Integral equation and simulation studies of the Heisenberg spin fluid
in an external magnetic field
F. Lado* and E. Lomba
Instituto de Quı ´mica Fı ´sica Rocasolano, CSIC, Serrano 119, 28006 Madrid, Spain
J. J. Weis
Laboratoire de Physique The ´orique et Hautes Energies, Ba ˆtiment 211, Universite ´ de Paris-Sud, 91405 Orsay Cedex, France
Received 26 February 1998
We develop a general method to study inhomogeneous liquids in an external field using orthogonal poly-
nomials tailored to the one-body density. The procedure makes integral equation calculations of these systems
no more difficult than those of ordinary homogeneous molecular fluids. We apply this method to the ferro-
magnetic Heisenberg spin fluid in an external magnetic field using both the reference-hypernetted chain closure
and a reference version of the Zerah-Hansen closure, with no further approximations. The calculation includes
a mapping of the two-phase region for several values of the external magnetic field. Comparison with Monte
Carlo simulation data shows the integral equation procedure yielding nearly exact results, in particular for
nonzero external fields. S1063-651X9815109-7
PACS numbers: 61.20.Gy, 64.60.-i, 75.10.-b, 75.30.-m
I. INTRODUCTION
The Gibbsian N -body density function of a Hamiltonian
H
N
that is rotationally and translationally invariant must it-
self be rotationally and translationally invariant, as must then
also be all reduced n -body density functions of this Hamil-
tonian. In particular, the one-body density is a constant and
the two-body density depends only on relative coordinates.
An external field destroys this homogeneity, producing an-
isotropy or nonuniformity 1,2, and so makes necessary the
joint calculation of the coupled one-body and two-body den-
sity functions. A striking if familiar example of the response
of a bulk system to an external field is ferromagnetism. In
this paper, we shall use this particular case to present a gen-
eral procedure 3 to compute the coupled one-body and two-
body density functions of an inhomogeneous classical fluid
in an external field. Remarkably, the procedure is no more
difficult to carry through than similar calculations for ordi-
nary homogeneous systems.
Perhaps the simplest model of a disordered continuum
system exhibiting ferromagnetic behavior is a fluid of hard
spheres with embedded Heisenberg spins described using
classical statistical mechanics 4–8. On the one hand, the
simplifications of this model compared to magnetic dipole-
dipole interactions are significant for simulation. The dipole-
dipole model presents substantial conceptual difficulties in
connection with the nonexistence of the thermodynamic limit
for orientationally ordered phases. Moreover, its simulation
results are strongly dependent on the boundary conditions.
Simulations for the Heisenberg spin model are free of these
problems. The integral equation formalism developed below
serves equally well for either potential. On the other hand,
disordered Heisenberg spin systems are of interest in their
own right. It has been known for some time that Co/P alloys
9 and Co/Au melts 10 have a tendency to form amor-
phous ferromagnets, although their ‘‘true liquid’’ nature has
been questioned because of the technical difficulties posed
by undercooling below the Curie temperature. While these
systems might be more representative of quenched spin flu-
ids, very recently Albrecht and co-workers 11 have finally
managed to undercool a Co
80
Pd
20
melt below its Curie tem-
perature at zero field, thus obtaining the first evidence of
ferromagnetic behavior in a liquid metal in conditions where
the Heisenberg exchange interaction absolutely dominates
the magnetic dipole-dipole contribution. The exchange inter-
action is the crucial term in the present paper and it is our
main aim to analyze how it models the phase behavior with
and without the presence of external fields.
The Heisenberg spin fluid in an external magnetic field B
0
is defined by the canonical partition function
Z =
1
N !
3 N
j =1
N
d r
j
d
j
exp
j
j
• B
0
-
i j
u
0
r
ij
-
i j
u
ss
r
ij
,
i
,
j
. 1
This can be factored into an ideal part and the excess, Z
=Z
id
Z
ex
, where
Z
id
=
1
N !
3 N
j =1
N
d r
j
d
j
exp
j
j
• B
0
=
1
N !
V
3
N
4
sinh B
0
B
0
N
, 2
*On leave from Department of Physics, North Carolina State Uni-
versity, Raleigh, NC 27695-8202.
PHYSICAL REVIEW E SEPTEMBER 1998 VOLUME 58, NUMBER 3
PRE 58 1063-651X/98/583/347812/$15.00 3478 © 1998 The American Physical Society