Systèmes ``Elastiques’’ désordonnés · S. Lemerle et al. PRL 80 849 (98) Other Interfaces...
Transcript of Systèmes ``Elastiques’’ désordonnés · S. Lemerle et al. PRL 80 849 (98) Other Interfaces...
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Systèmes Systèmes ````ElastiquesElastiques’’ ’’ désordonnésdésordonnés
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S. Lemerle (Orsay)J. Ferré (Orsay)
T. Klein (Grenoble)I. Joumard (Grenoble)
J.M. Triscone (Geneva)P. Paruch (Geneva)T. Tybell (Trondheim)
P. Le Doussal (ENS)
D. Carpentier (ENS Lyon)R. Chitra (Jussieu)E. Orignac (ENS)P. Chauve (Orsay/ENS)G. Schehr (ENS)A. Rosso (Geneva)A. Kolton (Geneva)
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GeneralGeneral ReferencesReferences on DESon DES• Classical systems :
G. Blatter et al. Rev. Mod. Phys 66 1125 (1994).
T.G. + P. Le Doussal, In ``Spin Glasses and Random Fields'', ed. A.P. Young, World Scientific 1998, cond-mat/9705096.
T. Nattermann and S. Scheidl Adv. Phys. 49 607 (2000)
TG + S. Bhattacharya, In ``High Magnetic Fields’’, ed. C. Berthier et al., Springer 2002, cond-mat/0111052.
• Quantum systems :
T.G. + E. Orignac In ``Theoretical Methods for Strongly Correlated Electrons", D.Senechal et al. ed, Springer (2004), cond-mat/0005220.
T.G. In ``Quantum phenomena in mesoscopic systems" (Varenna school CLI) IOS Press (2003), cond-mat/0403531.
And references therein..
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MagneticMagnetic domaindomain wallwall
S. Lemerle et al. PRL 80 849 (98)
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OtherOther InterfacesInterfaces
• Contact line [E. Rolley]
• Ferroelectrics [P. Paruch]
• Epitaxial growth
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ClassicalClassical crystalscrystals
Charged spheres: M. Saint Jean, GPS (Jussieu), 2000
Magnetic Bubbles: R. Seshadriet al.
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OtherOther ‘‘classicalclassical’ ’ CrystalsCrystals
• Vortex Lattice [K. van der Beek]
• Charge density waves
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Quantum Quantum systemssystems
I
V
• Strong repulsion : Wigner crystal
• Quantum fluctuations instead (in addition to) thermal fluctuations
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Wigner Wigner CrystalCrystal
E.Y. Andrei, et al PRL 60 2765 (1988)
R.L. Willett, et al. PRB 38 R7881 (1989)
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OtherOther quantum quantum CrystalsCrystals
• Spin density waves
• Luttinger liquids (1D interacting electrons)
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•Basic Features :
Disorder
(Thermal, quantum) fluctuations
`Elasticity’
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QuestionsQuestions
• Melting• Glassy phases
Competition ``Order’’ / ``Disorder’’
• Statics
• Dynamics
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StaticsStatics
T. Klein et al. Nature 413, 404 (2001)
P. Kim PRB 60 R12589 (99)
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Dynamics+
50 µm
-
(V. Repain et al. (Orsay))
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New type New type ofof physicsphysics
• Very controlled (e.g. magneticfield)
• Can pull on on the system
• Plunged in an external disorder
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HowHow to to modelmodel
Elastic description
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ElasticElastic description of description of crystalscrystals
R0i : crystal
ui : displacements
n=2 d=3 vortices
Elastic hamiltonian
∑∫ −=αβ
βααβ dqququqcH )()()(21
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Simplest elastic hamiltonian : c(q) = c q2
221 ))(( xucH ∫ ∂= βα
Long range forces ; bulk, shear and tilt
),,( zyxuz∂),,( zyxuαα∂ ),,( zyxuxy∂
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LimitationsLimitationsPeriodic
dislocations, etc.Interfaces
(overhangs, bubbles)
J. P Jamet, V. Repain M. Marchevsky, J. Aarts, P.H. Kes
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WhatWhat to to measuremeasure ((staticsstatics))
ur
2)]0()([)( ururB −=
Positional order
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K0 q
)()()( qqqS −= ρρ
Structure Structure FactorFactor
Fourier transform of:
)0()( 00)( uiKruiK eexC −=
Decorations Neutrons
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Thermal fluctuations : Thermal fluctuations : MeltingMelting
cTlu T ∝= 22
Lindemann criterion of melting :
2222 aClu LT == 2.01.0 −≈LC
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VorticesVortices
T
Hc2
Hc1
Meissner
Liquid
Normal
Tc
B
T
Solid Liquid
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ShouldShould wewe care about care about disorderdisorder??
