Numerical signs for a transition in the two-dimensional random field Ising model at T 0
Carlos Frontera and Eduard Vives
Departament d’Estructura i Constituents de la Mate `ria, Facultat de Fı ´sica, Universitat de Barcelona, Diagonal 647,
E-08028 Barcelona, Catalonia, Spain
and Escola Universita `ria Polite `cnica de Mataro ´, Avinguda Puig i Cadafalch 101-111, E-08303 Mataro ´, Catalonia, Spain
Received 8 January 1998; revised manuscript received 30 October 1998
Intensive numerical studies of exact ground states of the two-dimensional ferromagnetic random field Ising
model at T =0, with a Gaussian distribution of fields, are presented. Standard finite size scaling analysis of the
data suggests the existence of a transition at
c
=0.640.08. Results are compared with existing theories and
with the study of metastable avalanches in the same model. S1063-651X9950602-8
PACS numbers: 75.10.Nr, 05.40.-a
The study of systems with quenched disorder has been a
challenging problem for many years. The interplay between
thermal fluctuations and disorder has a great influence on the
existing phase transitions. Many systems are known to ex-
hibit such phase diagrams highly determined by the degree
of disorder vacancies, impurities, dislocations, etc. The
most typical examples can be found in magnetism, supercon-
ductivity, structural phase transitions, etc. For such systems
different models have been proposed. The Ising model with
quenched disorder is one of the simplest and it has the ad-
vantage that the pure model is well known. The disorder can
be of two types: i symmetry breaking terms, such as
random-fields or random magnetic impurities, and ii non-
symmetry breaking, such as random-bonds, vacancies, etc.
For all of the cases different probability distributions of dis-
order have been studied. Here we will focus on the study of
the random field Ising model RFIM in two dimensions
2D with a Gaussian distribution of fields. For many years
there has been discussion concerning the possibility of
whether a 2D model with symmetry breaking random fields
exhibits order at low temperatures. The initial studies lead to
a certain controversy: the Imry-Ma 1 argument suggests
that the lower critical dimension, below which ferromagnetic
order is destroyed, is d
l
2, with d =2 being the limiting
case. Renormalization group expansions 2 around d =6
lead to the ‘‘dimensional reduction’’ argument suggesting
that d
l
=3, discarding the possibility for ordering in the 2D
RFIM. It has also been suggested 3 that there are different
types of order for d 1. This controversy is probably due to
the difficulty in balancing the two ingredients of such mod-
els: disorder and thermal fluctuations.
More recently a different approach to disordered systems
has been proposed, namely the study of disordered systems
at T =0, i.e., without thermal fluctuations. From a theoretical
point of view this simplifies the problem without making it
trivial. Moreover, several experimental systems exhibit
phase transitions that can be catalogued into this ‘‘athermal’’
category: two examples are ferromagnetism at low tempera-
tures under an external magnetic field 4, and martensitic
transformations 5. Both systems present a first-order phase
transition that can be crossed by sweeping a control param-
eter and are greatly affected by the presence of quenched
disorder. We will concentrate on the study of the 2D RFIM
at T =0 for different values of the standard deviation of the
Gaussian distribution of fields. Our goal is to look for signs
of the existence of a phase transition at a certain
c
from a
ferromagnetic ordered state for
c
to a disordered state
for
c
. For the 3D RFIM at T =0, ground state studies
6,7 and renormalization group arguments 8 reveal the ex-
istence of such phase transition, but to our knowledge no
results for the 2D case have been published. Figure 1 sum-
marizes the finite size scaling study presented in this paper.
Data correspond to estimations of
cL
, obtained using dif-
ferent methods, as a function of the linear system size L
1/
where 1/ =0.5 is the exponent characterizing the correla-
tion length divergence. The standard extrapolation to L
→ , as will be discussed, renders
c
=0.640.08 different
from zero.
We consider the 2D RFIM on a L L square lattice with
periodic boundary conditions and with the Hamiltonian H
=-
i , j
nn
S
i
S
j
-
i =1
N
S
i
h
i
, where i and j are indices sweeping
the full lattice ( i , j =1, . . . , N =L L ), the sum refers to
nearest-neighbors nn pairs, S
i
=1 are spin variables, and
h
i
are independent random fields distributed according to the
Gaussian probability density with h =0 and h
2
=
2
. The
advantage of using a continuous distribution is that, for al-
most any configuration of fields h
i
the ground state is not
FIG. 1.
cL
versus L
-1/
. The results have been obtained using
MF at zero and higher orders, stars; exact solution of finite lattices,
, , , , and ; and studies of the metastability behavior,
black triangles from Ref. 15. Typical error bars are displayed. The
inset shows an example of the ground state of a L =64 system, with
=1.0
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