n exact diagonalization calculation of the t-J model on 2D square cluster has been studied for the ground state properties of HTSC. Effect of next-nearest-neighbour hopping and magnetic (both antiferromagnetic and ferromagnetic) interaction on $d_{x^{2} −y^{2}}$-wave pairing has been shown. Relative strength of the next-nearest-neighbour interaction with respect to that of near-neighbour interaction for the strongest $d_{x^{2} −y^{2}}$-wave pairing has been estimated. A schematic phase diagram is shown. It is shown that a two-sublattice model with antiferromagnetic interaction between them and a small intra-ferromagnetic-type interaction in one sublattice favours $d_{x^{2} −y^{2}}$-wave superconductivity and moderate negative type NNN hopping adds flavours to this phase.

An exact diagonalization calculation for a small cluster in the two-dimensional t-J model has been studied to calculate two-hole correlation. Calculations reveal dominant hole–hole correlation for holes sitting on next-nearest-neighbour (NNN) sites and critical coupling occurs at $J/t = 0.8$. With the increase in negative-type NNN hopping, correlation decreases at NNN sites whereas it increases at other sites. The thermodynamic properties such as entropy and specific heat are studied as functions of temperature with various NNN hopping strength. Results show that with the inclusion of negative NNN hopping, the system becomes more ordered. A qualitative transition temperature region has been estimated. It is shown that with the increase in NNN hopping strength, $T_{c}$ increases. Specific heat results show non-Fermi liquid-type behaviour of the system. All our calculations establish the importance of negative-type NNN hopping.

Role of correlated hopping is studied using extended Falicov–Kimball model in a small cluster. A discontinuous insulator-to-metal transition is observed at a critical 𝑓-level energy. Transition is sharper for larger correlated hopping. In the specific heat curves a two-peak structure consisting of a sharp peak followed by a Schottky-type broad peak is exhibited. In a limited parameter region, some heavy-fermion like characteristics have been observed

The two-dimensional (2D) extended Falicov–Kimball model has been studied to observe the role of nonlocal Coulomb interaction ($U_{\text{nc}}$) using an exact diagonalization technique. The f-state occupation ($n^f$), the f–d intersite correlation function ($c_{\text{fd}}$), the specific heat (𝐶), entropy (𝑆) and the specific heat coefficient (𝛾) have been examined. Nonlocal Coulomb interaction-induced discontinuous insulator-to-metal transition occurs at a critical f-level energy. More ordered state is obtained with the increase of $U_{\text{nc}}$. In the specific heat curves, two-peak structure as well as a singlepeak structure appears. At low-temperature region, a sharp rise in the specific heat coefficient 𝛾 is observed. The peak value of 𝛾 shifts to the higher temperature region with $U_{\text{nc}}$.