Table Of Content2-loop Functional Renormalization for elastic manifolds pinned by disorder in
N dimensions
Pierre Le Doussal and Kay J¨org Wiese
CNRS-Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure, 24 rue Lhomond, 75005 Paris, France.
(Dated: February 2, 2008)
5
0 WestudyelasticmanifoldsinaN-dimensionalrandompotentialusingfunctionalRG.Weextend
0
to N > 1 our previous construction of a field theory renormalizable to two loops. For isotropic
2
disorder with O(N) symmetry we obtain the fixed point and roughness exponent to next order in
n ǫ=4−d,wheredistheinternaldimension ofthemanifold. Extrapolation tothedirectedpolymer
a limit d=1 allows some handle on thestrong coupling phase of theequivalent N-dimensional KPZ
J growth equation, and eventually suggests an uppercritical dimension du ≈2.5.
3
1
Disordered elastic systems are under extensive study fourloop)onlyforthetoy-modellimitd=0,N =1[22].
] both theoretically and experimentally. They are of in- A case where ambiguities have been resolved from first
n
terest for a number of physical systems, such as CDW principles at T = 0 to 2-loop order, is the N = 1 depin-
n
[1], flux lattices [2, 3], wetting on disordered substrates ning transition [16, 18]. Finally, the FRG has also been
-
s [4], and magnetic interfaces [5], where the interplay be- solved in the large-N limit [20]. Its solution reproduces,
i
d tween the internal order and the quenched disorder of apparently with no ambiguity, the main results from the
. the substrate produces pinned phases with non-trivial replica-symmetry-breakingsaddle point of Ref. [25], and
t
a roughness and glassy features [6]. Typically they are de- also underlies the importance of specifying the system
m
scribedbyelasticobjects,withinternald-dimensionalco- preparation [20].
- ordinate x, parameterized by a N-component height, or
d In the more difficult case of the statics within the
displacement field u(x). Analytical methods are scarce,
n d = 4−ǫ expansion, detailed analysis to two and three
o anddevelopingafield-theoreticaldescriptionposesacon- loops [16, 19, 26] for the case of N = 1 have suggested
c siderable challenge. One reason is that naive perturba- several methods to construct a renormalizable field the-
[ tive methods fail, technically due to the breakdown of
ory. These methods give a unique finite β-function, with
the dimensional reduction phenomenon [7], and physi-
1 non-trivial anomalous terms. This β-function satisfies
v cally because describing the multiple energy minima in the potentiality constraint, with anomalous terms dis-
5 a glassseems to containsome non-perturbative features. tinctfromthose atdepinning, andafixedpointwiththe
1 One subset of these problems, the directed polymer (i.e.
same linear cusp non-analyticity as to one loop, hence
3 d=1)inarandompotential,mapsontotheKPZgrowth
confirming the consistency of the picture.
1
problem, well known to exhibit a strong coupling phase,
0 The aim of this paper is to extend these methods to
which is out of reach of standard perturbative methods
5 the N-component model. We show how an extended β-
0 [8]. It is thus important to obtain a field-theoretical de-
function can be obtained and point out the specific fea-
/ scription of this phase, since the value and even the ex-
t turesofthecaseN >1. ForthecaseofO(N)-symmetric
a istence of its upper critical dimension is still a matter of
m considerable debate [9, 10]. disorderwecomputethefixedpointandroughnessexpo-
nent ζ to next order in ǫ=4−d, where d is the internal
-
d One method which holds promise to tackle this class dimension of the manifold. We then study the extrapo-
n of problems is the functional renormalization group lations to the directed polymer limit d = 1, and discuss
o (FRG). Although it was introduced long ago, within a thevariousscenariosforthestrongcouplingphaseofthe
c 1-loop Wilson scheme [12, 13, 14], it is, not so sur- equivalent N-dimensional KPZ growth equation. In one
:
v prisingly, hampered with difficulties, and only recently of them, a value for the upper critical dimension is esti-
i attempts have been made to push the method further mated.
