SPEC/SPhT CEA emes hors de l Syst` Equilibre et Groupe de - PowerPoint PPT Presentation

SPEC/SPhT – CEA emes hors de l’´ Syst` Equilibre et Groupe de Renormalisation Non Perturbatif L´ eonie Canet Collaborateurs: e ∗ , B. Delamotte ∗∗ , O. Deloubri` ere † , I. Dornic ∗ , H. Chat´ noz ‡ et N. Wschebor †† H. Hilhorst # , M.A. Moore + , M.A. Mu˜ ∗ C.E.A. (Saclay), ∗∗ L.P.T.M.C. (Paris), # L.P.T. (Orsay), † Virginia Tech, + University of Manchester, ‡ University of Granada, †† University of MonteVideo L. C., C.E.A. Saclay

Outline • Introduction: Non-Equilibrium Systems • Non-Perturbative Renormalisation Group (NPRG) • Reaction-Diffusion Processes – Mean-Field - Field Theory – Directed Percolation (DP) – Branching and Annihilating Random Walks (BARW) • Growth Phenomena – The Kardar-Parisi-Zhang (KPZ) Equation – Kinetic Roughening and Strong Coupling Regime • Conclusion and Prospects L. C., C.E.A. Saclay 1

Introduction: Non Equilibrium Systems • existence of critical phenomena: phase transitions between non-equilibrium stationary states. − → challenge: identify the universality classes out of equilibrium. — no systematic analytical method — no conformal symmetry in d = 2 — no high order RG calculations available − → simple models: reaction-diffusion processes and growth phenomena. = ⇒ use of a non-perturbative approach: the Non Perturbative Renormalisation Group (NPRG) L. C., C.E.A. Saclay 2

Wilsonian Renormalisation Group discrete Spin-blocking: dk s| q continuum Separation of modes: s < | q<k s > | q>k q 0 k Λ e −S eff Z k [ s < ] = D s > e −S [ s > + s < ] Effective action: 0 δ 2 S eff δ S eff δ S eff 1 dd q Wilson-Polchinski : = 1 Z ∂k S eff k k k @ ∂kKk ( q ) δ s < ( q ) δ s < ( − q ) − δ s < ( q ) ∂ kKk ( q ) k A ( 2 π ) d 2 δ s < ( − q ) K.G. Wilson and J. Kogut, Phys. Rep. C 12 (1974) , J. Polchinski, Nucl. Phys. B 231 (1984) L. C., C.E.A. Saclay 3

Effective Average Action Γ [ψ ] = [ϕ] S k= Λ Λ Same procedure on Legendre transform: − → Γ k Gibbs free energy of fluctuation modes q > k Γ [ψ ] k k C. Wetterich, Phys. Lett. B 301 (1993) , T. Morris, Int. J. Mod. Phys. A 9 (1994) Γ [ψ ] = Γ[ψ] − → low-energy modes must be decoupled k=0 0 Z D φ e −S [ φ ] − ∆ S k [ φ ] + J.φ = ⇒ “mass” term in the partition function: Z k [ J ] = Z Legendre transform: Γ k = − ln Z k + J.ψ − ∆ S k [ ψ ] 2 R (q ) d d q ∆ S k [ φ ] = 1 Z k (2 π ) d φ ( q ) R k ( q ) φ ( − q ) 2 k 2  R k ( q 2 ) → 0 for k → 0 R k ( q 2 ) → ∞ for k → Λ k 2 q 2 L. C., C.E.A. Saclay 4

Exact Flow Equation for Γ k �� � ∂ k Γ k [ ψ ] = 1 � − 1 Γ (2) 2 Tr k [ ψ ] + R k ∂ k R k − → regularised in the IR and in the UV − → thermodynamical quantities from the limit k → 0 − → suitable for non-perturbative approximations: not restricted to critical dimension or weak couplings = ⇒ systematic and controlled approximation schemes. Applications at Equilibrium • frustrated antiferromagnets • Kosterlitz-Thouless transition M. Grater and C. Wetterich, Phys. Rev. Lett. 75 (1995) , B. Delamotte, D. Mouhanna and M. Tissier, Phys. Rev. B 69 G. von Gersdorff and C. Wetterich, Phys. Rev. B 64 (2001) (2004) L. C., C.E.A. Saclay 5

