Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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8.4 Spin-glass models 421
random bonds are frustrated since it is impossible to satisfy all bonds at the same time. The
dynamics for σ
i
(t) is given by the Langevin equation,
ν
−1
k
∂
∂t
σ
i
(t) =−
δ H
δσ
i
(t)
+ ς
i
(t), (8.4.5)
where ν
k
is a bare kinetic coefficient representing the microscopic time scales for the
dynamics. It is related to the variance of the Gaussian random noise by
ς
i
(t)ς
j
(t
)=2ν
−1
k
δ
ij
δ(t − t
). (8.4.6)
For studying the dynamics, the physical quantity of interest is the two-spin correlation
function
C
ij
(t − t
) =σ
i
(t)σ
j
(t
). (8.4.7)
The linear response function R
ij
to an external field h
j
, on the other hand, is defined as
R
ij
(t − t
) =
∂
∂h
j
(t
)
σ
i
(t) for t > t
. (8.4.8)
From the equation of motion (8.4.5) it is clear that the physical field h
i
which couples to
the spin σ
i
in the Hamiltonian H also appears as a driving field in the equation of motion.
The latter couples to the conjugate MSR field ˆσ
i
in the action. Hence the physical response
function is identical to the MSR function R defined above.
6
Following the MSR approach
outlined above, the value of the functional A[σ ] of the field σ which satisfies the field
equation (8.4.5) is obtained as
A[σ ]=
Dσ
A[σ
]δ(σ − σ
)
≡
Dσ A[σ ]J [σ, ς]δ
ν
−1
k
∂σ
i
(t)
∂t
+
δβ H
δσ
i
(t)
− ς
i
(t)
=
Dσ
D ˆσ A[σ ]J [σ, ς]exp
−
i
dt i ˆσ
i
3
ν
−1
k
∂σ
i
∂t
+
δ H
δσ
i
− ς
i
4
.
(8.4.9)
The functional Jacobian J, which is defined as
J = det
*
*
*
*
δς
i
(t)
δσ
j
(t)
*
*
*
*
, (8.4.10)
can be evaluated in the thermodynamic limit as
J = det
*
*
*
*
δ
δσ
j
(t)
ν
−1
k
∂
∂t
+
δ H
δσ
i
*
*
*
*
= det
*
*
*
*
*
δ
ij
ν
−1
k
∂
∂t
δ(t − t
) +
δ
2
H
δσ
2
i
δ(t − t
)
*
*
*
*
*
. (8.4.11)
6
See the discussion in Section 7.3.1 with respect to the fluctuation–dissipation theorem in MSR theory.
422 The ergodic–nonergodic transition
The Jacobian is added as an extra term in the action A
J
,
A
J
=−lnJ = G
0
(0)
dt
i
ν
k
δ
2
H
δσ
2
i
(t)
, (8.4.12)
where G
0
is the Green function satisfying
∂
∂t
δ(t − t
) = G
−1
0
(t − t
). (8.4.13)
We first note from (8.4.13)
δ(t − t
) =−
t
−∞
d
¯
tG
−1
0
(t −
¯
t)
=−
d
¯
tG
−1
0
(t −
¯
t)θ (t
−
¯
t)
≡
d
¯
tG
−1
0
(t −
¯
t)G
0
(
¯
t − t
), (8.4.14)
where θ represents the step function. Thus G
0
(t) =−θ(−t) satisfies the equation
∂
∂t
G
0
(t − t
) = δ(t − t
). (8.4.15)
The RHS of eqn. (8.4.11) is then obtained as,
J = det
*
*
*
*
*
δ
ij
ν
−1
k
G
−1
0
1 − G
0
ν
k
δ
2
H
δσ
2
i
*
*
*
*
*
. (8.4.16)
Using the result for (8.4.16), A
J
follows easily (up to a constant term in the action):
A
J
=−lnJ =
1
2
dt
i
ν
k
δ
2
H
δσ
2
i
(t)
, (8.4.17)
where we have taken the value θ(0) =
1
2
for the step function (Zinn-Justin, 2002). Note
that the contribution from the counter-term in A
J
is independent of the random interaction
J
i
1
···i
p
. By averaging expression (8.4.9) over the noise ς we obtain the result
A[σ ] =
Dσ
D ˆσ A[σ ]exp
−A
J
i
1
···i
p
[u, ˆu]−A
J
, (8.4.18)
with the action A
J
i
1
···i
p
[u, ˆu] being given by
A
J
i
1
···i
p
[u, ˆu]=
dt
i
ˆσ
i
(t)ν
−1
k
ˆσ
i
(t) + i ˆσ
i
(t)
3
ν
−1
k
∂σ
i
∂t
+
δ H
δσ
