(4+N)-Dimensional Elastic Manifolds in Random Media: a Renormalization-Group Analysis

18 Pages·0.23 MB·English

by  H. Bucheli

N
(4+ )-Dimensional Elastic Manifolds in Random Media:
A Renormalization-Group Analysis
H. Buchelia, O.S. Wagnera, V.B. Geshkenbeina,b,
A.I. Larkinb,c, and G. Blattera
aTheoretische Physik, ETH-H¨onggerberg, CH-8093 Zu¨rich, Switzerland
8 bL. D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia
9
cTheoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455
9
(August 1997)
1

n
a
J

6
Motivated by the problem of weak collective pinning of vortex lattices in high-

temperature superconductors, we study the model system of a four-dimensional elastic
]
n manifold with N transverse degrees of freedom (4+N-model) in a quencheddisorder en-
o vironment. We assume the disorder to be weak and short-range correlated, and neglect
c thermal effects. Using a real-space functional renormalization group (FRG) approach,
-
r we derive a RG equation for the pinning-energy correlator up to two-loop correction.
p The solution of this equation allows us to calculate the size R of collectively pinned
c
u
elastic domains as well as the critical force F , i.e., the smallest external force needed
c
.s to drive these domains. We find Rc ∝ δpα2 exp(α1/δp) and Fc ∝ δp−2α2 exp(−2α1/δp),
at where δp ≪ 1 parametrizes the disorder strength, α1 = (2/π)N/28π2/(N + 8), and
m α2 = 2(5N +22)/(N +8)2. In contrast to lowest-order perturbation calculations which
- we briefly review, we thusarrive at determining both α1 (one-loop) and α2 (two-loop).
d
n
o
c
[

1 I. INTRODUCTION
v
6 ThegenericproblemofaD-dimensionalelasticmanifoldwithN transversedegreesoffreedomsubject
3
toquenchedrandomimpuritiesthatcanpinthemanifoldhasattractedagreatdealofattentionformany
0
years. It has applications in numerous areas of physics such as dislocations, spin systems, polymers,
1
charge density waves, and vortex lattices in type-II superconductors. The case N = 1, for instance,
0
8 correspondstoD-dimensionalinterfacesseparatingtwocoexistingphasesin(D+1)-dimensionalsystems.
9 Onintermediatedistancesthefluxlatticeinsuperconductorsisbelievedtobehavelikeanelasticmanifold
/ with D =3 and N =2.
t
a The crucial quantity for defining order in these systems is the disorder-averagedsquare of the relative
m displacement u2(x) := [u(x) u(0)]2 , which describes the roughening of the N-component displace-
- ment field u(xh) of Di-dimhension−al suppoirt. In general, u2(x) depends on the distance R := x, the
d h i | |
dimensionalities D and N, and on the type of disorder. In the present paper, we choose the disorder to
n
be weak, short-range correlated, and of Gaussian type. Under these assumptions it has been shown [1]
o
c thatinlessthanfourdimensions (D 4)quencheddisorderalwaysdestroysthe translationallong-range
≤
: orderofthemanifold,breakingitupintocorrelateddomainsofextentR . Eachofthesedomainsbehaves
v c
elastically independently and is pinned individually. The length scale R over which short-range order
i c
X subsists is commonly referred to as the collective pinning radius and is usually defined by the criterion
r u2(Rc) ξ2, where ξ is the length scale describing the internal structure of the manifold [1,9]. For
a h i ∼
distancesRsmallerthanR (perturbativeregime),thegroundstateofthesystemisuniqueandthemean
c
square displacement u2(R) can be calculated perturbatively. On the other hand, for R > R (random
c
h i
manifold regime), there are many competing metastable minima inducing the so-called “wandering” of
the manifold as described by the scaling law u2(R) ξ2(R/R )2ζ(R R ). A lot of effort has gone
c c
h i ≈ ≫
into the determination of the wandering exponent ζ, using elaborate techniques such as the replica for-
malism combined with either variational approaches [2,3] or with the renormalization group [4–8] (RG).
