PHYSICAL REVIEW E
VOLUME 62, NUMBER 4
OCTOBER 2000
Polarizability fluctuations in dielectric materials with quenched disorder
Z. G. Yu,1 Xueyu Song,1 and David Chandler2
1
Department of Chemistry, Iowa State University, Ames, Iowa 50011
Department of Chemistry, University of California, Berkeley, California 94720
共Received 2 May 2000兲
2
We study a model of dielectric response for spatially disordered materials. In this model the local polarizability ␣ r is a quenched random variable. From a one-loop level renormalization-group analysis, we predict
that with increasing length scale L, the dimensionless fluctuation strength ¯␣ ␴ , where 1/¯␣ and ␴ 2 are the
average and the variance of the distribution for 1/␣ r , decays as 1/L 2 universally at large length scales. The
interplay of the random polarizability and the long-range dipole-dipole interaction is discussed.
PACS number共s兲: 05.50.⫹q, 77.22.⫺d, 05.40.⫺a
Polarization fluctuations significantly influence electron
transfer, energy transfer, and solvation dynamics in dielectric
materials. For simple polar solvents, both static and dynamic
manifestations of these fluctuations can be reasonably treated
in terms of simple generalizations of dielectric continuum
theory. In these approaches, the macroscopic frequencydependent dielectric constant is related to a homogeneous
local polarizability by the Clausius-Mossotti equation 关1–3兴.
For the spatially disordered or inhomogeneous systems such
as proteins and zeolites, however, this approach is no longer
valid at length scales comparable to the characteristic length
of the inhomogeneity. For example, a particular protein environment contains specific regions of high polarity and low
polarity. Thus, the dielectric response varies significantly
from one region to another in such systems. On the other
hand, when considering length scales much larger than the
inhomogeneity length, the effects of disorder are negligible,
and the homogeneous or continuum model is valid. This paper is concerned with the approach to this continuum limit,
analyzing how the effective inhomogeneity changes as a
function of length scale.
The particular model we consider is the disordered dielectric described in Ref. 关4兴. The local polarizability is random
and the randomness is ‘‘quenched’’ in the sense that its relaxation is ignored. Inhomogeneity is thus characterized by
the disorder length scale and strength. The former characterizes the density of disorder, and the latter is essentially the
variance of the local polarizability distribution. In Ref. 关4兴
we treated the dielectric response of this model with a plausible but approximate real-space renormalization procedure.
While the procedure seems to have general applicability, its
accuracy remains to be established. This paper takes a step in
that direction. An important result of Ref. 关4兴 is the prediction of nontrivial scaling in the approach to the continuum
limit. Here, we consider this scaling again, but with the aid
of field theoretic renormalization-group techniques, specifically with replicas 关5兴 and the ⑀ ⫽4⫺d expansion in d dimensions (d⫽3 in our case兲 关6兴. We obtain the recursion
equations of the polarizability fluctuation for general dipoledipole interaction strength and disorder strength. The dimensionless fluctuation strength ¯␣ ␴ decays as L ⫺ ␯ when increasing the length scale L, where 1/¯␣ and ␴ 2 are the
average and the variance of the 1/␣ r distribution, and ␣ r is
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PRE 62
the polarizability at position r. At the one-loop level in perturbation theory for the self-energy 关7兴, we show that ␯ varies from ␯ ⫽3/2, valid at small L, to ␯ ⫽2, valid at large L.
The former, ␯ ⫽3/2, is the ‘‘randomness dominated’’ result.
It follows from uncorrelated polarizability statistics. The latter, ␯ ⫽2, is the ‘‘interaction-dominated’’ result, which is
determined by the new fixed point due to the dipole-dipole
interaction.
We begin with the discrete model, which is defined on a
cubic lattice 关4兴. The unit cell with L⫽1 has size a. The
Hamiltonian reads
H⫽
1
2
兺r
m2r
␣r
⫺
1
2
兺
r⫽r⬘
mr•Trr⬘ •mr⬘ ,
共1兲
where mr is the polarizable dipole moment in the cell at
point r, ␣ r is the polarizability at that cell, and Trr⬘ is the
3⫻3 dipole-dipole tensor. In the continuum limit, a→0 ⫹ ,
Trr⬘ →3
共 r⫺r⬘ 兲共 r⫺r⬘ 兲
兩 r⫺r⬘ 兩
5
⫺
I
兩 r⫺r⬘ 兩 3
.
