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

SPEC/SPhT – CEA Syst` emes hors de l’´ Equilibre et Groupe de Renormalisation Non Perturbatif L´ eonie Canet

Collaborateurs:

• H. Chat´

e ∗, B. Delamotte ∗∗, O. Deloubri` ere †, I. Dornic ∗,

• H. Hilhorst #, M.A. Moore +, M.A. Mu˜

noz ‡ et N. Wschebor ††

∗ 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: continuum Separation of modes:

s<| q<k | q>k

>

s Λ k q dk s| q

Effective action:

e−Seff

k [s<] =

Z Ds> e−S[s> + s<]

Wilson-Polchinski :

∂kSeff k = 1 2 Z ddq (2π)d @∂kKk(q) δ2Seff k δs<(q) δs<(−q) − δSeff k δs<(q)∂kKk(q) δSeff k δs<(−q) 1 A 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

Same procedure on Legendre transform: − → Γk Gibbs free energy of fluctuation modes q > k

• C. Wetterich, Phys. Lett. B 301 (1993), T. Morris, Int. J. Mod. Phys. A 9 (1994)

− → low-energy modes must be decoupled

k=0 k= k k Λ Λ

Γ [ψ ] = Γ[ψ]

S

Γ [ψ ] = [ϕ] Γ [ψ ]

= ⇒ “mass” term in the partition function: Zk[J] =

Z Dφ e−S[φ] − ∆Sk[φ] + J.φ

Legendre transform:

Γk = − ln Zk+ Z J.ψ − ∆Sk[ψ] ∆Sk[φ] = 1 2 Z ddq (2π)dφ(q)Rk(q)φ(−q)  Rk(q2) → 0 for k → 0 Rk(q2) → ∞ for k → Λ

q2 k2 k2 (q )

R

k

2

• L. C., C.E.A. Saclay

4

Exact Flow Equation for Γk

∂kΓk[ψ] = 1 2Tr

• Γ(2)

k [ψ] + Rk

−1 ∂kRk

• −

→ 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

• Kosterlitz-Thouless transition
• M. Grater and C. Wetterich, Phys. Rev. Lett. 75 (1995),
• G. von Gersdorff and C. Wetterich, Phys. Rev. B 64 (2001)
• frustrated antiferromagnets
• B. Delamotte, D. Mouhanna and M. Tissier, Phys. Rev. B 69

(2004)

• L. C., C.E.A. Saclay

5

Non-Perturbative Approximations Schemes

• write an Ansatz for Γk dictated by the symmetries and physics of interest.

derivative expansion e.g. Ising model: Z2 symmetry = ⇒ invariant ρ = ψ2 Γk[ψ] =

• ddx
• U k (ρ) + 1/2 Zk(ρ) (∂ ψ)2

+ Z ddx n W a

k(ρ)

“ ∂2ψ ”2 + W b

k(ρ) (∂ ψ)2 “

ψ∂2ψ ” + W c

k(ρ)

h (∂ ψ)2i2 o

LPA (a) ∂2 (a) ∂4 (a′) 7-bcles (b) MC (c) ν 0.6506 0.6281 0.632 0.6304(13) 0.6297(5) η 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)

(b)

• R. Guida and J. Zinn-Justin, J. Phys. A 31 (1998), (c) M. Hasenbusch, Int. J. Mod. Phys. C 12 (2001)
• L. C., C.E.A. Saclay

6

Reaction-Diffusion Processes

O O A A

D

O

λ

2A O A 2A

σ

A

µ = ⇒ non-equilibrium phase transition between an active and an absorbing state.

• H. Hinrichsen, Adv. Phys. 49 (2000)
• critical exponents:

ns ∼ (p − pc)β ξ⊥[ξ] ∼ ˛ ˛p − pc ˛ ˛ν⊥[ν] z = ν ν⊥

Mean-Field ∂tn(t) = (σ−µ) n(t)−λ n(t)2 Stationary solutions: na = 0 and ns = (σ − µ)/(2 λ) ≡ ∆/(2 λ) Explicit solution: n(t) = n0 ns/[n0 + (ns − n0) exp(−∆t)] But processes are diffusion-limited = ⇒ role of density fluctuations

• L. C., C.E.A. Saclay

7

Field Theory

O O O A A

D

A 2A 2A

σ λ master equation

Z =

• DφD ˆ

φ e−S{λ,D,σ}[φ, ˆ φ] S{λ,D,σ}[φ, ˆ φ] =

• ddr dt
• ˆ

φ

• ∂t − D ∇2

φ + U {λ,σ}[φ, ˆ φ]