i
4
+
dt
i
[σ
i
u
i
+i ˆσ
i
ˆu
i
]. (8.4.19)
8.4 Spin-glass models 423
The average in eqn. (8.4.19) is therefore computed in terms of a generating functional
Z
J
i
1
···i
p
[u, ˆu]=
Dσ
D ˆσ exp
#
−A
J
i
1
···i
p
[u, ˆu]−A
J
$
. (8.4.20)
For all values of the random variable J
i
1
···i
p
we have the corresponding generating function
satisfying
Z
J
i
1
···i
p
[u
i
= 0, ˆu
i
= 0]=I
0
, (8.4.21)
where I
0
is a constant obtained after doing the trivial Gaussian integrals. The time corre-
lation of the spins averaged over the quenched random interactions is calculated using the
MSR field theory outlined earlier in Section 7.1. From the condition (8.4.21) it follows that
the disorder-averaged correlation functions can be obtained by averaging Z
J
i
1
···i
p
itself over
the random distribution (8.4.4). Since the random variable J
i
1
...i
p
is linear in the equation
of motion (see eqn. (8.4.19)), the corresponding MSR action functional is averaged out (De
Dominicis, 1978; Ma and Rudnick, 1978; Kirkpatrick and Thirumalai, 1987a, 1987b) with
respect to the Gaussian probability distribution for the J
i
1
...i
p
. The quenched averaging is
done directly on Z ,giving
¯
Z[u, ˆu]=
%
J
i
1
dJ
i
1
...i
p
P[J
i
1
,i
2
,...,i
p
]Z
J
i
1
...i
p
[u, ˆu]
≡
Dσ
D ˆσ exp
−A[σ, ˆσ ]−
dt
i
[σ
i
u
i
+ˆσ
i
ˆu
i
]
.
Using the standard field-theoretic techniques of Section 7.1 and assuming that J
i
1
...i
p
is
symmetric under exchange of the site indices, the action of the MSR theory is obtained in
terms of the spin σ and the corresponding conjugate field ˆσ as
A[σ, ˆσ ]=
dt
i
ˆσ
i
ν
−1
k
ˆσ
i
+i ˆσ
i
3
ν
−1
k
∂σ
i
∂t
+¯r
0
σ
i
+ 4uσ
3
i
− h
i
4
+ A
J
+ V [σ, ˆσ ], (8.4.22)
where the part V involving nonlinearities of cubic and higher order is obtained as
V (σ, ˆσ) =
dt
dt
μ
2N
p−1
i, j,i
3
,...,i
p
i ˆσ
i
(t)σ
j
(t)
×
i ˆσ
i
(t
)σ
j
(t
) + ( p − 1)i ˆσ
j
(t
)σ
i
(t
)
σ
i
3
(t)σ
i
3
(t
)...σ
i
p
(t)σ
i
3
(t
),
(8.4.23)
with μ = pβ
2
/2. The integration over the random interactions generates the 2p-spin cou-
plings (nonlocal in time) in the action functional.
424 The ergodic–nonergodic transition
Using a generalization (Kirkpatrick and Thirumalai, 1987a, 1987b) of the saddle-point
method (Gross and Mézard, 1984; Gardner, 1985) yields the following equation of motion
for the spin σ
i
(t) averaged over the quenched random interactions:
ν
−1
k
∂
∂t
σ
i
(t) +¯r
0
σ
i
+ 4uσ
3
i
(t) +
t
o
(t,
¯
t)σ
i
(
¯
t)d
¯
t = f
i
(t) + h
i
(t). (8.4.24)
The correlation of the renormalized noise f
i
in the new Langevin equation is obtained in
terms of a self-consistent expression involving the correlation function C,
f
i
(t) f
j
(t
)=δ
ij
(t − t
)
= 2ν
−1
k
δ(t − t
) + μC
p−1
(t − t
). (8.4.25)
The last term on the LHS of the renormalized equation (8.4.24) for the dynamics represents
a memory term involving the kernel . The latter is expressed self-consistently in terms of
the correlation and response functions as
(t − t
) = μ( p − 1)R(t − t
)C
p−2
(t − t
). (8.4.26)
In eqns. (8.4.25) and (8.4.26) the linear response function R and the correlation function
C are defined as
C(t, t
) =
1
N
N
i=1
σ
i
(t)σ
i
(t
),
R(t, t
) =
1
N
N
i=1
-
δσ
i
(t)
δς
i
(t
)
.
