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(Solved): (a) Consider the anisotropic Ising model in two dimensions, -\beta \mu =J_(x)\sum^x s_(i)s_(j)+J_(y ...
(a) Consider the anisotropic Ising model in two dimensions,
-\beta \mu =J_(x)\sum^x s_(i)s_(j)+J_(y)\sum^y s_(1)s_(j),s_(i)=+-1,where the sums are over nearest-neighbor pairs in the
xand
ydirections, respectively. Derive the duality transformation in
(J_(x),J_(y))space and locate the loci of self-duality. (b) Perform the duality transformation to the Ising model on the triangular lattice. By summing over every-other spin in the resulting lattice, complete the triangular-to-triangular mapping. (This called the "star-triangle transformation".) Find the self-mapping point. (c) Demonstrate that the duality transformation approach to locating the exact critical point fails for the spin-1 Ising model on the square lattice and for the spin-1/2 Ising model on the simple cubic lattice.
