Hidden dynamical symmetries in ageing phenomena Malte Henkel - PowerPoint PPT Presentation

Hidden dynamical symmetries in ageing phenomena Malte Henkel Laboratoire de Physique de Mat´ eriaux Universit´ e Henri Poincar´ e Nancy I, France collaborators: M. Pleimling F. Baumann, X. Durang, S.B. Dutta, M. Ebbinghaus, H. Grandclaude reviews: J. Phys. Cond. Matt. 19 , 065101 (2007); J. Stat. Mech. P07015 (2007) KIAS Seoul, 1 st of July 2008

Contents : I. Ageing phenomena physical ageing ; scaling behaviour and exponents II. Hidden dynamical symmetries Local scaling with z = 2 ; stochastic field-theory ; computation of response and correlation functions III. Local scale-invariance for z � = 2 Mass terms ; integrability ; test through responses and correlators in several models IV. Test case : 2 D disordered Ising model V. Conclusions

I. Ageing phenomena why do materials ‘look old’ after some time ? which (reversible) microscopic processes lead to such macroscopic effects ? physical ageing known since historical (or prehistorical) times systematic studies first in glassy systems Struik 78 a priori behaviour should depend on entire prehistory but evidence for reproducible and universal behaviour for better conceptual understanding : study ageing first in simpler systems (i.e. disordered ferromagnets) ageing : defining characteristics and symmetry properties : 1 slow dynamics (i.e. non-exponential relaxation) 2 breaking of time-translation invariance 3 dynamical scaling new evidence for larger, local scaling symmetries

Struik 78 1. observe slow relaxation after quenching PVC from melt to low T 2. creep curves depend on waiting time t e and creep time t 3. find master curve for all ( t , t e ) − → dynamical scaling → three defining properties of physical ageing

master curves of distinct materials are identical − → Universality ! good for theorists . . . Struik 78 conceptual confirmation in phase-ordering : Allen-Cahn equation

easier to study : ageing in simple systems without disorder consider a simple magnet (ferromagnet, i.e. Ising model) 1 prepare system initially at high temperature T ≫ T c > 0 2 quench to temperature T < T c (or T = T c ) → non-equilibrium state 3 fix T and observe dynamics Bray 94 competition : at least 2 equivalent ground states local fields lead to rapid local ordering no global order, relaxation time ∞ formation of ordered domains, of linear size L = L ( t ) ∼ t 1 / z dynamical exponent z universal Allen-Cahn equation v = − ( d − 1) K for domain walls

Snapshots of spin configurations in several 2 D / 3 D Ising models quenched to T < T c , for three different times t = 25 , 100 , 225. Left : pure Middle : disordered Right : 3 D spin glass

Scaling behaviour & exponents single relevant time-dependent length scale L ( t ) ∼ t 1 / z Bray 94, Janssen et al. 92, Cugliandolo & Kurchan 90s, Godr` eche & Luck 00, . . . � t � r � φ ( t , r ) φ ( s , 0 ) � = s − b f C correlator C ( t , s ; r ) := s , ( t − s ) 1 / z � � t � � δ � φ ( t , r ) � r � = s − 1 − a f R response R ( t , s ; r ) := s , � ( t − s ) 1 / z δ h ( s , 0 ) h =0 Surprise : scaling behaviour far away from the critical point, in the entire phase T < T c ?

How to understand these scaling forms → mean-field Langevin eq. for order parameter m ( t ) d m ( t ) = 3 λ 2 m ( t ) − m ( t ) 3 + η ( t ) , � η ( t ) η ( s ) � = 2 T δ ( t − s ) d t ole parameter λ 2 : contrˆ (1) λ 2 > 0 : T < T c , (2) λ 2 = 0 : T = T c , (3) λ 2 < 0 : T > T c two-time observables : response R ( t , s ), correlation C ( t , s ) � � R ( t , s ) = δ � m ( t ) � = 1 � 2 T � m ( t ) η ( s ) � , C ( t , s ) = � m ( t ) m ( s ) � � δ h ( s ) h =0 mean-field equation of motion : � � λ 2 − v ( t ) ∂ t R ( t , s ) = 3 R ( t , s ) + δ ( t − s ) � � λ 2 − v ( s ) ∂ s C ( t , s ) = 3 C ( t , s ) + 2 TR ( t , s ) � � λ 2 − v ( t ) with variance v ( t ) = � m ( t ) 2 � , v ( t ) = 6 ˙ v ( t )

se λ 2 ≥ 0 : fluctuations persist se λ 2 < 0 : fluctuations disappear in the long-time limit t , s → ∞ : ( t > s )   ; λ 2 > 0  2 min( t , s ) 1   � � ; λ 2 = 0 s / t R ( t , s ) ≃ s / t ; C ( t , s ) ≃ T s   ; λ 2 < 0  (3 | λ 2 | ) e − 3 | λ 2 | | t − s | e − 3 | λ 2 | ( t − s ) 1 fluctuation-dissipation ratio measures distance from equilibrium  ; λ 2 > 0 1 / 2 + O ( e − 6 λ 2 s )  X ( t , s ) = TR ( t , s ) ; λ 2 = 0 ∂ s C ( t , s ) ≃ 2 / 3  ; λ 2 < 0 1 + O ( e −| λ 2 | | t − s | ) relaxation far from equilibrium, when X � = 1, if λ 2 ≥ 0 ( T ≤ T c )