X
[15, 16, 17, 18, 19, 20, 21, 22]. The main problem is
We consider the model for an elastic N-component
r
that the effective action at zero temperature becomes
a manifold
non-analytic at a finite scale, the Larkin scale, where
metastability appears. Although fixed points are acces-
1
sible in a d=4−ǫ expansion, non-analyticity results in H= ddx (∇u)2+V(x,u) (1)
apparent ambiguities in the renormalized perturbation Z 2
theory at T = 0 [16, 19]. These problems are absent at
T > 0 [23, 24] (at least at leading order and for N = 1) in a random potential with second cumulant
but since temperature is dangerously irrelevant, the fi- V(x,u)V(x′,u′) = δd(x − x′)R(u − u′), where u = ui
nite temperature description is rather complicated [21]. is a N-component vector. We derive general equations,
Until now, it has lead to a complete first-principle solu- and later focus on the O(N) isotropic case, noting
tion of ambiguities (and calculation of the β-function to R(u)=h(r)withr=|u|. Introducingreplicasweobtain
2
the replicated action: a= Rab = Tδab
b 2T2 k2+m2
Hn = ddx 1 (∇u )2− 1 R(u −u ) (2) α β
T Z 2T a 2T2 a b
Xa Xab
We now carry perturbation theory in the disorder and a b c d e f
computethe one-loopandtwo-loopcorrectionstotheef-
fective action Γ[u]. We use the usual power counting of
the T = 0 theory, identical to the case N = 1. Infrared g h i j k l
divergencesford=4−ǫonlyoccurinthe2-replicaterm,
whichatzeromomentumdefines the renormalizeddisor- m n p q
der;thereisnocorrectiontothesinglereplicaterm. The
graphical rules are depicted in Fig. 1. We use functional
FIG.1: Graphicalrules,oneloopandtwoloopdiagrammatic
diagrams,andmass regularization. The method and no-
corrections to thedisorder
tationsareidenticalto[19], towhichwereferfor details.
Here we only stress the differences with the case N =1.
The 1-loop correction to disorder (graphs α and β in
The property of renormalizability amounts to cancella-
Fig. 2) reads:
tion of the 1/ǫ poles between the two last terms in (5)
using I˜= Nd and I˜ − 1I˜2 = N2(1 +O(ǫ0)) [16]. The
δ1R(u)= 1[∂ijR(u)]2−∂ijR(0)∂ijR(u) I . (3) β-function (ǫ5) is obtAaine2d from (d3),4ǫ(4) and (7) as:
(cid:18)2 (cid:19)
1
Summation over repeated indices is implicit everywhere, −m∂ R(u)=ǫR(u)+ [∂ R(u)]2−∂ R(0)∂ R(u)
and I = G2 = m−ǫI˜with G =(k2+m2)−1. We de- m 2 ij ij ij
fine the dRikmeknsionless functionkδ1(R) := mǫδ1R (recog- +(∂ R(u)−∂ R(0)) 1∂ R(u)∂ R(u)−α (8)
ij ij ikl jkl ij
nizable by the parenthesis around the argument R). For h2 i
later use we also denote the bilinear form δ1(R,R) :=
The cancellation works perfectly for the normal parts.
δ1(R). This yields the standard 1-loop FRG equations,
Anomalous parts, to which we turn now, produce the
recalledbelow,and ∂ R developsa cusp non-analyticity
ij last term.
