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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]. 

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 

and between rows by 

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. 

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 

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the Pauli spin matrices σ$_{x}$ and σ$_{z}$ 

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 

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 

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Given the fact that K ∗ 0 → ∞ as K$_{0}$ → 0, we find 

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where | 0 〉 is defined by 

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