Non-Perturbative Approximations Schemes • write an Ansatz for Γ k dictated by the symmetries and physics of interest. derivative expansion e.g. Ising model: Z 2 symmetry = ⇒ invariant ρ = ψ 2 � � U k ( ρ ) + 1 / 2 Z k ( ρ ) ( ∂ ψ ) 2 � d d x Γ k [ ψ ] = Z ” 2 ( ∂ ψ ) 2 i 2 o n “ k ( ρ ) ( ∂ ψ ) 2 “ ” h d d x W a ∂ 2 ψ + W b ψ∂ 2 ψ + W c + k ( ρ ) k ( ρ ) ∂ 4 ( a ′ ) LPA ( a ) ∂ 2 ( a ) 7-bcles ( b ) MC ( c ) ν 0.6506 0.6281 0.632 0.6304(13) 0.6297(5) η 0 0.0443 0.033 0.0335(25) 0.0362(8) (a) L. Canet, B. Delamotte, D. Mouhanna and J. Vidal, Phys. Rev. B 68 (2003) (a’) L. Canet, B. Delamotte, D. Mouhanna and J. Vidal, Phys. Rev. D 67 (2003) R. Guida and J. Zinn-Justin, J. Phys. A 31 (1998) , (c) M. Hasenbusch, Int. J. Mod. Phys. C 12 (2001) (b) L. C., C.E.A. Saclay 6

Reaction-Diffusion Processes D A O A O • critical exponents : σ 2A A ( p − p c ) β n s ∼ λ O 2A ˛ ν ⊥ [ ν � ] µ ˛ ˛ O A ξ ⊥ [ ξ � ] ∼ ˛ p − p c ν � = ⇒ non-equilibrium phase z = ν ⊥ transition between an active and an absorbing state. H. Hinrichsen, Adv. Phys. 49 (2000) ∂ t n ( t ) = ( σ − µ ) n ( t ) − λ n ( t ) 2 Mean-Field Stationary solutions: n a = 0 and n s = ( σ − µ ) / (2 λ ) ≡ ∆ / (2 λ ) Explicit solution: n ( t ) = n 0 n s / [ n 0 + ( n s − n 0 ) exp( − ∆ t )] processes are diffusion-limited = ⇒ role of density fluctuations But L. C., C.E.A. Saclay 7

Field Theory D A O A O master φ e −S { λ,D,σ } [ φ, ˆ � φ ] D φ D ˆ σ Z = 2A A λ equation O 2A � � � S { λ,D,σ } [ φ, ˆ ˆ φ + U { λ,σ } [ φ, ˆ d d r dt ∂ t − D ∇ 2 � � φ ] = φ φ ] M. Doi, J. Phys. A 9 (1976) , L. Peliti, J. Phys. (Paris) (1984) NPRG out of Equilibrium �� � − 1 � Γ (2) ∂ k Γ k [ ψ, ˆ ˆ k [ ψ, ˆ ψ ] + ˆ ∂ k ˆ ψ ] = 1 • exact flow equation for Γ k : 2 Tr R k R k • Ansatz for reaction-diffusion processes: � � Z k ∇ 2 − D k ∂ t � Γ k ( ψ, ˆ ˆ ψ + U k ( ψ, ˆ d d r dt � � ψ ) = ψ ψ ) L. Canet, B. Delamotte, O. Deloubri` ere and N. Wschebor, Phys. Rev. Lett. 92 (2004) specific model = ⇒ define Γ Λ and its symmetries L. C., C.E.A. Saclay 8

Directed Percolation (DP) D A O A O very large universality class σ A 2A λ ∼ Ising Model for non-equilibrium systems O 2A µ O A √ Z φφ 2 − φ ˆ φ 2 ) + λφ 2 ˆ h φ 2 i d d r dt U [ φ, ˆ ( µ − σ ) φ ˆ 2 λσ ( ˆ φ ] = φ + − → equivalence with the “Reggeon Field Theory” (J. Cardy and R. Sugar, J. Phys. A 13 (1980) ) • NPRG calculation (order ∂ 2 ) ⇒ critical exponents to order ǫ 2 but = 2 2 1.8 1.8 z − upper critical dimension d c = 4 . 1.6 1.6 − no exact solution in d = 1 . 1.4 1.4 numerics exponents exponents NPRG 1.2 1.2 ν 1 1 • rapidity symmetry: 0.8 0.8 n φ ′ ( x ′ , t ′ ) − ˆ = φ ( x, − t ) 0.6 0.6 β ˆ φ ′ ( x ′ , t ′ ) 0.4 0.4 = − φ ( x, − t ) 0.2 0.2 1 1 1.5 1.5 2 2 2.5 2.5 3 3 ⇒ invariants φ ˆ φ and φ − ˆ = φ d d L. Canet et al. Phys. Rev. Lett. 92 (2004) L. C., C.E.A. Saclay 9

