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

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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

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Overview :

1. Dynamical scaling & ageing : physical background
2. Form of the scaling functions &

Local Scale-Invariance (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¨
dinger invariance
8. Logarithmic ageing-invariance
9. Conclusions

ambition : argue for the applicability of lsi in non-equilibrium statistical mechanics & indicate some relevant mathematical structures

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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 → rb−1 Can one extend to local dynamical scaling, with z = 1 ? If z = 2, the Schr¨

dinger group is an example :

(Jacobi 1842), Lie 1881

t → αt + β γt + δ , r → Dr + vt + a γt + δ ; αδ − βγ = 1 ⇒ study ageing phenomena as paradigmatic example

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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

ccur 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 ?

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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 ≫ Tc > 0 2 quench to temperature T < Tc (or T = Tc)

→ 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) ∼ t1/z dynamical exponent z

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t = t1 t = t2 > t1

magnet T < Tc − → ordered cluster magnet T = Tc − → correlated cluster growth of ordered/correlated domains, of typical linear size L(t) ∼ t1/z dynamical exponent z : determined by equilibrium state

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illustration of statistical self-similarity for different times t1 < t2

Walter ’10

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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) two-time response R(t, s; r) := δ φ(t, r) δh(s, 0)

h=0

=

φ(t, r)

φ(s, r)

t : observation time

s : waiting time

causality condition t > s for responses

a) system at equilibrium : fluctuation-dissipation theorem

Nyquist 28, Kubo 66

R(t − s; r) = 1 T ∂C(t − s; r) ∂s , T : temperature b) far from equilibrium : C and R independent ! The fluctuation-dissipation ratio (fdr)

Cugliandolo, Kurchan, Parisi ’94

X(t, s) := TR(t, s) ∂C(t, s)/∂s measures the distance with respect to equilibrium : Xeq = X(t − s) = 1

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Dynamical scaling & ageing : 3D Ising model, T < Tc

no time-translation invariance dynamical scaling scaling regime : t, s ≫ τmicro and t − s ≫ τmicro

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Scaling regime : t, s ≫ τmicro and t − s ≫ τmicro C(t, s; r) = s−bfC t s , |r|z t − s

, R(t, s) = s−1−afR

t s , |r|z t − s

asymptotics : fC(y, 0) ∼ y−λC /z, fR(y, 0) ∼ y−λR/z for y ≫ 1

λC : autocorrelation exponent, λR : autoresponse exponent, z : dynamical exponent, a, b : ageing exponents

* exponents λC,R usually from independent renormalisations, * but initial conditions can imply scaling relations

Question : what about the form of these universal scaling functions ?

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Theoretical formulation

Langevin equation (model A of Hohenberg-Halperin 77) 2M∂φ ∂t = ∆φ − δV[φ] δφ + η

rder-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 ?

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Example : how to find these scaling forms → mean-field

Langevin eq. for magn. order parameter m(t)

drop spatial dependencies

dm(t) dt = 3λ2m(t) − m(t)3 + η(t) , η(t)η(s) = 2Tδ(t − s) contrˆ

le parameter λ2 :

(1) λ2 > 0 : T < Tc, (2) λ2 = 0 : T = Tc, (3) λ2 < 0 : T > Tc two-time observables : response R(t, s), correlation C(t, s) R(t, s) = δm(t) δh(s)

h=0

= 1 2T m(t)η(s) , C(t, s) = m(t)m(s) mean-field equation of motion (cumulants neglected) : ∂tR(t, s) = 3

λ2 − v(t)
R(t, s) + δ(t − s)

∂sC(t, s) = 3

λ2 − v(s)
C(t, s) + 2TR(t, s)

with variance v(t) = m(t)2, ˙ v(t) = 6

λ2 − v(t)
v(t)

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if λ2 ≥ 0 : fluctuations persist if λ2 < 0 : fluctuations disappear in the long-time limit t, s → ∞ : (t > s) R(t, s) ≃    1

s/t

e−3|λ2|(t−s) ; C(t, s) ≃ T      2 min(t, s) ; λ2 > 0 s

s/t

; λ2 = 0

1 (3|λ2|)e−3|λ2| |t−s|

; λ2 < 0 fluctuation-dissipation ratio measures distance from equilibrium

Cugliandolo, Kurchan, Parisi 94

X(t, s) = TR(t, s) ∂sC(t, s) ≃    1/2 + O(e−6λ2s) ; λ2 > 0 2/3 ; λ2 = 0 1 + O(e−|λ2| |t−s|) ; λ2 < 0 relaxation far from equilibrium, when X = 1, if λ2 ≥ 0 (T ≤ Tc)

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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,. . .

J [φ, φ] =

φ(2M∂t − ∆)φ +

φV′[φ]

J0[φ,

φ] : deterministic

−T

φ2 −
φt=0Cinit

φt=0

+ Jb[

φ] : noise (bruit)

φ : response field ;

C(t, s) = φ(t)φ(s), R(t, s) = φ(t) φ(s) averages : A0 :=

DφD

φ A[φ, φ] exp(−J0[φ, φ]) masses : Mφ = −M

φ

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Theorem : IF J0 is Galilei- and spatially translation-invariant,

then Bargman superselection rules

Bargman 54

φ1 · · · φn

φ1 · · · φm

0 ∼ δn,m

Illustration : computation of a response function

Picone & mh 04

R(t, s) =

φ(t)

φ(s)

=
φ(t)

φ(s)e−Jb[

φ]

=

φ(t)

φ(s)

0 = R0(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.

