Post on 07-Feb-2016
description
transcript
Statics and dynamics of elastic manifolds in mediaStatics and dynamics of elastic manifolds in mediawith long-range correlated disorderwith long-range correlated disorder
Andrei A. Fedorenko, Pierre Le Doussal and Kay J. Wiese
CNRS-Laboratoire de Physique Theorique de l'Ecole Normale Superieure, Paris, France
Outline:• Elastic manifolds in the nature• Models and their basic propertiesModels and their basic properties• Functional renormalization group Functional renormalization group • Fixed points and critical exponentsFixed points and critical exponents• Response to tilting force Response to tilting force • SummarySummary
AAF, P. Le Doussal, and K.J. Wiese, cond-mat/0609234
CompPhys06, 1st December 2006, Leipzig
Elastic Manifolds in the NatureElastic Manifolds in the Nature
Domain wall (DW) in an Ising ferromagnet with either Random Bond (RB) or Random Field (RF) disorder .An experiment on a thin Cobalt film (left)(S. Lemerle, et al 1998)
Cartoon of vortex lattice deformed by disorder.
A contact line for the wetting of a disordered substrate by Glycerine. Experimental setup (left). The disorder consists of randomly deposited islands of Chromium, appearing as bright spots (top right). Temporal evolution of the retreatingcontact-line (bottom right).
(S. Moulinet, et al 2002)
In all cases the configuration of manifold can be descibed by a displecment field
Elastic Manifolds in Disordered Media: Models
elasticity constant
Hamiltonian
random potential with zero mean and correlator
Universality classes
Random Bond (RB): are short-range functions
Random Field (RF) : for large
Random Periodic (RP): are periodicCDW, vortex lattice (Bragg glass)
Domain wall (DW) in random-bond magnets
DW in random-field magnets, depinning
Roughness exponent
Quantity of interest
SR disorder
Periodic systems
LR disorder
for extended defects Interface in a medium with planes of disorder with random orientation
(LR)
Driven dynamics
The typical force-velocity characteristics
Depinning transition ( , )
Creep ( , )
The equation of motion (overdamped dynamics):
driving force densityfriction,
pinning force correlator ( ) :
velocity:
dynamic exponent:
velocity:
Creep
Depinning
Flow
depinning transition
thermal rounding
Perturbation theory
Action
Observabales
Diagramatic rules
propagator
SR disorder vertex
LR disorder vertex
FRG for short-range correlated disorder
Fixed-point solution Depinning transition (T. Nattermann, S. Stepanow, et al 1992)
FRG equation to one-loop (D.S. Fisher, 1986)
has cusp above Larkin scale
Perturbation theory to all orders gives dimensional reduction (incorrect)
Imry – Ma gives
FRG to two-loop (P. Chauve, PLD, KJW, 2001)Exponents
RF RB
Depinning
Interfaces Periodic systems
(depinning)
FRG for system with LR correlated disorder
Correction to disorder
Flow equations in statics:
Flow equations in dynamics:
Critical exponents:
dot line - either SR disorder or LR disorder. a , b , and c contribute to SR disorder, d to LR disorder.
Correction to mobility and elasticity
New fixed points new universality classes
a b
c d
Double expansion in and
Random Bond Disorder
LR RB Fixed point for
Roughness exponent
Eigenfunctions computed at the LR RB FP
LR disorder at the LR RB FP is an analytic function, while SR disorder has a cusp, i.e.
LR RB FP is stable for SR RB FP controls the behavior for
Universal amplitude:
In constrast to SR disorder is preserved along RG flow
Stability analysisFixed point
corresponding eigenvalue is
(Exact to all orders!!!)
Random Field Disorder
LR RF Fixed point for
Depinning transitionRoughness exponent:
Universal amplitude ( ):
LR RF FP is stable for SR RF FP controls the behavior for
NOTE: that in fact this is a FP of mixed type:SR disorder is effectively RB and LR – RF !!!
Fixed point Stability analysisEigenfunctions computed at the LR RF FPcorresponding eigenvalue is
Random periodic
LR RP Fixed point
Universal amplitude (Bragg glass):
Depinning transition
Two first eigenvectors computed at the LR RP FP (only SR disorder is shown, LR )corresponding eigenvalue is , LR disorder
SR disorder for different
LR disorder at the LR RF FP is an analytic function, while SR disorder has a cusp, i.e.
LR RP FP is unstable with respect to non-potential perturbation corresponding to :
LR RP FP is stable for SR RP FP controls the behavior for
Fixed point Stability analysis
Tilting field: from linear response to transverse Meissner effectFlux lines in the presence of disorder (neglecting disclocations in flux lattice)
point-like disorder columnar disorder LR disorder(extended defects with random orientation)
Bragg glass Bose glass Weak Bose glass
Tilting force: No response to a weak transverse force
SR disorder:
LR disorder:
columnar disorder:
( -finite )
(L. Balents, 1993)
(transverse Meissner effect)
Localized Two-loop order:
SummarySummary
• We have derived the FRG equations which describe the large scale behavior of elastic manifolds in statics and near depinning transition in the presence of long-range correlated disorder.
• We have found 3 new fixed points which control the scaling behavior of Random Bond, Random Field and Periodic systems and identified the regions of their stability. In contrast to systems with only SR correlated random filed a mixed type of fixed point appears in systems with LR correlations. The static and dynamic critical exponents are computed to one-loop order.
• We have study the response of elastic manifold subjected to the tilting force in the presence of long-range correlated disorder. We argue existence of a new glass phase with properties interpolating between properties of the Bragg glass (point-like disorder) and Bose glass (columnar disorder).