Semjon Stepanow (~), Lei-Han Tang (~) and Heiko. Leschhorn. (~). (~). Institut fir. Theoretische. Physik,. Universitit zu. K61n, Z6lpicher Str. 77, D-5000. K61n 41,.
Phys.
J.
II
France
(1992)
2
1483-1488
1992,
AUGUST
1483
PAGE
Classification
Physics
Abstracts
47.55M
64.60A
75,10N
75.60
Communication
Short
Dynamics
of
Thomas
(~)
fir
in
disordered
a
medium
(~), Semjon Stepanow (~), Lei-Han Tang (~)
Nattermann
Institut
depinning
interface
Physik,
Theoretische
Universitit
K61n, Z6lpicher
zu
and
Heiko
Str.
77,
Leschhorn
D-5000
K61n
(~)
41,
Germany
(~)
Physik
Theoretische
III,
Bochum,
Ruhr-Universitit
102148,
Postfach
Bochum,
D-4630
Germany
(Recdved
16
June
1992, accepted
The dynamics analyzed. By a depinning transition are
Abstract
threshold the
z
=
8
The
=
=
driven
of the
motion
a
driven
interface
functional
in
disordered
a
renormalization
obtained
the
to
first
order
in
medium
scheme
group e
D
4
=
close
to
exponents
0,
>
depinning characterizing
the
where
D
is
the
dynamics is superdifusive with a dynamical exponent 2e/9 + O(e~), and the interface height difference over a distance L grows as Ll with 2 e/3 + O(e~). The interface velocity in the moving phase vanishes as (F Fc)~ with e/9 + O(e~) when the driving force F approaches its threshold value PC1
interface
(
of
is
1992)
June
22
dimension.
viscous
At the
motion
of
transition,
an
the
interface
in
ill,
medium
a
paradigms of condensed physics. matter of a magnetically or structurally ordered
This
system
pinning
random
with
problem arises,
e,g.,
random-bond
with
forces
in the or
is
one
domain-wall random-field
pushed through a porous or an medium [2]. Closely related problems include impurity pinning in type-II superconductors [3] and in charge-density-wave (CDW) systems [4]. Despite its importance this problem is largely unsolved although a number of attempts have been made in the past (see e.g. [5-10]). In this paper we focus on a simple realization of the problem, the motion of a D-dimensional interface profile z(x, t) obeying the following equation [5-8], disorder
Here
I is the
driving
force.
when
friction
The
interface
(or random
inverse
force
between
mobility) coefficient, q(x, z) is Gaussian
In(xo> zo)nlxo We is
a
immiscible
two
mainly concerned with the monotonically decreasing function
+ x> zo +
will be
z)1
random-field of
z
for
z
y is
the
distributed
=
is
stiffness with
constant,
(n)
=
and
F is
0 and
where
the
the
0 and
6~(x)Alz).
case
>
fluids
(2) correlator
decays rapidly
to
A(z) zero
over
A(-z)
=
a
finite
JOURNAL
1484
DE
PHYSIQUE II
N°8
(2) along the interface is correlator the length of problem. any As pointed out by Bruinsma and Aeppli [6], an important length of this model is Lc [y~a~/A(0)]~/~, where e D. For D < Dc 4 4, the interface is kept smooth (I.e., fluctuations in z is limited to a or smaller) on length scales L < Lc, but is able to explore the inhomogeneous force field on larger length scales. It follows that the maximum pinning force interface of linear dimension L > Lc is of the order of (L/Lc)D [A(0)Lfl]~/~, which on a piece of leads to the estimate Fc ci [A(0)/Lfl]~/~ A(0)~/~ for the critical driving force of a depinning transition steady-state For F Fc moving solution to (I) while for F < Fc expect 6]. > 16, we a the interface at long times is pinned at one of the presumably infinitely many locally stable configurations. The nature of the depinning transition at F Fc has not been studied in detail analytically. [The situation is different for D > 4, where pinning is essentially a small length mean-field theory is expected to be valid. This conclusion, which can scale phenomenon where relative be drawn by assessing the importance of the elastic and random force terrns in (I), distance
taken
Unless
a.
much
be
to
specified,
otherwise
smaller
the
width
of the
characteristic
other
than
=
=
=
-J
=
sets
ics
dimension the critical at Dc = 4 for Our main here is to develop a purpose
depinning
the
near
threshold
weak
disorder
[6].]
group approach to the critical dynamstraightforward extension of the perturbation vanishing mobility l~~ 0 at Lc, thereby freezes
renormalization
for D < 4.