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Correlation functions for z = 2

find C(t, s) = φ(t)φ(s) = φ(t)φ(s)e−Jb[

φ]0 from Bargman rule

C(t, s) = a0 2

RddR R(3)

0 (t, s, 0; R)

initial + T 2M ∞ du

RddR R(3)

0 (t, s, u; R) thermal

R(3)

0 (t, s, u; r)

=

φ(t, y)φ(s, y)

φ2(u, r + y)

sch-invariance fixes three-point R(3)

function up to an unknown scaling function Ψ = ⇒ how to obtain a prediction for fC(y) ?

Theorem : LSI with z = 2 =

⇒ λC = λR

Picone & MH 04

agrees with a different RG argument of Bray and with all models

Conclusion : concentrate on dynamical symmetries of ‘deterministic part’

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2. Form of the scaling functions & lsi

Question : ? Are there model-independent results

n the form of universal scaling functions ?

‘Natural’ starting point : try to draw analogies with conformal invariance at equilibrium * Equilibrium critical phenomena : scale-invariance * For sufficiently local interactions : extend to conformal invariance space-dependent re-scaling (angles conserved) r → r/b(r)

Bateman & Cunningham 1909/10, Polyakov 70

In two dimensions : ∞ many conformal transformations (w → β(w) complex analytic) ⇒ exact predictions for critical exponents, correlators, . . .

BPZ 84

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Hidden assumptions : 1) extension scale-invariance − → conformal invariance ?

formally : energy-momentum tensor symmetric & traceless Callan, Coleman, Jackiw ’70

but counterexamples :

   lattice animals

Miller & De Bell 93

hydrodynamics

Riva & Cardy 05

renormalised FT

Fortin, Grinstein, Stergiou 12

2) choice of so-called ‘primary’ scaling operators not all physical models are unitary minimal CFTs − → SLE 3) how do primary operators transform ? usual form φ′(w) = β′(w)∆φ(β(w)) alternative : logarithmic partner ψ

Gurarie 93, Khorrami et al. 97,. . .

ψ′(w) = β′(w)∆ [ψ(β(w)) + ln β′(w) · φ(β(w))]

Logarithmic conformal invariance has been found in, e.g. critical 2D percolation

Cardy 92, Watts 96, Mathieu & Ridout 07/08

disordered systems

Caux et al. 96

sand-pile models

Ruelle et al. 08-10

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What about time-dependent critical phenomena ?

Cardy 85

Characterised by dynamical exponent z : t → tb−z, r → rb−1 ? Can one extend to local dynamical scaling, with z = 1 ? If z = 2, the Schr¨

dinger group is an example :

Jacobi 1842, Lie 1881

t → αt + β γt + δ , r → Dr + vt + a γt + δ ; αδ − βγ = 1 ⇒ study ageing phenomena as paradigmatic example essential : (i) absence of tti & (ii) Galilei-invariance Transformation t → t′ with β(0) = 0 and ˙ β(t′) ≥ 0 and t = β(t′) , φ(t) = dβ(t′) dt′ −x/z d ln β(t′) dt′ −2ξ/z φ′(t′)

ut of equilibrium, have 2 distinct scaling dimensions, x and ξ .

mean-field for magnets : expect ξ = 0 in ordered phase T < Tc ξ = 0 at criticality T = Tc NB : if tti (equilibrium criticality), then ξ = 0.

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z = 2 : consider infinitesimal transformations (t′, r′) = (t, r) + ǫX(t, r)

X−1 = −∂t time translation X0 = −t∂t − 1 2r · ∂r − x 2 dilatation X1 = −t2∂t − tr · ∂r − M 2 r2 − xt special transformation Y−1/2 = −∂r spatial translations Y+1/2 = −t∂r − Mr Galilei transformations M0 = −M phase shift D rotations

close into Schr¨

dinger Lie algebra sch(d) =
X±1,0, Y±1/2, M0, D
[Xn, Xn′]

= (n − n′)Xn+n′ [Xn, Ym] = n 2 − m

Yn+m , [D, Ym] ⊂ Ym
Y (j)

m , Y (j′) m′

=

δj,j′(m − m′)M0 , [D, D] ∈ so(d) = ⇒ not semi-simple = ⇒ projective representations (‘mass’ M !)