The set of local polarizabilities, 兵 ␣ r其 , is chosen at random for
each realization of the system. The distribution of ␣ r is
quenched or ‘‘frozen.’’ In general, this distribution should be
chosen to imitate the actual physical system of interest. In
the simplest case considered here, we assume that the distribution is isotropic, we ignore spatial correlations, and for the
purpose of being concrete we assume each ␣ r can have only
one of two values. As such, the distribution is characterized
by the average 具 1/␣ r典 ⬅1/¯␣ and the variance 具 (1/␣ r) 2 典
⫺ 具 1/␣ r典 2 ⬅ ␴ 2 . The characteristic dimensionless fluctuation
strength is ␹ ⬅ ¯␣ ␴ . For this model, we address the following
question: when we increase the length scale to make every
unit cell larger and larger, if we still use Hamiltonian 共1兲 to
describe the dielectric response of the system with a new
length scale, how does the distribution of ␣ r evolve? Specifically, how does the dimensionless variance of this distribution ␹ change as increasing the length scale?
It is instructive to first answer this question in the trivial
case where the dipole-dipole tensor in Eq. 共1兲 is set to zero.
Let H t denote the resulting Hamiltonian. It describes noninteracting dipoles with random polarizability. Since each ran4698
©2000 The American Physical Society
PRE 62
POLARIZABILITY FLUCTUATIONS IN DIELECTRIC . . .
dom polarizability is independent of each other, ␹ (L) will
scale as L ⫺3/2. Thus, in this case, ␹ (2)⫽ ␹ (1)2 ⫺3/2. We can
derive this expected result by studying how the effective
Hamiltonian evolves with increasing the length scale. We
N
define m̃r⫽ 兺 i⫽1
mir as the coarse-grained dipole moment for
a cell with lattice size L⫽2. Here, there are N⫽2 3 of the
original unit cells in this larger cell centered at r, and mir is
the dipole of the ith unit cell in this larger cell. The partition
function at L⫽2, Z 2 ⫽ 兰 Dm̃r exp(⫺H̃t /k B T), should be the
same as that at L⫽1, Z 1 ⫽ 兰 Dmr exp(⫺Ht /k B T). Both can
be calculated assuming ␴ 2 is a small and keeping only the
leading term in ␴ 2 . Comparing both estimates thereby relates
¯␣ (2) and ␴ 2 (2) to ¯␣ (1) and ␴ 2 (1). The result of this
straightforward calculation is ¯␣ (2)⫽2 3 ¯␣ (1) and ␴ 2 (2)
⫽ ␴ 2 (1)/29 , giving the expected result ␹ (2)⫽ ␹ (1)2 ⫺3/2.
More generally, when the dipole-dipole interactions are
not omitted from the Hamiltonian 共1兲, we can analyze the
scaling with a well-known renormalization-group approach.
This approach requires that we work in a general
d-dimensional space. The scaling of ¯␣ follows from ¯␣
⫽ 兺 r,r⬘ 具 m̃r•m̃r⬘ 典 /N and the observation that spatial selfsimilarity requires 具 m̃r•m̃r⬘ 典 ⬃ 兩 r⫺r⬘ 兩 ⫺(d⫺2⫹ ␩ ) . Here,
具 ••• 典 denotes a thermal average, N is the number of the unit
cells in the whole system, i.e., N(L)⫽N(1)L ⫺d , and ␩ is
the anomalous dimension 关11兴. Dimensional analysis thus
shows that ¯␣ (1)⫽L 2⫺ ␩ ¯␣ (L). As such,
␹ 共 L 兲 ⫽L ⫺2⫹ ␩ ¯␣ 共 1 兲 ␴ 共 L 兲 .
共2兲
Once we find the recursion equation for ␴ , the flow behavior
of ␹ (L) can be obtained.
To find this equation, we use the replica technique to
transform the Hamiltonian with quenched random variables
into a translational invariant one with no random variables
关5兴. In particular, averaging the logarithm of the partition
function Z over the probability distribution of ␣ r , together
with the identity log Z⫽limn→0 (Z n ⫺1)/n, leads us to consider an effective Hamiltonian,
H eff⫽
m(r␮ ) •m(r␮ )
1
2
兺
r, ␮
⫺
1
8k B T
¯␣
兺
r, ␮ , ␯
⫺
1
2
兺
r⫽r⬘ , ␮
⬘ •m(r␮ )
m(r␮ ) •Trr
␴ 2 m(r␮ ) •m(r␮ ) m(r␯ ) •m(r␯ ) ⫹ . . . . 共3兲
Here ␮ and ␯ are replica indices summed from 1 to n, k B is
the Boltzmann constant, T is temperature. The replica dipole
moment variable m(r␮ ) has 3n components. Subsequent terms
not exhibited in Eq. 共3兲 involve higher orders of m(r␮ ) than
fourth order, which are irrelevant operators since, according
to the dimensional analysis, the coupling constants in these
terms have negative momentum dimensions, and have no
effect on the renormalization of the system 关7兴. To study
properties of disordered systems, the calculation for a general number of replicas, n, is carried out, and then the limit
n→0 is evaluated 关5,8–10兴. This replica representation can
be applied at different length scales L, in which case the
coefficients 1/¯␣ and ␴ in Hamiltonian 共3兲 are functions of L.
