Aperiodic Ising Models

Details Aperiodic Ising Models Aperiodic Ising Models

1996-04-18

arxiv.org/abs/cond-mat/9604116v1

The Mathematics of Long-Range Aperiodic Order, NATO ASI series C 489, edited by R.V. Moody (Kluwer, Dordrecht, 1997) pp. 199-2

Physics

Condensed Matter

37 pages, uuencoded gzipped PostScript file, to appear in: The Mathematics of Aperiodic Order, eds. R.V. Moody and J. Patera,

Scientific paper

We consider several aspects of non-periodic Ising models in one and two dimensions. Here we are not interested in random systems, but rather in models with intrinsic long-range aperiodic order. The most prominent examples in one dimension are sequences generated by substitution rules on a finite alphabet. The classical one-dimensional Ising chain as well as the Ising quantum chain with coupling constants modulated according to substitution sequences are considered. Both the distribution of Lee-Yang zeros on the unit circle in the classical case and that of fermion frequencies in the quantum model show characteristic gap structures which follow the gap labeling theorem of Bellissard. We also investigate the zero-field Ising model on two-dimensional aperiodic graphs, which are constructed from rectangular grids in the same spirit as the so-called Labyrinth. Here, duality arguments and exact solutions in the sense of commuting transfer matrices are used to gain information about the critical behaviour.

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