Logarithmic extensions of local scale-invariance Malte Henkel Groupe de Physique Statistique D´epartement de Physique de la Mati`ereet des de Mat´eriaux Institut Jean Lamour CNRS { Nancy - Universit´e,France J.D. Noh (KIAS & University of S´eoul- Cor´ee) M. Pleimling (Virginia Tech - E.U.A)´ arxiv:1009.4139, arxiv:1109.5022 Atelier: Logarithmic CFT and representation theory, Programme `Advanced conformal field-theory and application', Institut Henri Poincar´e,Paris, octobre 2011 Overview : 1. Critical phenomena 2. Ageing phenomena 3. Critical contact process (dp universality class) 4. Surface growth (kpz universality class) 5. Form of the scaling functions & lsi 6. Logarithmic conformal invariance 7. Logarithmic ageing invariance 8. Numerical experiments (dp and kpz classes in 1D) 9. Conclusions 1. Critical phenomena Equilibrium critical phenomena : scale-invariance For sufficiently local interactions : extend to conformal invariance space-dependent re-scaling (angles conserved) r 7! r=b(r) Bateman & Cunningham 1909/10, Polyakov 70 In two dimensions : 1 many conformal transformations (w 7! f (w) analytic) ) exact predictions for critical exponents, correlators, . BPZ 84 What about time-dependent critical phenomena ? Cardy 85 Characterised by dynamical exponent z : t 7! tb−z , r 7! rb−1 Can one extend to local dynamical scaling, with z 6= 1? If z = 2, the Schr¨odingergroup is an example : Jacobi 1842, Lie 1881 αt + β Dr + vt + a t 7! ; r 7! ; αδ − βγ = 1 γt + δ γt + δ ) study ageing phenomena as paradigmatic example 2. Ageing phenomena 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, . ) Three defining properties of ageing : 1 slow relaxation (non-exponential !) 2 no time-translation-invariance (tti) 3 dynamical scaling without fine-tuning of parameters Most existing studies on `magnets' : relaxation towards equilibrium Question : what can be learned about intrisically irreversible systems by studying their ageing behaviour? t = t1 t = t2 > t1 magnet T < Tc −! ordered cluster magnet T = Tc −! correlated cluster critical contact process =) cluster dilution voter model, contact process,. common feature : growing length scale L(t) ∼ t1=z z : dynamical exponent Two-time observables time-dependent order-parameter φ(t; r) two-time correlator C(t; s) := hφ(t; r)φ(s; r)i − hφ(t; r)i hφ(s; r)i δ hφ(t; r)i D E two-time response R(t; s) := = φ(t; r)φe(s; r) δh(s; r) h=0 t : observation time, s : waiting time Scaling regime : t; s τmicro and t − s τmicro t t C(t; s) = s−bf ; R(t; s) = s−1−af C s R s −λ =z −λ =z asymptotics (for y 1) : fC (y) ∼ y C , fR (y) ∼ y R λC : autocorrelation exponent, λR : autoresponse exponent, z : dynamical exponent, a; b : ageing exponents 3. Critical contact process (directed percolation) p 1 (a) contact process : A −! 2A, A −! ; + diffusion Harris 74 (b) percolation problem with preferred( k) direction Broadbent & Hammersley 57 (c) Reggeon field theory Cardy & Sugar 80 absorbing (= non-equilibrium) stationary state β −β/ν particle density: stat. a1 = hAi ∼ (p − pc) ; critical a(t) ∼ t k −νk −ν? relaxation time τ = ξk ∼ jp − pcj ; correlation length ξ? ∼ jp − pcj dynamical exponent z = νk/ν? Effective action at criticality Janssen, de Dominicis 70s-80s Z h 2 i J [φ,e φ] = dtdr φe D@t φ − r φ − κφe φe − φ φ rapidity-reversal symmetry : J is invariant under Grassberger 79, Janssen 81 t 7! −t ; φ(t; r) 7! −φe(−t; r) ; φe(t; r) 7! −φ(−t; r) active (ordered) phase : limt!1 a(t) = ρ1 > 0 absorbing (disordered) phase : limt!1 a(t) = 0 long-time behaviour in the active phase of the 1D contact process 8 9 < fast relaxation = contrast to magnets : no scaling in active phase : time-translation-inv. ; reason : single non-critical stationary state Enss, mh, Picone, Schollwock¨ 04 ageing and scaling for C(t; s): critical contact process main figures : 1D, insets : 2D Ramasco, mh, Santos, da Silva Santos 04 ; Enss et al. 04 8 < slow dynamics observe all 3 properties of ageing : no tti : dynamical scaling contrast to critical magnets : a 6= b =) no finite FDR ! autocorrelation exponent : λC = d + z + β/ν? Calabrese, Gambassi, Krzakala 06/07, Baumann & Gambassi 07 numerical values of some non-equilibrium exponents contact process (cp) A ! 