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ςLu ∝
S. Lemerle et al. PRL 80 849 (98)
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DisorderDisorder (point (point likelike defectsdefects))
)()(
)()(
0i
ii uRxx
dxxxVH
−−=
=
∑
∫
δρ
ρ
∑ −+∇−=K
xuxiKexux ))((000 )()( ρρρρ
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LossLoss of of translationaltranslational orderorder ((LarkinLarkin))
aRu a ≈)(
∫ ∇= rdrucH del
2))((2
rdrrVH ddis ∫= )()( ρ
22acRda−
02/ ρd
aVR)4/(1
20
2
2 dd
a VacaR
−
∝
ρ
No crystal belowfour spatial dimensions
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VeryVery difficultdifficult statstat--mechmechproblemproblem
• Optimization :
many solutions
Ε• Glass
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LarkinLarkin ModelModel
∫ ∇= rdrucH del
2))((2
rdrurfH ddis ∫= )()(
• Exactly solvableExponential lossof translationalorder drerC
−−≈4
)(
dth r
cBrB −∆
+= 42)(
• Not valid at large distance
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)()()())((0 xuxfxVeK
xuxiK ≈∑ −ρ
Not valid when : ξ≈)( cRu1≈uKMAX
• New length Rc
• Larkin model has no metastable states andpinning
• Rc is related to pinning 2c
c RcJ ξ
∝
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Interfaces: Interfaces: onlyonly oneone lengthlength
ξ≈)( cRu• Larkin length
R < Rc ; u(R) = R(4-d)/2
R > Rc ; u(R) = ?????
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CrystalsCrystals: : TwoTwo crucial crucial lengthscaleslengthscales
• Positional orderaRu a ≈)(
• Larkin lengthξ≈)( cRu
C(r)
rRa
)(qSdaR
qaR/1aR/1−
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TwoTwo types types ofof disorderdisorder
Random fieldRandom bond
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InterfacesInterfaces
∫ ∇= rdrucH del
2))((2
rdrurVH ddis ∫= ))(,(
)'()'()','(),( zzxxDxzVxzV −−= δδ22 −dLcu 2/2/2/1 md uLD −
md
RF
md
RB
Lu
Lu
−−
+−
∝
∝
44
44
Flory argument (mean field)
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ς
LuRB ∝ ζ : roughness exponent
d = 1 ; ζ = 2/3 (random bond)
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CrystalsCrystals
Identical to interfaces ? u ∼ Lζ
Above Rc
C(r) ∝ e−L2ζExpontential loss of positional order ??
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NaiveNaive vision of a D.E. vision of a D.E. crystalcrystal• Loss of translational order beyond Ra
• (Wrong) argument: disorder inducesdislocations at Ra
Ra Crystal brokenin crystallitesof size Ra
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PeriodicPeriodic systemssystems: new : new universalityuniversality classclass
2/1)(~ LLoguςLu ~
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B(r)
rRc Ra
dr −4
ν2r)log(rAd
Random AsymptoticmanifoldLarkin
• many metastable states ïglass !
• Quasi long range translational order !
• Power law Bragg peaks; Ad = 4-d
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Bragg GlassBragg Glass
K0 q
S(q)
•Existence of a thermodynamically stable glassy phase with quasi long range translational order (power law Bragg peaks) and perfect topological order (no defects)
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UnifiedUnified phase phase diagramdiagram
T
H,∆
No dislocations
Liquid T
H
Hc1
Hc2
BrGDislocations
Bragg glass
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B. Khaykovich et al. PRL 76 2555 (96)
Hardy et al. Physica C 232 347 (94)
K. Deligiannis et al. PRL 79 2121 (97)
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Y. Nonomura and X. Hu cond-mat 0002263
Y. Paltiel et al. cond-mat 0008092
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NeutronsNeutrons)(qS
daR
qaR/1aR/1−
No positional order Bragg Glass
ηη −∝ da RR exp
exp/1 Rexp/1 R−aR/1− aR/1
daR
ω0Kqz=
)(qS
• Collapse of intensity without broadening
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I. Joumard et al. PRL 82 4930 (99); T. Klein et al. nature (01)
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CDWCDW
A. Rosso + TG PRB 68 140201(R) (2003) + cond-mat
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HowHow to to solvesolve ??