Here, we make use of RG methods within the perturbative regime to determine the collective pinning
radius R . For that purpose, we will replace the estimate u2(R ) ξ2 by an improved definition of R
c c c
h i∼
1
basedonthedivergenceofthefourthderivativeofthepinning-energycorrelator,signalingtheappearance
of competing ground states at this length scale. The collective pinning radius R is not only a relevant
c
quantity for characterizing order,but also determines dynamic properties such as the minimum force F
c
needed to move the pinned manifold (critical force) and the activation barrier U for creep.
c
In this work we concentrate on the particularly interesting case D = 4, the upper critical dimension
of this problem, where the mean-square displacement shows a logarithmic behavior in the perturbative
regime [10], u2(R) lnR, (R < R ). This logarithmic situation motivates the use of renormalization
c
h i ∝
groupmethods. The (4+N)-dimensionalmodel system is stronglyrelatedto the problemof weak collec-
tive pinning of a (3+2)-dimensional manifold with dispersive elastic moduli, an issue which has received
a lotofinterestinconnectionwithpinning andcreepbehaviorofvorticesinhigh-T superconductors. In
c
a future paper [20], the RG technique developed below will be applied to that topic (see also Ref. [19]).
The main objects of interest in our investigations are the collective pinning radius R and the critical
c
force density F . In section II, we introduce the non-dispersive (4+N)-model and define the static and
c
dynamicGreen’sfunctionsaswellastherelevantcorrelationfunctions. Forsimplicity,weneglectthermal
effect,i.e.,wesetT =0. Then,insectionIII,wederiveR andF bymeansoflowest-orderperturbation
c c
calculations combined with scaling techniques and the dynamic approach, and discuss the problem with
thesemethods. SectionIVisdevotedtotherenormalizationgrouptreatmentofthepinningproblem. We
construct the RG transformation and derive a functional RG equation for the pinning energy correlator
in one and two-loop approximation. We show how R can be obtained by solving this equation and
c
determine the critical force density F using simple scaling relations. We discuss various aspects making
c
this technique superior to those presented in section III. In a second step, we apply these results to the
dynamical situation, where the system is driven with an external force. In particular, we determine the
behavior of the friction coefficient η under the action of the RG and give an alternative derivation of F .
c
Finally, in section VI we summarize and present our conclusion.
II. MODEL
A. Non-dispersive (4+N)-model
We considera four-dimensionalelastic manifoldina (4+N)-dimensionalspace inthe continuumlimit:
A point ofthe manifold is representedby a four-componentvector x (internal degreesof freedom), while
the displacement relative to the equilibrium position at that point is characterized by the continuous
N-component vector field u(x) (transverse degrees of freedom). The elastic part of the free energy is
given by the isotropic expression
1
[u]= C d4x uα(x) uα(x),
el
F 2 ∇ ·∇
Z
whereα=1,2,...,N;asusual,indicesappearingtwiceareimplicitlysummedover. Inthepresentwork,
the elastic modulus C is taken to be a constant (non-dispersive model).
The elastic manifold is embedded in a random environment which we model as a Gaussian random
pinning potential U (x,s) with zero mean and short-range correlations,
pin
U (x,s)U (x′,s′) =γ δ4(x x′)δN(s s′). (1)
pin pin U
h i − −
The parameter γ is a measure of the disorder strength and is assumed to be small. Angular brackets
U
... denote the averageoverallpossible realizationsofdisorder. Note that U depends onthe internal
pin
h i
as well as on the transverse degrees of freedom of the system. For a given configuration u(x) and a
random, but fixed disorder, the energy density arising from the interaction of the manifold with the
disordered potential is the convolution of the potential U with the form factor p(s):
pin
E (x,u(x))= dNsU (x,s)p(s u(x)). (2)
pin pin
| − |
Z
Thefunctionp(s)characterizestheinternalstructureofthemanifold. Inthefollowing,wewilladoptthe
simple expression
2
s2
p(s)=exp , (3)
−ξ2
(cid:18) (cid:19)
where ξ is chosen as the smallest resolvable length scale of the system. The effect of the form factor is
to smear the random potential U over a length ξ, so that the typical effective distance between two
pin
‘valleys’ in the energy landscape is of order ξ.
