Nontrivially Topological Phase Structure of Ideal Bose Gas System within Different Boundary Conditions

The phase structure of ideal Bose gas system within different boundary conditions, i.e., the periodic boundary condition and Dirichlet boundary condition in this work, in an infinite volume, is investigated. It is found that the ground states of ideal Bose gas within those two boundary conditions are both topologically nontrivial, which can not be classified by the traditional symmetry breaking theory. The ground states are different topological phases corresponding to those two boundary conditions, which can be distinguished by the off--diagonal particle number susceptibility. Moreover, this result is universal. The boundary condition may play an important role in pining the critical endpoint of QCD diagram on the approach of the lattice simulations and the computation of some solvable statistical models .
Comment: I found some mistakes in this version manuscript