=
1
N
N
i=1
σ
i
(t)i ˆσ
i
(t
), (8.4.27)
with the overbar indicating the average over the quenched randomness and the angular
brackets in the expression above stand for the average over the noise ς . Note that the
renormalizing contributions both in eqn. (8.4.25) and in eqn. (8.4.26) result from the
non-Gaussian part of the action V (σ, ˆσ). The first term on the RHS of eqn. (8.4.23) renor-
malizes the bare kinetic coefficient ν
k
which determines the correlation of bare noise ς ,
while the second term contributes to the memory term . With p =2 this model would
also correspond to the equations obtained by Sompolinsky and Zippelius (1982).
The above expressions for the kernel functions and are related in equilibrium. For
fluctuations around equilibrium TTI holds and the correlation and response functions are
related through the fluctuation–dissipation theorem (FDT)
R(t − t
) =−θ(t − t
)
∂
∂t
C(t − t
). (8.4.28)
From the above equations, it follows quite straightforwardly that the correlation function
C(t) normalized with respect to its equal-time value, i.e., φ(t) =C(t)/C(t =0), satisfies
the dynamical equation
d
0
˙
φ(t) + φ(t) + λ
p
t
0
ds φ
p−1
˙
φ(s) = 0, (8.4.29)
8.4 Spin-glass models 425
where d
0
=(¯r
0
ν
k
)
−1
and λ
p
=μ¯r
2−p
0
with the equal-time correlation being given by
C(t =0) =¯r
−1
0
.Forp =3, eqn. (8.4.29) is identical to the mode-coupling eqn. (8.1.8)
(apart from an inertial term involving second derivatives with respect to time) of the
structural-glass problem with the memory function H =c
2
φ
2
. Keeping both p =2 and
p =3 gives rise to a model similar to the φ
12
model discussed in Section 8.1.1.Wehave
demonstrated earlier in this chapter that these models undergo a dynamic transition at a
low enough temperature T =T
c
. The corresponding behavior of the normalized density
correlation φ(t ) is given by
φ(t) =
p − 2
p − 1
+ At
−a
. (8.4.30)
In the long-time limit φ(t) goes to a nonzero value. The power-law exponent a satisfies
(for p =2) the equation
2
(1 − a)
(1 − 2a)
=
1
2
(8.4.31)
in terms of the function. This is the same as the equation satisfied by the power-law
exponent in the Leutheusser model with solution a =0.395. The renormalized kinetic
coefficient is
d
R
(ω) = d
0
+ λ
p
∞
o
e
iωt
φ
p−1
(t). (8.4.32)
In the zero-frequency limit this satisfies
d
R
(T → T
c
) ∼ (T − T
c
)
−γ
(8.4.33)
with γ =1.765. The similarity of these results to those of MCT has led to the speculation
that this model is in the same universality class as the structural glass transition along the
lines of ideas introduced in connection with the second-order phase-transition point (Ma,
1976). It is worth noting here that the dynamic transition predicted for the p-spin models
is removed in finite dimensions due to nucleation processes or activation over finite free-
energy barriers (Kirkpatrick and Thirumalai, 1989; Parisi et al., 1999; Drossel et al., 2000).
This is similar to what happens in MCT of structural glass, where finally the ergodicity is
maintained and the dynamic transition is absent. Kirkpatrick and Thirumalai (1987a)have
argued from consideration of the statics of the p-spin models that at a temperature T
s
< T
d
there is a thermodynamic transition. The static transition was identified with the one-step
replica-symmetry-breaking transition.