at u = 0 beyond the Larkin length scale L . For the
c We start with the anomalous part of the repeated
O(N) model one has ∂ R = h′δ + uˆ uˆ (h′′ − h′) =
ij r ij i j r counter-term:
h′′(0)δ + 1h′′′(0)r(δ +uˆ uˆ )+O(r2) and thus h′′′(0)
ij 2 ij i j
becomes non-zero at Lc (uˆ=u/|u|). δa1,1(R)=−(µij +νij)∂ijR(u)I˜2 , (9)
The 2-loopcorrections to disorder can be decomposed
into a “normal” part, which is the complete result when where we denote the limits of small argument v →0:
R(u) is analytic [15], and an “anomalous” part which
arises from non-analyticity. The normal part reads: µij :=∂iklR(v)∂jklR(v)|v→0 (10)
δ2R(u)=(∂ R(u)−∂ R(0))∂ R(u)∂ R(u)I (4) νij :=∂ijklR(v)(∂klR(v)−∂klR(0))|v→0 (11)
n ij ij ikl jkl A
+ 1∂ R(u)(∂ R(u)−∂ R(0))(∂ R(u)−∂ R(0)) I2. which, in general, are direction dependent. For a O(N)
ijkl ik ik jl jl
h2 i model, the third derivative tensor:
The first line stems from diagrams b and a of Fig. 1 re-
∂ R(v)=A(r)(δ vˆ +δ vˆ +δ vˆ)+B(r)vˆvˆ vˆ (12)
spectively and the second from g,h,i,j. One has I = ijk ij k ik j kj i i j k
A
k1,k2Gk1Gk2G2k1+k2 =m−2ǫI˜Aandwedenoteinanalogy withvˆ=v/|v|,A(r)=(rh′′−h′)/r2 andB(r)=(r2h′′′−
tRo δ1(R) the dimensionless function δ(2)(R) := m2ǫδ2R. 3rh′′+3h′)/r2,hasavˆ-dependentsmallv limit(12)with
The FRG β-function is then: A(0)=−B(0)=h′′′(0)/2. This yields:
−m∂mR|R0 =ǫ[R+δ1(R)+2δ2(R)−δ1,1(R)] , (5) µ =h′′′(0)2(1δ + N +1vˆvˆ ) (13)
ij ij i j
where the repeated 1-loop counter-term δ1,1(R) := 2 4
2δ1(R,δ1(R,R)) arises when reexpressing the bare dis- and, similarly one finds ν = N+1h′′′(0)2(δ −vˆvˆ ).
order R in (2), in terms of the dimensionless renormal- ij 4 ij i j
0 Let us first superficially examine the structure of the
ized one,defined as mǫR, as detailed in [19]. From(3) it
2-loop graphs, following the discussion in [19]. As for
reads:
N = 1, one can discard c = d = 0 from parity and
δ1,1(R)=[(∂ R−∂ R(0))∂ δ1(R) similarly set m+n = 0 and p+q = 0. One can then
ij ij ij
write:
−∂ δ1(R)| ∂ R]I˜2 (6)
ij u=0 ij
∂ δ1(R)=∂ R(∂ R−∂ R(0))+∂ R∂ R (7) δ(2)(R)=−(µ˜ I˜ +ν˜ I˜2)∂ R(u) (14)
ij ijkl kl kl ikl jkl a ij A ij ij
3
wherethefirsttermcomesfromgraphse(moreproperly, N ζ1 ζ10 ζ2 ζ20
from the sum of all graphs a to f) and the second from
1 0.2082980 0.2 0.0068573 0
graphsk+l (from the sum ofgraphsito l). Globalcan-
2 0.1765564 0.166667 0.17655636 -0.00555556
cellationofthe1/ǫpoleintheβ-functionworksprovided
2.5 0.1634803 0.153846 -0.000417 -0.00782058
µ˜ +2ν˜ =µ +ν . This then produces α =µ˜ /2in
ij ij ij ij ij ij
the FRG equation above. 3 0.1519065 0.142857 -0.0029563 -0.00971817
We can now use the methods introduced in [19] to an- 4.5 0.1242642 0.117647 -0.009386 -0.013583
