VOLUME 79, NUMBER 13 PHYSICAL REVIEW LETTERS 29 SEPTEMBER 1997
Phase Diagram of Coupled Glassy Systems: A Mean-Field Study
Silvio Franz
1
and Giorgio Parisi
2
1
ICTP Strada Costiera 11, P.O. Box 563, 34100 Trieste, Italy
2
Università di Roma “La Sapienza,” Piazzale A. Moro 2, 00185 Rome Italy
(Received 8 January 1997)
In the example of the spherical p -spin model, we study the phase diagram of glassy systems in the
presence of an attractive coupling with a quenched configuration. We find competition among two
phases, separated by a coexistence line terminating in a critical point, as in ordinary first-order phase
transitions. We argue that these results are not an artifact of the mean-field approximation, and may be
observed in numerical simulations of realistic glassy models. [S0031-9007(97)04049-0]
PACS numbers: 64.70.Pf
The transition from liquid to glasses presents in many
materials highly universal features. These may be quali-
tatively understood in the framework of the Gibbs-
DiMarzio scenario and its generalizations [1]. Roughly
speaking, the picture is the following. Still in the liquid
phase, when the temperature is smaller than a crossover
value (T
d
), the system may be trapped for a long time in
one of the exponentially large number of local minima of
the free energy. In this region the large time dynamics
becomes extremely slow because it is dominated by
transitions among different local minima. The number
(N ) of these local minima is related to the complexity
(or configurational entropy) ST by the formula N
expN ST , N being the number of particles. The total
entropy S is the sum of two contributions: the entropy
of each minimum and the complexity. The complexity is
supposed to vanish linearly at a lower temperature (i.e., at
a temperature T
c
, T
d
), where the height of the typical
barriers becomes infinite. The correlation time diverges
at T
c
, and one can argue in favor of a Vogel-Fulcher law.
This scenario is exactly implemented in a large class
infinite range models, with the only difference that the
lifetimes of local equilibrium states (and the correspond-
ing free-energy barriers) diverge when the volume of the
system goes to infinity [2,3]. In fact, the correlation time
diverges at T
d
, as can be seen in the mode coupling ap-
proximation which is exact for these models. On the
contrary, in short range systems T
d
signals a change in
behavior, but we cannot assign to it any sharply defined
value.
In this Letter we will show that if we generalize the
models by introducing two coupled replicas of the same
system [4–8], we find that T
d
corresponds to the edge
of a metastable region. In the same way the complexity
is related to the difference of a free-energy in the
stable and in the metastable phase. Now, in short range
models the properties of a metastable phase can only be
approximately computed because of the finite mean life
of metastable states, and in the mean-field approximation
metastable states have an infinite mean life. It is now clear
that the complexity and T
d
can be sharply defined only in
the framework of the mean-field approximation. We will
see that, as in ordinary first-order phase transition, a kind
of Maxwell construction allows us to extract qualitative
features of the phase diagram of real systems.
Let us describe the construction in the case of a
system composed by only one type of particles with
coordinates x
i
, for i 1, N ; the generalization to many
kind of particles is trivial. We consider two replicas of
the same system, with coordinates x and y , respectively,
in an asymmetric relation. The replica y is a typical
configuration distributed according to the Boltzmann-
Gibbs law with the original Hamiltonian of the system
[i.e., H y ] at a temperature T
0
, and does not feel any
influence from the replica x . The replica x , instead, feels
the influence of the replica y , and for a fixed value of y ,
thermalizes at a temperature T with a Hamiltonian
H
e
x j y H x 2e
X
i ,k 1,N
w x
i
2 y
k
. (1)
The function w is different from zero only at short
distance; an example is w x 1 if jx j , a and w x
0 if jx j , 1. An interesting behavior is present when the
value of a is smaller than the typical interatomic distance
(e.g., a 0.3 atomic distances). The quantity q
N
21
P
i ,k 1,N
w x
i
2 y
k
measures then the similarity of
the two configurations, and would be close to one when
the two replicas stay in similar configurations. For
positive coupling e the x variables feel a potential which
pushes them near to the y variables. We can define a free
energy for the x variables in the presence of the quenched
y variables as
F T , e, y N b
21
ln
√
Z
dx exp2bH
e
x j y
!
, (2)
a quantity that should be self-averaging with respect to the
distribution of the y and can therefore be computed as
F
Q
T , T
0
, e
R
dy exp2b
0
H y F T , e, y
R
dy exp2b
0
H y
, (3)
The temperature T
0
of the reference configuration y
can be equal or different from that of the x configura-
tion (T ).
2486 0031-9007 97 79(13) 2486(4)$10.00 © 1997 The American Physical Society