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 solutionDepinning 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 FP

corresponding 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 effect

Flux 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).