alyzethetotal2-loopcontributiontotheeffectiveaction, 6 0.1043517 0.1 -0.0135901 -0.0155556
includingpossiblyambiguousgraphs. Onefirstcomputes 8 0.0856120 0.0833333 -0.0162957 -0.016572
Γ[u] in a region of u where no ambiguity is present, us-
10 0.0725621 0.0714286 -0.016942 -0.0166517
ing excluded replica sums, and constraints valid in the
12.5 0.0610692 0.0606061 -0.0165154 -0.0161654
zero-temperature theory (the so-called sloop elimination
15 0.0528216 0.0526316 -0.01564 -0.0154217
method, Section V.B in [19]). One finds that extraction
17.5 0.046595 0.0465116 -0.0147 -0.014608
of the 2-replica part yields µ˜ = µ and 2ν˜ = ν ,
ij ij ij ij
i.e. it works as for N = 1. This is equivalent to renor- 20 0.0417 0.0416667 -0.0138 -0.013804
malizability diagram by diagram, and thus it satisfies 0.2 0.01
the global renormalizability condition. The background 0.005
0.15
method also yields that result ([19], Section V.C). The
5 10 15 20
end result for the β-function, α =µ , although unam- 0.1 -0.005
ij ij
biguousforN =1,needsfurtherspecificationforN >1, 0.05 -0.01
since the limit in (13) is direction dependent. -0.015
5 10 15 20
Another important consideration for the resulting β-
function is the issue of the “super-cusp”. For N = 1 it
FIG. 2: Numerical results for the exponents ζ1 and ζ2 for
wasfound that the β-function is suchthat the cuspnon-
′′ different values of N (top). Numerical plots of ζ1(N) (bot-
analyticity of R (u) at u = 0 does not become worse at tom/left) andζ2(N)(bottom/right),inbluewiththenumer-
twoloops. Thatbyitselfconstraintstheamplitudeofthe ical values from the table as dots. The red curves (no dots)
anomalousterm, since any other choice yields a stronger represent the asymptotic expansion.
singularity[28]. Wenowpointoutthatifv andu,in(9),
(10), (14), are colinear, i.e. µ (v)=µ (u) then there is
ij ij
originanddecaysto0atinfinityfasterthanapowerlaw,
no super-cusp. Indeed the result:
thus corresponding to short range (SR) disorder. Find-
1 ingthe associatedζ =ζ ǫ+ζ ǫ2+O(ǫ3)is aneigenvalue
α (uˆ)= lim ∂ R(ruˆ)∂ R(ruˆ) (15) 1 2
ij r→02 ikl jkl problem, which has to be solved order by order in ǫ fol-
obviouslyyieldscancellationofthelinearterminuin(8) lowing[16,18,19]. Ourresultsforζ1 andζ2 aregivenon
(although it is not the only possibility [27]). Colinearity Fig. 2. Although for SR disorder no analytical expres-
of v = ua −uc and u = ua −ub is natural if one com- sion can be found for ζ1 and ζ2, their large-N behavior
putes the effective action in a background configuration canbeobtainedfromanasymptoticanalysisof(16). Let
breaking the rotational symmetry, which appears to be us extend the analysis of Balents and Fisher (BF) [13].