• M. Doi, J. Phys. A 9 (1976), L. Peliti, J. Phys. (Paris) (1984)

NPRG out of Equilibrium

• exact flow equation for Γk:

∂kΓk[ψ, ˆ ψ] = 1

2Tr

• ˆ

Γ(2)

k [ψ, ˆ

ψ] + ˆ Rk −1 ∂k ˆ Rk

• Ansatz for reaction-diffusion processes:

Γk(ψ, ˆ ψ) =

• ddr dt
• ˆ

ψ

• Zk ∇2 − Dk ∂t
• ψ + U k(ψ, ˆ

ψ)

• L. Canet, B. Delamotte, O. Deloubri`

ere and N. Wschebor, Phys. Rev. Lett. 92 (2004)

specific model = ⇒ define ΓΛ and its symmetries

• L. C., C.E.A. Saclay

8

Directed Percolation (DP)

O O A A

D

O

λ

2A O A 2A

σ

A

µ

very large universality class ∼ Ising Model for non-equilibrium systems

U[φ, ˆ φ] = Z ddr dt h (µ − σ)φ ˆ φ + √ 2λσ( ˆ φφ2 − φ ˆ φ2) + λφ2 ˆ φ2i − → equivalence with the “Reggeon Field Theory”

(J. Cardy and R. Sugar, J. Phys. A 13(1980))

= ⇒critical exponents to order ǫ2 but

− upper critical dimension dc = 4. − no exact solution in d = 1.

• rapidity symmetry:

n φ′(x′, t′) = − ˆ φ(x, −t) ˆ φ′(x′, t′) = −φ(x, −t) = ⇒ invariants φ ˆ φ and φ − ˆ φ

• NPRG calculation (order ∂2)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 1.5 2 2.5 3 exponents d numerics NPRG 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 1.5 2 2.5 3 exponents d

z ν β

• L. Canet et al. Phys. Rev. Lett. 92 (2004)
• L. C., C.E.A. Saclay

9

Branching and Annihilating Random Walks (BARW)

O O O A A

D

A

λ σ

k

kA

m

(m+1)A

no spontaneous decay process

O A

• M. Bramson and L. Gray, Z. Wahrsch. verw. Gebiete 68, (1985)
• Mean-Field: system always active for σm = 0.
• simulations: evidence of absorbing phase transitions in low dimensions.

− → two critical behaviours

• dd m:

DP 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:

• dd-BARW:

O A 2A 2A

σ λ

even-BARW:

O A 2A

σ λ

3A

RG performed at σ ∼ 0 and around dc = 2

dc = 2 — critical dimension of the pure annihilation fixed point.

• L. C., C.E.A. Saclay

10

Odd-BARW

O A 2A 2A

σ λ

U[φ, ˆ φ] = −λ(1− ˆ φ2)φ2 + σ(1− ˆ φ)φˆ φ

• 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)

µR λ σ

at a rate

( , ) A

O

µ

A

A+A O

σ λ

generated by renormalisation

= ⇒ effective action ∼ SDP ∆R = σR − µR = ⇒ fluctuations can induce an absorbing transition only for d ≤ 2.

σ

active

λ

D D

σ

D

λ

D

abs. active

σ λ

D

active abs.

D e

π λ

4 D D

σ

D 2 D

π λ σ

2

1 2 3 4 d

non−trivial transition Mean−Field

• L. C., C.E.A. Saclay

11

Odd-BARW: NPRG and Numerical Simulations

− → NPRG calculation of the phase diagrams − → Monte Carlo simulation

• f the Master Equation:
• hypercubic lattice with 2d neighbours (N ∼ 224)
• no restriction of occupation number
• time continuous limit =

⇒ ∆t ≪ 1

critical rates: non-universal quantities = ⇒ accessible through integration

• f the flow.

0.001 0.01 0.1 1 10 1e+04 1e+06 n(t) t n(t) = t-β/ν||

• L. Canet, H. Chat´

e and B. Delamotte,

• Phys. Rev. Lett. 92 (2004)
• L. C., C.E.A. Saclay

12

Odd-BARW: Mean-Field

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D

• L. C., C.E.A. Saclay

13

Odd-BARW: Perturbative RG

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D

d=2 d=1 d=3

absorbing phase active phase

• L. C., C.E.A. Saclay

14

Odd-BARW: NPRG

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D

d=6 d=5 d=4 d=3 d=2 d=1

absorbing phase active phase

• L. C., C.E.A. Saclay

15

Odd-BARW: NPRG + Simulations

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D

d=6 d=5 d=4 d=3 d=2 d=1

absorbing phase active phase 0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D

d=6 d=5 d=4 d=3 d=2 d=1

absorbing phase active phase

• L. C., C.E.A. Saclay

16

Odd-BARW: NPRG

− → Phase diagram of odd-BARW

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D d=6 d=5 d=4 d=3 d=2 d=1 absorbing phase active phase 0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D d=6 d=5 d=4 d=3 d=2 d=1 absorbing phase active phase

= ⇒ non-perturbative threshold for the emergence of an inactive phase for d > 2. − → threshold vs dimension

4 6 8 10 3 4 5 6 λth / D d fit simulations NPRG

Existence of a DP absorbing phase transition in all finite dimensions.

