Aperiodicity and Disorder — Do They Play a Role?

Geometric State Sum Models from Quasicrystals

“…Essentially quasicrystals [19][20][21][22][23] are structures that exhibits a new kind of order − aperiodic order − which lies between disorder and periodicity. Statistical mechanics or quantum mechanics models defined in lattices can be straightforward generalized to quasicrystals [24]. The choice of Z 6 root lattice and its associated 3DPT quasicrystal provides a toy model for the conformal symmetry associated with the D 6 root system and for the grand unified gauge theory associated with the exceptional Lie algebra E 8 .…”

Properties of 1D Classical and Quantum Ising Models: Rigorous Results

“…One-dimensional classical Ising chains. The classical Ising spin chain in one dimension with quasi-periodic interaction (nearest neighbor) and magnetic field have also been investigated both numerically and analytically (see, for example, [2,3,5,[27][28][29]33,[77][78][79]81], and references therein). A particularly curious problem is the analyticity of the free energy function in the case the interaction strength couplings as well as the external field are modulated by a quasi-periodic sequence (Fibonacci, say).…”

Partition Function Zeros of Aperiodic Ising Models

“…Therefore, the influence of an aperiodic order in the underlying structure on the phase transition of the Ising model has been investigated by various means, including numerical simulations, series expansions, and zeros of the partition function; an overview on the results and a comprehensive lists of references on the subject can be found in recent review articles. 1,2 The most general prediction stems from heuristic scaling arguments, adapted from a relevance criterion for disordered Ising models. 2 It yields an inequality involving a characteristic exponent that describes the fluctuations in the disordered or aperiodically ordered system, the correlation critical exponent ν of the pure system, and the space dimension of the fluctuation.…”

“…1,2 The most general prediction stems from heuristic scaling arguments, adapted from a relevance criterion for disordered Ising models. 2 It yields an inequality involving a characteristic exponent that describes the fluctuations in the disordered or aperiodically ordered system, the correlation critical exponent ν of the pure system, and the space dimension of the fluctuation. Planar quasiperiodic graphs obtained by cut-and-project methods 3 have low fluctuations, because they are flat sections through higher-dimensional peri- odic lattices.…”