2A; A !;, parity-conserved model (pc) A $ 3A; 2A !;, diffusion-coagulation (dc)2 A ! A d a b λC =z λR =z cp 1 −0:68(5) 0:32(5) 1:85(10) 1:85(10) tmrg [1] −0:57(10) 0:3189 1:9(1) 1:9(1) mc [2] −0:6810 1:76(5) mc [3] −0:6810 0:3189 1:7921 1:7921 scal [5] 2 0.3(1) 0:901(2) 2:8(3) 2:75(10) mc [2] −0:198(2) 0:901(2) 2:58(2) 2:58(2) scal [5] 0:9(1) 2:5(1) exp [6] > 4 d=2 − 1 d=2 d=2 + 2 mf [2] pc 1 −0:430(4) 0:570(4) 1:9(1) 1:9(2) mc [4] −0:430(4) 0:570(4) 1:86(1) 1:86(1) scal dc 1 −1=2 1 2 2 exact [7] [1] Enss et. al. 04 ; [2] Ramasco et. al. 04 ; [3] Hinrichsen 06 ; [4] Odor´ 06 ; [5] Baumann & Gambassi 07 ; [6] Takeuchi et. al. 09 ; [7] Durang, Fortin, mh 11 in the contact process 1+ a = b : (= rapidity-reversal symmetry of stationary state of cp ) specific property ! why does 1 + a = b also hold in the pc class ? =) try new form of FDR ! Enss et. al. 04 ; Baumann & Gambassi 07 R(t; s) fR (t=s) Ξ(t; s) := = ; Ξ1 := lim lim Ξ(t; s) C(t; s) fC (t=s) s!1 t!1 1 universal function, Ξ 6= 0 measures distance to stationary state in d = 4 − " dimensions, from an one-loop calculation B & G 07 119 π2 Ξ = 2 1 − " − + O("2) 1 480 120 quantitatively consistent with tmrg estimateΞ 1 = 1:15(5) in 1D. NB : 1 + a = b invalid in other non-equilibrium universality classes ) need different forms of fdr ! Baumann et. al. 05 ; Durang & mh 09, Durang et al. 11 4. Surface growth deposition (evaporation) of particles on a surface ! height profile h(t; r) generic situation : RSOS (restricted solid-on-solid) model Kim & Kosterlitz 89 p = deposition prob. 1 − p = evap. prob. here p = 0:98 some universality classes : 2 µ 2 (a) KPZ @t h = νr h + 2 (rh) + η Kardar,Parisi,Zhang 86 2 (b)EW @t h = νr h + η Edwards,Wilkinson 82 4 (c)MH @t h = −νr h + η Mullins,Herring 63 ; Wolf & Villain 80 η is a gaussian white noise with hη(t; r)η(t0; r0)i = 2νT δ(t − t0)δ(r − r0) d −d P Family-Viscek scaling on a spatial lattice of extent L : h(t) = L j hj (t) Ld 1 X D 2E L2ζ ; if tL−z 1 w 2(t; L) = h (t) − h(t) = L2ζ f tL−z ∼ Ld j t2β ; if tL−z 1 j=1 β : growth exponent, ζ : roughness exponent, ζ = βz two-time correlator : t r C(t; s; r) = hh(t; r)h(s; 0)i − h(t) h(s) = s−bF ; C s s1=z with ageing exponent : b = −2β Kallabis & Krug 96 two-time integrated response : * sample A with deposition rates pi = p ± i , up to time s, * sample B with pi = p up to time s ; then switch to common dynamics pi = p for all times t > s Z s L *h(A) (t; s) − h(B) (t)+ z 1 X j+r j+r −a t jrj χ(t; s; r) = du R(t; u; r) = = s Fχ ; L j s s 0 j=1 Effective action of the KPZ equation : Z h 2 µ 2 2i J [φ, φe] = dtdr φe @t φ − νr φ − (rφ) − νT φe 2 =) Very special properties of KPZ in d = 1 spatial dimension ! Exact critical exponents β = 1=3, ζ = 1=2, z = 3=2, λC = 1 kpz 86 ; Krech 97 @φ @ v Special KPZ symmetry in1 D : let v = @r , φe = @r ep + 2T Z h ν µ i J = dtdr p@ v − (@ v)2 − v 2@ p + νT (@ p)2 e t 4T r 2 r e r e is invariant under time-reversal Lvov, Lebedev, Paton, Procaccia 93 ; Frey, Tauber,¨ Hwa 96 t 7! −t ; v(t; r) 7! −v(−t; r) ; ep 7! +ep(−t; r) 2 ) fluctuation-dissipation relation for t sTR (t; s; r) = −@r C(t; s; r) 2 find ageing exponents : λR = λC = 1,1+ a = b + z 1D relaxation dynamics, starting from an initially flat interface 8 < slow dynamics observe all 3 properties of ageing : no tti : dynamical scaling confirm expected exponents b = −2=3, λC =z = 2=3 N.B. : this confirmation is out of the stationary state Kallabis & Krug 96 ; Krech 97 ; Bustingorry et al. 07-10 ; Chou & Pleimling 10 ; D'Aquila & Tauber¨ 11 relaxation of the integrated response,1D 8 < slow dynamics observe all 3 properties of ageing : no tti : dynamical scaling exponents a = −1=3, λR =z = 2=3, as expected from FDR N.B. : numerical tests for 2 models in KPZ class Simple ageing is also seen in space-time observables 2=3 t r 3=2 correlator C(t; s; r) = s FC s ; s 1=3 t r 3=2 integrated response χ(t; s; r) = s Fχ s ; s confirm expected value of dynamical exponent z = 3=2 Values of some growth and ageing exponents in 1D model z a b λR = λC β ζ KPZ 3=2 −1=3 −2=3 1 1=3 1=2 exp 0:336(11) 0:50(5) EW 2 −1=2 −1=2 1 1=4 1=2 MH 4 −3=4 −3=4 1 3=8 3=2 Takeuchi, Sano, Sasamoto, Spohn 10/11 Two-time space-time responses and correlators consistent with simple ageing for 1D KPZ Similar results known for EW and MH universality classes Roethlein, Baumann, Pleimling 06 5.