A
theory of Efetov and Larkin [4] yields a dynamics on larger length scales. However, as we demonstrate below, this difficulty can be renormalization of A(z) which by considering the functional becomes singular at the overcome origin, thus opening the door to a systematic expansion in D dimensions. 4 We present e the first results so obtained for the critical characterizing the depinning transition. exponents calculation will be presented elsewhere. Details of our perturbation theory consists of expanding n(x, z) at a flat interface (or, for that The usual other reference configuration), and solving the resulting equation order by order matter, any disorder [4, 5, 7]. Such a procedure is justified when deviations in the strength of the from the reference interface position are of order a or smaller. This is indeed the case for a fast moving (For D < 2 the related discussion. interface with D > 2 which is the starting point of our equation ill] yields a rough interface. that this is the origin Edwards-Wilkinson We believe for the break down of perturbation theory as observed by Koplik and Levine [7].) self consistently determined For a moving interface, we write z(x, t) vi + h(x, t), where u is condition (h(x, t)) from the 0. Equation (I) now takes the form =
=
=
=
l~
yT7~h
=
Due
coupling
response
equation
diffusion
lowest
the
to
system's
order
to
a
with
corrections
n(x,
Au +
+ F
vi +
h(x,t)).
through the random force term, the perturbation is described by a modified I + bl and 7e~ parameters few y + b?. At large u the be easily found from the perturbation theory, can different
among
modes
Fourier
long-wavelength, slow-varying
external
=
=
b7(~)
=
7gjL[
-lg~L[
bl(~) =
Lv
=
(7a/ul)"~
is
the
diffusion
length
over
a
=
l~~l
"
~~
(4«)D/2 (D
period a/v
time
1/(8x~7~). Here and coupling constant, with c identical to a previous one by Feigel'man who considered due to the pinning force [5]. The width of the interface
is the
(4) ,
,
where
(3)
below the to
this
II ~~
+
and
g
=
cA"(0)
bl(~) the velocity to order is given by
e > 0. correction
°~~~
Result
for
+
in u
=
O(e) (4) is F/I
l~~
N°8
DYNAMICS
equation (4)
From
origin,
the
at
OF
calculation
and the
to
we
that next
INTERFACE
that
see
b7/7
bl(~l is given
where
that
apparent
the
both
by order yields l~fl
DEPINNING
is
I
=
(I
corrections a
obtained
DISORDERED
related
are
e/D
factor
smaller
order
in
terms
perturbation
from
the
to
1485
derivative
second
Extending
A(z)
of
the
above
),
+
each
in
e
MEDIUM
bl/I.
than
bl(~l/1+ 2(bl(~l /1)~
+
(4). Only leading
in
series
IN A
(6)
order
theory, while
valid
shown
are
in
large
for
(6).
It is
be
cannot
u,
pinning threshold, where u 0. near The usual e-expansion scheme allows one to sum up the series via renormalization group flow equations. Specifically, we consider 1, 7, F renormalizable quantities Au, and A to be which depend on the cut-off length, L Lv. The flow equations can be immediately upper used
directly
read
off
the
-
=
(4)
from
equation
The
last
and
Efetov
the
coefficients
functional
(6),
and
also
can
d
L
~(Z)
equation (7d)
system by
density-waves in Equation (7c)
Daniel
go
assumed
"
at
can
g(Lo).
from
be
A
Lo, leads
~~
~ 2
also
appears
This
~~2 in
negative
go,
set
~2(~)
the
[13], and in his medium [14].