The total free energy of the manifold in the presence of disorder is eventually
1
[u]= [u]+ [u]= d4x C uα(x) uα(x)+E (x,u(x)) , (4)
el pin pin
F F F 2 ∇ ·∇
Z (cid:20) (cid:21)
with the pinning energy correlator
E (x,u(x))E (x′,u′(x′)) =δ4(x x′)K (u(x) u′(x′)), (5)
pin pin ξ
h i − −
where
π N u2
K (u)=γ ξN 2 exp , u:= u , (6)
ξ U 2 −2ξ2 | |
(cid:16) (cid:17) (cid:18) (cid:19)
asobtainedfromEqs.(1)–(3). TheexponentialinthecorrelatorK (u)isdirectlyrelatedtotheexpression
ξ
for the form factor, Eq. (3).
The equilibrium configuration u(x) of the system results from the competition between the elastic
energy the system has to pay for a distortion and the pinning energy the system can win by accommo-
dating to the impurity potential. Unfortunately, determining these configurations reveals an impossible
undertaking because of the randomness of E . One must therefore content oneself with calculating
pin
disorder-averagedquantities, relying on the system’s assumed property of self-averaging.
B. Green’s functions
Minimizing the free-energy functional (4) with respect to u(x),
δ
[u]=0, α=1,...,N, (7)
δuα(x)F
defines the equation of state,
C 2uα(x)=Fα (x,u(x)), (8)
− ∇ pin
where F stands for the pinning force density,
pin
δ ∂
Fα (x,u(x)):= [u]= E (x,u(x)). (9)
pin −δuα(x)Fpin −∂uα pin
The equation of state (8) can equivalently be written in the form
uα(x)= d4yGαβ(x y)Fβ (y,u(y)), (10)
− pin
Z
where
1
Gαβ(x)=δαβG(x), G(x)= , x:= x , (11)
(2π)2Cx2 | |
is the static Green’s function of the system defined by the relation
C 2G(x)=δ4(x).
− ∇
3
For the dynamic approach in section III we need the time-dependent Green’s function Gαβ(x,t). We
assume a dissipative dynamics for our elastic manifold which we characterize by the friction coefficient
η . With the equation of motion
◦
∂
C 2+η uα(x,t)=Fα (x,u(x,t)),
− ∇ ◦ ∂t pin
(cid:18) (cid:19)
the dynamic Green’s function takes the form Gˆαβ(k,ω) = δαβ/(Ck2 iη ω) in Fourier representation.
◦
−
Transforming back to real space, one readily obtains
Θ(t)η η x2
Gαβ(x,t)=δαβG(x,t), G(x,t)= ◦ exp ◦ . (12)
16π2C2t2 −4Ct
(cid:18) (cid:19)
The Heaviside-function Θ(t) reflects the causality of the Green’s function.
C. Correlation functions
For later use we define the derivatives of E ,
pin
∂ ∂
Eα1···αl(x,u(x)):= E (x,u(x)), α ,...,α =1,...N, l =1,2,3,... ,
pin ∂uα1 ···∂uαl pin 1 l
and the corresponding correlators,
Eα1···αl(x,u(x))Eβ1···βn(x′,u′(x′)) =( 1)nδ4(x x′)Kα1···αlβ1···βn(u(x) u′(x′)), (13)
h pin pin i − − ξ −
where naturally Kαβ···(u) := (∂/∂uα)(∂/∂uβ) K (u). Each partial derivative ∂/∂u′α contributes a
ξ ··· ξ
minus sign, thus giving rise to the factor ( 1)n. In particular, the force-force correlator is given by
−
π N uαuβ u2
Kαβ(u)= γ ξN−2 2 δαβ exp . (14)
ξ − U 2 − ξ2 −2ξ2
(cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19)
Of special interest in section IV will be the curvature of the force at the origin,
Kαβγδ(0)=Γ δαβδγδ+δαγδβδ+δαδδβγ =:Γ ∆αβγδ, (15)
ξ ξ ξ
(cid:0) (cid:1)
N
with Γ :=γ ξN−4 π 2.