4699
With Fourier transforms, we can rewrite and generalize
the Hamiltonian 共3兲 as 关10,12,13兴.
H eff⫽H 0 ⫹H int ,
H0
1
⫽⫺
k BT
2
␬⑀
H int
⫽⫺
k BT
4!
冕冋
q
共 r 0 ⫹q 2 兲 ␦ i j ⫹g 0
冕冕冕冕
q1
q2
q3
q4
q iq j
q2
共4兲
册
␾ ␣i 共 q 兲 ␾ ␣j 共 ⫺q 兲 ,
共5兲
i jkl
i jkl
i jkl
共 v 0 F ␣␤␥
␦ ⫹u 0 S ␣␤␥ ␦ ⫹2w 0 T ␣␤␥ ␦ 兲
冉兺 冊
4
⫻ ␾ ␣i 共 q 1 兲 ␾ ␤j 共 q 2 兲 ␾ ␥k 共 q 3 兲 ␾ l␦ 共 q 4 兲 ␦
i
qi ,
共6兲
where ␾ ␣i (q) denotes the ith Cartesian component of the
Fourier transform of m(r␣ ) , 兰 q denotes the integration
兰 d d q/(2 ␲ ) d truncated at a large wave-vector cutoff, 兩 q 兩
⭐q c ⬃1/a, and ␬ has the dimension of momentum. The
quantity g 0 is the bare strength of the dipole-dipole interaction, r 0 is the bare ‘‘mass,’’ and u 0 ⫽⫺3 ␴ 2 (1)/(k B T) 2 . The
u 0 term comes directly from the basic model, Eq. 共3兲. We
also include the other two terms with coefficients v 0 and w 0 ,
since these terms will be generated by subsequent iterations.
Repeated subscript and superscript indices are to be summed,
i jkl
i jkl
i jkl
and the tensors F ␣␤␥
␦ , S ␣␤␥ ␦ , and T ␣␤␥ ␦ are given in terms
ij
of Kronecker deltas ␦ ␣␤ and ␦ as follows:
1 i j kl
i jkl
ik jl
il jk
F ␣␤␥
␦ ⫽ 共 ␦ ␦ ⫹ ␦ ␦ ⫹ ␦ ␦ 兲 ␦ ␣␤ ␦ ␤␥ ␦ ␥ ␦ ,
3
1 i j kl
i jkl
ik jl
il jk
S ␣␤␥
␦ ⫽ 共 ␦ ␦ ␦ ␣␤ ␦ ␥ ␦ ⫹ ␦ ␦ ␦ ␣␥ ␦ ␤ ␦ ⫹ ␦ ␦ ␦ ␣ ␦ ␦ ␤␥ 兲 ,
3
i jkl
T ␣␤␥
␦⫽
冉
␦ ik ␦ jl ⫹ ␦ il ␦ jk
␦ i j ␦ kl ⫹ ␦ il ␦ jk
1
␦ ␣␤ ␦ ␥ ␦
⫹ ␦ ␣␥ ␦ ␤ ␦
3
2
2
⫹ ␦ ␣ ␦ ␦ ␤␥
␦ i j ␦ kl ⫹ ␦ ik ␦ jl
2
冊
.
In this formulation, the unperturbed or reference Green’s
ij
(q)⫽ 具 ␾ ␣i (q) ␾ ␤j (⫺q) 典 0 , where 具 ••• 典 0 defunction is G ␣␤
notes the average with the weight functional
⬀exp(⫺H0 /kBT). Because of the long-range interaction, the
Green’s function has longitudinal and transverse components,
ij
G ␣␤
共 q 兲 ⫽ 关 G L0 共 r 0 ,g 0 ,q 兲 P Lij ⫹G T0 共 r 0 ,q 兲 P Tij 兴 ␦ ␣␤ ,
where G L0 ⫽1/(r 0 ⫹g 0 ⫹q 2 ), G T0 ⫽1/(r 0 ⫹q 2 ), and P Tij ⫽ ␦ i j
⫺q i q j /q 2 and P Lij ⫽q i q j /q 2 are the projection operators for
transverse and longitudinal components, respectively.