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dynamical symmetry of Schr¨

dinger equation S = 2M∂t − ∂r · ∂r :

Lie 1881 Niederer 72

[S, X0] = −S , [S, X1] = −2tS −

x − d

2

M0

= ⇒ fixes scaling dimension of solution of Sφ = 0, x = xφ = d/2 co-variant two-point (response) function :

mh 93, mh & Unterberger 03

φ1(t1, r1)φ2(t2, r2) ∼

Causality

Θ(t1 − t2) ·

δ(M1 + M2)

Bargman superselection rule

· δx1,x2 × (t1 − t2)−x1 exp

−M1

2 (r1 − r2)2 t1 − t2

non-relativistic AdS/CFT correspondence, cold atoms, . . .

since 2008

− → classification of non-relativistic conformal symmetries, but for pure vector fields only

Duval & Horv´ athy 09

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*Ageing : no time-translation-invariance = ⇒ leave out X−1 ! go over to ageing algebra age(d) :=

X1,0, Y±1/2, M0, D
X0

= −t∂t − 1 2r · ∂r − x 2 dilatation X1 = −t2∂t − tr · ∂r − M 2 r2 − (x + ξ)t special Y−1/2 = −∂r spatial translations Y+1/2 = −t∂r − Mr Galilei transformations M0 = −M phase shift

two scaling dimensions x,ξ of a scaling operator φ Schr¨

dinger operator : S = 2M∂t − ∂2

r +2M

x + ξ − d

2

t−1

[S, X0] = −S , [S, X1] = −2tS physical example : kinetic spherical model

(model A dynamics)

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Example for the t−1-term in Langevin eq. : spherical model

continuous spins Si ∈ R, constraint

i∈Λ S2 i = N = |Λ|

hamiltonian H = −

(i,j) SiSj

Berlin & Kac 1953

for d > 2 phase transitions Tc > 0, exponents not mean-field if d < 4 standard trick : consider mean spherical model

i∈Λ S2

i

= N = |Λ|

Lewis & Wannier 1953

Langevin equation, with Lagrange multiplier z(t) & centered gaussian noise ηi(t)

∂Si ∂t = −δH δSi + z(t)Si + ηi , ηi(t)ηj(s) = 2Tδi,jδ(t − s)

set g(t) := exp

2

t

0 dt′ z(t′)

, spherical constraint gives Volterra eq.

g(t) = f (t) + 2T t dτ f (t − τ)g(τ) , f (t) =

e−4tI0(4t)

d find for T ≤ Tc : g(t) t→∞ ∼ t−̥ ⇔ z(t)∼ ̥

2 t−1

Godr` eche & Luck 00 Picone & mh 04

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Co-variant two-time response function : a = 1

z (x +

x) − 1 , a′ − a = 2

z

ξ +

ξ

,

λR z

= x + ξ

R(t, s; r) = δφ(t, r) δh(s, 0)

h=0

= R(t, s) exp

−M1

2 r2 t − s

R(t, s)

∼ s−1−a t s 1+a′−λR/z t s − 1 −1−a′

N.B. : holds true for z = 2, autoresponse R(t, s; 0) can also be used for generic z

Ex : kinetic Glauber-Ising model T = Tc

Picone & mh 04 ; mh, Enß, Pleimling 06

1D a′ − a = − 1

2

2D a′ − a ≃ −0.17(2) 3D a′ − a ≃ −0.022(5)

a′ = a fits lattice data ; 2nd-order ε-expansion disagrees

Calabrese & Gambassi 03 should one re-sum the ε-expansion series, as at equilibrium, for consistency with simulations ?

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3. Large-distance behaviour & Causality

link with AdS/CFT correspondence ?

Question : space-dependent part of co-variant two-point function was sch(d) : F ∼ exp

−M

2 r2 t − s

, cga(d) : F ∼ exp
−2 γ · r

t − s

? how to guarantee that |F| → 0 for large distances |r| → ∞ ?

Three ingredients :

1 dualisation of mass/rapidity → dual coordinate ζ 2 extension to parabolic sub-algebras 3 analyticity in dual space

? in which setting are co-variant n-point functions analytic ?

SLIDE 26

Schr¨

dinger algebra sch(d) :

id´ ee : treat the mass M as a variable, define ‘dual’ coordinate ζ

Giulini 96

φ(t, r) = φM(t, r) = 1 √ 2π

R

dζ e−iMζ φ(ζ, t, r) trade projective representation for ‘true’ representation in dual space Xn = in(n + 1) 4 tn−1r2∂ζ − tn+1∂t − n + 1 2 tnr · ∂r − (n + 1)x 2tn Ym = i

m + 1

2

tm−1/2r∂ζ − tm+1/2∂r

Mn = itn∂ζ

mh & Unterberger 03

Generators live at the boundary of (d + 3)-dim. Lorentzian space

e.g. Minic & Pleimling 08, Fuertes & Moroz 09, Leigh & Hoang 09,. . .

The Schr¨

dinger/heat equation becomes S

φ = 0, explicitly S φ = 2i ∂2 φ ∂ζ∂t + ∂2 φ ∂r2 = 0

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Parabolic subalgebras of B2

Parabolic subalgebra : the Cartan subalgebra h ⊕ the ‘positive roots’. positive roots : all roots to the right of a straight line through h Classification of parabolic subalgebras of B2 ∼ = conf(3)C : mh & Unterberger 03

1. extended ageing

age(1) := age(1) + CN

= minimal standard parabolic subalgebra

2. extended Schr¨
dinger

sch(1) := sch(1) + CN

3. extended conformal Galilean (altern)

cga(1) = alt(1) := alt(1) + CN

Havas & Plebanski 78, Brown & Henneaux, mh 97, Negro et al. 97, Ovsienko & Roger 98, Lukierski et al. 06/07, Barnich et al. 06 . . . les cordistes 09. . .