required for the present theory to hold. Define h=1/Nhˆ, y =r2/2 and hˆ(r)=Q(y). For y ≫1
We now specialize to the O(N) model. Starting from the FRG equation can be linearized:
(8) and further rescaling h(r) → m−4ζh(rmζ), using ζ
′ ′′ ′′ ′′′
(ǫ−2ζ)Q +2ζyQ −(A+3B)Q −2ByQ =0 (17)
we obtainthe followingFRG flow-equationto two loops:
−m∂mh(r)=(ǫ−4ζ)h(r)+ζrh′(r) with A = (1 − N1)Q′0 + N4N−21hˆ′′′(0)2 and B = N1Q′0 +
1 N+3ˆh′′′(0)2, hˆ′′(0) = Q′(0) = Q′. BF noted that there
+ h′′(r)2−h′′(0)h′′(r) 8N2 0
2 is an overlapping region 1 ≫ y ≫ N where the solution
N −1h′(r) h′(r) can also be found perturbatively by expansion in 1/N,
′′
+ −2h (0) yielding for Q a pure exponential. It is indeed an exact
2 r (cid:18) r (cid:19)
solutionof(17),withauniquevalueforζ ,theBFresult
1 1
+ (h′′(r)−h′′(0)) h′′′(r)2 ζ1 ≈ ζ10 with ζ10 = 1/(4+N) (i.e. the result from the
2
replica variational method [25]). The corrections (which
N−1(h′(r)−rh′′(r))2(2h′(r)+r(h′′(r)−3h′′(0))) arise from the neglected non-linear terms) are shown to
+
2 r5 be exponentially small; a more accurate estimate being
N +3 N −1h′(r) ζ ≈ζ1 withζ1 =ζ0+(N+2)2/(N+4)22−(N+2)/2/(4e).
−h′′′(0)2 h′′(r)+ . (16) 1 1 1 1
(cid:20) 8 4 r (cid:21) To next order we find similarly the approximation to ζ2
[29]:
where the last line arises from the anomalous term (15).
ThisFRGequationadmitsforanyN anon-trivialattrac- (N2−1)(2+N)
tive fixed point such that h′′(r) has a linear cusp at the ζ20 =−2(4+N)3(3+N) , (18)
4
where we havenotattempted to estimate further correc-
tions, presumably again exponentially small at large N.
We note that ζ0 arises from the anomalous terms only.
2 [1] G. Gru¨ner, Rev. Mod. Phys. 60, 1129 (1988). A. Rosso
These estimatesarelistedandplottedonFig.2together
and T. Giamarchi, Phys.Rev.B 70, 224204 (2004).
with the numerical solution of (16). The quality of the [2] G. Blatter et al., Rev.Mod. Phys. 66, 1125 (1994).
large-N analysis is quite remarkable. [3] T.Giamarchi and P. Le Doussal, Phys. Rev. B 52, 1242
We now discuss the extrapolation of our result to the (1995). T. Nattermann and S. Scheidl, Adv. Phys. 49,
directed polymer (DP) case d = 1, ǫ = 3, plotted in 607 (2000).
Fig. 3. We see that the 2-loop corrections are rather big [4] S. Moulinet, C. Guthmann, E. Rolley, Eur. Phys. J. A8
437 (2002)
at large N, so extrapolation down to ǫ = 3 is difficult.
[5] S. Lemerle et al. Phys.Rev.Lett. 80, 849 (1998).
However both 1- and 2-loop results as well as the Pade-
[6] See reviews in Spin glasses and random fields Ed. A. P.
(1,1) reproduce well the two known points on the curve:
Young, World Scientific, Singapore, 1998.
ζ = 2/3 for N = 1 [8] and ζ = 0 for N = ∞ [20]. This [7] K. Efetov, A.Larkin,Sov.Phys.JETP 45, 1236 (1977).
branch in Fig. 3 corresponds to zero temperature and a [8] M.Kardar,G.Parisi,andY.-C.Zhang,Phys.Rev.Lett.
continuummodel. Ontheotherhandwefindthatforall 56, 889 (1986). M. Kardar, Nucl. Phys. B 290, 582
curves in figure 3 the roughness ζ becomes smaller than (1987).
the thermal ζ = 1 at N = N ≈ 2.5. This naturally [9] B.M. Forrest and L.-H. Tang, Phys. Rev. Lett. 64, 1405
th 2 uc (1990); J.M. Kim, M.A. Moore, and A.J. Bray, Phys.
suggests the scenario that at non-zero temperature ζ =
Rev. A 44, 2345 (1991);
1/2forN ≥Nuc,i.e. Nuc isthe upper criticaldimension [10] E. Marinari, A. Pagnani, and G. Parisi, J. Phys. A 33,
[2]. Thesameargumentgivesanuppercriticaldimension 8181 (2000).
