14 Then, the argument of the hyperbolic sine in (A23) be- comes The sum of interest is in fact, which means that as the temperature passes through the critical temperature, the hyperbolic sine function changes sign. This is the source of the “sgn” term in (3.16) and (A1). The calculations utilized to obtain Ω$_{µ}$$_{0}$ can be adapted to the evaluation of the sum over q in Ω$_{µ}$ 1 $_{2}$. After a set of tedious but straightforward calculations, we obtain for that result which can be obtained by integrating the sum in (A13) with respect to a . Making use of the fact that d da ln $^{(}$a/b + $^{√}$( a/b ) 2 − 1 ) = 1 / √ a 2 − b $^{2}$, we find the an- tiderivative of the last line of (A16) with respect to a M 2 ln $^{[}$1 + 4 τ + 8 τ 2 + O $^{(}$τ 3 )] = 2 Mτ (A25) Ω$_{µ}$ 1 2 = N − 1 ∏ p =0 cosh$^{(}$ M 2 γ$_{p}$$^{)}$ (A26) M − 1 ∑ q =0 ln$^{(}$ a − b cos 2 πq M ) (A17) 2 ln$^{[}$sinh$^{(}$ M 2 ln Γ$^{)]}$ (A18) Finally, if we define γ$_{p}$ = ln Γ$_{p}$ we have a result that be- gins to look like a contribution to the expressions utilized by Ferdinand and Fisher in the case of periodic boundary conditions. To complete the connection, we note that in this case Appendix B: Contributions of the terms in (3.16) to the partition function in the case of antiperiodic boundary conditions We first focus on the sums involving hyperbolic cosines. We have N − 1 ∑ r =1 ln cosh 1 2 Mγ$_{2}$$_{r}$$_{+1}$ = 1 2 πi ∫ ln cosh M 2 γ ( z ) Nz N − 1 z N + 1 dz (B1) N − 1 ∑ r =1 ln cosh 1 2 Mγ$_{2}$$_{r}$ = 1 2 πi ∫ ln cosh M 2 γ ( z ) Nz N − 1 z N − 1 dz (B2) a = (1 + t $^{2}$)2 − 2 t (1 − t $^{2}$) cos 2 π ( p + µ ) N (A19) b = 2 t (1 − t $^{2}$) (A20) Γ$_{p}$ = (1 + t $^{2}$)2 t (1 − t $^{2}$) − cos 2 π ( p + µ ) N +$^{√}$$^{(}$ (1 + t $^{2}$)2 t (1 − t $^{2}$) − cos 2 π ( p + µ ) N ) 2 − 1 = cosh 2 βJ coth 2 βJ − cos 2 π ( p + µ ) N +$^{√}$$^{(}$cosh 2 βJ coth 2 βJ − cos 2 π ( p + µ ) N ) 2 − 1 (A21) Then, and The contours of integration in (B2) shown in Fig. 4; the contours of integration in (B1) are identical except for a shift along the unit circle. The contours enclose the roots of the function 1 / ( z N − 1), as depicted on the left hand side of that figure. They are then deformed to lie just inside and outside the unit circle—see the right hand side of Fig. 4. We have already established in Section III A that the other singularities in the integrands in (B1) and (B2) lie along the real axis. They are branch cuts which are close to the unit circle in the vicinity of the critical temperature. We now concentrate on the asymptotic expression in the vicinity of the critical point and perform expansions like those in Section III A and in Eq. (A25) and find the following expression Eq. (A18) corresponds to the logarithm of the product over q of Ω$_{µ}$$_{0}$. Reconstituting the product over p , we have Let’s focus on one term in the product for Ω$_{00}$: the p = 0 term. It is Ω$_{µ}$$_{0}$ = N − 1 ∏ p =0 sinh$^{(}$ M 2 γ$_{p}$$^{)}$ (A22) sinh$^{(}$ M 2 ln$^{(}$ cosh 2 βJ coth 2 βJ − 1+ √ (cosh 2 βJ coth 2 βJ − 1) 2 − 1$^{))}$ (A23) The function cosh 2 βJ coth 2 βJ has a minimum at βJ = with φ$_{p}$ = 2 π ( p + 1 / 2) /N in case of Eq. (B1) and with φ$_{p}$ = 2 πp/N for Eq. (B2). This holds when the complex variable z is in the im- mediate vicinity of z = 1, where the unit circle passes through the real axis. In fact, φ = ± i ( z − 1). The branch ∑ φ$_{p}$ ln cosh M 2 $^{√}$4 τ 2 + φ 2 $_{p}$, (B3) cosh − 1 ( √ 1 2 $^{(}$1 + √ 2 ) ) . We write βJ = cosh − 1 ( $^{√}$1 2 $^{(}$1 + √ 2 ) ) + τ (A24)