ξ U 2
(cid:0) (cid:1)
III. PERTURBATION THEORY
In this section we presentthe usual ways to define and determine the collective pinning radius R and
c
the criticalforcedensityF . First,wecalculateR by alowest-orderperturbationcalculation,andassess
c c
F by dimensional estimates [11]. Second, we use the dynamic approach [12,13] to determine F in an
c c
alternative manner. We also point out the disadvantages of these simple methods.
A. Dimensional estimates
The collective pinning radius R is usually derived by computing the fluctuations of the displacement
c
field u(x) induced by the random environment,
u2(R) := [u(x) u(0)]2 , R:= x ,
h i h − i | |
in the absence of an external force, F =0. The condition u2(R ) ξ2 defines the collective pinning
ext c
h i≃
radius R which represents the extension of a collectively pinned elastic domain. For small fluctuations,
c
4
u2(R) ξ2, one can use a perturbative approach[11]. The starting point is the integral equation (10).
h i≤
We expand the pinning force, F (y,u(y)) = F (y,0)+O(u(y)), and only retain the lowest-order
pin pin
term, so that Eq. (10) reads (Gαβ =δαβG)
uα(x)= d4yG(x y)Fα (y,0). (16)
− pin
Z
From this expression and by means of Eqs. (9) and (13), we obtain
uα(x)uα(x′) = Kαα(0) d4yG(x y)G(x′ y)+O(γ2).
h i − ξ − − U
Z
Next, we write the Green’s functions in Fourier representation;the mean square of the relative displace-
ment then takes the form
d4k
u2(R) = 2Kαα(0) 1 cos(k x) Gˆ(k)2+O(γ2). (17)
h i − ξ (2π)4 − · U
Z (cid:16) (cid:17)
Surprisingly enough, this lowest-order perturbation expression is true to all orders in γ [14] (this can
U
be checkedby keepingu(y) on the rhs.of Eq.(16)andgoing throughthe subsequentcalculations). Two
elements areresponsible for this striking fact: the short-rangecorrelation δ4(x y) in E E and
pin pin
∝ − h i
the implicit assumption made in this derivation that the system is in a unique state, which is only valid
in the perturbative regime. The integralin Eq.(17) is invariantunder space rotations;one cantherefore
choose x parallel to the x -axis and introduce cylindrical coordinates (K2 :=k2+k2+k2),
4 1 2 3
2Kαα(0) ∞ ∞ 1 cos(k R)
u2(R) = ξ 4πK2dK dk − 4 .
h i −(2π)4C2 4 (K2+k2)2
Z0 Z−∞ 4
Performing the K-integration yields
Kαα(0) ∞ 1 cos(k R)
u2(R) = ξ dk − 4 . (18)
h i − 4π2C2 4 k
Z0 4
The k -integration produces a log-contribution which diverges for k . For this reason, one has to
4 4
→ ∞
introduce an upper cutoff given by the inverse of the smallest length scale in the problem which is ξ.
With Kαα(0)= γ ξN−2N(π/2)N/2, Eq. (14), one finally obtains
ξ − U
N π N R
u2(R) 2 δ ln , R ξ, (19)
h i≈ 4π2 2 p ξ ≫
(cid:16) (cid:17) (cid:18) (cid:19)
where δ := γ ξN−4/C2 is the dimensionless disorder parameter. The collective pinning radius defined
p U
through the relation u2(R ) ξ2 then is
c
h i≃
N
4π2 2 2 1
R ξ exp , (20)
c
≃ N (cid:18)π(cid:19) δp!
whichisexponentiallylargeinthelimitofweakpinning,δ 1. WiththeuppercutoffinEq.(18)being
p
≪
given only up to a factor of order unity, the constant of proportionality in Eq. (20) is not unequivocally
determined. Much more relevant, though, is the uncertainty in the numerical factor in the exponential
function, which has its origin in the criterion u2(R ) ξ2, where the constant of proportionality is
c
h i ≃
assumed to be of order unity.
