Rugged Free Energy Landscapes Common Computational Approaches to Spi
This collection of lectures and tutorial reviews by renowned experts focusses on the common computational approaches in use to unravel the static and dynamical behaviour of complex physical systems at the interface of physics, chemistry and biology. Parad
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Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton Edinburgh, EH14 4AS, Scotland [email protected] Faculty of Physics, Adam Mickiewicz University, 61-614 Pozna´ n, Poland [email protected] Department of Mathematics, University of Sri Jayewardenepura, Gangodawila, Sri Lanka [email protected]
Abstract. We review a class of 3D lattice spin models in which planar Peierls boundaries between + and − spins can be created at zero energy cost. These socalled Gonihedric Ising models have (in general) specially tuned nearest neighbour, next-to-nearest neighbour and plaquette interactions, which endow the models with some novel properties both in and out of equilibrium. After reviewing the genesis of the models in string theory, we discuss investigations of both their equilibrium and non-equilibrium behaviours by various analytical and numerical means. The purely plaquette variant of the model displays all the standard indications of glassy behaviour without any recourse to quenched disorder, whilst still possessing a crystalline low-temperature phase in equilibrium.
7.1 (Pre-)History of the Model The Gonihedric 3D Ising model is a lattice spin model in which planar Peierls boundaries between + and − spins can be created at zero energy cost. Instead of weighting the area of Peierls boundaries as the case for the usual 3D Ising model with nearest-neighbour interactions, the edges, or “bends” in an interface are weighted, a concept which is related to the intrinsic curvature of the boundaries in the continuum. The model is a generalised Ising model living on a cubic 3D lattice with nearest neighbour, next to nearest-neighbour and plaquette interactions. The ratio between the couplings of these three terms is fixed to a one-parameter family which endows the model with unusual properties both in and out of equilibrium. Of particular interest for the discussion here will be that the model manifests all the indications of glassy behaviour without any recourse to quenched disorder, whilst still possessing a crystalline low-temperature phase in equilibrium.
D. A. Johnston et al.: The Gonihedric Ising Model and Glassiness, Lect. Notes Phys. 736, 173–199 (2008) c Springer-Verlag Berlin Heidelberg 2008 DOI 10.1007/978-3-540-74029-2 7
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D. A. Johnston et al.
In these notes, we follow a roughly chronological order by first reviewing the background to the formulation of the model, before moving on to the elucidation of the equilibrium phase diagram by various means and then to the investigation of the non-equilibrium, glassy behaviour of the model. We apologise in advance for our narrow focus on things Gonihedric at the expense of other lattice models with glassy behaviour since the aim is to concentrate on giving an overview of the Gonihedric Ising model in 3D. The model has an unusual genesis since it was originally introduced as a potential discretisation of string theory. The Nambu–Goto [1] action (or Hamiltonian in statistical mechanical language) in b
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