Boundary Boltzmann Weight for the Eight-Vertex SOS Model: Vertex-IRF Correspondence

In this paper, we build on the biquasiles and dual graph diagrams introduced in [7]. We introduce biquasile Boltzmann weights that enhance the previous knot coloring invariant defined in terms of finite biquasiles and provide examples differentiating links with the same counting invariant, demonstrating that the enhancement is proper. We identify conditions for a linear function [Formula: see text] to be a Boltzmann weight for an Alexander biquasile [Formula: see text].

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Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

Journal of Statistical Physics ◽

10.1007/bf01014383 ◽

1984 ◽

Vol 35 (3-4) ◽

pp. 193-266 ◽

Cited By ~ 618

Author(s):

George E. Andrews ◽

R. J. Baxter ◽

P. J. Forrester

Keyword(s):

Sos Model

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Study of the island morphology at the early stages of Fe/Mo(110) MBE growth

Open Physics ◽

10.2478/s11534-008-0124-z ◽

2009 ◽

Vol 7 (2) ◽

Cited By ~ 1

Author(s):

Dmytro Goykolov ◽

Miroslav Kotrla

Keyword(s):

Low Temperature ◽

Aspect Ratio ◽

Theoretical Study ◽

Morphological Changes ◽

Nearest Neighbors ◽

Sos Model ◽

Mbe Growth ◽

Calculated Dependence ◽

Average Aspect Ratio ◽

Higher Temperature

AbstractWe present theoretical study of morphology of Fe islands grown at Mo(110) surface in submonolayer MBE mode. We utilize atomistic SOS model with bond counting, and interactions of Fe adatom up to third nearest neighbors. We performed KMC simulations for different values of adatom interactions and varying temperatures. We have found that, while for the low temperature islands are fat fractals, for the temperature 500 K islands have faceted rhombic-like shape. For the higher temperature, islands acquire a rounded shape. In order to evaluate qualitatively morphological changes, we measured average aspect ratio of islands. We calculated dependence of the average aspect ratio on the temperature, and on the strength of interactions of an adatom with neighbors.

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GIBBS MEASURES FOR SOS MODELS ON A CAYLEY TREE

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025706002494 ◽

2006 ◽

Vol 09 (03) ◽

pp. 471-488 ◽

Cited By ~ 24

Author(s):

U. A. ROZIKOV ◽

Y. M. SUHOV

Keyword(s):

Ising Model ◽

Normal Subgroup ◽

Finite Index ◽

Cayley Tree ◽

Nearest Neighbor ◽

Gibbs Measures ◽

Natural Generalization ◽

Mirror Reflection ◽

Sos Model ◽

Translation Invariant

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,…, m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, [Formula: see text] such that [Formula: see text], there exists a unique symmetric TISGM μ* and [Formula: see text] there are exactly three symmetric TISGMs: [Formula: see text] (a "bottom" symmetric TISGM), [Formula: see text] (a "middle" symmetric TISGM) and [Formula: see text] (a "top" symmetric TISGM). For [Formula: see text] we also construct a continuum of distinct, symmertric SGMs which are non-TI. Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM).

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