content_image/387281.jpg content_image/387283.jpg content_image/387303.jpg content_image/387300.jpg content_image/387299.jpg content_image/387298.jpg content_image/387284.jpg content_image/387288.jpg content_image/387289.jpg content_image/387291.jpg content_image/387294.jpg
where γ ( k ) is also non-negative for real argument. The angle δ $^{′}$( k ) is defined by 

<img src='content_image/387281.jpg'>

and 

<img src='content_image/387283.jpg'>

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: 

<img src='content_image/387303.jpg'>

Since τ $^{z}$( k ) | 0 〉 = −| 0 〉 and for M even, exp( iπM ) = 1, evaluating the matrix element in (4.8) is an easy matter: 

<img src='content_image/387300.jpg'>

Checking the behavior of δ $^{′}$( π ), we see that this equation can be written as 

<img src='content_image/387299.jpg'>

<img src='content_image/387298.jpg'>

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$_{M}$( − ). 

## B. Periodic and antiperiodic boundary conditions 

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 

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]: 

<img src='content_image/387284.jpg'>

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: 

<img src='content_image/387288.jpg'>

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 

<img src='content_image/387289.jpg'>

The resulting Casimir force is denoted f$_{cas}$ ( cyl. ) and is just: 

<img src='content_image/387291.jpg'>

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: 

<img src='content_image/387294.jpg'>

This implies the same scaling function as in (3.15). The integral in (4.55) has the value π $^{2}$/ 12 at x = 0. 