• L. Canet, H. Chat´

e and B. Delamotte,

• Phys. Rev. Lett. 92 (2004)
• L. C., C.E.A. Saclay

17

Odd-BARW: Effective Destruction Mechanisms

• Large diffusion D ≫ 1:

σ D λ D

= ⇒ Mechanism suppressed in d > 2

t x

• f ‘‘stranger’’

annihilation particles

• Small diffusion D ≪ 1:

σ D λ D

= ⇒ Mechanism independent of the dimension

t

‘‘self−annihilation’’

x

• L. C., C.E.A. Saclay

18

Odd-BARW: ‘Sites’ Mean-Field

• Derivation of an alternative Mean-Field limit for reaction-diffusion

0.5 1 1.5 2 2.5 3 2 4 6 8 10 12 σ/D λ/D d=6 d=5 d=4 d=3 d=2 d=1 absorbing phase active phase

=5/2 ° absorbing phase σD λ D active phase ∆

d d+1

=2 α

• L. Canet and H. Hilhorst, submitted to J. Stat. Phys. (2006)

Exact calculation of the transitions lines in d > 2.

• L. C., C.E.A. Saclay

19

Even-BARW

O A 2A

σ λ

3A

U[φ, ˆ φ] = −λ(1 − ˆ φ2)φ2 + σ(1 − ˆ φ2)ˆ φφ − → number of particles conserved modulo 2. = ⇒ Z2 symmetry (invariants φ2, ˆ

φ2 and φ ˆ φ)

• simulations: absorbing transition in d = 1 of a

new universality class named ‘PC’ (parity conserving).

• P. Grassberger, F. Krause, and T. von der Twer, J. Phys. A 17, (1984), I. Jensen, Phys. Rev. E 50, (1994)
• perturbative RG: performed at σ ∼ 0 and around dc = 2

i.e. in the vicinity of the pure annihilation fixed point FA. − → σ becomes irrelevant (and FA stable) for d < 4/3. = ⇒ suggests the emergence of an inactive phase for d < 4/3.

• J. Cardy and U. C. T¨

auber, Phys. Rev. Lett. 77 (1996)

• L. C., C.E.A. Saclay

20

Even-BARW: NPRG

− → NPRG calculation (LPA)

• existence of 3 Fixed Points:

– Gaussian FG = {σ = 0, λ = 0} – Annihilation FA = {σ = 0, λ = 0} – Non-Perturbative FPC = {σ = 0, λ = 0} = ⇒ FPC governs an absorbing phase transition in d = 1. − → flow diagram of even-BARW in d = 1

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.5 1 1.5 2

active phase absorbing phase

FPC FA FG

σ λ

• L. Canet, H. Chat´

e, B. Delamotte, I. Dornic and M. A. Mu˜ noz, Phys. Rev. Lett. 95 (2005)

• L. C., C.E.A. Saclay

21

Even-BARW: NPRG

− → stability of the fixed points

• 2
• 1

1 2 0.8 1 1.2 1.4 1.6 1.8 2 eigenvalues d 4/3 FA FPC

• 2
• 1

1 2 0.8 1 1.2 1.4 1.6 1.8 2 eigenvalues d 4/3

• L. Canet, H. Chat´

e, B. Delamotte, I. Dornic and M. A. Mu˜ noz, Phys. Rev. Lett. 95 (2005)

= ⇒ FPC drives a transition in d < 4/3. NPRG LPA : ν = 2.0 ± 0.1 Monte Carlo: ν = 1.85 ± 0.1 = ⇒ FPC becomes unphysical for d > 4/3. = ⇒ FPC is not gaussian in any dimension.

Existence of a non-perturbative fixed point FPC governing the PC transition in d = 1.

• L. C., C.E.A. Saclay

22

Interface Growth and Roughening Transition

• roughness:

D [h(x, t)-h(x-L, t-τ)]2E ∼ L2χ f( τ Lz)

• Kardar-Parisi-Zhang equation:

∂th(x, t) = ν ∇2h(x, t) + λ 2 ` ∇h(x, t) ´2 + η(x, t). ˙ η(x, t)η(x′, t′) ¸ = 2 D δd(x − x′) δ(t − t′).

