Non-relativistic variants of conformal invariance and physical - PowerPoint PPT Presentation

Non-relativistic variants of conformal invariance and physical ageing Malte Henkel Groupe de Physique Statistique, Institut Jean Lamour (CNRS UMR 7198) Universit´ e de Lorraine Nancy , France Atelier ‘Renormalization in statistical physics and lattice field theories’ Institut Montpell´ erian Alexandre Grothendieck, aoˆ ut 2015 mh, J.D. Noh and M. Pleimling , Phys. Rev. E85 , 030102(R) (2012) mh , Nucl. Phys. B869 , 282 (2013); mh & S. Rouhani , J. Phys. A46 , 494004 (2013) mh , Springer Proc. Math. Stat. 85 , 511 (2014); mh & S. Stoimenov , idem 111 , 33 (2015) mh & X. Durang , J. Stat. Mech. P05022 (2015) & work in progress

Overview : 1. Dynamical scaling & ageing : physical background 2. Form of the scaling functions & L ocal S cale- I nvariance ( lsi ) 3. Long-distance behaviour & Causality parabolic sub-algebras, analyticity 4. Proposal for local scale-transformations for z � = 2 5. Simple magnets and growing interfaces : analogies 6. Numerical experiments 7. Logarithmic conformal & Schr¨ odinger invariance 8. Logarithmic ageing-invariance 9. Conclusions ambition : argue for the applicability of lsi in non-equilibrium statistical mechanics & indicate some relevant mathematical structures

1. Dynamical scaling & ageing : physical background Equilibrium critical phenomena : scale-invariance For sufficiently local interactions : extend to conformal invariance space -dependent re-scaling (angles conserved) r �→ r / b ( r ) Polyakov 70 In two dimensions : ∞ many conformal transformations ( w �→ f ( w ) analytic) ⇒ exact predictions for critical exponents, correlators, . . . BPZ 84 What about time -dependent critical phenomena ? Characterised by dynamical exponent z : t �→ tb − z , r �→ r b − 1 Can one extend to local dynamical scaling, with z � = 1 ? If z = 2, the Schr¨ odinger group is an example : (Jacobi 1842), Lie 1881 t �→ α t + β , r �→ D r + v t + a ; αδ − βγ = 1 γ t + δ γ t + δ ⇒ study ageing phenomena as paradigmatic example

Ageing phenomena why do materials look old after some time ? known & practically used since prehistoric times (metals, glasses) systematically studied in physics since the 1970s Struik ’78 occur in widely different systems (structural glasses, spin glasses, polymers, simple magnets, . . . ) The three defining symmetry properties of ageing : 1 slow relaxation (non-exponential !) 2 no time-translation-invariance ( tti ) 3 dynamical scaling ‘Magnets’ : no disorder, no frustration − → more simple to understand Interfaces : out of equilibrium, many analogies Question : what is the current evidence for larger, local scaling symmetries ?

for symmetry analysis : 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 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

t = t 1 t = t 2 > t 1 magnet T < T c − → ordered cluster magnet T = T c − → correlated cluster growth of ordered/correlated domains, of typical linear size L ( t ) ∼ t 1 / z dynamical exponent z : determined by equilibrium state

illustration of statistical self-similarity for different times t 1 < t 2 Walter ’10

Two-time observables : analogy with ‘magnets’ time-dependent order-parameter φ ( t , r ) two-time correlator C ( t , s ; r ) := � φ ( t , r ) φ ( s , 0 ) � − � φ ( t , r ) � � φ ( s , r ) � � � � � R ( t , s ; r ) := δ � φ ( t , r ) � � φ ( t , r ) � two-time response = φ ( s , r ) � δ h ( s , 0 ) h =0 � t : observation time causality condition t > s for responses s : waiting time a) system at equilibrium : fluctuation-dissipation theorem Nyquist 28, Kubo 66 R ( t − s ; r ) = 1 ∂ C ( t − s ; r ) , T : temperature T ∂ s b) far from equilibrium : C and R independent ! The fluctuation-dissipation ratio ( fdr ) Cugliandolo, Kurchan, Parisi ’94 TR ( t , s ) X ( t , s ) := ∂ C ( t , s ) /∂ s measures the distance with respect to equilibrium : X eq = X ( t − s ) = 1

