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Functional renormalization
• instead of a few coupling constants, follow fonction(s) under RG
wetting transition, interacting fermions,.. ,
• in systems with quenched disorder
may be necessary follow proba distribution
- strong disorder FP of disordered quantum spin chains
- freezing transitions in some 2D models disordered Coulomb gas
RSRG
- FRG for disordered elastic systems and random field problems
Windsor Summer School, August 2004
Pierre Le Doussal, CNRS-LPTENS Paris
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FRG for disordered elastic systems
Lectures I and II: • statics
Lecture III • dynamics: depinning and creep
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program
• model of everything
• simpler model, with and without
• perturbation theory
• functional RG
• results and physics
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Bragg Glass:
Periodic object + weak disorder Abrikosov vortex lattice
No dislocationNo translational order
Neutron diffraction
decoration
Klein et al. KBaBiO
divergent Bragg peaks
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Ingredients
• field
• elastic energy
• disorder energy
interactions
deformation
substrate impurities
describe: Gibbs equilibrium at temperature T
want: roughness
correlation functions
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field
target space
internal space
displacement
deformation
heigth
is flat ground state
x
d=1 directed polymer in dimension N
particle in dimension N d=0
u(x)
x1
x2
d=2, N=1 interfacemagnetic domain wall in 3D
magnetic DW in 2D film: d=1 N=1
embedding space
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Elastic energy
N=1
CDW: N=1
vortex lattice: N=2 (line cristal)
periodic objects (cristals)
in d=3
general
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Thermal roughness
N=1
flat d>2
d=2
free field power counting
freepropagator
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Disorder energy
u(x)
x1
x2
random potential
• short range in internal space: point impurities
interfaces
random bond
random field
CDW, vortex lattice(Bragg glass)
is • long range
magnetic DWthe function
• short range
• periodic
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Random Bond vs Random Field
unsatisfied bonds
• random fieldsx
u
u
• random bonds
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u(x)
x1
x2
starting model: summary
• random bond
• random field
force has SR correlations
• random periodic we want:
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directed polymer
random variables on each bond
path energy of
find optimal path
of minimal energy
Exact results for N=1
Bethe Ansatz n to 0
Probability theory
DP mapped to KPZ growth in N dimension
and Burgers equation
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perturbation around thermal FPpower counting assume
PT around thermal FP choose
if AND• SR disorder irrelevant
• SR disorder relevant
directed polymer, KPZ growth
• STRONG disorder phase optimization
FLORY
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Replicas!!!
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Replica averages
n=0 implicit
n=0 implicit
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Replicas: summary
off diagonal correlations(disorder)
connected (thermal)
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A simpler model I
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Cumulants
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Replica matrix inversion
if
cyclic matrix
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A simpler model II
disorder: thermal:
unchanged by disorder
will remain true in full model
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Simpler model: without
in given disorder realization:
no barrier
no pinning
responds elastically
single minimum
zero T fixed point
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Full model: replicated action
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Cumulants
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Program..• perturbation theory
• divergences
power countingUV cutoff
IR cutoff
here:harmonic well
• renormalization • Wilson
integrate out fast modes
rescale to get fixed form
field theoretic
compute renormalized vertices
from effective action
• HERE:
obtain in terms of
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Perturbation theory I T=0
inversiondone before
quadratic part of action is Larkin random force model
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power counting
invariant if
is it stable ?Larkin RF model T=0 fixed point
All relevant d<4!!
should flow away from Larkin’s random force model
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surprise!
to ALL orders in perturbation theory !!
DIMENSIONAL REDUCTION:
observables at T=0 are the same as the pure model in d’ = d-2 to all orders in PT
• Random field Ising model
• CDW: Larkin Efetov 77
in
here:
,etc..
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Toy DimRed
minimum
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Sophisticated DimRed
solution of
Parisi-Sourlas-Cardy
represented as:
Written as supersymmetric theory
Correlations identical (non perturbatively) to pure theory in d-2 at T >0
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Why is DimRed wrong here ?
• single mode approximation (d=0)
H
u
changes sign
several local extrema (typically)
increase V
decrease m
H
u
Beyond the Larkin length: GLASS
For
• more general
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What does DR compute?
number of unstable directions
• DR computes average over all extrema with weigth
• we want average over absolute minimum only
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Is there hope ?
- the problem is highly non linear
- infinity of operators seem relevant
- perturbation theory is TRIVIAL so it is not clear HOW to handle them
Who said glass physics was simple ?
