2 near the critical point. With an eye toward developing a greater depth of understanding of effects of boundary conditions and dimensionality, we have looked closely at one system for which exact results can be obtained: an Ising model in one and two dimensions. With regard to the one-dimensional Ising model relevant results follow almost immediately from a simple transfer matrix anal- ysis. In the case of the two dimensional Ising model, we have been able to perform an analysis of that model that yields the critical Casimir force for four different bound- ary conditions: periodic, antiperiodic, free (the appro- priate surrogate for Dirichlet boundary conditions in the Ising model) and fixed-spin. The results of this analysis are consistent with those we have extracted from various expressions already in the literature. A key conclusion that we draw is that boundary con- ditions alone do not determine the relative amplitudes of critical Casimir forces. Indeed, we find that as dimen- sionality is reduced, periodic (and antiperiodic) bound- ary conditions give rise to a critical Casimir force that is increasingly stronger than the force resulting from free boundary conditions. In fact, the critical Casimir force for a one dimensional Ising model with free boundary conditions is effectively non-existent. The paper is organized as follows. In section II we calculate the Casimir force for the one-dimensional Ising model as a prelude to the discussion of the two dimen- sional version. We obtain the Casimir forces with free, fixed, periodic and anti-periodic boundary conditions. We observe significant differences in the forces in these four cases. In section III, we present our results for the two dimensional Ising model and compare them with the one dimensional results. The forces are derived in two different ways. First, we show how they follow from ex- pressions already in the literature. We also demonstrate how they arise from an unified analysis of the two di- mensional Ising model making use of an approach along the lines of the method developed by Schultz, Mattis and Lieb [15]. In section V, we discuss our findings, their im- plications, and summarize our conclusions. For clarity of exposition, details of our calculations are relegated to appendices. When periodic boundary conditions hold, we calculate the partition function by evaluating the trace of T N 1 over a complete set of bounding spin states. Given (2.5) and (2.6) we have immediately Therefore, the free energy of this system is Z$_{periodic}$ ( N ) = 2 N $^{[}$cosh $^{N}$( βJ ) + sinh $^{N}$( βJ ) $^{]}$, (2.7) F$_{periodic}$ ( N ) = − Nk$_{B}$T ln (2 cosh( βJ )) − k$_{B}$T ln $^{[}$1 + tanh( βJ ) N ] (2.8) In the thermodynamic limit the second term of (2.8) be- comes negligible, so the bulk free energy per unit length (alternatively per spin) is f$_{B}$ = − k$_{B}$T ln 2 − k$_{B}$T ln cosh( βJ ) (2.9) The correlation function, C ( l$_{1}$, l$_{2}$ ), and the associated correlation length, ξ , in the one-dimensional Ising model are defined as follows [16] C ( l$_{1}$, l$_{2}$ ) = tanh( βJ ) l$_{2}$ − l$_{1}$ ≡ exp [ − ( l$_{2}$ − l$_{1}$ ) /ξ ] (2.10) ξ = − 1 / (ln tanh βJ ) (2.11) which leads to Taking the derivative of the free energy (2.8) with respect to N and subtracting the bulk free energy per unit length, we are left with − ∂ F ( N ) ∂N + f$_{B}$ = − k$_{B}$T ξ e − N/ξ 1 + e N/ξ = − k$_{B}$T N $^{{}$N ξ e − N/ξ 1 + e N/ξ } = F Cas 1 d per (2.12) H = − J ∑ i,j s$_{i}$s$_{j}$, (2.1) A. periodic boundary conditions II. ONE DIMENSION The quantity on the right hand side of (2.3) is most con- veniently evaluated making use of the eigenvector decom- position of T$_{1}$ where β = 1 /k$_{B}$T , a spin state at the left boundary of the N -spin array of the form 〈 l | and a spin state | r 〉 at the right boundary of that array, the partition function is equal to The spin variables s$_{i}$ can assume the values ± 1. Given the transfer matrix T$_{1}$ =$^{(}$ e βJ e − βJ e − βJ e βJ ) (2.2) Z = 〈 l | T N 1 | r 〉 (2.3) T$_{1}$ = | e 〉 2 cosh( βJ ) 〈 e | + | o 〉 2 sinh( βJ ) 〈 o | (2.4) where | e 〉 = 1 √ 2$^{(}$ 1 1$^{)}$ (2.5) | o 〉 = 1 √ 2$^{(}$ 1 − 1$^{)}$ (2.6) The partition function of the one dimensional Ising model is well known to be derivable from the two-by-two transfer matrix of that model. The Ising model possesses a global Z$_{2}$ symmetry and is described by the following Hamiltonian where J > 0 is the spin-spin coupling constant, and the sum is over nearest neighbor bonds i, j on the lattice.