**10**, 71-90 (1999)

https://doi.org/10.1007/s100510050831

## Surface effects on phase transitions of modulated phases and at Lifshitz points: A mean field theory of the ANNNI model

Institut für Physik,
Johannes Gutenberg-Universität Mainz,
55099 Mainz, Staudinger Weg 7, Germany

Corresponding author: ^{a}
binder@chaplin.physik.uni-mainz.de

Received:
28
July
1998

Published online: 15 July 1999

The semi-infinite axial next nearest neighbor Ising (ANNNI) model in the disordered phase is
treated within the molecular field approximation, as a prototype case for surface effects in
systems undergoing transitions to both ferromagnetic and modulated phases. As a first step, a
discrete set of layerwise mean field equations for the local order parameter
*m*_{n} in the
*n*th layer
parallel to the free surface is derived and solved, allowing for a surface field *H*_{1}
and for
interactions *J*_{S} in the surface plane which differ from the interactions
*J*_{0} in the bulk, while only in
the *z*-direction perpendicular to the surface competing
nearest neighbor ferromagnetic exchange
(*J*_{1}) and next nearest neighbor antiferromagnetic exchange
occurs. We show that for
and temperatures in between
the critical point of the bulk and
the disorder line the decay of
the profile is exponential with two competing lengths
with
while
stays finite at *T*_{cb}. The amplitudes of these
exponentials
(*a* is the lattice spacing) are obtained from boundary conditions that
follow from the molecular field equations.
For but , as well as at the Lifshitz
point and
in the modulated region , we obtain a modulated profile
,
where again the amplitude *A* and the phase *Ψ* can be found
from the boundary conditions.
As a further step,
replacing differences by differentials we derive a
continuum description, where
the familiar differential equation in the bulk (which contains both
terms of order
and here)
is supplemented by two boundary conditions, which both contain
terms up to order . It is shown that the solution of the continuum theory reproduces
the lattice model only when both
the leading correlation length
( or *ξ*, respectively) *and* the
second characteristic length (
or the wavelength of the modulation
, respectively) are
very large.
We obtain for a surface transition, with a two-dimensional ferromagnetic order
occurring at a transition
exceeding the transition of the bulk, and calculate the associated
critical exponents within mean field theory.
In particular, we show that at the Lifshitz point
with
while for the crossover exponent is .
We also consider the "ordinary transition"
and obtain the critical
exponents and associated critical amplitudes
(the latter are often singular when ). At the
Lifshitz point, the exponents of the surface layer and
surface susceptibilities take the values
,
while from scaling relations the surface
"gap exponent" is
found to be and the surface order
parameter exponents are . Open
questions and possible applications are discussed briefly.

PACS: 05.50.+q – Lattice theory and statistics (Ising, Potts, etc.) / 05.70.Jk – Critical point phenomena / 64.60.Cn – Order-disorder transformations; statistical mechanics of model systems

*© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 1999*