• M. Kardar, G. Parisi and Y.-C. Zhang, Phys. Rev. Lett. 56, (1986).

− → directed polymer in random media − → stirred fluids (noisy Burgers equation) − → flame-front propagation (Kuramoto-Sivashinski equation) − → magnetic flux lines in superconductors

• d = 1: interface always rough.

exact exponents: χ =1/2, z =3/2.

• d ≥ 2: roughening phase transition

λ ≪ 1: smooth phase (Edwards-Wilkinson) λ ≫ 1: rough phase

− → statistical properties? − → upper critical dimension?

• L. C., C.E.A. Saclay

23

Field Theory

Langevin equation = ⇒ Janssen-de Dominicis response functional

• H. K. Janssen, Z. Phys. B 23, (1976), C. de Dominicis, J. Phys. (Paris) C 1, (1976).

S[h, ˜ h] = Z ddx dt  ˜ h » ∂th − ν ∇2h − λ 2 (∇h)2 – − D ˜ h2 ff .

• Perturbative RG:

β-function exact to all orders in d = 2 + ǫ.

• K. J. Wiese, J. Stat. Phys. 93 143 (1998)
• Mode Coupling Theory:

surprisingly accurate exponents but ad-hoc approximation.

• E. Frey et al., Phys. Rev. E 53 4424 (1996)

1 2 3 4 gaussian stable FP rough phase:

d

roughening transition:

?

unstable FP

χ=0 χ=1/2

g = λ

D

ν3

2

• L. C., C.E.A. Saclay

24

NPRG Analysis

• Ansatz:

Γk(ψ, ˜ ψ) =

• x,t
• ˜

ψ

• µk ∂t − νk ∇2

ψ − λk 2

• ∇ψ

2 ˜ ψ − Dk ˜ ψ2.

• Galilean symmetry:

 ψ′(x′, t′) = ψ(x, t) + v.x ˜ ψ′(x′, t′) = ˜ ψ(x, t) together with  x′ = x − λk v.x t′ = t

• Galilean invariance:

 µk = 1 ∂sλk = ⇒ satisfied at all scale k and for any ( ˜ ψ, ψ).

• Fluctuation-dissipation

theorem:  ∂sDk = ∂sνk in d = 1 ⇒ satisfied at all scale k and for any ( ˜ ψ, ψ).

• Perturbative results:

 d ≥ 2 transition line with χ = 0 d = 1 fixed point with χ = 1/2 ⇒ recovered exactly.

• L. C., C.E.A. Saclay

25

Non Perturbative Phase Diagram

1 2 3 4

d

stable unstable

perturbative strong coupling smooth phase rough phase rough phase

g = Dλ2 ν3

• L. Canet, submitted to Europhys. Lett. cond-mat/0509541 (2005)

= ⇒ d > 2

Unstable Fixed Point: phase transition χ = 0 in all d > 2. Stable Fixed Point: rough phase

= ⇒ d < 2

Stable Fixed Point: rough phase χ = 1/2 in d = 1. Strong coupling Fixed Point in all dimensions.

• L. C., C.E.A. Saclay

26

Strong Coupling Phase

• d ≤ 2
• d ≥ 2

20 40 60 80 100 120 140

• 0.06 -0.04 -0.02

0.02 0.04 0.06 0.08 gKPZ φ d=0.5 d=1.0 d=1.5 d=1.9 d=2.0 20 40 60 80 100 120 140

• 0.06 -0.04 -0.02

0.02 0.04 0.06 0.08 gKPZ φ 100 200 300 400 500 600 700 800

• 0.06
• 0.04
• 0.02

0.02 0.04 gKPZ φ d=4.0 d=3.0 d=2.5 d=2.1 d=2.0 100 200 300 400 500 600 700 800

• 0.06
• 0.04
• 0.02

0.02 0.04 gKPZ φ

Two rough phases of different natures on either sides of d = 2.

• L. C., C.E.A. Saclay

27

Conclusion

• Phase diagram of odd-BARW

= ⇒ existence of a non-perturbative threshold for the transition.

• Flow diagram of even-BARW

= ⇒ existence of a genuine non-perturbative fixed point.

• Rough phase of the KPZ equation above d = 2

= ⇒ strong coupling fixed point.

Prospects

• Other reaction-diffusion processes:

− → PCPD (Pair Contact Process with Diffusion) − → role of quenched disorder in DP

• Other non-equilibrium systems:

− → growth phenomena with quenched disorder − → ageing and glassy dynamics

• L. C., C.E.A. Saclay

28