Another standard choice for regularization of the model preventing σ
i
from exploding in
an unstable direction of the random coupling J
i
1
...i
p
of the Hamiltonian would be to impose
a constraint
1
N
N
i=1
σ
2
i
(t) = 1, (8.4.34)
426 The ergodic–nonergodic transition
which also makes C(t, t) =1. For the p-spin Hamiltonian this constraint gives what is
called the spherical p-spin model (Crisanti and Sommers, 1992; Crisanti et al., 1993).
8.4.2 MCT and mean-field theories
In spite of the similar mathematical structure of the MCT for structural glass and the mean-
field p-spin glass models, there are some obvious differences. A subtle point to note here
is that the natures of the nonlinearities in the corresponding dynamical equations giving
rise to the mode-coupling models in the respective cases of structural and spin glasses
are quite different. For the structural glasses, we have seen in the earlier sections of the
present chapter that the driving nonlinearities are in the reversible part of the dynamics.
They are generated from a Gaussian Hamiltonian and are essentially of dynamic origin.
This is due to the special Poisson-bracket structure of the relevant hydrodynamic variables
contributing to the reversible part of the equations of motion (Das et al., 1985b). In the
infinite-range p-spin models, on the other hand, the relevant nonlinearities are dissipative
and come from the non-Gaussian part of the effective Hamiltonian.
The analogy of the structural glasses with the infinite p-spin models was extended by
Kirkpatrick and Thirumalai (1989) to propose for the structural case a density-functional
form of the free-energy functional in units of k
B
T ,
F
ST
=
1
2
dx
1
dx
2
δρ (x
1
)χ
−1
0
(x
1
− x
2
)δρ(x
1
)
+
1
3
g
3
dx
1
[δρ (x
1
)]
3
+
1
4
g
4
dx
1
[δρ (x
1
)]
4
− μ
dx
1
ρ(x
1
), (8.4.35)
where μ is the chemical potential. The constants g
3
and g
4
are coupling constants in
the non-Gaussian free energy. These terms are related to three- and four-point correla-
tions in the fluid. The equation of motion for the time dependence of the density fluctua-
tions δρ (x, t) is chosen in the form of a relaxational dynamics for the conserved quantity
(density),
−1
0
∂
∂t
δρ(x, t ) =−∇
2
δF
ST
[δρ]
δρ (x, t)
+ ξ(x, t), (8.4.36)
which is similar to the form of eqn. (8.4.5) in the p-spin model. ξ is the Gaussian white
noise. The mode-coupling model constructed for the above dynamics shows a transition
similar to the ENE transition described in the case of the p-spin model. On the other hand,
multiple solutions of the density-functional theory appear and each metastable solution is
given a canonical weight P[δρ ] for the corresponding density configuration. A static order
parameter is defined in terms of the average of the equal-time density correlation
Q(x
1
, x
2
) =δρ (x
1
)δρ(x
2
), (8.4.37)
8.4 Spin-glass models 427
where the angular brackets denote the average with weight P[δρ ]. It was shown that Q(k)
satisfies the nonlinear integral equation
Q(k) = χ
0
(k)
I
†
(k)
1 + I
†
(k)
, (8.4.38)
where
I
†
(k) = 2g
3
dk
Q(k − k
)Q(k
).
From the mode-coupling equations corresponding to the nonlinear Langevin equations
given by (8.4.35) and (8.4.36) it was shown (Kirkpatrick and Thirumalai, 1989) that the
nonergodicity parameter q
EA
(k), i.e., the long-time limit of the density–density correlation
function δρ(k, t)δρ(−k, 0), satisfies the same integral equation, (8.4.38),asQ(k) does.
Both these order parameters are proportional to the lowest-order non-Gaussian coupling,
i.e., g
3
. The equation of motion for the density fluctuations here (eqn. (8.4.36))isvery
different from those considered in the previous sections and is rather similar to what has
been considered for the mean-field spin models.