Dynamical scaling & ageing : 3 D Ising model, T < T c dynamical scaling no time-translation invariance scaling regime : t , s ≫ τ micro and t − s ≫ τ micro

Scaling regime : t , s ≫ τ micro and t − s ≫ τ micro � t � � t � s , | r | z s , | r | z C ( t , s ; r ) = s − b f C , R ( t , s ) = s − 1 − a f R t − s t − s asymptotics : f C ( y , 0) ∼ y − λ C / z , f R ( y , 0) ∼ y − λ R / z for y ≫ 1 λ C : autocorrelation exponent, λ R : autoresponse exponent, z : dynamical exponent, a , b : ageing exponents * exponents λ C , R usually from independent renormalisations, * but initial conditions can imply scaling relations Question : what about the form of these universal scaling functions ?

Theoretical formulation Langevin equation (model A of Hohenberg-Halperin 77 ) ∂ t = ∆ φ − δ V [ φ ] 2 M ∂φ + η δφ order-parameter φ ( t , r ) non-conserved M : kinetic co´ efficient V : Landau-Ginsbourg potential η : gaussian thermal noise, centered and with variance � η ( t , r ) η ( t ′ , r ′ ) � = T / M δ ( t − t ′ ) δ ( r − r ′ ) fully disordered initial conditions (centered gaussian noise) ? interesting symmetries of NOISY Langevin equations ?

Example : how to find these scaling forms → mean-field Langevin eq. for magn. order parameter m ( t ) drop spatial dependencies 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 (cumulants neglected) : � � 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 )

if λ 2 ≥ 0 : fluctuations persist if 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 s / t R ( t , s ) ≃ s / t ; C ( t , s ) ≃ T   ; 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 Cugliandolo, Kurchan, Parisi 94  ; 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 )

Stochastic field-theory Langevin equations do not have non-trivial dynamical symmetries ! Galilei-invariance is broken by interactions with the thermal bath compare results of deterministic symmetries to stochastic models ? go to stochastic field-theory, action Janssen 92, de Dominicis,. . . � � � φ 2 − J [ φ, � φ (2 M ∂ t − ∆) φ + � � � φ t =0 C init � � φ V ′ [ φ ] φ ] = − T φ t =0 � �� � � �� � J 0 [ φ, � φ ] : deterministic + J b [ � φ ] : noise ( b ruit) � C ( t , s ) = � φ ( t ) φ ( s ) � , R ( t , s ) = � φ ( t ) � φ : response field ; φ ( s ) � � D φ D � φ A [ φ, � φ ] exp( −J 0 [ φ, � averages : � A � 0 := φ ]) masses : M φ = −M � φ

Theorem : IF J 0 is Galilei- and spatially translation-invariant, then Bargman superselection rules Bargman 54 � � φ 1 · · · φ n � φ 1 · · · � φ m 0 ∼ δ n , m Illustration : computation of a response function Picone & mh 04 � � � φ ] � φ ( s ) e −J b [ � φ ( t ) � φ ( t ) � R ( t , s ) = φ ( s ) = 0 � � φ ( t ) � = φ ( s ) 0 = R 0 ( t , s ) Bargman rule = ⇒ response function does not depend on noise ! left side : computed in stochastic models right side : local scale-symmetry of deterministic equation Comparison of results of assumed deterministic age ( d )-symmetry with explicit stochastic models/experiments justified .

Correlation functions for z = 2 find C ( t , s ) = � φ ( t ) φ ( s ) � = � φ ( t ) φ ( s ) e −J b [ � φ ] � 0 from Bargman rule � a 0 R d d R R (3) C ( t , s ) = 0 ( t , s , 0; R ) initial 2 � ∞ � + T R d d R R (3) d u 0 ( t , s , u ; R ) thermal 2 M 0 � � R (3) φ ( t , y ) φ ( s , y ) � φ 2 ( u , r + y ) 0 ( t , s , u ; r ) = 0 sch -invariance fixes three-point R (3) function up to an unknown 0 ⇒ how to obtain a prediction for f C ( y ) ? scaling function Ψ = Theorem : LSI with z = 2 = ⇒ λ C = λ R Picone & MH 04 agrees with a different RG argument of Bray and with all models Conclusion : concentrate on dynamical symmetries of ‘deterministic part’