7 The cylinder is assumed to couple to itself, so the ac- tual depiction of the system ought to be as a torus, as shown in Fig. 3. Free boundary conditions are enforced the Pauli spin matrices σ$_{x}$ and σ$_{z}$ σ x = ( 1 0 0 − 1 ) (4.4) σ y = ( 0 − i i 0 ) (4.5) σ z = ( 0 − 1 − 1 0 ) (4.6) (4.7) FIG. 2: The scaling contribution in the two dimensional Ising model to the Casimir force for antiperiodic boundary conditions (solid curve), periodic boundary conditions (short dashed curve) and free boundary conditions (long dashed curve) and fixed spin boundary conditions (alternatively long and short dashed curve). FIG. 3: The M × N lattice with M = 9 and N = 7, shown as a torus to indicate the cyclical nature of the vertical bond arrangement. The vertical bonds between the seven rows are shown dashed. by changing the strength of one array of vertical bonds to K$_{0}$ and then by taking the limit K$_{0}$ → 0. The partition function Z$_{f}$ , is then given by and between rows by V$_{2}$ = exp$^{}$ $_{}$K$_{2}$ M ∑ j =1 σ x $_{j}$σ x j +1   (4.1) V$_{1}$ = exp$^{}$ $_{}$− K ∗ 1 M ∑ j =1 σ z j   (4.2) In (4.1) and (4.2), K$_{n}$ with n = 1 , 2 are, respectively, the vertical and horizontal couplings, both positive for the ferromagnetic case, and K ∗ n is the dual of K$_{n}$ , i.e. tanh K ∗ n = exp [ − 2 K$_{n}$ ] (4.3) Writing V$_{1}$ in the form (4.2) requires that we accompany it by a factor (2 sinh(2 K$_{1}$ )) M/ 2 whenever it appears in an expression generating the partition function. In this article we make use of the following representations of Denoting the partition function for free boundary con- ditions by Z$_{f}$ , our first step is to write Z$_{f}$ in terms of a transfer matrix along the axis of a cylinder with circumference M and length N . In the representation with two dimensional Pauli spin matrix σ x $_{j}$, diagonal, j = 1 , . . . , M , the transfer operator within a row is given by riodic or antiperiodic boundary conditions. In this way, we encompass all the boundary conditions that were con- sidered above. Additionally, we will demonstrate by a duality transformation that the critical Casimir force for free boundary conditions is simply related to the Casimir force for fixed boundary conditions obtained previously by Evans and Stecki [20]. 4 2 2 4 0.4 0.2 0.2 0.4 - - - - Z$_{f}$ = (2 sinh(2 K$_{1}$ )) M ( N − 1) / 2 lim K$_{0}$ → $_{0}$(2 sinh 2 K$_{0}$ ) M/ 2 × Tr$^{(}$( V$_{2}$V$_{1}$ ) N − $^{1}$V$_{2}$ exp$^{[}$ − K ∗ 0 M ∑ m =1 σ z m ]) (4.8) lim K$_{0}$ → 0 ( e − MK ∗ 0 exp$^{[}$ − K ∗ 0 M ∑ m =1 σ z m ]) = | 0 〉〈 0 | (4.9) σ z $_{j}$| 0 〉 = −| 0 〉 , j = 1 , . . . , M (4.10) Given the fact that K ∗ 0 → ∞ as K$_{0}$ → 0, we find where | 0 〉 is defined by