The possibility of a sharp ENE transition in the various mode-coupling models must
be viewed carefully. The multi-spin model involves an infinite-dimensional mean-field
approach since every spin is interacting equally with every other spin. The existence of
the dynamic transition in this case is generally associated with the appearance of many
metastable minima (Thirumalai et al., 1989). Ergodicity breaks when the system gets
trapped in one minimum for infinitely long times. Since the barrier between the metastable
states is infinite, reaching the true equilibrium state is forbidden. In analogy with the p-spin
model, the mode-coupling model for a real liquid (with short-range forces) is often termed
a mean-field theory. If we consider only the schematic form of the MCT for the structural-
glass case, with the one-dimensional nonlinear integral equation then such an analogy is
possibly justified. However, in the structural case the uncorrelated binary-collision contri-
butions to the transport coefficients are already included in the bare or short-time part and
the mode-coupling contribution in fact represents a correction taking into account corre-
lated motions in the many-particle system. Although these are accounted for only at the
level of the Kawasaki approximation or one-loop approximation in the simple MCT, it is
still a correction beyond the uncorrelated dynamics.
Another important aspect of this analogy between mean-field spin models and MCT
of structural glasses is the often-cited reference to activated hopping processes. Indeed,
the dynamic transition predicted for the p-spin models is removed in finite dimensions
due to nucleation processes or activation over finite free-energy barriers (Kirkpatrick and
Thirumalai, 1989; Parisi et al., 1999; Drossel et al., 2000). In analogy with p-spin models,
if we term MCT a mean-field theory where the system is caught in a single free-energy
minimum, then restoration of the ergodic behavior will come from activated jumps over
the barriers. Hence the absence of the sharp ENE transition in a supercooled liquid (which
is also a fact supported by results from experiments and simulations) is often ascribed
to “hopping processes” or activated hopping over energy barriers. However, as we have
428 The ergodic–nonergodic transition
already seen in Section 8.3, the origin of the restoration of ergodicity in the various models
of liquid-state dynamics (Das and Mazenko, 1986; Schmitz et al., 1993) arises from a
simple Hamiltonian, without inclusion of hopping processes.
Finally, let us consider a toy model (Kawasaki and Kim, 2001) involving an N -component
fluid consisting of N density variables ρ
i
, i =1, 2,...,N and M momentum variables g
α
,
α =1, 2,...,M with (M < N ). This model is characterized by a built-in quenched disor-
der. This has been done in the spirit of an earlier description of fluids due to Kraichnan
(1959). The equations of motion for the density and momentum variables are written in the
form
˙ρ
i
= K
iα
g
α
+
ω
√
N
J
ijα
ρ
j
g
α
, (8.4.39)
˙g
α
=−ω
2
ρ
j
K
jα
−
ω
√
N
J
ijα
{ω
2
ρ
i
ρ
j
− k
B
T δ
ij
}−γ
0
b
α
+ f
α
, (8.4.40)
with the noise correlation given by
f
α
(t) f
β
(t
)=2k
B
T γ
0
δ(t − t
). (8.4.41)
ω, γ
0
, and k
B
T are constants for the model. The above equations follow from the formalism
developed in Chapter 7 for obtaining the nonlinear Langevin equations. To construct the
equations we define the nonzero Poisson brackets between the density and momentum
variables as
{ρ
i
, g
α
}=−
K
iα
+
ω
√
N
J
ijα
ρ
j
, (8.4.42)
where J
ijα
denote the quenched variables assumed to follow a Gaussian distribution. The
driving free-energy functional for the stationary solution of the Fokker–Planck equation
corresponding to the above Langevin equation is of the form exp[−F/(k
B
T )] with the
free-energy functional given by
F[{ρ
i
}, {g
α
}] =
1
2
M
α=1
g
2
α
+
N
i=1
ω
2
ρ
2
i
. (8.4.43)
The above nonlinear Langevin equations (8.4.39) and (8.4.40) follow directly from eqns.
(6.1.48) and (6.1.71). The nonlinear terms (quadratic in the fields) in the dynamical equa-
tions are proportional (by construction) to the quenched variables J. In the present model
an important difference from the dissipative dynamics of the earlier examples of p-spin
models (Section 8.4.1) and the DFT model (Section 8.4.2) is that the nonlinearities in
eqns. (8.4.39) and (8.4.40) are all in the reversible part of the dynamics, as can easily be
seen by doing a time reversal in the above equations. From the analysis of this model it
was shown (Kawasaki and Kim, 2001) that for the case of N = M no sharp transition is
seen in this toy model, even in the limit N →∞. The ideal glass transition is recovered in
this model only when the number N of density variables is chosen to be different from the
number of momentum components (M) (with N > M). In this case δ = M/N < 1, which
restricts the available phase space and presumably facilitates the sharp dynamic transition.
