Dresden 2000 – scientific programme

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 4: Statistische Mechanik 1

MP 4.1: Talk

Monday, March 20, 2000, 16:30–16:45, W A317

Aperiodic Ising Models — •Uwe Grimm — Institut für Physik, Technische Universität Chemnitz, 09107 Chemnitz, Germany

Recent results on the critical properties of Ising quantum chains with coupling constants modulated according to aperiodic substitution sequences [1,2] and of classical Ising models defined on planar quasiperiodic graphs [1,3] are reviewed. For the Ising quantum chain in a transverse field, exact real-space renormalization transformations prove the Harris-Luck relevance criterion [4] for substitution systems. Furthermore, critical exponents can be calculated exactly, including coupling-dependent exponents for marginal aperiodic modulations. For classical aperiodic Ising models, certain solvable cases [1], series expansions [3] and partition function zeros [1,3] yield information about the critical behaviour. The results are in accordance with Monte-Carlo simulations [5] and generically consistent with the Harris-Luck criterion [4] based on scaling arguments.

[1] U. Grimm, M. Baake, Aperiodic Ising models, in: The Mathematics of Long-Range Aperiodic Order, ed. R. V. Moody (Kluwer, Dordrecht, 1997) pp. 199–237.

[2] J. Hermisson, U. Grimm, M. Baake, Aperiodic Ising quantum chains, J. Phys. A 30 (1997) 7315–7335.

[3] P. Repetowicz, U. Grimm, M. Schreiber, High-temperature expansion for Ising models on quasiperiodic tilings, J. Phys. A 32 (1999) 4397–4418.

[4] J. M. Luck, A classification of critical phenomena on quasi-crystals and other aperiodic structures, Europhys. Lett. 24 (1993) 359–364.

[5] O. Redner, M. Baake, Invaded cluster algorithm for critical properties of periodic and aperiodic planar Ising models, cond-mat/9910136.