We show that the standard textbook description of (restricted) Hartree-Fock theory for (Fermi) Hubbard-like models is in need of an update, and we present such an update allowing us to correct basic and established results in the condensed matter physics literature that are qualitatively wrong. Our update amounts to adding a test that reliably checks the thermodynamic stability of solutions of Hartree-Fock equations. This stability test makes it possible to detect, by simple means and with certainty, regions in phase space where the model exhibits mixed phases where two conventional phases coexist and translation invariance is broken in complicated ways; in such a mixed phase, unconventional physics is to be expected. Our results show that mixed phases are ubiquitous in Hubbard-like models in arbitrary dimensions.
We show that the standard textbook description of (restricted) Hartree-Fock theory for (Fermi) Hubbard-like models is in need of an update, and we present such an update allowing us to correct basic and established results in the condensed matter physics literature that are qualitatively wrong. Our update amounts to adding a test that reliably checks the thermodynamic stability of solutions of Hartree-Fock equations. This stability test makes it possible to detect, by simple means and with certainty, regions in phase space where the model exhibits mixed phases where two conventional phases coexist and translation invariance is broken in complicated ways; in such a mixed phase, unconventional physics is to be expected. Our results show that mixed phases are ubiquitous in Hubbard-like models in arbitrary dimensions.
We study an extension of the 2D Fermi--Hubbard model, which was recently introduced in [Das et al., Phys. Rev. Lett. 132, 263402 (2024)] and shown to describe altermagnetism that can be studied in cold atom systems. Using an updated Hartree--Fock method that can detect instabilities towards phase separation, we show that the model is in a mixed phase in large parts of the parameter regime at half-filling. We argue that the occurrence of a mixed phase is an indication of exotic physics which, in this model, occurs in parameter regimes accessible in cold atom experiments.
We propose an analytical method for the construction of Hartree-Fock phase diagrams for the (fermion) Hubbard model and various generalizations thereof. Such phase diagrams are traditionally constructed numerically, but we argue that, by using asymptotic techniques, it is possible to obtain analytic formulas approximating the curves separating the different phases to very high accuracy. To illustrate the new method, we apply it to the two-dimensional Hubbard model on the square lattice at zero temperature. This yields formulas for the Hartree-Fock phase boundaries that agree with, but also improve on, earlier numerical results. In particular, our results provide the first rigorous proof of the existence of mixed phases in this model.
Models of strongly interacting Fermi systems are sufficiently intractable that simple mean-field approximations (typically Hartree-Fock) are often employed to obtain qualitative insight into their properties, and also to set parameter values by comparing with band-structure calculations. However, if the correlations are accounted for with use of more sophisticated (but still mean-field-like) methods, qualitative changes in charge densities and magnetic properties can result. We compare four ``mean-field'' approximations for an extended Hubbard model and show that they yield quite different results. Implications for the choice of model for high-temperature superconductors are discussed.
Abstract. A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by 〈Si · Sj〉, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorderinteraction phase diagram such physics may be important. PACS numbers: 71.27.+a,71.23.-k,71.10.-wCritiquing HF for the Anderson-Hubbard Model 2 1.
The students interested in the study of strongly correlated electron systems normally feel, in the beginning of their studies, a great difficulty in understanding the complex correlations of these systems. We consider that for these students it is very important to have texts that analyze pedagogically the theoretical research methods of the subject. We show how to determine the thermodynamic properties of the Hubbard model for the two-dimensional square lattice, utilizing the Hartree-Fock approximation. All the procedures and peculiarities which lead to the determination of the thermodynamic properties and to the comprehension of the meaning of the Hartree-Fock approximation applied to models of strongly correlated electron systems are pedagogically presented in detail. We obtain the magnetization, the magnetic susceptibility, the internal energy, the specific heat, and also a magnetic phase diagram of the studied system. Our approach clearly shows that the use of the Hartree-Fock approximation, in this case, is very simple.
We show that the extended attractive Hubbard model on a square lattice allows for a variety of superconducting phases, including exotic mixed-symmetry phases with [Formula: see text] and [Formula: see text] symmetries, and a novel [Formula: see text] state. The calculations are performed within the Hartree-Fock Bardeen-Cooper-Schrieffer framework. The ground states of the mean-field Hamiltonian are obtained via a minimization scheme that relaxes the symmetry constraints on the superconducting solutions, hence allowing for a mixing of s-, p- and d-wave order parameters. The results are obtained within the assumption of uniform-density states. Our results show that extended attractive Hubbard model can serve as an effective model for investigating properties of exotic superconductors.