16 The plot of this difference is shown in Fig. 6. Note that FIG. 6: The difference between the exponents in (B17) and (B18), plotted versus the scaled reduced temperature, t . it is always negative. Appendix C: Duality and boundary conditions 5 10 15 20 25 30 - 2.0 - 1.5 - 1.0 - 0.5 exp$^{[}$ − N $^{√}$4 τ 2 + ( π/M ) 2 + N 2 π ∫ ( ln cosh M 2 $^{√}$4 τ 2 + φ 2 ) dφ$^{]}$ (B15) z N = (1 − Φ) N = e N ln(1 − Φ) → e − N Φ = e − $_{N}$√$_{(}$$_{π/M}$$_{)}$$_{2}$$_{+Δ}$$_{2}$ (B14) have to consider the factor Exponentiating the sums and expanding the exponen- tials with respect to the very small (in fact infinitesi- mally small in the thermodynamic limit N → ∞ ) terms, we find that there is a remainder after the subtraction that goes as Establishment of the duality between the two dimen- sional Ising lattice with free boundary conditions and the lattice with fixed boundary conditions can be achieved in two ways: by comparison of high and low temperature expansions for the partition function or with the use of automorphisms in the Kauffman operators. We will re- view both approaches in this Appendix The second term in the exponential comes from the inte- gral around the unit circle. There are also multiplicative terms, and they may depend non-trivially on key vari- ables. However, they have only marginal effect on the overall amplitude of the expression in (B15). In the case of the hyperbolic sine sums, the analysis is basically the same, up to the point at which we look for the branch point. Here, we look for the values of z at which the hyperbolic sine goes through zero. This happens when 1. Duality via high and low temperature expansions The first demonstration of Kramers-Wannier duality relation between the Casimir force for fixed spin and free boundary conditions [25, 26] utilizes a standard argu- ment based on high and low temperature expansions [27]. Figure 7 shows the Ising lattce subject to fixed bound- ary conditions along with its dual. The fixed-boundary- condition lattice has bonds represented as solid black lines, and the sites at which the spins sit are shown as black circles. The expansion that we consider for the par- tition function of this lattice is the low temperature one, in which we assume a “ground state” in which all spins are up, and we expand with respect to down spins. This generates a graphical expansion for the partition function containing terms such as the one displayed in Fig. 8. Figure 9 shows one of the sections of reversed spin in 8. The red dashed lines denote the boundary of that region. As indicated in the figure, the strength of the horizontal bonds is K$_{1}$ and the strength of the vertical bonds is K$_{2}$ . The contribution to the partition function of a portion of the boundary intersecting a horizontal bond is e − 2 K$_{1}$ corresponding to the alteration of the Boltzmann factor associated with the mismatch of a pair of spins; when The value of Φ for which this occurs that is closest to the intersection between the unit circle and the real axis ( z = 1) is The same set of steps taken above leads us to a remainder if the exponentiated hyperbolic sine sums are subtracted from each other that goes as The term in (B18) dominates the term in (B17). To see this, let’s focus on the partition function in the region of interest, writing Then, if we take the difference between the exponents in (B17) and (B18), we have N M $^{[}$1 π ∫ ∞ −∞ ln coth $^{√}$t 2 + Ω 2 d Ω − $^{√}$π 2 + 4 t 2 + 2 | t |$^{]}$ (B21) τ = t M (B19) φ → 2 Ω M (B20) exp$^{[}$ − N | 4 τ | + N 2 π ∫ ( ln sinh M 2 $^{√}$4 τ 2 + φ 2 ) dφ$^{]}$ (B18) Φ = ± 2 τ (B17) sin M 2 $^{√}$Φ 2 − 4 τ 2 = 0 (B16)