A rough estimate for the critical force density F can be gained by scaling. The typical elastic energy
c
of a collectively pinned domain is
2
ξ
U C V ,
c c
∼ R
(cid:18) c(cid:19)
where V :=R4 is the collective pinning volume. The energy gain arising from the action of an external
c c
force F shifting the domain by ξ is
ext
U F V ξ.
ext ext c
∼
5
Comparison between both energy scales gives a scaling estimate for the critical force density,
C ξ 2 C 8π2 2 N2 1
F exp . (21)
c
∼ ξ (cid:18)Rc(cid:19) ≃ ξ − N (cid:18)π(cid:19) δp!
B. Dynamic approach
In the dynamic approach [12,13] the critical force F is determined directly without any reference to
c
thepinningradiusR . Tobeginwith,thesystemisdrivenbyalargeexternalforcedensityF F . In
c ext c
≫
this regime, the elastic manifold does not noticeably feel the pinning potential and F is only opposed
ext
by the friction force F which we describe by the phenomenological expression F = η v, with
frict frict ◦
−
theviscosityη amaterialconstant. Theflowvelocityvofthemanifoldisdeterminedbythesteady-flow
◦
condition
1
F +F =0 = v=v := F .
frict ext ◦ ext
⇒ η
◦
If F is decreased, the system begins to sensibly interact with the randomly distributed impurities via
ext
the pinning force F . This becomes noticeable through fluctuations in the velocity field, leading to a
pin
reduction of the average velocity, v =v δv with v δv >0. The average of F can be interpreted
◦ ◦ pin
− ·
as an effective friction force with a velocity-dependent viscosity δη(v),
F = δη(v)v. (22)
pin
h i −
As before, the steady-flow velocity is given by the relation
F + F +F =0 = η(v)v:= η +δη(v) v=F . (23)
frict pin ext ◦ ext
h i ⇒
The applied force for which the relative fluctuations of the vi(cid:0)scosity are(cid:1)of order unity then furnishes a
useful criterion for the critical force density:
δη(F )
ext
=1. (24)
η
◦ (cid:12)Fext=Fc
(cid:12)
(cid:12)
Alternatively, a criterion can be formulated mak(cid:12)ing use of the reduction in the flow velocity, Fpin =
h i
η δv, and the condition δv(v )=v .
◦ c c
−
We now calculate the friction coefficient δη due to pinning within a perturbative approach up to first
order in δ . We start from Eq. (22),
p
vαδη = Fα (x,vt+u(x,t)) ,
−h pin i
and expand the pinning force into a Taylor series, keeping the first two terms,
vαδη Fα (x,vt) Fαβ(x,vt)uβ(x,t) .
≃−h pin i−h pin i
The first term vanishes; in the second one, we use the integral equation (10) generalized to the time-
dependent case and expanded to lowest order,
uα(x,t)= d4ydsG(x y,t s)Fα (y,vs).
− − pin
Z
Remembering Eq. (13) and the definition Fα = Eα , one obtains
pin − pin
vαδη dtG(0,t)Kαββ(vt), (25)
≃ ξ
Z
6
where we made use of the time-translation symmetry. Inserting the explicit expressionfor the correlator
Kαββ and for the time-dependent Green’s function (12), we find:
ξ
δη(v) N +2 π N ∞ dt v2t2
2 δ exp +O(δ2).
η ≃ 16π2 2 p t − 2ξ2 p
◦ (cid:16) (cid:17) Z0 (cid:18) (cid:19)
The integration produces a log-divergence for t 0. A lower cutoff is given by the time t the elastic
ξ
→
manifoldneedstorecoverfromadistortiononthesmallestlengthscaleξ. Balancingelasticanddynamic
terms in the Green’s function G(x,t), Eq. (12), on the scale ξ, we obtain
η ξ2
◦
t . (26)
ξ
≃ 4C
On the other hand, the exponential factor in the integrand, which originates from the correlator Kαββ,
ξ
provides an upper cutoff,
√2ξ
t , (27)
v
≃ v
describing the time scale for the interaction between the system and the pinning centers at velocity v.
