Abstract
A technique for obtaining exact thermodynamic and correlation functions of regular fractals is illustrated by a development for the Ising system on the triangular Sierpińsky gasket. A generalised hyperscaling statement is obtained. The technique is then applied to the fully frustrated antiferromagnetic case, where the exact ground state entropy per spin is shown to be S(0)=0.4930…. This result shows that the introduction of (regularly disposed) holes into the triangular lattice does not relieve frustration, as is commonly supposed. Relations completely determining the correlation functions are obtained. That for the zero field pair correlation function of vertex spins at separation r is G−1(2r)=G−2(r)−G−+1, together with a relation which sets the correlation- length scale. Exact asymptotic expressions are derived for ferromagnetic correlations in both zero and non-zero field. The zero-field, zero-temperature correlation length of the fully frustrated system is shown to be ξ=0.79, representing extremely rapid decay of correlations.
Published Version
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