10 vanish, which turns out to be the case on detailed cal- culation. The summand in (1.45) becomes Nγ , and the sum may be implemented as a contour integral, following the methods of [23, 24]: where γ ( k ) is also non-negative for real argument. The angle δ $^{′}$( k ) is defined by cosh(2 K$_{2}$ ) = cosh(2 K ∗ $_{1}$) cosh( γ ( k )) − sinh(2 K ∗ $_{1}$) sinh( γ ( k )) cos( δ $^{′}$( k )) (4.45) and sin( δ $^{′}$( k )) sinh( γ ( k )) = sin k sinh(2 K$_{2}$ ) (4.46) lim N $_{→∞}$N − $^{1}$ln Z$_{f}$ = − M 4 π ∮ C dω 1 + e $_{iMω}$γ ( ω ) (4.51) where C is the rectangle in the complex ω plane with sides which are segments of the lines Re( ω ) = ± π, Im( ω ) = ± ϵ . The number ϵ is chosen small enough so that the only singularities of the integrand inside the con- tour are the zeros of the denominator. Equation (1.46) can then be written as: Notice that the term multiplying sinh( γ ( k )) in (4.43) is itself a spin operator in the paired subspace. Raising (4.43) to the N th power is, then, straightforward: The first term on the right hand side is the bulk term. The Casimir contribution is contained in the second term on the right hand side of (4.52). The function γ has branch cuts on the imaginary axis symmetrically posi- tioned about the real axis. Using the 2 π periodicity of the integrand in (4.52), the contour in (4.52) will be de- formed into one surrounding the branch cut in the upper half plane. It is convenient to make the transformation ω = i ˆ γ ( u ) and interchange K 1 and K 2 in Eq. (4.44). Expanding the denominator, integrating by parts and re- summing gives the Casimir free energy from (4.52) as lim N $_{→∞}$N − $^{1}$ln Z$_{f}$ = M 4 π ∫ − π dω γ ( ω ) − M 2 π ∫ π + iϵ − π + iϵ dω e iMω 1 + e $_{iMω}$γ ( ω ) (4.52) F$_{cas}$ = − k$_{B}$T π ∫ π 0 du ln $^{[}$1 + exp $^{(}$− M ˆ γ ( u ) )] (4.53) The resulting Casimir force is denoted f$_{cas}$ ( cyl. ) and is just: f$_{cas}$ ( cyl. ) = − k$_{B}$T 4 π ∫ π − π du ˆ γ ( u ) [1 − tanh ( M ˆ γ ( u ) / 2)] (4.54) Note that this expression is invariant under interchange of K ∗ 1 and K$_{2}$ ; that is, there is a dual symmetry; further, this force is attractive . The scaling form (with K$_{1}$ = K$_{2}$ ) is accessed by noting that ˆ γ ( u ) ∼ [ˆ γ (0) 2 + u $^{2}$]1 / 2 for small u and ˆ γ (0). We then introduce scaling variables y = Mu/ 2 and x = M ˆ γ (0) / 2 and take the scaling limit, giving: This implies the same scaling function as in (3.15). The integral in (4.55) has the value π $^{2}$/ 12 at x = 0. M 2 ¯ f$_{cas}$ ( cyl. ) → − k$_{B}$T π ∫ ∞ −∞ dy ( x 2 + y $^{2}$)1 / 2 $^{[}$1 − tanh $^{(}$( x 2 + y $^{2}$)1 / 2 )] (4.55) The limit as N → ∞ is easily taken, since M < ∞ implies that the sum is finite. Another way of examin- ing this is to go back to (4.12). As N → ∞ , we expect only the contribution of the maximum eigenvector to sur- vive, this provided its scalar product with | 0 〉 does not B. Periodic and antiperiodic boundary conditions This is in fact correct whether M is odd or even. The result for Z a $_{f}$, the case in which antiperiodicity is enforced in each M -spin row, is a simple variation on (4.50). One simply replaces the requirement k ∈ Ω$_{M}$(+) in the sum by k ∈ Ω$_{M}$( − ). ln Z$_{f}$ = 1 2 ∑ k ∈ Ω$_{M}$(+) ln (cosh( Nγ ( k )) + sinh( Nγ ( k )) cos( δ $^{′}$( k ))) (4.50) Z$_{f}$ = ∏ k ∈ Ω$_{M}$(+) { cos( Nγ ( k )) + sinh( Nγ ( k )) cos( δ $^{′}$( k )) } 1 / 2 (4.49) Thus Z$_{f}$ = exp[ N ( K ∗ 1 + K$_{2}$ )] × ∏ 0 <k<π { cosh( Nγ ( k )) + sinh( Nγ ( k )) cos( δ $^{′}$( k )) } (4.48) Checking the behavior of δ $^{′}$( π ), we see that this equation can be written as Since τ $^{z}$( k ) | 0 〉 = −| 0 〉 and for M even, exp( iπM ) = 1, evaluating the matrix element in (4.8) is an easy matter: ( V $^{′}$( k )) N = cosh( Nγ ( k )) − sinh( Nγ ( k ))(cos( δ $^{′}$( k )) τ $^{z}$( k ) − sin( δ $^{′}$( k )) τ $^{y}$( k )) (4.47)