Specific heat and partition function zeros for the dimer model on the checkerboard B lattice: Finite-size effects

There are three possible classifications of the dimer weights on the bonds of the checkerboard lattice and they are denoted as checkerboard $A$, $B$, and $C$ lattices [Phys. Rev. E 91, 062139 (2015)]. The dimer model on the checkerboard $B$ and $C$ lattices has much richer critical behavior compared to the dimer model on the checkerboard $A$ lattice. In this paper we study in full detail the dimer model on the checkerboard $B$ lattice. The dimer model on the checkerboard $B$ lattice has two types of critical behavior. In one limit this model is the anisotropic dimer model on rectangular lattice with algebraic decay of correlators and in another limit it is the anisotropic generalized Kasteleyn model with radically different critical behavior. We analyze the partition function of the dimer model on a $2M\ifmmode\times\else\texttimes\fi{}2N$ checkerboard $B$ lattice wrapped on a torus. We find very unusual behavior of the partition function zeros and the specific heat of the dimer model. Remarkably, the partition function zeros of finite-size systems can have very interesting structures, made of rings, concentric circles, radial line segments, or even arabesque structures. We find out that the number of the specific heat peaks and the number of circles of the partition function zeros increases with the system size. The lattice anisotropy of the model has strong effects on the behavior of the specific heat, dominating the relation between the correlation length exponent $\ensuremath{\nu}$ and the shift exponent $\ensuremath{\lambda}$, and $\ensuremath{\lambda}$ is generally unequal to $1/\ensuremath{\nu}$ ($\ensuremath{\lambda}\ensuremath{\ne}1/\ensuremath{\nu}$).