content_image/396042.jpg content_image/396038.jpg content_image/396051.jpg content_image/396039.jpg content_image/396036.jpg content_image/396037.jpg content_image/396040.jpg content_image/396041.jpg content_image/396035.jpg
from (3.8) to find the following Casimir force 

<img src='content_image/396042.jpg'>

To obtain the scaling form of the Casimir force, we start with looking at the behavior of the function γ ( e $^{iφ}$) in the vicinity of the critical point at which 

The solution to the equation (3.10) is given by 

as given in Ref.[17]. We now introduce the reduced tem- perature, τ : 

Then for small τ and φ , c ( e $^{iφ}$) = 1+2 τ $^{2}$+ φ 2 2 , and for the corresponding expansion of the function γ ( c ( e $^{iφ}$)), we are left with 

If we set φ = 2 ω and replace ω by Ω /M and τ by x/M , we find the following expression for the Casimir force t 

<img src='content_image/396038.jpg'>

The universal function ϑ ( x ) that emerges from (3.14) is 

<img src='content_image/396051.jpg'>

This result is displayed as the short dashed curve in Fig. 2. 

## C. The two dimensional Ising model with antiperiodic boundary conditions 

The exact partition function for a two dimensional Ising model on M × N square lattice with anti-periodic- 

<img src='content_image/396039.jpg'>

<img src='content_image/396036.jpg'>

<img src='content_image/396037.jpg'>

## B. The scaling limit 

<img src='content_image/396040.jpg'>

antiperiodic boundary conditions is 

<img src='content_image/396041.jpg'>

The derivation of the above partition function is given in Appendix A and is based on the expression calculated by Wu and Hu in Ref. [18]. Note that the last term in (3.16) changes sign at the critical temperature; see Appendix A for more detail. As previously, quantity M is the width of the strip, while N is its extent. Again, we will take the limit N → ∞ before allowing M to become large. 

The difference between periodic and antiperiodic boundary conditions lies in the sign relationships be- tween the terms in Eqs. (3.1) and (3.16). Consider the two terms with the log of the hyperbolic cosine in Eq. (3.16). They appear with opposite signs, and as we will see, this leads to near-cancellation of their contributions to the partition function. On the other hand, the other two terms in (3.16) may or may not add up. In order to assess the full partition function, we need to look at what remains after all near cancellations are taken into account. 

Appendix B contains an analysis of the contributions of the various terms in (3.16) to the partition function. The end-result is that the hyperbolic sine terms now domi- nate. We take a derivative with respect to M , the film thickness, to obtain the Casimir force. Performing the same scaling analysis as in Section III A, we find for the Casimir force in the case of antiperiodic boundary con- ditions. 

<img src='content_image/396035.jpg'>

For comparison, see (3.14) for the Casimir force in the case of periodic boundary conditions. Figure 2 shows the Casimir forces for periodic, antiperiodic and free bound- ary conditions. Note that when antiperiodic boundary conditions apply, the critical Casimir force is repulsive, leading to film thickening, as was the case for the one dimensional Ising model. 

To obtain the Casimir force for a two dimensional Ising model in the case of free boundary conditions, we use the expressions used in Li, et. al. [19]. After some manipula- tions, we find the asymptotic scaling form of the critical 

## D. Free boundary conditions 

