Pierre Le Doussal (LPTENS) Kay J. Wiese (LPTENS) Florian Kühnel (Universität Bielefeld ) Long-range correlated random field and random anisotropy O(N) 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, (2006). AAF and F. Kühnel, Phys. Rev. B 75, (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 groupFunctional renormalization group Phase diagrams and critical exponentsPhase diagrams and critical exponents Outline

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 disorderLR 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 ofperiodic 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, 2006 andwith 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

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 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 Summary 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.