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Ising model in half-space: A series of phase transitions in low
magnetic fields
A. G. Basuev St. Petersburg State University of Technology and Design
Abstract:
For the Ising model in half-space at low temperatures and for the “unstable
boundary condition,” we prove that for each value of the external magnetic
field
$\mu$, there exists a spin layer of thickness
$q(\mu)$ adjacent to the
substrate such that the mean spin is close to
$-1$ inside this layer and
close to
$+1$ outside it. As
$\mu$ decreases, the thickness of the
$(-1)$-spin layer changes jumpwise by unity at the points
$\mu_q$, and
$q(\mu)\to\infty$ as
$\mu\to+0$. At the discontinuity points
$\mu_q$ of
$q(\mu)$, two surface phases coexist. The surface free energy is piecewise
analytic in the domain
$\operatorname{Re}\mu>0$ and at low temperatures. We consider the
Ising model in half-space with an arbitrary external field in the zeroth
layer and investigate the corresponding phase diagram. We prove Antonov's
rule and construct the equation of state in lower orders with the precision
of
$x^7$,
$x=e^{-2\varepsilon}$. In particular, with this precision, we find the
points of coexistence of the phases
$0,1,2$ and the phases
$0,2,3$, where
the phase numbers correspond to the height of the layer of unstable spins
over the substrate.
Received: 29.09.2006
Revised: 20.03.2007
DOI:
10.4213/tmf6136
Fulltext:
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