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Conformal Galilean algebra cga(1) : id´ ee : treat the rapidity γ as a variable, define ‘dual’ coordinate ζ

simplify here : d = 1

φ(t, r) = φγ(t, r) = 1 √ 2π

R

dζ e−iγζ φ(ζ, t, r) explicit representation in dual space Xn = in(n + 1)tn−1r∂ζ − tn+1∂t − n + 1tnr∂r − (n + 1)xtn Yn = i (n + 1) tnr∂ζ − tn+1∂r N.B. in d > 1 dimensions, rotation generator(s) intrisically rotate also components of ζ

Cherniha & mh 10

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extra generator N for maximal parabolic sub-algebras :

H. & U. 03

mh 13/14

sch(1) :

N = ζ∂ζ − t∂t + ξ

cga(1) :

N = −ζ∂ζ − r∂r − ν

ξ : 2nd scaling dimension in age(d), ν : scalar

Find co-variant dual two-point function F = φ1 φ2,

x1 = x2 ζ± = 1

2(ζ1±ζ2), t = t1 − t2, r = r1 − r2

sch(1) :

F(ζ−, t, r) =

f0|t|−x1f 2ζ−t + ir2 |t|

cga(1) :
F(ζ+, t, r) =

f0|t|−2x1f 2ζ+t + ir t

extra information from N-covariance :
sch(1) :

f (u) = f0u−x1−ξ1−ξ2

cga(1) :

f (u) = f0u−ν1−ν2

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I) for sch(1) : use ζ = ζ1 − ζ2, η = ζ1 + ζ2, invert dualisation

F = 1 2π

R2dζ1dζ2 e−iM1ζ1−iM2ζ2 |t|−x f

2(ζ1 − ζ2)t + ir 2 |t|

=

|t|−x 4π

R

dη e−i(M1+M2)η/2

4πδ(M1+M2)
R

dζ e−i(M1−M2)ζ/2 f

2sign (t)
ζ + i

2 r 2 sign (t) |t|

=

δ(M1 + M2)|t|−x f0

R

dζ e−iM1ζ (2sign (t))−x−ξ

ζ +

ir 2 2sign (t)|t| −x−ξ = δ(M1 + M2) (2sign (t))−x−ξ Mx+ξ−1

1

|t|−x f0

R+ iM1

2 r2 t

dζ e−iζζ−x−ξ

I (0)

± (x+ξ)

e− M1

2 r2 t

= δ(M1 + M2)|t|−x 2−x−ξMx+ξ−1

1

f0I (0)

+ (x + ξ)

=: F0

e− M1

2 r2 t Θ(t)

if x + ξ > 0

M1 t

> 0 & physical convention M1 > 0 ⇒ causality condition t = t1 − t2 > 0 co-variant F should be interpreted as (causal) reponse function !

extensions to logarithmic representations and age(d) possible

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II) for cga(1) : use ζ± = 1

2(ζ1 ± ζ2), invert dualisation.

H+ : upper complex half-plane

Definition. g : H+ → C, is in the Hardy class H+

2 , if it is holomorphic in H+ and

M2 = sup

v>0

R

du |g(u + iv)|2 < ∞ Lemma [Akhiezer]. If g ∈ H+

2 , there is a function G+ ∈ L2(0, ∞) such that for v > 0

g(w) = g(u + iv) = 1 √ 2π ∞ dγ eiγw G+(γ)

fix λ := r/t, write f (ζ+ + iλ) = fλ(ζ+). show that : if ν > 1

4 and λ > 0, then fλ ∈ H+ 2 . From Lemma :

f (ζ+ + iλ) =

1 √ 2π

R

dγ+ Θ(γ+) ei(ζ++iλ)γ+ F+(γ+) A) for λ > 0, derive a two-point function F which decays for λ → ∞ : F = |t|−2x1 π √ 2π

R2dζ+dζ− e−i(γ1+γ2)ζ+e−i(γ1−γ2)ζ−
R

dγ+ Θ(γ+) F+(γ+)e−γ+λeiγ+ζ+ = |t|−2x1 π √ 2π

R

dγ+ Θ(γ+) F+(γ+)e−γ+λ

R

dζ− e−i(γ1−γ2)ζ−

R

dζ+ ei(γ+−γ1−γ2)ζ+ = δ(γ1 − γ2)Θ(γ1)F0,+(γ1)e−2γ1λ|t|−2x1

SLIDE 32

show that : if ν > 1

4 and λ < 0, then fλ ∈ H− 2 .

