On the Phase Structure of the Asymmetric Three-State Potts Model
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ON THE PHASE STRUCTURE OF THE ASYMMETRIC THREE-STATE POTTS MODEL
G. v. GEHLEN Physikalisches Institut der Universit~t Bonn, Nussallee 12, D-5300 Bonn 1, West Germany
ABSTRACT Finite-size scaling is metric Z3-Potts model.
applied to the Hamiltonian version of the asym-
Results for the phase boundary of the commensurate
region and for the corresponding critical index v are presented. gued that there is
no Lifshitz point,
It
is
ar-
the incommensurate phase extending
down to small values of the asymmetry parameter.
INTRODUCTION The melting of an ordered phase into an incommensurate floating phase has recently attracted considerable interest.
Experimentally such a melting
has been observed in many adsorbed monolayers
[I],
graphite [2],
Xe on Cu [31,
and H on Fe [4].
e.g. in monolayer Kr on
On the theoretical side two-
dimensional spin models have been proposed [5],
[6],
[7],
[8], which are ex-
pected to describe the main features of the melting, since they exhibit, apart from an ordered and a disordered phase,
also a floating third phase,
or even infinitely many phases. A particulary simple model of this kind, which assumes only nearest neighbour interactions, [6] and Huse [7],
is
the asymmetric Potts model introduced by Ostlund
and in its
and Rittenberg [8].
It
Hamiltonian quantum version, by Centen, Marcu,
has spin variables at each site which take three
values a = 0,1,2 and assumes y-directions,
the interaction to be different in
violating parity invariance in
the y-direction.
the x- and
The classical
action is:
S =- E
{g cos(
l(ox+,a, -xy (1)
(x,y+l + +gyOS g Cos (2i(a nx+
The parameter
Cit is (
-
x,y)-)} 0' )
&
sufficient to study 00 : 5
S 600)
tunes the asymmetry
produced in the y-direction. Consider some features of the action, The first
term takes the values gx9 -gx/2,
respectively,
for ax+I,y-ax,y = 0,1,2,
so that aligned neighbouring spins are favoured.
peratures this gives rise Aay = axy+-a
eq.(i): -gx/2
= 0 is
m xyml x,y Mat. Ree. Soc. Symp. Proc. Vol.
to ferromagnetic ordering. favoured over La
In
At low tem-
the second term
= I by y
21 (1984) CElsevier Science Publishing Co.,
Inc.
28
)-
g y( cos
0
cos(120
-
4) }
=
Vg y Sin(600
4))5
g
(2)
-1 Since gy 'xT for
0
4>
(gy is
the coupling strength times 0 = 1/kT),
this means that
we have to go to lower temperatures in order to get the same ali-
gment force as at
4
The boundary of the ordered commensurate phase
= 00.
should therefore roughly follow a curve T n sin(60
4),An
distorted since for large At the special value
4 =600,
y
Aa
0
-
4).
= 2 looses importance
= 0 is
yno
This is
somewhat
compared to An = 1.
y
longer favoured over Aay = 1,
so that in the y-direction spin sequences like 0000000 are no longer preferred to e.g. 0012220 ...
etc. We see that for
4t
00 there is
a chance
to obtain a floating phase.
PROBLEMS OF THE ASYMMETRIC POTTS MODEL UNDER DISCUSSION The application of different approaches to the action eq.(1), Monte Carlo calculations
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