With these two cutoffs, the ratio between δη and η becomes
◦
δη(F ) N +2 π N 4√2C
ext 2 δ ln +const. +O(δ2). (28)
η◦ ≃ 16π2 2 p " ξFext ! # p
(cid:16) (cid:17)
This expression is valid in the non-linear regime, i.e., for forces F <4√2C/ξ (so that t <t ), where
ext ξ v
the effective friction coefficient η(F ) = F /v(F ) deviates from η . The critical force F is found
ext ext ext ◦ c
from the result (28), using the criterion δη(F )/η 1 (for weak disorder δ 1, we expect F to be
c ◦ p c
≃ ≪
much smaller than C/ξ, so that the constant in Eq. (28) can be neglected). We obtain
N
4√2C 16π2 2 2 1
F exp , (29)
c
≃ ξ −N +2 (cid:18)π(cid:19) δp!
whichisindeedexponentiallysmall. Asinthepreviousmethod,boththeconstantofproportionalityand
the numerical factor in the exponential function can only be ascertained up to a number of order unity.
IV. RENORMALIZATION GROUP ANALYSIS
As we have seen in the previous section, the leading term in the fluctuations of the displacement field
u(x) shows a logarithmic behavior, u2 lnR2, cf. Eq. (19). The logarithmic dependence weights all
h i ∝
scales equally such that all length scales are relevant in the final result. This provides the motivation to
apply the renormalization group (RG). Our guiding line for this analysis will be the work of Efetov and
Larkin [14] dealing with the pinning problem of charge-density waves.
A. Construction of the RG transformation
Thefollowingreal-spacerenormalizationprocedurewassuggestedbyKhmel’nitskiiandLarkin[15]for
the problem of friction between two rough surfaces. In a recent publication [6], Balents, Bouchaud, and
M´ezardgave a precise formulation of this method within the framework of the momentum-shell RG and
used it to investigate the random manifold regime (large scales R > R ) of the ((4-ǫ)+N)-model; here,
c
we are interested in the perturbative regime (small scales R<R ) with the aim to determine R .
c c
Assume that we have renormalized the displacement up to a scale R > ξ; let us denote the corre-
1
sponding field by u (x). The associated free-energy functional reads
(1)
1
[u ]= d4x C uα (x) uα (x)+E (x,u (x)) .
F (1) 2 ∇ (1) ·∇ (1) pin,(1) (1)
Z (cid:20) (cid:21)
7
Next, we go over to a larger scale R >R and separate u (x) into a far and a near-field contribution,
2 1 (1)
u (x)=u (x)+w(x), with
(1) (2)
uα (x)= d4yGαβ(x y)Eβ (y,u (y)),
(2) − − pin,(1) (1)
Z
|x−y|>R2
wα(x)= d4yGαβ(x y)Eβ (y,u (y)), (30)
− − pin,(1) (1)
ZΩ
where Ω:= x,yR < x y <R . The free energy can then be written as
1 2
{ | | − | }
1
[u ]= d4x C uα (x) uα (x)+E (x,u (x))
F (2) 2 ∇ (2) ·∇ (2) pin,(2) (2)
Z (cid:20) (cid:21)
with the renormalized pinning energy density,
1
E (x,u (x))=E (x,u (x))+ C wα(x) wα(x)+2 uα (x) wα(x) .
pin,(2) (2) pin,(1) (1) 2 ∇ ·∇ ∇ (2) ·∇
h i
The mixedterm uα wα is zero whenintegratedoverand canbe omitted. Integratingthe quadratic
∇ (2)·∇
term by parts, one finds
1
E (x,u (x))=E (x,u (x)) wα(x)C 2wα(x). (31)
pin,(2) (2) pin,(1) (1) − 2 ∇
Inthelimitofzerotemperature,outofallthepossiblenear-fieldcontributionsw(x)onlythoseminimizing
the energy are relevant (in a statistical sense),
∂E (x,u (x))
pin,(2) (2)
=0.
∂wα
Combining this condition with Eq. (31), we can relate w(x) to the pinning force Eα ,
pin,(1)
C 2wα(x)= Eα (x,uα (x)). (32)
− ∇ − pin,(1) (1)
(It is interesting to note that by differentiating Eq. (31) with respect to uα , one shows that for the
(2)
pinning force
∂w
Eα (x,u (x))=Eα (x,u (x))+O ,
pin,(2) (2) pin,(1) (1) ∂u
(cid:18) (2)(cid:19)
holds, which tells us that the force correlator Kαβ(u=0) will not change under the RG.)
