4 In (3.1) where with and γ$_{l}$ = ln$^{[}$ c$_{l}$ + $^{√}$c 2 l − 1$^{]}$ (3.2) c$_{l}$ = cosh 2 K coth 2 K − cos lπ/n = cosh 2 K coth 2 K − 1 2 ( z$_{l}$ + z − 1 l ) ≡ c ( z$_{l}$ ) (3.3) K = J k$_{B}$T (3.4) z$_{l}$ = e ilπ/n (3.5) Given that we are interested in the limit N → ∞ , corresponding to the thermodynamic limit for the width of the of the two dimensional strip with finite thickness M , we can replace the sums in the exponents of (3.1) by integrals. This means that the partition function in question reduces to 2 exp$^{[}$ N 2 π ∫ 2 π 0 ln sinh$^{(}$ M 2 γ ( e $^{iφ}$)$^{)}$ dφ$^{]}$ + 2 exp$^{[}$ N 2 π ∫ 2 π 0 ln cosh$^{(}$ M 2 γ ( e $^{iφ}$)$^{)}$ dφ$^{]}$ (3.6) Our starting point in the calculation of periodic bound- ary conditions is the formula utilized by Ferdinand and Fisher [17] for the partition function of a M × N square Ising lattice wrapped on a torus. A. Periodic boundary conditions exp$^{[}$$^{N}$$^{−}$$^{1}$ ∑ r =1 ln cosh 1 2 Mγ$_{2}$$_{r}$$_{+1}$$^{]}$ + exp$^{[}$$^{N}$$^{−}$$^{1}$ ∑ r =1 ln sinh 1 2 Mγ$_{2}$$_{r}$$_{+1}$$^{]}$ + exp$^{[}$$^{N}$$^{−}$$^{1}$ ∑ r =1 ln cosh 1 2 Mγ$_{2}$$_{r}$$^{]}$ + exp$^{[}$$^{N}$$^{−}$$^{1}$ ∑ r =1 ln sinh 1 2 Mγ$_{2}$$_{r}$$^{]}$ (3.1) which can be rewritten as follows: 2 exp$^{[}$ N 2 π ∫ 2 π 0 ln cosh$^{(}$ M 2 γ ( e $^{iφ}$)$^{)}$ dφ$^{]}$ ×$^{(}$ 1 + exp$^{[}$ N 2 π ( ∫ 2 π 0 ln sinh$^{(}$ M 2 γ ( e $^{iφ}$)$^{)}$ dφ −$^{∫}$ 2 π 0 ln cosh$^{(}$ M 2 γ ( e $^{iφ}$)$^{)}$ dφ$^{)])}$ = 2 exp$^{[}$ N 2 π ∫ 2 π 0 ln cosh$^{(}$ M 2 γ ( e $^{iφ}$)$^{)}$ dφ$^{] {}$ 1 + exp$^{[}$ N 2 π ∫ 2 π 0 ln tanh$^{(}$ M 2 γ ( e $^{iφ}$)$^{)}$ dφ$^{]}}$ (3.7) Taking the log and then the derivative with respect to m , we end up with the following integral Making use of the fact that lim$_{x}$$_{→∞}$ tanh x = 1, we cal- culate the force for the bulk system which we subtract − ∂ F ∂M = k$_{B}$T π ∫ 2 π 0 γ ( e $^{iφ}$) tanh Mγ ( e $^{iφ}$) dφ (3.8) Because tanh x < 1 for any positive x , the function ln tanh x will be negative. Thus the integral on the last line of (3.7) produces a negative number, which is multi- plied by N , the transverse extent of the strip, the analog of the area of a three dimensional film. We are interested in the limit n → ∞ , so the second term in curly brackets on the last line of (3.7) will be negligible compared to unity, and the sinh contribution to the partition function in (3.6) can be discarded without error. In this section, we consider an Ising model on a two dimensional square lattice consisting of M rows and N columns. The Hamiltonian of this model is the same as given in Eq. (2.1) but the sum is over a two dimensional lattice. We will begin by focusing on periodic and an- tiperiodic boundary conditions, first demonstrating how the critical Casimir force follows from expressions for the partition function already in the literature. Then, we perform an analysis of the two dimensional Ising model based on the fermion variable approach introduced by Schultz, Mattis and Lieb that leads to precisely the same results. Finally, we make use of this same approach to rederive existing expressions for the critical Casimir force in the case of free and fixed boundary conditions. As has already been noted, the Casimir forces in the two di- mensional Ising model for free and fixed-spin boundary conditions are connected by a duality relationship. III. TWO DIMENSIONS