arXiv subjects

Michael Baake

Publications and source records attributed to Michael Baake.

166 records · Page 10Linked to original sources

arXiv1998

Random Tilings: Concepts and Examples

We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile densities. In this context, and under rather mild assumptions, we prove a generalization of the first random tiling hypothesis which connects the maximum of the entropy with the symmetry of the ensemble. Explicit examples are obtained through the re-interpretation of several exactly solvable models. This also leads to a counterexample to the analogue of the second random tiling hypothesis about the form of the entropy function near its maximum.

cond-mat.stat-mech↗

arXiv1997

Aperiodic Ising Quantum Chains

Some years ago, Luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In this article, we show how Luck's criterion can be derived within an exact renormalisation scheme for Ising quantum chains with coupling constants modulated according to substitution rules. Luck's conjectures for this case are confirmed and refined. Among other outcomes, we give an exact formula for the correlation length critical exponent for arbitrary two-letter substitution sequences with marginal fluctuations of the coupling constants.

cond-mat.stat-mech↗

arXiv1997

Coordination sequences for root lattices and related graphs

The coordination sequence s(k) of a graph counts the number of its vertices which have distance k from a given vertex, where the distance between two vertices is defined as the minimal number of bonds in any path connecting them. For a large class of graphs, including in particular the classical root lattices, we present the coordination sequences and their generating functions, summarizing and extending recent results of Conway and Sloane. A possible application to the theory of critical phenomena in lattice models is outlined.

cond-mat.stat-mech↗

arXiv1996

Aperiodic Ising Models

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.

cond-mat↗