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

1. CompPhys07, 30th November 2007, Leipzig 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 Pierre Le Doussal (LPTENS) Kay J. Wiese (LPTENS) Florian Kühnel (Universität Bielefeld ) AAF and F. Kühnel, Phys. Rev. B 75, 174206 (2007) AAF, P. Le Doussal, and K.J. Wiese, Phys. Rev. E 74, 061109 (2006).

2. Outline • Random field and random anisotropy O(N) models • Long-range correlated disorder • Functional renormalization group • Phase diagrams and critical exponents

3. 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 RFIM, • vortex phases in impure superconductors RF, A. A. Abrikosov,   1957 Bragg peaks Decoration Disorder destroys the true long-range order P.Kim et al, 1999 Klein et al ,1999 A.I. Larkin, 1970 Bragg glass: no translational order, but no dislocations T. Giamarchi, P.Le Doussal, 1995 • disordered liquid crystals • amorphous magnets • He-3 in aerogels RA,

4. Random field and random anisotropy O(N) symmetric models Random field model Hamiltonian - component spin - quenched random field Random anisotropy model Dimensional reduction. Perturbation theory suggests that the critical behavior of both models is that of the pure models in . Dimensional reduction is wrong! - strength of uniaxial anisotropy - random unit vector

5. Correlated disorder Real systems often contains extended defects in the form of linear dislocations, planar grain boundaries, three-dimensional cavities, fractal structures, etc. • Long-range correlated disorder • dimensional extended defects with random orientation Correlation function of disorder potential Probablity that both points belong to the same extended defect A. Weinrib, B.I Halperin, 1983 Another example: systems confined in fractal-like porous media (yesterday talk by Christian von Ferber)

6. Phase diagram and critical exponents The true long-range order can exist only above the lower critical dimension (A.J. Bray, 1986) The ferromagnetic-paramagnetic transition is described by three independent critical exponents Connected two-point function Disconnected two-point function Schwartz-Soffer inequality: (RF) Generalized Schwartz – Soffer inequality T.Vojta, M.Schreiber, 1995 The divergence of the correlation length is described by • 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

7. The “minimal ” model There is infinite number of relevant operators (D.S. Fisher, 1985) Hamiltonian - random potential SR disorder LR disorder Replicated Hamiltonian are arbitrary in the RF case and even in the RA case

8. FRG for uncorrelated RF and RA models FRG equation in terms of periodic for RF and for RA D.S. Fisher, 1985 We have to look for a non-analytic fixed point! 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) such that and with M.Tissier, G.Tarjus, 2006 TT FP gives exponents corresponding to dimensional reduction

9. 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 FRG to two-loop order P. Le Doussal, K.Wiese, 2006 M. Tissier, G. Tarjus, 2006 M.Tissier, G.Tarjus, 2006 RA model has a similar behavior with the main difference that and M.Tissier, G.Tarjus, 2006

10. FRG for long-range correlated RF and RA O(N) models One-loop flow equations Critical exponents

11. Long-range correlated random field O(N) model above Stability regions of various FPs Critical exponents: Positive eigenvalue LR disorder modifies the critical behavior for

12. Long-range correlated random field O(N) model below There is no true long-range order. However, there are two quasi-long-range ordered phases Phase diagram Generalized Schwartz – Soffer inequality T.Vojta, M.Schreiber, 1995 is satisfied at equality

13. Long-range correlated random anisotropy O(N) model below There is no true long-range order. 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

14. 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 . 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