content_image/432338.jpg content_image/432336.jpg content_image/432334.jpg content_image/432322.jpg content_image/432335.jpg content_image/432327.jpg content_image/432328.jpg
the portion of the boundary intersects a vertical bond the contribution is e − 2 $^{K$_{2}$}$. 

An important feature of this expansion is that fixed boundary conditions, in which the spins at the periph- ery of the system are constrained to point up, guarantee 

<img src='content_image/432338.jpg'>

<img src='content_image/432336.jpg'>

<img src='content_image/432334.jpg'>

that all contributions to the low temperature expansion consist of complete closed boundaries 

The next step in the development is to generate a high temperature expansion for the partition function of the spin system on the dual lattice. The spins on this lattice interact via nearest neighbor bonds that are both hori- zontal and vertical as in the original lattice. The strength of the horizontal bonds is K ∗ 1 and of the vertical bonds is K ∗ $_{2}$. The high temperature expansion is based on the decomposition 

The expansion is in the second term in parentheses on the last line of (C1). Because there is precisely one such term for each pair of spins and because each spin must appear an even number of times in the expansion, the high temperature expansion is represented graphically by figures like the border shown in Fig. 9, this time with the portions of the figure standing for bonds between spins on the dual lattice, as shown in Fig. 10. The portion is shown in Fig. 10. Each segment of the portion yields a 

factor tanh K ∗ 1 if it is horizontal and tanh K ∗ $_{2}$. Perfect correspondence is achieved if 

This is the standard set of duality relations. 

Here, in order that the high temperature expansion also consist of closed contours, we must allow all spins on the dual lattice to vary without restrictions. That is, the boundary conditions on the dual lattice must be free. This establishes the complete duality between the two versions of the two dimensional Ising model. 

<img src='content_image/432322.jpg'>

<img src='content_image/432335.jpg'>

<img src='content_image/432327.jpg'>

<img src='content_image/432328.jpg'>