B) for λ < 0, derive a two-point function F which decays for |λ| → ∞ : F = δ(γ1 − γ2)Θ(−γ1)F0,−(γ1)e2γ1|λ||t|−2x1

Combine λ > 0 and λ < 0, and extend to d ≥ 1 dimensions, assume rotation-invariance & continuity in r F(t, r) = δx1,x2δ(γ1 − γ2) |t|−2x1 exp

−2
γ1 · r

t

F0(γ2

1)

co-variant F should be interpreted as correlation function ! different representations give qualitatively distinct results simple analytic structure in dual variables

SLIDE 33

4. Proposal : local scale-transformations for z = 2

Extend known cases z = 1, 2 = ⇒ axioms of LSI :

MH 97/02 1 M¨

bius transformations in time (generator Xn)

t → t′ = αt + β γt + δ ; αδ − βγ = 1 require commutator : [Xn, Xn′] = (n − n′)Xn+n′

2 Dilatation generator : X0 = −t∂t − 1

z r · ∂r − x z

Implies simple power-law scaling L(t) ∼ t1/z (no glasses !).

3 Spatial translation-invariance → 2e family Ym of generators. 4 Xn contain phase terms from the scaling dimension x = xφ 5 Xn, Ym contain further ‘mass terms’ (Galilei !) 6 finite number of independent conditions for n-point functions. 7 use ‘local’ generators (e.g. no fractional derivatives) mh, Nucl. Phys. B641, 405 (2002)

SLIDE 34

Theorem : LSI without ‘masses’

MH 02

Commutators [Xn, Xn′] = (n − n′)Xn+n′, [Xn, Ym] = n

z − m

Yn+m

with n, n′ ∈ Z and m ∈ Z − 1/z have only the realisations : z Xn = −tn+1∂t − n+1

z tnr∂r − (n+1)x z

tn − n(n+1)

2

B10tn−1rz Yk−1/z = −tk∂r − z2

2 kB10tk−1r−1+z

2 Xn = −tn+1∂t − 1

2(n + 1)tnr∂r − 1 2(n + 1)xtn

− n(n+1)

2

B10tn−1r2 − (n2−1)n

6

B20tn−2r4 Yk−1/2 = −tk∂r − 2kB10tk−1r − 4

3k(k − 1)B20tk−2r3

1 Xn = −tn+1∂t − A−1

10 [(t + A10r)n+1 − tn+1]∂r

−(n + 1)xtn − n+1

2 B10 A10 [(t + A10r)n − tn]

Yk−1 = −(t + A10r)k∂r − k

2B10(t + A10r)k−1

free parameters (two in each case) : z, A10, B10, B20

SLIDE 35

Three distinct algebras emerge :

1. generic z :

[Xn, Xn′] = (n − n′)Xn+n′ , [Xn, Ym] = n z − m

Yn+m

with n ∈ Z and m ∈ Z − 1/z. Only if B10 = 0 : = ⇒ [Ym, Ym′] = 0. In this case, if z = 2/N and furthermore N ∈ N, then finite-dimensional subalgebra

X±1,0, Y−N/2,−N/2+1,...,N/2
mh, Phys. Rev. Lett. 78, 1940 (1997)

called nowadays by string theorists ‘spin-l algebra’ or ‘N-cga’

But if B10 = 0 : difficult closure problem !

SLIDE 36

2. z = 2. The Schr¨
dinger algebra.
r N = 1

then have two dimensionful parameters B10 and B20 Find closed infinite-dimensional extension of sch(1) :

mh ’02

define three families of charges Z (0)

n

:= −2tn, Z (1)

m

:= −2tm−1/2r and Z (2)

n

:= −ntn−1r2 [Ym, Ym′] = (m − m′)

4B20Z (2)

m+m′ + B10Z (0) m+m′

[Xn, Z (0,2)

n′

] = −n′Z (0,2)

n+n′ , [Xn, Z (1) m ] = −(n/2 − m)Z (1) n+n′

[Ym, Z (1)

m′ ]

= −Z (0)

m+m′ , [Ym, Z (2) n ] = −nZ (1) m+n

Recover Schr¨

dinger-Virasoro algebra

mh 94

sv(1) = Xn, Ym, Mnn∈Z,m∈Z+ 1

2

⊃ sch(1) for B20 = 0 and B10 = M/2.

? physical applications of these infinite-dimensional symmetries ?

SLIDE 37

3. z = 1. Around the conformal galilean algebra or N = 2.

mh ’07,’02

Then [Yn, Yn′] = A10(n − n′)Yn+n′ ,

in d = 1 dimensions.

* If A10 = 0, then isomorphic to vect(S1) × vect(S1) ∼ = conf(2). Xn = ℓn + ℓn , Yn = A10ℓn * Invariant Schr¨

dinger operator S = −A10∂t + ∂r. (with x = B10/2A10)

* Set A10 =: µ and B10 =: 2γ. Quasi-primary operator φi characterised by the triplett (xi, µi, γi). φ1φ2 = δx1,x2δµ1,µ2δγ1,γ2 f0 t−2x1

12

1 + µ1

r12 t12 −2γ1/µ1

φ1φ2φ3 = f123 t

−x13,2 13

t

−x23,1 23

t

−x12,3 12

1 + µ

r13 t13 −γ13,2/µ 1 + µ r23 t23 −γ23,1/µ 1 + µ r12 t12 −γ12,3/µ

and Bargman rule µ1 = µ2 = µ3 =: µ universal constant.

