13 Ω$_{µν}$ = N − 1 ∏ p =0 M − 1 ∏ q =0 [ (1 + t $^{2}$)2 − 2 t (1 − t $^{2}$) cos 2 π ( p + µ ) N − 2 t (1 − t $^{2}$) cos 2 π ( q + ν ) M ] 1 / 2 (A8) Thus, the p = q = 0 term in the product for Ω$_{00}$ is proportional to | θ − θ$_{c}$ | . The term going as the sign of that expression guarantees that the product goes as θ − θ$_{c}$ . It is a straightforward exercise to verify that all other factors in the product are free of singularities in the vicinity of the critical point. The quantity Ω$_{0}$$_{,}$$_{0}$ in Eq. (A2) needs special attention. Consider the p = q = 0 term in the product for Ω$_{0}$$_{,}$$_{0}$, [(1 + t $^{2}$)2 − 4 t (1 − t $^{2}$)]1 / 2 = [(1 − 2 t − t $^{2}$)$^{2}$]1 / 2 = | 1 − 2 t − t $^{2}$| (A9) | 1 − 2 tanh βJ − tan $^{2}$βJ | = 1 cosh $^{2}$βJ | cosh $^{2}$βJ − sinh $^{2}$βJ − 2 cosh βJ sinh βJ | = 1 cosh $^{2}$βJ | 1 − sinh 2 βJ | (A10) 1. Sum over the variable q Making use of (A3), we have for the last line of (A9) Now we focus on the sum over the variable q in Eq. (A8). There are two cases of interest: ν = 0 and ν = 1 / 2. We will analyze the case ν = 0 making use of the method outlined in Section III C. The generic sum of interest is M − 1 ∑ q =0 1 a − b cos 2 πq M , (A13) cosh βJ coth βJ = 2 , (A11) sinh 2 βJ = 1 (A12) According to (3.10),the critical temperature is at which can be rewritten as which is equivalent to the integral 1 2 πi ∫ 1 a − b ( z + 1 /z ) / 2 Mz M − 1 z M − 1 dz = − 1 2 πi 2 b ∫ 1 z 2 − 2 az/b + 1 Mz N z M − 1 dz = − 1 2 πi 2 b ∫ 1 ( z − a/b − $^{√}$( a/b ) 2 − 1)( z − a/b + $^{√}$( a/b ) 2 − 1) × z M z M − 1 dz (A14) Setting a/b + $^{√}$( a/b ) 2 − 1 = Γ and using the fact that a/b − $^{√}$( a/b ) 2 − 1 = ( a/b + $^{√}$( a/b ) 2 − 1) − $^{1}$, we can write (A15) as follows The integral is around the contours shown in Fig. 4. This integral is evaluated by closing it around the two poles in the first term in the intregrand on the third line of (A14). We find after evaluating residues that the result of this integration is M √ a 2 − b 2 Γ M + 1 Γ M − 1 = M √ a 2 − b 2 Γ M/ 2 + Γ − M/ 2 Γ M/ 2 − Γ − M/ 2 = M √ a 2 − b 2 coth$^{(}$ M 2 ln Γ$^{)}$ (A16) M √ a 2 − b 2 ( ( a/b + $^{√}$( a/b ) 2 − 1) M ( a/b + $^{√}$( a/b ) 2 − 1) M − 1 − ( a/b − $^{√}$( a/b ) 2 − 1) M ( a/b − $^{√}$( a/b ) 2 − 1) M − 1$^{)}$ (A15)