integrated
diagrammatic technique of Larkin equations one of us [12] obtained for of equations be expressed in the can
the
of flow
set
Fisher
random
a
~~e
~
~(~)
of
treatment
work
recent
=
i +
which
(31e)(]~~e
(~ ~) equilibrium interface in a Narayan on sliding charge-
an
with
be
to
appears
is
choice
natural
a
diverging g(L) and hence A"(0) a (7b) then yields an infinite I at L result
(8)
~j~>
to
Lc. Inserting (8) into possible. On the face of it~ this
(~)j
give
to
~~~~ where
=
Taylor expansion for A(z).
a
~ ~~ ~
Interestingly,
=
L
(7a) (7b) (7c)
O(e~), -gL~, -3g~L~.
=
directly of the
one
form
~
random
L
obtained
be
[4], and is in fact only in
7/d In In I Id In dg/dln
d In
at ci
if
an
length L a Lc, beyond which finite
analytic A(z)
(e/3(go()"~
m
dynamics
no
is t
is
clearly unphysical.
diverging behavior has actually been noted earlier in the study of impurity pinning a of charge-density-waves by Efetov and Larkin [4], and in other related problems, and its implication remains controversial. One opinion is to dub the pole an artifact of the one-loop approximation bearing no real physical significance. The second and much more interesting proposal is to accept the divergence as a real phenomenon associated with the nonanalytic behavior of A(z) at the origin, and try to continue the renormalization procedure [13]. In the remaining part of the paper we explore the of latter approach and show that it consequences Such
indeed The
(7d)
leads
to
a
divergence
is still
solution,
we
well
make
consistent
renormalization
of g corresponds to defined from z away the
scaling
scheme
singularity
a =
0 and
can
and
of
to
A(z)
thus
be
at
fruitful the
results.
origin.
followed.
To
Nevertheless, look
for
a
equation point
fixed
ansatz
A(L, z)
=
c~~A~/~L~~A)(zA~~/~L~l),
(9)
JOURNAL
1486
limi-co A)(y)
where is
an
function
even
A*(y),
=
2()A* (v)
(e
Examining the behavior possible. In the first case derivative
second
y
as
a
e
=
argument.
of its
(10)
of
0, thus is
-
not
A*(v)
al (II)
with
type order
for
a
depends
the
for A
references
in
a2
=
lo
for A, A*
=
oo
-
II
(lo)
0.
=
types of singular
two
O((y().
yields,
behavior
are
form yields a diverging The second possibility is
This
difficulty.
(lo
+
=
=
driving force, cutoff Ao
lower and
F
Ci
f
-
x/Lo
identifying Lo
Fc.
F
=
of the
an
Fc
Here
momentum
Lc yields
with
integration. Using
space
Fc in
-(16x~7)~~Af~~A'(Lo,0+)
"
the
with
agreement
the
estimate
scaling given
[5] and [6].
(II) yields
and
>(L) where
that A*(0) 1. As taking the limit L
that
Integrating (7b) using (9)
the
+
ja~y2
(e
find
the
of the
on
(9)
form
A(z),
with
in e,
reduction
of the
i + ai (v( +
=
that
(v("~
out
way
such
and
()/3. Here A*"(0+) Using an expansion of the a2 is finite. lbut equations and (7a) (7b) remain valid to the first (7c)] not we cA"(0+). The singular term (z(, however, yields understanding that g
2( and
e
=
=
a
shows
y
I + al
N°8
II
A*"(v)lA* (v)
lA*'(v)l~
small
at
A*(y)
have
we
2c, and A is chosen Inserting (9) into (7d)
(vA*~(v)
+
PHYSIQUE
DE
>(Lo)-
"
scale
As
before,
transformation
lb~~~
x
-
bx,
=
7T7~h + b~~t f
(
t
(12)
>o(L/Lo)-~~-
scale dependence to the first no blh, equation (3) b~t, and h
has
7
=
order can
-
-
Au) + b~~tn(bx, vb~t +
in
be
Performing
e.
rewritten
as
blh).
