Phase diagrams of the Bose-Fermi-Hubbard model: Hubbard operator approach

Abstract

The phase transitions in the Bose-Fermi-Hubbard model are investigated. The boson Green's function is cal- culated in the random phase approximation (RPA) and the formalism of the Hubbard operators is used. The regions of existence of the superuid and Mott insulator phases are established and the (; t) (the boson chemical potential vs. hopping parameter) phase diagrams are built at different values of boson-fermion in- teraction constant (in the regimes of x ed chemical potential or x ed concentration of fermions). The effect of temperature change on this transition is analyzed and the phase diagrams in the (T; ) plane are constructed. The role of thermal activation of the ion hopping is investigated by taking into account the temperature depen- dence of the transfer parameter. The reconstruction of the Mott-insulator lobes due to this effect is analyzed. parameters of the BFHM which describes these objects can be tuned in experiments and results of theoretical calculations can be tested in practice. Dieren t theoretical approaches were used to investigate the BFHM: mean eld theory (2), strong coupling approach (3), density matrix renormalization group (4), quantum Monte Carlo (5) and others. Another example of systems which can be described by the Bose-Fermi-Hubbard-type Hamil- tonian are intercalated crystals (for example, TiO2 intercalated by Li ions). In such systems the impurity ions and electrons play the role of the bosons and fermions, respectively. A lattice gas type models are also used for the description of ionic conductors (7{9). In (10) the pseudospin-electron model of intercalation was formulated and thermodynamics of the model was investigated in the mean eld approximation. It was shown that due to the presence of electrons the eectiv e inter- action between ions is generated and the character of this interaction depends on the lling of the electron band (when the chemical potential of the electrons is near the band edges, the rst order phase transition between uniform phases or phase separation in the regime of the xed electron concentration takes place; such an eect is important when constructing batteries (11)). In this work we consider the BFHM and propose a method for calculation of correlation func- tions and average values of boson and fermion concentrations. The method is based on the intro- duction of Hubbard operators acting in the on-site basis of states and is similar to the composite fermion approach used in (2) (the composite fermions are formed by a fermion and bosons or bosonic holes) but is more general and universal. Introducing the on-site Hubbard operators we can employ a known technique developed for the calculation of a correlation (Green's) function built on these operators, for example, equation of motion method (12) or diagrammatic approach based on the corresponding Wick's theorem (13). In this paper we use the rst of them. As a rst