Nuc for the KPZ-equation of non-linear surface growth [11] M. L¨assig and H. Kinzelbach, Phys. Rev. Lett. 78, 903
[8, 11]. On the other hand, simulations on discretized (1997),M.L¨assig,Phys.Rev.Lett.80,2366-2369(1998).
models of both the directed polymer (at T = 0) and [12] D.S. Fisher, Phys.Rev.Lett. 56, 1964 (1986).
[13] L.BalentsandD.S.Fisher,Phys.Rev.B48,5959(1993).
the KPZ equation [9, 10] suggest that ζ > 1/2 in all
[14] T. Nattermann et al., J. Phys. (Paris) 2, 1483 (1992).
dimensions,butshouldbe takenwithcaution[30]. Since
O. Narayan and D. S. Fisher, Phys. Rev. B 46, 11520
the FRG is a systematic expansion in ǫ = 4−d, such a
(1992).
scenario seems reconcilable with our above results only [15] H. Bucheli et al., Phys.Rev.B 57, 7642 (1998).
through non-perturbative corrections in ǫ, possibly non- [16] P. Chauve, P. Le Doussal and K.J. Wiese, Phys. Rev.
analytic at ǫ=2. Lett. 86, 1785 (2000).
To conclude we have obtained for the N-component [17] S. Scheidl, unpublished; S. Scheidl and Y. Dincer,
cond-mat/0006048;Y.Dincer,Diplomarbeit,K¨oln1999.
model a FRG description at 2-loop order. Various stud-
[18] P. Le Doussal, K.J. Wiese and P. Chauve, Phys. Rev. B
ies, including at large N, are under way to obtain a bet-
66, 174201 (2002).
ter understanding of the structure of the theory. For the
[19] P. Le Doussal, K.J. Wiese and P. Chauve, Phys. Rev. E
KPZgrowthandthedirectedpolymerwehaveimproved 69, 026112 (2004).
the determination of the possible upper critical dimen- [20] P. Le Doussal and K.J. Wiese, Phys. Rev. Lett. 89,
sion. Further numerics, in particular for the directed 125702 (2002); Phys.Rev.B68, 174202 (2003); Nuclear
polymer at T =0 would be helpful. Physics B 701, 409 (2004).
[21] L. Balents and P. Le Doussal, Europhys. Lett. 65, 685
(2004).
ζ [22] L.Balents and P.LeDoussal, cond-mat/0408048, toap-
pear in Adv.in Physics.
0.8 [23] P. Chauve et al., Phys. Rev.B 62, 6241 (2000).
[24] P. Chauve and P. Le Doussal, Phys. Rev. E. 64, 051102
(2001).
0.6 [25] M. M´ezard and G. Parisi, J. Phys. I 1, 809 (1991).
[26] K.J. Wiese and P.Le Doussal, in preparation.
[27] For the O(N) model one must have α = αδ +βuˆ uˆ
0.4 ij ij i j
and the absence of super-cusp reads α(1+N)+2β =
1+Nh′′′(0)2.
2
0.2 [28] An alternative scenario is that a non-linear cusp with
1-loop dimension-dependent exponent develops, e.g. h′′(r) ∼
2-loop direct Pade r1+Aǫ. It may require deviations from the simplest sce-
N nario of a uniform density of shocks of codimension one.
5 10 15 20 [29] Weusethe1-looprelation(ǫ−2ζ)h′′(0)+N+3h′′′(0)2 =0.
4N
[30] Note that theUV-cutoffof theRSOSmodel in [10] is in
FIG.3: Resultsfortheroughnessζ at1-and2-looporder,as the middleof thescaling plots for high dimensions.
a function of the numberof components N. Weboth show a
direct extrapolation and thePade (1,1): ζPade = 1−ǫǫζζ21/ζ1.