Consequences : If λ 2 > 0 : free random walk, the system never reaches equilibrium ! If λ 2 = 0 : slow relaxation, because of critical fluctuations In both situations : observe 1 slow dynamics (non-exponential relaxation) 2 time-translation-invariance broken 3 dynamical scaling behaviour − → the conditions for physical ageing are all satisfied if T ≤ T c − → the system remains out of equilibrium If λ 2 < 0 : rapid relaxation, with finite relaxation time τ rel ∼ 1 / | λ 2 | , towards unique equilibrium state

Return to scaling forms : � t � r s − b f C correlator C ( t , s ; r ) = s , ( t − s ) 1 / z � t � r s − 1 − a f R response R ( t , s ; r ) = s , ( t − s ) 1 / z values of exponents : equilibrium correlator → classes S and L � exp( −| r | /ξ ) � class S � a = 1 / z C eq ( r ) ∼ = ⇒ = ⇒ | r | − ( d − 2+ η ) class L a = ( d − 2 + η ) / z if T < T c : z = 2 and b = 0 if T = T c : z = z c and b = a for y → ∞ : f C , R ( y , 0 ) ∼ y − λ C , R / z , λ C , R independent exponents Question : general arguments to find form of scaling functions ?

Tests of dynamical scaling : 3 D Ising model, T < T c dynamical scaling no time-translation invariance C ( t , s ) : autocorrelation function, quenched to T < T c scaling regime : t , s ≫ τ micro and t − s ≫ τ micro

Fluctuation-dissipation theorem Ageing goes on far from equilibrium ! No fluctuation-dissipation theorem : R ( t , s ; r ) � = T ∂ C ( t , s ; r ) /∂ s rather use fluctuation-dissipation ratio to measure distance from equilibrium Cugliandolo, Kurchan, Parisi 94 X ( t , s ) := TR ( t , s ) ∂ s C ( t , s ) At equilibrium : X ( t , s ) = 1. Otherwise, X ( t , s ) � = 1. Experimentalists often use effective temperature T eff := T / X ( t , s ) T eff is not a thermodynamic ensemble quantity, since it may depend on the observable Calabrese & Gambassi 04

Experimental examples for the breaking of the FDT I spin glass CdCr 1 . 7 In 0 . 3 S 4 , quenched to T / T c = 0 . 8 Herisson & Ocio 02 � t trace susceptibility χ ZFC ( t , s ) = s d u R ( t , u ) over against C ( t , s ) sub-ageing corretions to scaling ? for C ≈ 1, straight line with slope − 1 / T in χ − C plot

Experimental examples for the breaking of the FDT II mechanical response of a colloidal suspension of PMMA measure diffusive motion and drift (under an external perpendicular field – 2 D sample) find subageing with truly small µ ∼ 0 . 48 Makse et al 06

II. Hidden dynamical symmetries A) Langevin equation (model A of Hohenberg & Halperin 77 ) 2 M ∂φ ∂ t = ∆ φ − δ V [ φ ] + η δφ order-parameter φ ( t , r ) non-conserved M : kinetic co´ efficient V : Landau-Ginsbourg potential η : gaussian noise, cantered and with variance � η ( t , r ) η ( t ′ , r ′ ) � = 2 T δ ( t − t ′ ) δ ( r − r ′ ) fully disordered initial conditions (centred gaussian noise) B) master equation e.g. Glauber 63 i.e. kinetic Ising model with heat-bath dynamics random initial state → relaxation towards equilibrium stationary states

Local scaling with z = 2 → LSI Question : extended dynamical scaling for given z � = 1 ? MH 92, 94, 02 motivation : 1. conformal invariance in equilibrium critical phenomena, z = 1 2. Schr¨ odinger-invariance of simple diffusion, z = 2 Lie 1881, Niederer 72, Hagen 71, Kastrup 68 γ t + δ , r �→ R r + v t + a α t t �→ , αδ = 1 γ t + δ Lie algebra age 1 := � X 1 , 0 , Y ± 1 / 2 , M 0 � generators : (no TTI !) � x � X n = − t n +1 ∂ t − n + 1 t n r ∂ r − n ( n + 1) M t n − 1 r 2 − t n 2( n + 1) + n ξ 2 4 � � m + 1 Y m = − t m +1 / 2 ∂ r − M t m − 1 / 2 r 2 M n = − t n M also contains ‘phase changes’ in the wave function ! (projective)

commutators in root diagramme odinger operator : S = 2 M ∂ t − ∂ 2 Schr¨ r odinger equation : S φ = 0 Schr¨ Schr¨ odinger-invariance : [ S , X 0 ] = −S , [ S , Y − 1 / 2 ] = 0 and [ S , X 1 ] = − 2 t S + 2 M ( x + ξ − 1 2 ) = ⇒ if x + ξ = 1 / 2, solutions of the 1 D Schr¨ odinger/diffusion equation mapped by age 1 onto another solution → local dynamical symmetry co-variant two-point function R 12 := � φ 1 ( t , r ) φ 2 ( s , 0 ) � : X R 12 = 0 with X ∈ age d ⊂ conf d +2 MH 94, MH & Unterberger 03, MH et al 06 � � R 12 ∼ s − 1 − a � t � 1+ a ′ − λ R / 2 � t � − 1 − a ′ r 2 −M 1 s − 1 exp 2 t − s s , a ′ − a = ξ 1 + ξ 2 , λ R = 2( x 1 + ξ 1 ), M 1 + M 2 = 0 with 1 + a = x 1 + x 2 2 causality condition t > s : R 12 is a response function ! reproduced in some ageing systems with z = 2 WHY ? ?