Eq. (31), together with Eqs. (30) and (32), is the starting point for our further considerations. It will
allow us to derive a functional renormalization group (FRG) equation for the pinning energy correlator
K (u), the subscript R denoting the scale of renormalization. We will then solve this equation by
R
expanding the relevant correlators around u = 0. It turns out that the fourth derivative, Kαβγδ(0), as
R
wellashigher-orderevenderivativesdivergeatafinitescalewhichweidentify withthecollectivepinning
radius R . Indeed, we know that on length scales smaller than R (perturbative regime), the system is
c c
in a unique state and the pinning energy as well as the related correlators are analytic functions in the
field u. At R the manifold starts to probe different energy ‘valleys’ and our calculation, which rests on
c
the analyticity of the pinning energy, breaks down, reflected by the singular behavior of the curvature of
K(u). Inwhatfollowswewilltakethe divergenceofKαβγδ(0)asanunambiguousdefinitionofR which
R c
allows one to go beyond the simple estimate u2(R ) ξ2.
c
h i≃
B. One-loop FRG equation
We first construct a formal expansion in w of E , Eq. (31), which we will then use to derive a
pin,(2)
perturbation series in δ of E E . Thereby, we will relate the energy correlatorsat renormal-
p pin,(2) pin,(2)
h i
ization scales R and R , K and K , respectively; this will directly lead to the FRG equation for
2 1 R2 R1
8
K (u). To simplify the notation, we will omit the subscript ‘pin’ and use the abbreviations
R
E¯ :=E (x ,u (x )), Eαβ···:=Eαβ··· (x ,u (x )),
j pin,(2) j (2) j j pin,(1) j (2) j
E¯′ :=E (x ,u′ (x )), E′αβ···:=Eαβ··· (x ,u′ (x )),
j pin,(2) j (2) j j pin,(1) j (2) j
K¯ :=K (u (x ) u′ (x )), Kαβ···:=Kαβ···(u (x ) u′(x )).
ij R2 (2) i − (2) j ij R1 2 i − 2 j
Let us furthermore define the operator
Gαβ( ):= d4x Gαβ(x x )( ).
ij ··· j i− j ···
ZΩ
We begin by expanding the first term on the rhs. of Eq. (31) with respect to w(x) into a power series:
∞
1
E (x ,u (x ))=E (x ,u (x )+w(x ))= Eα1···αlwα1(x ) wαl(x ).
(1) 0 (1) 0 (1) 0 (2) 0 0 l! 0 0 ··· 0
l=0
X
Next, we replace everywhere w by its implicit definition, Eq. (30). The force function Eβ under the
(1)
integralis in turnwritten as a Taylorexpansionas above. We proceediteratively, thereby eliminating w
from our equation. Grouping the different terms with increasing number of Green’s functions, we arrive
at
1
E (x ,u (x ))= E +EµGµνEν EµGµνEνρGρσEσ EµρGµνEνGρσEσ+
− (1) 0 (1) 0 − 0 0 01 1 − 0 01 1 12 2 − 2! 0 01 1 02 2
1
+EµGµνEνρGρσEσκGκλEλ+ EµGµνEνρκGρσEσGκλEλ+
0 01 1 12 2 23 3 2! 0 01 1 12 2 13 3
2 1
+ EµκGµνEνρGρσEσGκλEλ+ EµρκGµνEνGρσEσGκλEλ+... .
2! 0 01 1 12 2 03 3 3! 0 01 1 02 2 03 3
In a diagrammatic language this can be written as
0 0 1 0 1 2 0 µ(cid:8)ν(cid:8)u1 0 1 2 3
u + u u + u u u + Hu(cid:8) + u u u u +
µ ν µ ν ρ σ H µ ν ρ σ κ λ
ρ Hu
σ 2
0 1 ρ(cid:8)σ(cid:8)u2 0 µ(cid:8)ν(cid:8)1uρ σ2u 0 µ(cid:8)ν(cid:8)u1
+ u uH(cid:8) + 2 (cid:8)uH + H(cid:8)u u2 + ... .
