We analyze the transformation from insulator to metal induced by thermal fluctuations within the Falicov-Kimball model. Using the Dynamic Mean Field Theory (DMFT) formalism on the Bethe lattice we find rigorously the temperature dependent Density of
States ($DOS$) at half filling in the limit of high dimensions. At zero temperature (T=0) the system is ordered to form the checkerboard pattern and the $DOS$ has the gap $Delta$ at the Fermi level $varepsilon_F=0$, which is proportional to the interaction constant $U$. With an increase of $T$ the $DOS$ evolves in various ways that depend on $U$. For $U>U_{cr}$ the gap persists for any $T$ (then $Delta >0$), so the system is always an insulator. However, if $U < U_{cr}$, two additional subbands develop inside the gap. They become wider with increasing $T$ and at a certain $U$-dependent temperature $T_{MI}$ they join with each other at $varepsilon_F$. Since above $T_{MI}$ the $DOS$ is positive at $varepsilon_F$, we interpret $T_{MI}$ as the transformation temperature from insulator to metal. It appears, that $T_{MI}$ approaches the order-disorder phase transition temperature $T_{O-DO}$ when $U$ is close to 0 or $ U_{cr}$, but $T_{MI}$ is substantially lower than $T_{O-DO}$ for intermediate values of $U$. Having calculated the temperature dependent $DOS$ we study thermodynamic properties of the system starting from its free energy $F$. Then we find how the order parameter $d$ and the gap $Delta $ change with $T$ and we construct the phase diagram in the variables $T$ and $U$, where we display regions of stability of four different phases: ordered insulator, ordered metal, disordered insulator and disordered metal. Finally, we use a low temperature expansion to demonstrate the existence of a nonzero DOS at a characteristic value of U on a general bipartite lattice.
Regular pattern formation is ubiquitous in nature; it occurs in biological, physical, and materials science systems. Here we propose a set of experiments with ultracold atoms that show how to examine different types of pattern formation. In particula
r, we show how one can see the analog of labyrinthine patterns (so-called quantum emulsions) in mixtures of light and heavy atoms (that tend to phase separate) by tuning the trap potential and we show how complex geometrically ordered patterns emerge (when the mixtures do not phase separate), which could be employed for low-temperature thermometry. The complex physical mechanisms for the pattern formation at zero temperature are understood within a theoretical analysis called the local density approximation.
