12 Thus which is a manifestation of duality. In this paper, we exploit the exact solutions for the partition function of the Ising model in one and two di- mensions to examine the impact of boundary conditions and dimensionality on the Casimir forces. Our results are effectively summarized in Figs. 1 and 2. As both fig- ures unambiguously illustrate, boundary conditions alone do not suffice to predict the relative strengths of critical Casimir forces. This is perhaps most strikingly evident in the case of the forces associated with periodic and free boundary conditions. While free boundary condi- tions give rise to a vanishing critical Casimir force in one dimension, in the two dimensional case the same bound- ary conditions result in a significant attractive force. By contrast, periodic boundary conditions yield a non-zero critical Casimir force in one dimension–that is, a force that is infinitely larger than the one associated with free boundary conditions. On the other hand, in two dimen- sions, the Casimir force generated by the same boundary conditions is smaller in amplitude than the Casimir force resulting from free boundary conditions. In the case of anti-periodic boundary conditions, the critical Casimir force is repulsive in both one and two We are pleased to acknowledge useful and important discussions with Professor Mehran Kardar. Financial support from NSF grant numbers DMR-06-45668 and DMR-07-04274 is greatly acknowledged. Here, we make use of results in Ref. [18] for the reduced partition function of the two dimensional square Ising model with antiperiodic boundary conditions in both di- rections (see Eq. (79) in Ref. [18] ) The methods presented in this paper can also be extended to calculate Casimir forces with mismatched boundary conditions. dimensions. An additional contrast between periodic and antiperiodic boundary conditions is evident in the limit T = 0 in one dimension; at that point, the Casimir force is zero for the case of periodic boundary but is finite in the anti-periodic case. V. CONCLUSION Acknowledgments Given for a square lattice with uniform interaction strength J , we find and thus, f ++ $_{cas}$( t ) = f free cas ( − t ) (4.66) A$_{0}$ = (1 + t $^{2}$)2 (A4) A$_{1}$ = 2 t (1 − t $^{2}$) (A5) A$_{2}$ = 2 t (1 − t $^{2}$) , (A6) A$_{3}$ = 0 (A7) t = tanh βJ (A3) Ω$_{µ,ν}$ = N − 1 ∏ p =0 M − 1 ∏ q =0 [ A$_{0}$ − A$_{1}$ cos 2 π ( p + µ ) N − A$_{2}$ cos 2 π ( q + ν ) M − A$_{3}$ cos$^{(}$ 2 π ( p + µ ) N − 2 π ( q + ν ) M ) ] 1 / 2 (A2) Q aa = 1 2 [ − Ω 1 $_{2}$, 1 2 + Ω 1 $_{2}$, 0 + Ω$_{0}$$_{,}$ 1 2 + sgn$^{(}$ θ − θ$_{c}$ θ$_{c}$ ) Ω$_{0}$$_{,}$$_{0}$$^{]}$ (A1) with θ = k$_{B}$T/J and Appendix A: Derivation of exact partition functions for the Ising Model with antiperiodic-antiperiodic boundary conditions