential, for different values of temperature and Coulomb potential, has been compared with both numerical and Bethe Ansatz results with an overall good agreement for high temperatures and a discrepancy for low temperatures simply due to the different size of the charge gap [22]. The charge susceptibility has been also computed as a function of the filling. The peculiar behavior we found (i.e., the double peak structure, the large renormalization of the charge fluctuations as the system approaches the metal-insulator transition and the vanishing at exactly half-filling) well reproduce the results found by the finite temperature Bethe Ansatz technique [28,29]. Namely, we find that the system overcomes a non-trivial metal-insulator transition in the sense that the charge susceptibility is largely enhanced in the vicinity of half-filling [21,22]. These results show that the method is a reasonable framework to study the one-dimensional Hubbard model.