Distinct from the two- and three-point functions of conformal invariance. = ⇒ extension to dynamical symmetries of Vlassov equation

Stoimenov & mh 15

SLIDE 38

In the limit A10 = µ → 0, contraction to altern-Viraosoro algebra av(1) ⊃ alt(1) ≡ cga(1) ≡ bms3 .

r ‘full conformal galilean algebra’

Havas & Plebanski ’78, Brown & Henneaux 86, mh ’97, Negro et al. ’97, Martelli & Tachikawa 09, . . .

In d space dimensions, generators of av(d) ⊃ cga(d) ≡ alt(d) (γ ∈ Rd) Xn = −tn+1∂t − (n + 1)tnr · ∇ − (n + 1)tnx − n(n + 1)tn−1γ · r Y (j)

n

= −tn+1∂j − (n + 1)tnγj R(jk) = −

rj∂k − rk∂j
−
γj∂γk − γk∂γj
;

j = k Cherniha & mh ’10 with abbreviations ∂j =

∂ ∂rj .

Representation characterised by : (i) scaling dimension x (ii) rapidity γ. Non-vanishing commutators :

Xn, Xm
= (n − m)Xn+m ,
Xn, Y (j)

m

= (n − m)Y (j)

n+m ,

R(jk)

, Y (ℓ)

m

= δj,ℓ Y (k)

m

− δk,ℓ Y (j)

m

* two Virasoro-like independent central charges

Ovsienko & Roger 98

SLIDE 39

* contract two- & three-point functions (limit µ → 0), find

mh ’02 ; Martelli & Tashikawa 09, Bagchi, Mandal & Gopakumar 09, Hosseiny & Rouhani 10,. . .

φ1φ2 = δx1,x2δγ1,γ2 f0 t−2x1

12

exp

−2γ1 · r12

t12

φ1φ2φ3 = f123 t

−x13,2 13

t

−x23,1 23

t

−x12,3 12

exp

−γ12,3 · r12

t12 − γ23,1 · r23 t23 − γ31,2 · r31 t31

with xij,k := xi + xj − xk and γij,k := γi + γj − γk.

* For d = 2 so-called exotic central extension of cga(2), but

incompatible with ∞-dim. extension of cga(2) ⊂ av(2)

Lukierski, Stichel, Zakrewski 06/07

known (conditionally) invariant non-linear hydrodynamic equations (= Navier-Stokes)

Zhang & H´

rvathy ’09, Cherniha & mh ’10

* similar classification from a geometric point of view, using the Newton-Cartan formalism

Duval & H´

rvathy 09

SLIDE 40

5. Simple magnets and growing interfaces : analogies

Common properties of critical and ageing phenomena : * collective behaviour,

very large number of interacting degrees of freedom

* algebraic large-distance and/or large-time behaviour * described in terms of universal critical exponents * very few relevant scaling operators * justifies use of extremely simplified mathematical models

with a remarkably rich and complex behaviour

* yet of experimental significance

SLIDE 41

Interface growth

deposition (evaporation) of particles on a substrate → height profile h(t, r) slope profile u(t, r) = ∇h(t, r)

p = deposition prob. 1 − p = evap. prob.

Questions : * average properties of profiles & their fluctuations ? * what about their relaxational properties ? * are these also examples of physical ageing ? ? does dynamical scaling always exist ? are there extensions ?

SLIDE 42

Magnets thermodynamic equilibrium state

rder parameter φ(t, r)

phase transition, at critical temperature Tc variance :

(φ(t, r) − φ(t))2

∼ t−2β/(νz) relaxation, after quench to T ≤ Tc autocorrelator C(t, s) = φ(t, r)φ(s, r)c Interfaces growth continues forever height profile h(t, r)

same generic behaviour throughout

roughness : w(t)2 =

h(t, r) − h(t)

2 ∼ t2β relaxation, from initial substrate : autocorrelator C(t, s) =

h(t, r) − h(t)

h(s, r) − h(s)

ageing scaling behaviour :

when t, s → ∞, and y := t/s > 1 fixed, expect, with waiting time s

bservation time t > s

C(t, s) = s−bfC (t/s) and fC(y)

y→∞

∼ y−λC /z b, β, ν and dynamical exponent z : universal & related to stationary state autocorrelation exponent λC : universal & independent of stationary exponents

SLIDE 43

Magnets exponent value b =

;

T < Tc 2β/νz ; T = Tc

Interfaces exponent value b = −2β models : (a) gaussian field H[φ] = − 1

2

dr (∇φ)2

(b) Ising model H[φ] = − 1

2

dr
(∇φ)2 + τφ2 + g

2φ4

such that τ = 0 ↔ T = Tc

dynamical Langevin equation (Ising) : ∂tφ = −D δH[φ] δφ + η = D∇2φ + τφ + gφ3 + η (a) Edwards-Wilkinson (ew) : ∂th = ν∇2h + η (b) Kardar-Parisi-Zhang (kpz) : ∂th = ν∇2h + µ

2(∇h)2 + η

η(t, r) is the usual white noise, η(t, r)η(t′, r′) = 2Tδ(t − t′)δ(r − r′)

phase transition exactly solved d = 2 relaxation exactly solved d = 1

Onsager ’44, Glauber ’63, . . .

growth exactly solved d = 1

Sasamoto & Spohn ’10 Calabrese & Le Doussal ’11, . . .

