Partition functions of Polychronakos like spin chains associated with polarized spin reversal operators

We construct polarized spin reversal operator (PSRO) which yields a class of representations for the $BC_N$ type of Weyl algebra, and subsequently use this PSRO to find out novel exactly solvable variants of the $BC_N$ type of spin Calogero model. The strong coupling limit of such spin Calogero models generates the $BC_N$ type of Polychronakos spin chains with PSRO. We derive the exact spectra of the $BC_N$ type of spin Calogero models with PSRO and compute the partition functions of the related spin chains by using the freezing trick. We also find out an interesting relation between the partition functions of the $BC_N$ type and $A_{N-1}$ type of Polychronakos spin chains. Finally, we study spectral properties like level density and distribution of spacing between consecutive energy levels for $BC_N$ type of Polychronakos spin chains with PSRO.