Singularities of the renormalization group flow for random elastic manifolds

3 Pages·0.09 MB·English

by  D. A. Gorokhov

Singularities of the renormalization group flow for random
elastic manifolds
D. A. Gorokhov and G. Blatter
9 Theoretische Physik, ETH-Ho¨nggerberg, CH-8093 Zu¨rich, Switzerland
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9 We consider the singularities of the zero temperature renormalization group flow for random
1 elastic manifolds. When starting from small scales, this flow goes through two particular points l∗
andlc,wheretheaveragevalueoftherandomsquaredpotentialhU2iturnesnegative(l∗)andwhere
n
a thefourth derivativeof thepotentialcorrelator becomes infiniteat theorigin (lc). Thelatter point
J sets the scale where simple perturbation theory breaks down as a consequence of the competition
0 between manymetastablestates. Weshowthatunderphysically welldefinedcircumstanceslc <l∗
2 and thustheapparent renormalization of hU2i tonegative values does not takeplace.

] PACS numbers: 05.20.-y, 11.10.Hi, 74.60.Ge, 75.60.Ch, 82.65.Dp
h
c Consider an elastic manifold with d internal degrees of freedom embedded into a (d+N)-dimensional space in the
e presence of a random potential U(u,r). The free energy of the manifold takes the form
m
- C ∂u 2
at F[u]= ddr 2 ∂r +U(u,r) , (1)
st Z " (cid:18) (cid:19) #
.
t with C the elasticity. The random potential U(u,r) is assumed to be gaussian with an isotropic correlator
a
m hU(u,r)U(u′,r′)i = K(|u−u′|)δ(r−r′). The model Hamiltonian (1) describes a large class of disordered systems
including random magnets, dislocations in metals, and vortices in superconductors [1].
-
d In a recent paper [2] the functional renormalizationgroup(FRG) approachhas been used in order to calculate the
n critical force for the depinning of a (4+N)-dimensional elastic manifold in the presence of a weak randompotential.
o Thecollective-pinningscalehasbeenidentifiedwiththe FRGflowpoint,wherethefourthderivativeofthe correlator
c of the randompotential at the originK(4)(0) becomes infinite. By induction it is possible to show that all the higher
[
even derivatives also become singular. However, if we look at the equation for the correlator itself, it turns out that