• Average over disorder (replica trick)
• Two main methods :
Variational approach
Renormalization (functional RG)
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ReplicasReplicas
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AverageAverage overover disorderdisorder
∑∫∫ −+∇=K
xuxiKdc xVexduH )()( ))((0
22 ρ
• Classical systems
xdxuxuKxducH d
ba Kba
a
da ∑∑∫∑ ∫ −∆−∇=
,0
2 )))()((cos()( ρ
• Quantum problem (disorder is time independent)
')))',(),((cos(
)(
,0
12
ττττρ dxddxuxuK
xducS
d
ba Kba
a
da
∑∑∫
∑ ∫−∆−
∇= +
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VariationalVariational MethodMethod
Find the best quadratic Hamiltonian∑∑∫ −−
− ΦΦ=ab i
dbqi
aqinab
n
nnqdqiGS
ωωωω ,,
10 ),(
Gab is a 0x0 matrixMinimize
0),(
var =qidG
dF
nab ω0)( 00var S
SSFF −+=
∫ ∑∫++=− .....221 ..),( abGnnab eqqiG ωω
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0x0 0x0 limitlimit• Replica symetric
Unstablebaaa GG ≠
• Replica symetry broken solution
Hierarchical structure
a,b continuous in [0,1]
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RSBRSB
• Signals metastability and Glassyproperties
• Disordered elastic system = glass
• RSB from d=4 to d=2 (or 1+1) above a lengthscale Rc
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FRG FRG methodmethod
xdxuxuK
xduH
d
ba KbaT
a
daT
c
∑∑∫
∑ ∫−−
∇=
∆
,0
2
)))()((cos(
)(
2ρ
β
....)( 42 +++≈∆ cubuauIn usual RG :
Needs only to keep b and c (higherpowers are irrelevant)
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DisorderedDisordered SystemSystem
dLTLu
−∝
∝2
0∑ ∫ ∇a
daT
c xdu 2)(
xdxuxuK d
ba KbaT ∑∑∫ −∆−
,
1 )))()(((2
Needs to keep thewhole function
dL −∝∆ 4
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∆’’ ∆’’
uu
• Nonanalyticity at a finite lengthscaleRc such that u(Rc) ~lc• Cusp signals metastability and glassystates
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DynamicsDynamics
• Competition between disorder and elasticity: glassy properties
• Dynamics ?
Large v:
Nature of moving phase ?
Depinning:( )βcF-Fv ∝
Fc F
v T≠0
T=0
Creep: v=????
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PinningPinning ((FcFc) ) andand LarkinLarkin lengthlength ((RcRc))
2c
c RcF ξ
=
∫ ∇= rdrucH del
2))((2 ∫= rdrFuH d
el )(
22ξ−dccR ξdcFR
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Large VLarge VInterfaces reorder at large V
Rv
Thermal Roughening for R > Rv
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DepinningDepinning
ζ for F ∼ Fc differs from ζ for F=0
Only RF universality class
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HowHow to to studystudy
Martin-Siggia-Rose; Keldysh
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TAFF vs TAFF vs CreepCreep
FFc
T≠0
T=0
v
∆
x
• TAFF : typical barrier
• Linear response Fev ∆−∝ β
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CreepCreep• Glassy system
• Slow dynamics determined by statics qty
∫ ∇= rdrucH del
2))((2 ∫= rdrFuH d
el )(
222
)(2
)( −−+
−−
≈
≈≈
ςς
βς
d
opt
LUopt
FLU
evFL opt
ς22+−dcR ζ+dFR
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S. Lemerle et al. PRL 80 849 (98)
D.T. Fuchs et al. PRL 81 3944 (98)
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FerroelecticsFerroelectics
T. Tybell et al. PRL 89 097601 (02)
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FRG: other scales thannaive version of creep
Molecular dynamics; (A. Kolton)
Peli.mng
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CrystalCrystal vs Interfacesvs Interfaces• Disorder remains in perp. direction
• Motion via static channels
• Moving glass
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S(q)
Coupled channels: Moving BrGS(q)
Decoupled channels:Smectic
S(q)
No channels:Plastic
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F. Pardo et al. Nature, 396 348 (1998)A. Kolton et al PRL 83 3061 (1999)
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canales.mpeg
Molecular dynamics (A. Kolton)
smectic.mpegplastic.mpeg transol.mpeg
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DynamicalDynamical Phase Phase DiagramDiagram
F
T
H
Fc
Liquid
Moving Bragg glass
Smectic
PlasticVortex Glass
Bragg glass
D=3
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PeakPeak EffectEffectV
JJ1
BrG
J2 (disordered)
• Peak at « melting » of the Bragg glass :
• second peak in magnetization
• Peak effect in transport
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Transverse critical forceTransverse critical forceCrystal without disorder Moving glass
Fy
vy
FcyFx
Fy
Fy
vy
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Absence of Hall voltageAbsence of Hall voltage
• Flor < Ftran : no hall voltage
• Compatible with theexistence of a transverse threshold
F. Perruchot et al. Physica B 256 587 (1998)
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PandoraPandora BoxBox
• Dislocations
3D: lucky (Bragg glass)
2D: dislocations control dc transport
Melting
• Dynamics ??????
• Aging and other glassy goodies
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SpecializedSpecialized RefsRefs..•Bragg glass :T.G. + P. Le Doussal Phys. Rev. B 52 1242 (1995).T.G. + P. Le Doussal Phys. Rev. B 55 6577 (1997).P. Le Doussal + T.G. Physica C 331 233 (2000).
Moving glass:T.G. + P. Le Doussal Phys. Rev. Lett. 76 3408 (1996).P. Le Doussal + T.G. Phys. Rev. B 57 11356 (1998).
• Creep from RG:P. Chauve + T.G. + P. Le Doussal Phys. Rev. B 62 624 (2000).
•Quantum problems:T.G. + P. Le Doussal Phys. Rev. B 53 15206 (1996).R. Chitra + T.G. + P. Le Doussal Phys. Rev. B 65 035312 (2001)T.G. + P. Le Doussal + E. Orignac Phys. Rev. B 64 245119 (2001)