Branching and Annihilating Random Walks (BARW) D no spontaneous decay process A O A A O O σ m (m+1)A A M. Bramson and L. Gray, Z. Wahrsch. verw. Gebiete 68 , (1985) λ k O kA • Mean-Field: system always active for σ m � = 0 . • simulations: evidence of absorbing phase transitions in low dimensions. � odd m : DP − → two critical behaviours even m : new universality class (PC) • perturbative RG: J. Cardy and U. C. T¨ auber, Phys. Rev. Lett. 77 (1996) − → lowest order processes are the most relevant: σ σ 2A 3A A A odd-BARW: even-BARW: λ λ O O 2A 2A RG performed at σ ∼ 0 and around d c = 2 d c = 2 — critical dimension of the pure annihilation fixed point . L. C., C.E.A. Saclay 10

Odd-BARW σ 2A φ 2 ) φ 2 + σ (1 − ˆ A U [ φ, ˆ φ ] = − λ (1 − ˆ φ ) φ ˆ φ λ O 2A • simulations: existence of a phase transition in the DP class. H. Takayasu and A. Yu. Tretyakov, Phys. Rev. Lett. 68 (1992) , I. Jensen, J. Phys. A 26 (1993) • perturbative RG: J. Cardy and U. C. T¨ auber, Phys. Rev. Lett. 77 (1996) σ λ A = ⇒ effective action ∼ S DP A+A O generated by renormalisation ∆ R = σ R − µ R µ R λ σ ( , ) µ at a rate A O = ⇒ fluctuations can induce an absorbing transition only for d ≤ 2 . non−trivial transition Mean−Field σ σ σ D D D active active 2 π 4 D σ λ σ λ e π D 2 D active D λ λ λ abs. abs. D D D d 1 2 3 4 L. C., C.E.A. Saclay 11

Odd-BARW: NPRG and Numerical Simulations critical rates: − → NPRG calculation of non-universal quantities = ⇒ accessible through integration the phase diagrams of the flow. 10 n(t) = t - β / ν || − → Monte Carlo simulation 1 0.1 of the Master Equation: n(t) 0.01 0.001 • hypercubic lattice with 2d neighbours ( N ∼ 2 24 ) 1e+04 1e+06 • no restriction of occupation number t L. Canet, H. Chat´ e and B. Delamotte, • time continuous limit = ⇒ ∆ t ≪ 1 Phys. Rev. Lett. 92 (2004) L. C., C.E.A. Saclay 12

Odd-BARW: Mean-Field 3 2.5 2 σ /D 1.5 1 0.5 0 0 2 4 6 8 10 12 λ /D L. C., C.E.A. Saclay 13

Odd-BARW: Perturbative RG d=1 d=2 d=3 3 active phase 2.5 2 absorbing phase σ /D 1.5 1 0.5 0 0 2 4 6 8 10 12 λ /D L. C., C.E.A. Saclay 14

Odd-BARW: NPRG d=1 d=2 d=3 d=4 d=5 d=6 3 active phase 2.5 2 absorbing phase σ /D 1.5 1 0.5 0 0 2 4 6 8 10 12 λ /D L. C., C.E.A. Saclay 15

Odd-BARW: NPRG + Simulations d=1 d=1 d=2 d=2 d=3 d=3 d=4 d=4 d=5 d=5 d=6 d=6 3 3 active phase active phase 2.5 2.5 2 2 absorbing phase absorbing phase σ /D σ /D 1.5 1.5 1 1 0.5 0.5 0 0 0 0 2 2 4 4 6 6 8 8 10 10 12 12 λ /D λ /D L. C., C.E.A. Saclay 16