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Physical review. B, Condensed matter

Hartree theory for the negative- U extended Hubbard model. II. Finite temperature

S. Robaszkiewicz, R. Micnas, K. A. Chao

Published Oct 1, 1982• 68 citations
Publisher

Abstract

The zero-temperature Hartree theory is extended to finite temperature to investigate the phase diagram, the order parameters, and the chemical potential of the negative-$U$ extended Hubbard model in the weak-coupling regime. All the transitions between the charge-ordered, the singlet-superconducting, the mixed, and the nonordered phases are second order. In a special region of the parameter space, we discover the heat-charge-order process, which is the main source for the qualitative difference between the results of the present weak-coupling regime and the strong-coupling regime investigated earlier. Phenomena associated with the heat-charge order are compared to other work in which an effect similar to the heat-charge order exists.

DOI
10.1103/physrevb.26.3915

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The zero-temperature Hartree theory is extended to finite temperature to investigate the phase diagram, the order parameters, and the chemical potential of the negative-UUU extended Hubbard model in the weak-coupling regime. All the transitions between the charge-ordered, the singlet-superconducting, the mixed, and the nonordered phases are second order. In a special region of the parameter space, we discover the heat-charge-order process, which is the main source for the qualitative difference between the results of the present weak-coupling regime and the strong-coupling regime investigated earlier. Phenomena associated with the heat-charge order are compared to other work in which an effect similar to the heat-charge order exists.
We have investigated the ground-state properties of the negative-UUU extended Hubbard model in the weak-coupling regime with the Hartree theory formulated in terms of the Bogoliubov variational approach. While the pure phases of charge order (CO), singlet superconducting (SS), and nonordered can be solved analytically with a model density of states, the mixed phase of CO and SS requires numerical computation. The ground-state phase diagram is derived, and the order parameters and the chemical potential of the ground state are analyzed. The characteristic difference between the behaviors of the SS and the CO order parameters at very weak coupling suggests a complicated competition between various interactions. Present results are compared with those derived earlier for the strong-coupling regime.
We consider the finite-temperature properties of the extended Bose-Hubbard model realized recently in an ETH experiment [Nature 532, 476 (2016)]. Competing short- and global-range interactions accommodate fascinating collective phenomena. We formulate a self-consistent mean-field theory to describe the behaviors of the system at finite temperatures. At a fixed chemical potential, we map out the distributions of the superfluid order parameters and number densities with respect to the temperatures. For a charge density wave, we find that the global-range interaction enhances the charge order by increasing the transition temperature at which the charge order melts out, while for a supersolid phase, we find that the disappearance of the charge order and the superfluid order occurs at different temperature. At a fixed number-density filling factor, we extract the temperature dependence of the thermodynamic functions such as internal energy, specific heat and entropy. Across the superfluid phase transition, the specific heat has a discontinuous jump.
We study the Holstein-Hubbard model at half filling to explore ordered phases including superconductivity (SC), antiferromagnetism (AF), and charge order (CO) in situations where the electron-electron and electron-phonon interactions are strong (comparable to the electronic bandwidth). The model is solved in the dynamical mean-field approximation with a continuous-time quantum Monte Carlo impurity solver. We determine the superconducting transition temperature Tc{T}_{c}Tc​ and the SC order parameter and show that the phonon-induced retardation or the strong Coulomb interaction leads to a significant reduction and shift of the Tc{T}_{c}Tc​ dome against the effective electron-electron interaction Ueff{U}_{\text{eff}}Ueff​ given by the Hubbard UUU reduced by the phonon-mediated attraction in the static limit. This behavior is analyzed by comparison to an effective static model in the polaron representation with a renormalized bandwidth. In addition, we discuss the superconducting gap $\ensuremath{\Delta}$ and $2\ensuremath{\Delta}/{T}_{c}$ to reveal the effect of the retardation and the Coulomb interaction. We also determine the finite-temperature phase diagram including AF and CO. In the moderate-coupling regime, there is a hysteretic region of AF and CO around Ueff=0{U}_{\mathrm{eff}}=0Ueff​=0, while the two phases are separated by a paramagnetic metal in the weak-coupling regime and a paramagnetic insulator in the strong-coupling regime.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.