Incollection,

Ising Model in Half-space. Layering Transitions

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Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)

Abstract

The extension of Pirogov-Sinai theory is developed. Results applicable for bulk and surface phases of lattice models are proved, and state equation is constructed. The region of first-order phase transition is extended in external fields space to $C^\Phi ,\Phi$ -- the phase set of a model. It is proved the next extension of Lee-Yang theorem: the partition functions with stable boundary condition have no zeros in external fields space $C^\Phi.$ For Ising model in half-space with small values of temperature and mixing boundary condition it is proven for each external field $$ the existence of the spin layer with the thickness $q(\mu)$ over the bottom boundary. In this layer the average of spin is approximately -1 and outside one is about +1. With the decreasing of the external field $$ in points $_q$ the thickness $q()$ changes stepwise with unit magnitude and $q(\mu )\rightarrowınfty$ when $\mu+0.$ In points $\mu_q$ there is the coexistence of two surface phases. The free surface energy is proven the piecewise analytical function in region Re $\mu>0$ and small values of temperature. It is considered also the model with the external arbitrary field $_0$ in the zero-layer, the external field outside the zero-layer $>0.$ In the latter case the phase diagram of layering transitions is also constructed (see figure 1, outside ABCD). The Antonov's rule is proved. Using the surface state equation the points $B_0$ and $B_1$ of coexistence phases $\0,1,2\$ and $\0,2,3\$ with the accuracy of $x^7,x=\exp(-2\varepsilon)$ are constructed.

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