The renormalization behavior of this model is controled
by the Callan-Symanzik equations 关15,7兴, which can be studied most conveniently by using the dimensional regularization and minimal subtraction procedure of ’t Hooft and Veltman 关16兴. We use this procedure and notation as outlined, for
example, in Amit’s text 关7兴. In particular, we use the ⑀ expansion to obtain the ␤ functions and renormalization con-
Z. G. YU, XUEYU SONG, AND DAVID CHANDLER
4700
PRE 62
stants up to one-loop level. The singular part of the one-loop
ij
(k)G ␥lm␦ (k⫹p), at the symmetry point (p 2
diagram, 兰 k G ␣␤
2
⫽ ␬ ) is ␦ ␣␤ ␦ ␥ ␦ J i jlm , where
J i jlm ⫽
冋
册
1 11
1
⫺ ln共 1⫹g 0 / ␬ 2 兲 ␦ i j ␦ lm ⫹ ln共 1⫹g 0 / ␬ 2 兲
⑀ 48
48
⫻ 共 ␦ il ␦ jm ⫹ ␦ im ␦ jl 兲 .
From this result and the renormalization condition at the
symmetry point, where the four-point vertex function 关7兴 is
i jkl
i jkl
i jkl
␬ ⑀ ( v F ␣␤␥
␦ ⫹uS ␣␤␥ ␦ ⫹2wT ␣␤␥ ␦ ), we obtain
冉
␤ v⬅ ␬
冉
⳵
v
⳵␬
冊冏
冉
⫽⫺ ⑀ v ⫹ 2⫺
0
冊
7
1
v2
12 1⫹ ␬ 2 /g
冊 冉
冊
1
1
2
5
v u⫹ 6⫺
v w,
3 1⫹ ␬ 2 /g
3 1⫹ ␬ 2 /g
⫹ 2⫺
共7兲
冉
␤ u⬅ ␬
冉
冉
⳵
u
⳵␬
⫹ 2⫺
冊冏
⫽⫺ ⑀ u⫹
0
冉
冊
4 17
1
u2
⫺
3 36 1⫹ ␬ 2 /g
冊 冉
冊 冉
FIG. 1. Exponent ␯ as a function of length scale. Solid, shortdashed, long-dashed, and dot-dashed lines correspond to g⫽10⫺4 ,
10⫺3 , 10⫺2 , and 100 , respectively. The initial fluctuation strength is
fixed with u(1)⫽⫺1.0.
Up to one-loop level, g(l)⫽g 关13兴. Since initially, v 0
⫽0 in our model, according to Eq. 共7兲, v will remain zero
with changing the length scale. Thus we just need to solve
two coupled equations
冊
冊
10 13
1
1
1
⫺
uw
v u⫹
2
2 1⫹ ␬ /g
3 18 1⫹ ␬ 2 /g
l
7
2 1
1
1
w⫹
2⫺
w 2,
⫹ ⫺
v
3 9 1⫹ ␬ 2 /g
12 1⫹ ␬ 2 /g
共8兲
冉
␤ w⬅ ␬
冉
⳵
w
⳵␬
⫹ 2⫺
⫹
冊冏
⫽⫺ ⑀ w⫹
0
冉
冊
8 9
1
⫺
w2
3 12 1⫹ ␬ 2 /g
冊 冉
⫹
冉
冉
冉
冊
l
共9兲
The symbol 兩 0 indicates that all derivatives are to be taken at
fixed bare parameters. When g goes to infinity, the model we
consider here is essentially the same as that studied by Aharony 关10兴. In that limit, we reproduce Aharony’s results regarding critical exponents in the limit g→⬁, though the
fixed points we find in this limit are different from those in
Ref. 关10兴 due to different coefficients for u 0 , v 0 , and w 0 .
There are six nontrivial fixed points altogether, which may
be divided into two groups of three. The first three all have
v * ⫽0, and only one of them with u 0* ⫽⫺3.055⑀ and w *
0
⫽2.504⑀ is stable and physically reachable 关10,14兴.
The recursion relations for the coupling constants in the
model, u, v , and w, are determined by the above ␤ functions.
When we change the length scale in real space, the momentum scale changes accordingly. If we use parameter l so that
␬ (l)⫽ ␬ l, the functions v (l), u(l), and w(l) will obey the
following equations l 关 d v (l)/dl 兴 ⫽ ␤ v , l 关 du(l)/dl 兴 ⫽ ␤ u ,
and l 关 dw(l)/dl 兴 ⫽ ␤ w .