µ ν κHHu · κHHu κHHu
λ 3 λ λ
3 3
A bullet with subscript ‘j’ and l legs α ,...,α stands for the expression ( Eα1···αl)/l!, whereas a solid
1 l − j
line with the indices ‘µ’ and ‘ν’ joining two bullets ‘i’ and ‘j’ represents Gµν, i.e., the Green’s function
ij
Gµν(x x ), plus an integration over x Ω. The factor 2 in front of the penultimate term is the
i j j
− ∈
symmetry factor which accounts for the fact that the expansion gives the two equivalent diagrams
(cid:8)u u (cid:8)u
(cid:8) (cid:8)
(cid:8)uH = (cid:8)uH .
H H
Hu Hu u
In a similar way, we expand the second term on the rhs. of Eq. (31), making use of Eq. (32):
1 1
wα(x )C 2wα(x )= wα(x )Eα (x ,u (x )+w(x ))=... .
−2 0 ∇ 0 −2 0 (1) 0 (2) 0 0
Adding the series for E (x ,u (x )) and 1wα(x )C 2wα(x ) we finally obtain
(1) 0 (1) 0 −2 0 ∇ 0
9
(cid:8)u
(cid:8)
E¯ = u+ 1 u u+ 1 u u u + 1 u u u u + 1 u (cid:8)Hu
− 0 2 2 2 2 HHu
(cid:8)u (cid:8)u
(cid:8) (cid:8)
1 uH(cid:8) u + 1 u u u u u + 1 u u uH(cid:8)
− 2 HHu 2 2 HHu
u
(cid:8)(cid:8)u u (cid:8)(cid:8)u (cid:8)(cid:8)u u (cid:16)(cid:17)(cid:16)(cid:17)u
+ u uH(cid:8)HHu + 12 u H(cid:8)u HHuu − 23 uH(cid:8)HHuu − QP(cid:16)u(cid:17)QPQPu+... .
(33) u
Let us now turn to the energy-energy correlation function, hE¯0E¯0′′i, where xj′ := x′j. We write both
energy functions in an expansion as done above. Using the linearity of the averaging procedure, we
obtain a seriesof n-pointcorrelatorsof the form E(′)α1β1··· E(′)αnβn··· . Assuming Gaussiandisorder,
h ··· i
we can make use of Wick’s theorem: n-point functions with odd n vanish and those with even n can be
decomposed into a sum of products of 2-point correlators. For instance, application of Wick’s theorem
transforms the second-order term
Gµ01νGρ0′σ1′hE0µE1νE0′ρ′E1′σ′i
into the expression
Gµ01νGρ0′σ1′ hE0µE0′ρ′ihE1νE1′σ′i+hE0µE1′σ′ihE1νE0′ρ′i+hE0µE1νihE0′ρ′E1′σ′i ,
h i
diagrammatically represented by
0 1 0 1 0 1
u u u u u u
.
u u u u u u
0′ 1′ 0′ 1′ 0′ 1′
Dotted lines stand for 2-point correlation functions; the number of these lines indicates the order in δ
p
of the corresponding diagram.
The number of diagrams is considerably reduced by the special properties of the correlations. First,
the correlator K (u), being an even function in u, has all its odd derivatives vanishing at u = 0,
R1
Kα1···αl(0) = 0 (l = odd). Therefore, each term containing an odd correlator connecting two points ‘i’
R1
and ‘j’ on the same ‘tree’ (i.e., already linked by a solid line) vanishes,
Eα1···αlEβ1···βn =( 1)nδ4(x x )Kα1···αlβ1···βn(u (x ) u (x ))
h i j i − i− j R1 (2) i − (2) j
=( 1)nKα1···αlβ1···βn(0)=0 (l+n=odd).
− R1
Furthermore, terms with a correlation between two neighboring points ‘i’ and ‘j’ joined by a solid line,
Eα···Eβ··· δ4(x x ), give no contribution, because the delta function has no overlap with the
h i j i ∝ i − j
domain of integration Ω. An example is given by the last diagram above. The remaining non-vanishing
diagrams up to second order in δ (zero and one-loop diagrams) are
p
10