(13)
z(x,t). dynamical exponent to be distinguished from the interface coordinate equations (2), (9), and (12) that (13) becomes scale invariant at u f 0 upon ()/3. A finite u, however, changes the character of the noise correlator the choice z 2 (e v~~/(~~l), as can be seen by comparing the two terms in the second above a length scale Lv argument of n in (13). Physically, Lv serves as the correlation length of the net pinning (or driving) force along the interface. Stop the I(Lv)u which in renormalization at Lv yields f Here
It
is the
z
follows
from
=
=
=
-J
=
gives
turn
~
'~
Lv where
we
alone
final
Our consider I* 1*
=
from
used
(12)
the
task
is to
integrals IA
and
the
relation
0
determine
"~~~~
f~",
with
relation =
v(z
the
~' v
between
()
=
z
which is
exponent
I~$~ A(L, z)dz,
"
~
1/(2
(),
and
(. [The
satisfied
( from (10).
which is
an
oo;
if
we
the
~~~~~
(14b) Lv
condition
in the For
invariant
fZ~ A*(y)dy.
=
(10)
have
yields the scaling
-J
~~'
this
ci
u~~/(~~l)
present case.] purpose
of the flow
it is
useful
to
equation (7d), and
random-field disorder, we have I) ( < e/3, For IA > 0, which is true for CA~~IA; and iii) ( > e/3, 1* e/3, 1* 0; Case I) is inconsistent with bounded demand A*(y) to be for all y and vanish at infinity. One can also show flow equation (7d) that, if A(L, z) is initially positive everywhere and decays to zero
it) (
=
=
=
DYNAMICS
N°8
sufficiently fast
it)
case
there
OF
large actually
at
is
INTERFACE
z, a
Inserting ( the
=
e/3
exponents
that
equations (14)
to
the
first
order
and
exp(-I
=
MEDIUM
1487
in the
(IS)
jy~).
expression for
z
have
we
the
following
results
in e,
zci2-~e, disorder,
randoli~bond
for
exp(-A°)
in
(ce ~e, Note
DISORDERED
IN A
the limiting form A* has no negative parts thus excluding iii). In unique solution with exponential tails given implicitly by [13]
A*
for
DEPINNING
where
q
can
uci)+~e.
~e,
b~l-
be
written
as
the
(16)
derivative
of
a
random
short-range correlations, IA be quite different 0 and hence the exponents can from those given in (16). scheme Let us conclude by recapturing the main steps that led to a successful renormalization for the interface dynamics at and above the depinning transition. We have shown that a simple of the perturbation theory carried out to the lowest order extension into difficulty on the runs length scale Lc where pining efsects become significant. When the procedure renormalization correlator in the moving direction), the is extended function A(z) (random force to the whole divergent behavior of the coupling constant be attributed nonanalyticity of A(z) to the can the divergence can be formally removed and a at the origin. By isolating the singular term consistent renormalization scheme is found at the transition F Fc. The interface roughness e/3 + O(e2) so obtained difsers, to the first order in e, from the value e/2 from exponent ( potential
with
=
=
=
perturbation
though
we
corrections. a
length
theory [4, 8], but coincides here for no a pliori reason A finite interface velocity v
see
scale Lv
(F
m~
Fc)~~,
above
random-field problem, that of the equilibrium Iinry-Ma type argument ii] to exclude higher order (F Fc)' interrupts the renormalization at process
with an m~
which
one
crosses
the
to
over
regime where the
random
independently on the moving interface as in the Edwards-Wilkinson equation. It is act interesting to note that, using our expressions (16) at D yields the temporal roughness I (/z 3/4, fl in surprisingly good with simulation results of Parisi [8] exponent agreement (I). of mention here seT-consistency similar lattice version We that argument to the on a a Harris equilibrium lib] yields given by for disordered inequality systems one an forces
=
=
=
1Iv a
sharp
order
in
if
On
is to
be
assumed.