冊
冊
10 13
1
u共 l 兲w共 l 兲
⫺
3
18 1⫹l 2 /g
⫹ 2⫺
13
2 5
1
1
uw⫹ ⫺
vw
18 1⫹ ␬ 2 /g
3 18 1⫹ ␬ 2 /g
1
1
u 2.
36 1⫹ ␬ 2 /g
冉
4 17
1
du 共 l 兲
⫽⫺ ⑀ u 共 l 兲 ⫹ ⫺
u共 l 兲2
dl
3 36 1⫹l 2 /g
冊
冊
1
7
w共 l 兲2,
12 1⫹l 2 /g
共10兲
8
9
1
dw 共 l 兲
⫽⫺ ⑀ w 共 l 兲 ⫹ ⫺
w共 l 兲2
dl
3 12 1⫹l 2 /g
冉
⫹ 2⫺
冊
13
1
1
1
u
l
w
l
⫹
u共 l 兲2,
兲
兲
共
共
18 1⫹l 2 /g
36 1⫹l 2 /g
共11兲
with initial conditions u(1)⫽⫺ ␴ 2 and w(1)⫽u 2 (1)/2.
Here, we take ␬ equal to one.
Having obtained solutions u(l) and w(l) to Eqs. 共10兲 and
共11兲, we can compute the exponent ␯ (L) for the scaling of
polarizability fluctuation strength ␹ (L)⬃L ⫺ ␯ (L) . Noting l
⬃1/L, ␯ (L) can be expressed as
␯ 共 L 兲 ⬅⫺
d ln ␹ 共 L 兲
d ln L
⫽ 共 2⫺ ␩ 兲 ⫹
1 d ln u 共 l 兲
2 d ln l
1
⫽ 共 2⫺ ␩ 兲 ⫹ u ⫺1 共 l 兲 ␤ u 共 l 兲 .
2
共12兲
PRE 62
POLARIZABILITY FLUCTUATIONS IN DIELECTRIC . . .
4701
This expression is valid to all orders of the loop expansion.
At one-loop level, the anomalous dimension ␩ ⫽0 and ␤ u (l)
is given by Eq. 共10兲.
Figure 1 depicts how exponent ␯ (L) for the dimensionless width of polarizability distribution flows as increasing
the length scale. We fix the initial fluctuation strength u(l
⫽1)⫽⫺1.0 and vary the dipole-dipole interaction strength
g. We can see that when the length scale becomes very large,
␯ (L) approaches 2. This is because when L is very large 共or
l→0), u(l) will be at its fixed point and ␤ u ⫽0, thus ␯ (L)
⫽2⫺ ␩ , and at the one loop level, ␩ ⫽0.
For the noninteracting model H t , there is no symmetry
breaking and only the u term in the Hamiltonian 共6兲 is nonzero. Moreover, there is no divergence for the diagrams at
any order of u. Therefore ␤ u ⫽⫺ ⑀ u, showing that the noninteracting model has only the trivial zero fixed point. Because in this case ␩ ⫽0, from Eq. 共12兲, ␯ (L)⫽2⫺ ⑀ /2⫽3/2.
This is the result we noted earlier that follows from the uncorrelated statistics for the independent random variables.
In Fig. 1, when the length scale is not very large, the
deviation from this universal exponent may be significant. At
small length scales, exponent ␯ (L) is around 3/2, indicating
that in this regime, the behavior of polarizability fluctuations
is close to the result for uncorrelated random variables. In
this regime the randomness dominates over the dipole-dipole
interactions. At large scales, since the effective strength of
dipole-dipole interactions is renormalized as g(l)/ ␬ 2 (l)
⬃g/l 2 ⬃gL 2 , the dipole-dipole interactions become more
and more significant as increasing the length scales. In this
regime the interaction dominates over the randomness, giving rise to a universal behavior. We see from the figure,
when the strength of dipole-dipole interactions is weak, exponent ␯ (L) increases slowly as increasing length scales,
and at some regions ␯ (L) becomes greater than 2 and eventually goes to 2. When the strength of dipole-dipole interactions is strong, ␯ (L) increases fast from 3/2 and reaches 2
monotonically. The different behaviors of ␯ (L) for different
strengths of ␴ 2 and g reflect competition and cooperation
between long-range dipole-dipole interactions and local polarizability fluctuations as changing length scales.
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D.C. has been supported by the Office of Basic Energy
Sciences, Chemical Sciences Division of the U.S. Department of Energy, Grant No. DE-FG03-87ER13793.