SLIDE 44

Family-Viscek scaling on a spatial lattice of extent Ld : h(t) = L−d

j hj(t)

Family & Viscek 85

w2(t; L) = 1 Ld

Ld

j=1
hj(t) − h(t)

2 = L2αf

tL−z

∼

L2α

; if tL−z ≫ 1 t2β ; if tL−z ≪ 1

β : growth exponent, α : roughness exponent, α = βz two-time correlator :

limit L → ∞

C(t, s; r) =

h(t, r) −
h(t)

h(s, 0) −

h(s)
= s−bFC

t s , r s1/z

with ageing exponent : b = −2β

Kallabis & Krug 96

expect for y = t/s ≫ 1 : FC(y, 0) ∼ y−λC /z autocorrelation exponent rigorous bound : λC ≥ (d + zb)/2

Yeung, Rao, Desai 96 ; mh & Durang 15

KPZ class, to all orders in perturbation theory λC = d , if d < 2

Krech 97

SLIDE 45

1D relaxation dynamics, starting from an initially flat interface

bserve all 3 properties of ageing :

   slow dynamics no tti dynamical scaling confirm simple ageing for the 1D kpz universality class confirm expected exponents b = −2/3, λC/z = 2/3

pars pro toto

Kallabis & Krug 96 ; Krech 97 ; Bustingorry et al. 07-10 ; Chou & Pleimling 10 ; D’Aquila & T¨ auber 11/12 ; mh, Noh, Pleimling 12 . . .

SLIDE 46

A spherical model of interface growth : the Arcetri model

? kpz − → intermediate model − → ew ?

preferentially exactly solvable, and this in d ≥ 1 dimensions

inspiration : mean spherical model of a ferromagnet

Berlin & Kac 52 Lewis & Wannier 52

Ising spins σi = ±1

bey

i σ2 i = N = # sites

spherical spins Si ∈ R spherical constraint

i S2

i

= N

hamiltonian H = −J

(i,j) SiSj − λ i S2 i

Lagrange multiplier λ

exponents non-mean-field for 2 < d < 4 and Tc > 0 for d > 2 kinetics from Langevin equation ∂tφ = −D δH[φ]

δφ

+ z(t)φ + η time-dependent Lagrange multiplier z(t) fixed from spherical constraint all equilibrium and ageing exponents exactly known, for T < Tc and T = Tc

Ronca 78, Coniglio & Zannetti 89, Cugliandolo, Kurchan, Parisi 94, Godr` eche & Luck ’00, Corberi, Lippiello, Fusco, Gonnella & Zannetti 02-14 . . .

SLIDE 47

consider RSOS/ASEP-adsorption process :

rigorous : continuum limit gives KPZ

Bertini & Giacomin 97

use not the heights hn(t) ∈ N on a discrete lattice, but rather the slopes un(t) = 1

2 (hn+1(t) − hn−1(t)) = ±1

RSOS

? let un(t) ∈ R, & impose a spherical constraint

nun(t)2 !

= N ? ? consequences of the ‘hardening’ of a soft ew-interface by a ‘spherical constraint’ on the un ?

SLIDE 48

Arcetri model : precise formulation & simple ageing

slope u(t, x) = ∂xh(t, x) obeys Burgers’ equation,

mh & Durang 15

replace its non-linearity by a mean spherical condition = ⇒ ∂tun(t) = ν (un+1(t) + un−1(t) − 2un(t)) + z(t)un(t) +1 2 (ηn+1(t) − ηn−1(t))

n
un(t)2

= N

ηn(t)ηm(s) = 2Tνδ(t − s)δn,m

Extension to d ≥ 1 dimensions :

z(t) Lagrange multiplier

define gradient fields ua(t, r) := ∇ah(t, r),

a = 1, . . . , d :

∂tua(t, r) = ν∇r · ∇rua(t, r) + z(t)ua(t, r) + ∇aη(t, r)

r

d

a=1
ua(t, r)2

= dNd interface height : ua(t, q) = i sin qa h(t, q)

; q = 0

in Fourier space

SLIDE 49

exact solution :

ω(q) = d

a=1(1 − cos qa),

q = 0

h(t, q) =

h(0, q)e−2tω(q)

1

g(t) + t dτ η(τ, q)

g(τ)

g(t) e−2(t−τ)ω(q) in terms of the auxiliary function g(t) = exp

−2

t

0 dτ z(τ)

,

which satisfies Volterra equation g(t) = f (t) + 2T t dτ g(τ)f (t − τ) , f (t) := d e−4tI1(4t) 4t

e−4tI0(4t)

d−1 * for d = 1, identical to ‘spherical spin glass’, with T = 2TSG : hamiltonian H = − 1

