EXPRESSION OF ENTROPY FOR SINGLE TUNNELING MODEL

We study the many body localization aspects a single-particle mobility edges in fermionic systems. We investigate incommensurate lattices and random disorder Anderson models. Many body localization and quantum nonergodic properties are studied by comparing entanglement and internal entropy and by calculating the scaling of subsystem particle number fluctuations respectively. Every crystalline structure shows some deviations or the other from the regular atomic arrangement, as prescribed by the symmetry and structure of the respective unit cell. These deviations from the ideal crystal structure are called imperfections. The translational symmetry in a crystal is consistent with laws of thermodynamics, since these laws are applied to describe the growth of crystals all of which have some imperfections. Any increase in the defect concentration raises the entropy, which in term lowers the free energy at a finite temperature. In the equilibrium state there is a finite concentration of imperfection in the crystal. The concentration of a particular type of imperfection depends on the type of the crystal lattice, the binding energy of the lattice and structure of the imperfection itself. Entropy is the measure of the disorder of a system, the greater the disorder, the higher is the entropy. In the magnetic field the moments will be partially ordered, so that the entropy is lowered by the field. The relatively small heat capacity associated with the lattice vibrations of solids at temperature near and below 1oK makes this region interesting in connection with an evaluation of contribution of the conduction electrons to the heat capacity of the metals. In the present study we limit ourselves to find out expression of entropy for tunneling model. For this purpose we have developed first the defect contribution to the specific heat for tunneling model.