We will discuss ordering structure in inhomogeneous systems and time-dependent systems. The former is motivated by the problem to find in what condition the super-solid state appears. So far, the condition has been studied in uniform system. However, by the recent development of nanoscale design of material and also of the technique of optical lattice, we may control inhomogeneous structure of the lattice. Here we will study a system where the system parameters change in space and study the local ordering patterns. In particular, we explore the possibility to find the coexistence of the spatial density order (DLRO) and the super-fluidity (ODLRO). We study the following spin model (Matsubara-Matsuda model). The latter is motivated by the experiment on the optical lattice where the system is changed from a solid state due to the periodic optical potential to a super-fluid state (BEC). We will discuss the metastable or spinodal phenomena in pure quantum mechanical systems.
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