1 this equation also exhibits a singularity: at some length l∗ the average value of the random potential squared Kl∗(0)
v becomesnegative. This situationis,ofcourse,unphysical. The goalofthis note isto showthatthe collective-pinning
0 scale l is always smaller than the length l∗ at which the average value of the pinning potential squared becomes
0 c
negative. We briefly review the method of the calculation of the collective-pinning length used in Ref. [2] for the
2
∗ ∗
1 (d+N)-dimensional problem, determine the two length lc and l and show that lc <l for a physical situation.
0 Theone-loopzerotemperatureFRGequationfora(d+N)-dimensionalelasticmanifoldcanbewritteninthe form
9 [3,4]
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/ ∂K (u) 1
t l =(4−d−4ζ)K (u)+ζu Kµ(u)+I Kµρ(u)Kµρ(u)−Kµρ(u)Kµρ(0) , (2)
a ∂l l µ l 2 l l l l
m (cid:20) (cid:21)
- where ζ is the wandering exponent, I =A /C2Λ4−d (A =2πd/2/Γ(d/2) and Λ−1 is the short scale cutoff ), and the
d d d
upper indices µ and ρ denote the derivative with respect to the cartesian coordinates µ and ρ. Differentiating this
n
o equation four times with respect to u and substituting u=0, we obtain [5]
c
v: ∂Kl(4)(0) =(4−d)K(4)(0)+ I(N +8)K(4)2(0), (3)
i ∂l l 3 l
X
ar where Kl(4)(0)=∂4Kl/∂u4µ|u=0. IntegratingEq. (3) we find that the function Kl(4)(0) becomes infinite atthe scale lc
defined by
1 3(4−d)
l = ln 1+ . (4)
c 4−d I(N +8)K(4)(0)!
0
The scale l defines the collective-pinning radius R =elc/Λ.
c c
On the other hand, differentiating Eq. (2) twice with respect to u and setting u=0 we obtain
1
∂K(2)(0)
l =(4−d−2ζ)K(2)(0). (5)
∂l l
This equation again can be easily solved and substituting the expression for K(2)(0) into Eq. (2) with u=0 we find
l
that K(0) becomes negative at the scale [6]
1 2(4−d)K(0)
∗
l = ln 1+ . (6)
c 4−d INK(2)(0)2 !
∗
The scheme used in Ref. [2] for the determination of the collective pinning radius R is valid only if l < l . Thus,
c c
∗
we have to prove that indeed in a physical situation l > l . Taking into account Eqs. (4) and (6) we can write this
c
inequality in the form
3N K (0)K(4)(0)
< 0 0 . (7)
2(N +8) K(2)(0) 2
0
h i
Next,letusshowthattheinequality(7)issatisfiedforanyphysicalcorrelatorK(u)withapositiveFouriertransform
K(k) = duK(u)exp(iku) [7] ; the distribution of the Fourier components of the potential U is then given by the
tphraotduKct(uRP)(cUakn)b∝e repkreexsepnt−edUik2n/2thKe(kfo)rmand is well defined as long as K(k) > 0. The condition K(k) > 0 implies
Q (cid:0) (cid:1)
K(u)= du′P(u−u′)P(u′). (8)
Z
Taking into account that ∆K(u = 0) = NK(2)(0) and ∆2K(u = 0) = [N(N +2)/3]K(4)(0), with ∆ the Laplace
operator in the N-dimensional space, we can rewrite the right hand side of the inequality (7) in the form
K (0)K(4)(0) 3N ∆2K(u=0)K(u=0)
0 0 = . (9)
K(2)(0) 2 N +2 [∆K(u=0)]2
0
h i
Using the Schwarz inequality (y ,y ) ≤ ||y ||||y || and Eq. (8), with the scalar product and the norm defined as
1 2 1 2
(y ,y )= duy (u)y (u)and||y ||= duy2(u) 1/2 (inparticular,K(0)=||P||2,∆K(0)=(P,∆P),and∆2K(0)=
1 2 1 2 1 1
||∆P||2), we arrive at the result
R (cid:2)R (cid:3)
K (0)K(4)(0) 3N ∆2K(u=0)K(u=0) 3N
0 0 = ≥ . (10)
K(2)(0) 2 N +2 [∆K(u=0)]2 N +2
0
h i
We then can reformulate the condition (7) to read
3N 3N
≤ , (11)
2(N +8) N +2
which is always true and hence l∗ >l for any physical correlatorK (u). At the point l the third derivative K(3)(0)
c 0 c lc
exhibits a jump [3] and the FRG equations (3) and (5) break down as we have used the fact that all odd derivatives
of the correlator vanish at the point u = 0 in their derivation. The appearance of new terms in the FRG equations
will then prevent the function K (0) from taking negative values at scales beyond l .
l c
[1] for a general rewiew see G. Forgacs, R. Lipowsky, and Th. M. Nieuwenhuizen, in “Phase Transitions and Critical Phe-
nomena”, Vol. 14, edited by C. Domb and J. Lebowitz (Academic Press, London, 1991); G. Blatter, M.V. Feigel’man,
V.B. Geshkenbein, A.I. Larkin, and V.M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994); T. Halpin-Healy and Y.-C. Zhang,
254, 215 (1995) and references therein.
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[2] H. Bucheli, O. S. Wagner, V.B. Geshkenbein, A.I. Larkin, and G. Blatter, Phys. Rev.B 57, 7642 (1998).
[3] D. S.Fisher, Phys. Rev.Lett. 56, 1964 (1986).
[4] L. Balents and D. S.Fisher, Phys.Rev.B 48, 5949 (1993).
[5] We start with the correlator K0(u) which has zero odd derivatives at the point u = 0. From Eq. (2) we see that all odd
derivatives remain zero (at least as long as l<lc).
[6] We want to point out that theresult (6) remains trueeven if ζ dependson l.
[7] We wish to point out that there is no proof that Kl(k) remains positive for any l undertheRGtransformation.
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