Pierre Le Doussal (LPTENS) Kay J. Wiese (LPTENS)Florian Kühnel (Universität Bielefeld )

Long-range correlated random field Long-range correlated random field and random anisotropy and random anisotropy O(N)O(N) models models

Andrei A. Fedorenko

CNRS-Laboratoire de Physique Theorique de l'Ecole Normale Superieure, Paris, France

AAF, P. Le Doussal, and K.J. Wiese, Phys. Rev. E 74, 061109 (2006).

AAF and F. Kühnel, Phys. Rev. B 75, 174206 (2007)

CompPhys07, 30th November 2007, Leipzig

• Random field and random anisotropy O(N) modelsRandom field and random anisotropy O(N) models

• Long-range correlated disorderLong-range correlated disorder

• Functional renormalization group Functional renormalization group

• Phase diagramsPhase diagrams and critical exponents and critical exponents

OutlineOutline

Examples of random field and random anisotropy systems

Def.: N-component order parameter is coupled to a random field. Random Field (RF) : linear coupling Random Anisotropy (RA) : bilinear coupling

• diluted antiferromagnets in uniform magnetic field

• vortex phases in impure superconductors

• disordered liquid crystals • amorphous magnets • He-3 in aerogels

RFIM,

A. A. Abrikosov,   1957

Bragg glass: no translational order, but no dislocations

Decoration

T. Giamarchi, P.Le Doussal, 1995

P.Kim et al, 1999 Disorder destroys the true long-range order A.I. Larkin, 1970

RF,

Klein et al , 1999 Bragg peaks

RA,

Random field and random anisotropy O(N) symmetric models

- component spin

Hamiltonian

- quenched random field

Random field model

Random anisotropy model

- strength of uniaxial anisotropy

- random unit vector

Dimensional reduction.

Perturbation theory suggests that the critical behavior of both models is that of the pure models in .

Dimensional reduction is wrong!

Correlated disorder

Correlation function of disorder potential

• Long-range correlated disorder

dimensional extended defects with random orientation

Probablity that both points belong to the same extended defect

Real systems often contains extended defects in the form of linear dislocations, planar grain boundaries, three-dimensional cavities, fractal structures, etc.

Another example: systems confined in fractal-like porous media (yesterday talk by Christian von Ferber)

A. Weinrib, B.I Halperin, 1983

Phase diagram and critical exponents

The ferromagnetic-paramagnetic transition is described by three independent critical exponents

Connected two-point function

The true long-range order can exist only above the lower critical dimension (A.J. Bray, 1986)

Below the lower critical dimension only a quasi-long-range order is possible:

• order parameter is zero

• infinite correlation length, i.e., power law decay of correlations

Disconnected two-point function

The divergence of the correlation length is described by

Schwartz-Soffer inequality: (RF)

Generalized Schwartz – Soffer inequality

T.Vojta, M.Schreiber, 1995

The “minimal ” model

Hamiltonian

- random potential

There is infinite number of relevant operators (D.S. Fisher, 1985)

Replicated Hamiltonian

SR disorder LR disorder

are arbitrary in the RF case and even in the RA case

FRG for uncorrelated RF and RA models

We have to look for a non-analytic fixed point!

D.S. Fisher, 1985

FRG equation in terms of periodic for RF and for RA

D.E. Feldman, 2002

RF model above the lower critical dimension

Singly unstable FP exists for

For there is a crossover to a weaker non-analytic FP (TT-phenomen)

M.Tissier, G.Tarjus, 2006and with

........

such that

TT FP gives exponents corresponding to dimensional reduction

FRG for uncorrelated RF and RA O(N) models

RF model below the lower critical dimension ( ) for There is a stable FP which describes a quasi-long-range ordered (QLRO) phase

RA model has a similar behavior with the main difference that and

M.Tissier, G.Tarjus, 2006

P. Le Doussal, K.Wiese, 2006

FRG to two-loop order

M. Tissier, G. Tarjus, 2006

M.Tissier, G.Tarjus, 2006

FRG for long-range correlated RF and RA O(N) models

One-loop flow equations

Critical exponents

Long-range correlated random field O(N) model above

Stability regions of various FPs

Positive eigenvalue

LR disorder modifies the critical behavior for

Critical exponents:

There is no true long-range order.

Long-range correlated random field O(N) model below

However, there are two quasi-long-range ordered phases

Phase diagram

Generalized Schwartz – Soffer inequality

T.Vojta, M.Schreiber, 1995

is satisfied at equality

There is no true long-range order.

Long-range correlated random anisotropy O(N) model below

There are two quasi-long-range ordered phases

Two different QLRO has been observed

in NMR experiments with He-3 in aerogel???

V.V. Dmitriev, et al, 2006

Open questionsOpen questions

• Metastability in TT region with subcusp non-analyticity: corrections to scaling, distributions of observables

• Equilibrium and nonequilibrium dynamics of the RF and RA models, aging

SummarySummary

• Correlation of RF changes the critical behavior above the lower critical dimension and modifies the critical exponents. Below the lower critical dimension LR correlated RF creates a new LR QLRO phase.

• LR RA does not change the critical behavior above the lower critical dimension for , but creates a new LR QLRO phase below the lower critical dimension .