Tricritical behavior and phase diagrams of the Bari model for Mott insulator-superconductor transitions

Assuming that the Mott insulator (semoconductor) represents the normal state of an electronic system, Bari investigated the instability of this normal state to a Cooper-pairing interaction of superconductivity. The Bari model Hamiltonian is given by $H=U\Sigma i^{}{n}_{i\uparrow }{n}_{i\downarrow }-\left(\frac{I}{N}\right)\Sigma i^{}\Sigma j^{}{C}_{i\uparrow }^{†}{C}_{j\downarrow }^{†}{C}_{j\downarrow }{C}_{j\uparrow }$, where $C_{i\sigma }^{†}$ and $C_{i\sigma }$ are, respectively, the creation and destruction operators of an electron with spin $\sigma (\uparrow or \downarrow )$ on lattice site $i$, $n_{i\sigma }=C_{i\sigma }^{†}{C}_{i\sigma }$, $U>\mathrm{}0\mathrm{}$ is the strength of the intrasite Coulomb repulsion between electrons having antiparallel spins, $N$ is the total number of Wannier sites, and $\frac{I}{N}>0\mathrm{}$ is the Cooper-pairing strength. The model is generalized so as to contain a longitudinal magnetic field and a symmetry-breaking field conjugate to the offdiagonal long-range order of superconductivity. Due to the form of Hamiltonian, the model can be solved exactly in the thermodynamic limit for arbitrary band populations by using mean-field (Weiss) theory and all results are valid regardless of the dimensionality of the system. Depending upon the choice of Hamiltonian parameters and band population, the nature of the phase transition is found to change from a continuous to a discontinuous type resulting in a tricritical point. In the case of a half-filled band, critical and tricritical exponents and amplitudes of various thermodynamic quantities are calculated exactly. For the cases of arbitrary band populations, phase diagrams are displayed which illustrate some sensitive changes in nature and shape compared to the corresponding phase diagrams of the half-filled-band case, e.g., the appearance of disconnected regions of Mott insulating states is found for electron populations exceeding that of the half-filled band.