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Hard Squares for z = −1
R.J. Baxter
Mathematical Sciences Institute, Australian National University, ACT 0200, Australia
Annals of Combinatorics 15 (2) pp.185-195 April, 2011
AMS Subject Classification: 05A15, 82B20
Abstract:
The hard square model in statistical mechanics has been investigated for the case when the activity z is −1. For cyclic boundary conditions, the characteristic polynomial of the transfer matrix has an intriguingly simple structure, all the eigenvalues x being zero, roots of unity, or solutions of x3 = 4cos2m/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orientations. We remark that they are all unexpectedly simple and that for the rotated lattice with free or fixed boundary conditions there are obvious likely generalizations to any lattice size.
Keywords: statistical mechanics, lattice models, transfer matrices

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