In
(D
our
+ case
() /2, (17)
ii) is
fulfilled
as
an
equality
to
the
first
e.
the
difserent
threshold
Fc yields a calculation shows, Fc is formally related to the amplitude As our response. A(z) at the origin. It is believed that (e.g. Ref. [9]), of the singular part of the correlator critical in the that a small perturbation on a length Fc, the system becomes at F sense scale L > Lc may provoke an arbitrarily large sandpile model of Bak, Tang in the response, as and Wiesenfeld [16]. Our finding of a dynamical exponent z 2e/9 < 2 suggests that 2 the dynamics at the depinning transition is indeed superdifsusive. It would be interesting to explore the use of functional renormalization approach to other systems characterized group
I.e.,
responds
it
to
an
=
=
by
a
threshold
dynamics.
JOURNAL
1488
PHYSIQUE
DE
II
N°8
Acknowledgements. We
would
like
Narayan yond Lc.
thank
to
Daniel
kindly
who
Fisher
pinning of charge-density-waves,
on
The
is
research
forschungsbereich
166 and
supported by the
sent
preprints
us
of his
work
recent
with
inspired our analysis ofthe dynamics beForschungsgemeinschaft through Sonder-
which Deutsche
237.
References
ii]
For
reviews
recent
equilibrium
closely related
the
on
B 3 (1989) 1597; R. and Nieuwenhuizen
P., Inn. J. Mod. Phys. Forgacs G., Lipowsky
problem M.,
Th.
in
Nattermann
e-g-,
see,
Phase
and
transitions
Rujan
and
T.
critical
phe-
(Academic Press, London, 1991) p.135. Rubio M. A., Edwards C. A., Dougherty A. and Gollub J. P., Phys. Rev. Lent. 63 (1989) 1685. Larkin A. I, and Ovchinikov Yu. N., J. Low Temp. Phys. 34 (1979) 409. JETP 45 (1977) 1236. Efetov K. B. and Larkin A. I., Sov. Phys, Feigel'man M. V., Sov. Phys. JETP 58 (1983) 1076. Bruinsma Lent. 52 (1984) 1547. R. and Aeppli G., Phys. Rev. Koplik J. and Levine H., Phys. Rev, B 32 (1985) 280; Kessler D. A., Levine H. and Tu Y., Phys. Rev. A 43 (1991) 4551. Lent. 17 (1992) 673. Parisi G., Europhys. Robbins M. O., Phys. Lent, Martys N., Cieplak M. and Rev. 66, (1991) 1058; Martys N., Robbins M. O, and Cieplak M., Phys. Rev. B 44 (1991) 12294. Tang L.-H. and Leschhom H., Phys. Rev. A 45 (1992) R8309; C.
nomena,
[2] [3] [4] [5] [6] [7] [8] [9]
[lo]
Domb
Buldyrev S-V-,
(1992)
A 45 Edwards
[11] [12] [13] [14] [15] [16]
S-F-
and
Lebowitz
J. L.
Barab£si
A.-L.,
Eds.
Caserta
F.,
Vol
added
Following exponent
of
and
A,
H-E-
and
T., Phys.
Vicsek
Rev.
Wilkinson
D-R-,
Proc.
R.
London,
Sac.
Ser.
A
(1992) 3615;
68
Lent.
59
(1982)
381
Harvard
(1987)
17.
preprint (1992).
381.
proof:
in
similar the
arguments
interface
equilibrium problem equations (14) and the function
S., Stanley
Havlin
R8313.
Stepanow S., submitted to Ann. Physik. Fisher D-S-, Phys. Rev. Lent. 56 (1986) 1964. Narayan O. and Fisher D-S-, Phys. Rev. Lent. Barris A, B,, J. Phys. C 7 (1974) 1671. Wiesenfeld Bak P., Tang C. and K., Phys. Rev.
Note
14
for
discussed relation
in
as
the
random-bond
previously by z
2 =
(e
random-field disorder
Fisher
( )/3,
is
case,
[13]. which
Other are
found
we
given by f
0.2083 =
critical
valid
for
that e,
exponents different
roughness
the same
as
follow
choices
in
the from
of the