• T=0 effective action remains non trivial
• T>0 perturbation theory is non-trivial
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Effective action functional
• action
generates connected correlations
• effective action
renormalized (proper) vertices
correlations functionsare tree diagrams from
cannot be disconnected by cutting a line
1-particle irreducible diagrams
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Property
is unchanged by disorder
c,m are uncorrected
one replica part
set c=1invariance of disorder by
Definition :
renormalized disorder defined from q=0
below bare disorder denoted
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perturbation theory for
3-replica
vanishes at T=0 loop
higher power in only left:
Log divergent in d=4
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Details
3 replica term
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FRG
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FRG via Wilson: mode elimination
split fast and slow
notations
q
q
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FRG via Wilson: rescaling
stop RG at
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T=0
start with R(u) analytic
recover DR
Analysis of FRG equation
non analytic FP possible
R(u) becomes non-analytic at u=0 beyond Larkin scale
O(ε)
cusp
with
D.Fisher 86
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T=0FRG fixed points
1) Random PERIODIC disorder:
choose
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Physics of the Cusp: metastability
• “Toy RG”Balents, Bouchaud, Mezard
Veff(u)
u“shock”u+uf
V(u+uf) degeneratestates
ujumps
• Linear cusp in –R’’(u) in many limits: large N, d=0 “toy model”,-relations to Burger’s turbulence (d=1)
• temperature T>0 smoothes the cusp• Mode minimization (T=0) produces shocks in effective action
is finite if shock between and
occurs with probability (uniform proba density )
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T=0FRG fixed points
2) RANDOM FIELD disorder:
must choose
3) RANDOM BOND (SR) disorder:
shooting method
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Higher loops
• random field
• random bond
one loop two loop exact
PLD,K.Wiese,P.Chauve 2001
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N components and RSB
• Mezard Parisi:
- Replica Gaussian variational approx. (RSB)
- solution exact for LR disorder and
- for SR disorder
• Balents Fisher:
• PLD Wiese- obtain FRG equation for all
solution yields CUSP in one to one correspondence with (full) RSB in Mezard-Parisi
to order any
and
recover MP with no need for spontaneous RSB
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Conclusion of statics
- problems with contructing perturbative RG for disordered elastic systems
- one loop FRG at T=0 appears as a way to solve them (evade dimensional reduction)
- physical results reasonable, compares well with replica RSB
- higher loops, finite T approach andfull consistency of the method
are not yet fully resolved
- similar approach for random field spin models
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Dynamicsvelocity v
T=0appliedforce f
creep
depinningstatics
• depinning transition
• equilibrium dynamics
‘‘near equilibrium’’ dynamics: creep
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Equation of motion
friction elastic force randompinning force
thermalnoise
externalforce
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Scaling picture of depinning
critical blocked configuration moving
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expected renormalisations
corrections to friction
uncorrectedcorrections topinning force
correlator
yields
correctionsproduce
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Dynamical action
take thermal and disorder averages
correlation
response
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perturbation theory
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Dynamical FRG
Narayan Fisher conjecture is EXACT
assume it goes to random field…
Are there several univ class at depinning ?
thus to one loop
to all orders if R(u) analytic
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dynamical quantities
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one loop theory is NOT consistent
for quasi-static depinning
β• -function for force correlatorsame as statics
where is irreversibility ?
random bond• conjectures random field
how many univ class?
Narayan FisherOne loop: Nattermann et al. 92
Narayan Fisher
numerics ?
cusp
one loop depinning : summary
predicts a threshold force
BUT:
good points: scaling, exponents
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Two loop depinning
• different from statics: irreversibility recovered
• single FP for RB,RF
PLD,K.Wiese,P.Chauve 2001
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numerics
Roters and Usadel
new high precision algorithm by Rosso and Krauth
Find exact critical string configuration on cylinder
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T>0 : creepvelocity v
appliedforce f
creep
depinningstatics
• equilibrium dynamics
‘‘near equilibrium’’ dynamics: creep
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Qualitative argument for creep
• small f: limited by typical nucleation event
• near equilibrium, activated dynamics over optimal barrier
Assume:
Q: What happens after jump ?
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T>0 statics, equilib. Dynamics
except in thermal boundary layer
Balents,Chauve,TG,PLDremains analytic
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Exponential growth of time scales
remains analytic
but curvature is blowing up
(not rescaling time)
CREEP LAWbarriers grow as
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one loop FRG at non zero velocity and temperature
Chauve,Giamarchi,PLD
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one loop FRG, v>0 and T>0 : creep physics
follow RG flow of
• new depinning regime
Chauve,TG,PLD 98
• derive creep law
suggest fast deterministic motion
• statistics of nucleation events• which scales equilibrate ?
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Conclusion
• T=0 FRG allows to describe statics and depinning
• non analyticity of effective action crucial to obtain
correct T=0 physics (glass, pinning, depinning threshold)
• statics and v=0+ depinning differ only at two loop
feedback of non-analytic terms crucial to distinguish both
• temperature formally irrelevant but plays crucial role for
rounding of the cusp: determines barriers and thermal creep
• open questions, application to quantum problems
• quantitative predictions for large class of experimentally relevant systems
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FRG for quantum problems
• Balents
• disorder is independent of time
correlated in ‘‘time direction’’
imaginary time, Matsubara, path integral
Europhys. Lett. 24 489 1993
• TG, PLD, Orignac
Keldysh
• Ghorokhov, Blatter, Fisher cond-mat/0205416
cond-mat/0104583
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further works on FRG
• connection with droplet theory of glasses
• ultra-broad distribution of relaxation times, barrier fluctuations
consistency of T>0 FRG (to all loops)
calculation of frequency dependence of response function
Balents, PLD, cond-mat/0408048
Balents, PLD, Phys. Rev. E 69 061107 (2004)