8.4 Spin-glass models 429
The ergodicity-restoring process in the present case also does not involve any hopping
processes. One of the basic ingredients which are common to these models (in which the
sharp dynamic transition is cut off) is that they all have momentum fluctuations taken into
account in the fluctuating-hydrodynamics description.
It is instructive in this respect to consider the explicit form of the Hamiltonian in the two
cases. In the p-spin case every spin interacts with every other spin, as a result of which the
resulting model is of infinite-dimensional mean-field type. In comparison, the liquid or the
structural glass is a finite-dimensional system, since the interaction part of its Hamiltonian
is usually a sum of short-ranged two-body interactions, H
liq
=
5
N
i=j =1
U (|R
i
− R
j
|),
where R
i
represents the position of the ith particle. At low temperatures the individual
particles merely vibrate around their mean positions on a random lattice structure, which
corresponds to a local minimum of the Hamiltonian. The disorder is self-generated in
the amorphous solid-like state since each particle experiences a different environment.
By defining vibrations around an amorphous lattice H
liq
can be mapped to a multi-spin
interaction-type Hamiltonian (Kühn and Horstmann, 1997; Singh and Das, 2009) involv-
ing soft spins. Imposing the frozen structure automatically implies the system being caught
in a single minimum and that it is of mean-field nature.
Those subtle differences notwithstanding, the analogy of the p-spin interaction spin
models with the MCT for structural glasses has led to the assumption that there is a
dynamic similarity between the two types of systems with and without intrinsic disorder.
The landscape studies of prototype fragile liquids have further strengthened the analogy
between structural and p-spin models. Similar observations have been made in different
studies. For example, a finite energy-density interval where local minima dominate over-
whelmingly in number over saddles occurs in mean-field models (Cavagna et al., 1998),
which is similar to the case of simple atomic systems. The analogy is also extended to
the nonequilibrium dynamics. The off-equilibrium solutions of the spherical p-spin model
(discussed in the next chapter) agree well with the MD-simulation data on the nonequilib-
rium state of the structural glass. As we will see from this point onwards in the remaining
chapters of this book, the analogy of the mean-field spin-glass models with the molecular
systems has been often appealed to in order to aid understanding of the behavior of the
supercooled liquids. This ranges from nonequilibrium dynamics (discussed in Chapter 9)
to the scenario of an underlying phase transition at low temperatures (Chapter 10).
Appendix to Chapter 8
A8.1 Calculation of the spring constant
Let us consider the density function in the form of a sum of Gaussian profiles around a set
of lattice points {R
α
},
n(x) =
α
π
3/2
R
i
e
−α(x−R
i
)
2
≡
N
i=1
φ
α
(|x − R
i
|), (A8.1.1)
where φ
i
(r) =(α/π)
3/2
exp(−αr
2
). We want to compute here the response due to a dis-
placement δR
1
of a single lattice site R
1
(say). We write the density as
n(x) = φ
α
(|x − R
1
− δR
1
|) +
N
i=2
φ
α
(|x − R
i
|)
=
˜
φ
α
(|x − R
1
) +
N
i=2
φ
α
(|x − R
i
|), (A8.1.2)
where we have introduced the notation
˜
φ
α
(|x −R
1
|) ≡ φ
α
(|x −R
1
−δR
1
|). The ideal-gas
part of the free energy remains the same under this displacement. However, the interacting
part is changed. We compute this as
β F
in
=−
1
2
dx
dx
n(x)c(|x −x
|)n(x
)
=−
1
2
dx
dx
˜
φ
α
(|x − R
1
|) +
N
i=2
φ
α
(|x − R
i
|)
× c(|x − x
|)
˜
φ
α
(|x
− R
1
|) +
N
i=2
φ
α
(|x
− R
i
|)
=−
1
2
dx
dx
c(|x −x
|)
×
˜
φ
α
(|x − R
1
|)
˜
φ
α
(|x
− R
1
|) +
˜
φ
α
(|x − R
1
|)
N
i=2
φ
α
(|x
− R
i
|)
430