2

i,j JijSiSj ; Jij random matrix, its eigenvalues

distributed according to Wigner’s semi-circle law

Cugliandolo & Dean 95

* also related to distribution of first gap of random matrices Perret & Schehr 15/16 * for 2 < d < 4, scaling functions identical to the ones of the critical bosonic pair-contact process with diffusion, with rates Γ[2A → (2 + k)A]=Γ[2A → (2 − k)A] = µ

k = 1, 2

Howard & T¨ auber 97 ; Houchmandzadeh 02 ; Paessens & Sch¨ utz 04 ; Baumann, mh, Pleimling, Richert 05

SLIDE 50

phase transition : long-range correlated surface growth for T ≤ Tc 1 Tc(d) = 1 2 ∞ dt e−dtt−1I1(t)I0(t)d−1 ; Tc(1) = 2, Tc(2) = 2π π − 2 Some results : always simple ageing upper critical dimension d∗ = 2

1. T = Tc, d < 2 :

rough interface, width w(t) = t(2−d)/4 = ⇒ β = 2−d

4

> 0 ageing exponents a = b = d

2 − 1, λR = λC = 3d 2 − 1 ; z = 2

exponents z, β, a, b same as ew, but exponent λC = λR different

2. T = Tc, d > 2 :

smooth interface, width w(t) = cste. = ⇒ β = 0 ageing exponents a = b = d

2 − 1, λR = λC = d ; z = 2

same asymptotic exponents as ew, but scaling functions are distinct

3. T < Tc :

rough interface, width w2(t) = (1 − T/Tc)t = ⇒ β = 1

2

ageing exponents a = d

2 − 1, b = −1, λR = λC = d−2 2

; z = 2

SLIDE 51

Illustration : Shape of the height Fluctuation-Dissipation Ratio,

T = Tc

Cugliandolo, Kurchan, Parisi 94

X(t, s) := TR(t, s) ∂C(t, s) ∂s = X t s t/s→∞ − → X∞ =

d/(d + 2)

; 0 < d < 2

d/4

; 2 < d

limit FDR X∞ is universal

Godr` eche & Luck 00

distinct from XEW,∞ = 1/2 for all d > 0

green line : XEW for d = 4

SLIDE 52

Example for the t−1-term in Langevin eq. : Arcetri model

continuous slopes ui ∈ Rd, constraint

i∈Λ u2 i = dN

for d > 0 phase transition Tc(d) > 0, exponents not mean-field if d < 2 spherical constraint :

i∈Λ u2

i

= dN

Langevin equation, with Lagrange multiplier z(t) & centered gaussian noise ηi(t)

∂ua(t,r) ∂t = ν∆ua(t,r) + z(t)ua(t,r) + ∇aη(t,r) ,

η(t, r)η(s, r′)
= 2νTδ(t − s)δ(r − r′)

set g(t) := exp

2

t

0 dt′ z(t′)

, spherical constraint gives Volterra eq.

g(t) = f (t) + 2T t dτ f (t − τ)g(τ) , f (t) = de−4tI1(4t) 4t

e−4tI0(4t)

d−1 find for T ≤ Tc : g(t) t→∞ ∼ t−̥ ⇔ z(t)∼ ̥

2 t−1

quite analogous to spherical model of a ferromagnet

Godr` eche & Luck 00 Picone & mh 04

SLIDE 53

6. Some numerical experiments

(A) Kardar-Parisi-Zhang (KPZ) (B) majority voter/Glauber models, at T = Tc (C) directed percolation (DP)

simple ageing of the correlators and responses, especially C(t, s) = s−bfC t s

, R(t, s) = s−1−afR

t s

fC(y)

∼ y−λC /z , fR(y) ∼ y−λR/z

y ≫ 1

values of the non-equilibrium exponents & scaling relations KPZ in 1D : λC = λR = 1, 1 + a = b + 2

z , b = −2β = − 2 3, z = 3 2

Glauber in 2D : λC = λR, a = b = 2β/νz

DP : λC = λR = d + z + β

ν⊥ , 1 + a = b = 2β ν

what can be said on the form of the scaling function of the auto-response ?

N.B. : Galilei-invariance for KPZ is kept under renormalisation, unusual form

SLIDE 54

(A) assumption : R(t, s) =

ψ(t)

ψ(s)

1D KPZ equation/RSOS model

good collapse ⇒ no logarithmic corrections ⇒ x′ = x′ = 0 no logarithmic factors for y ≫ 1 ⇒ ξ′ = 0 ⇒ only ξ′ = 1 remains fR(y) = y−λR/z

1 − 1

y −1−a′ h0 − g0 ln

1 − 1

y

− 1

2f0 ln2

1 − 1

y

use specific values of 1D KPZ class λR

z − a = 1

find integrated autoresponse χ(t, s) = s

0 du R(t, u) = s1/3fχ(t/s)

fχ(y) = y1/3

A0
1 −
1 − 1

y −a′ +

1 − 1

y −a′ A1 ln

1 − 1

y

+ A2 ln2
1 − 1

y

with free parameters A0, A1, A2 and a′

SLIDE 55

h.n.p. 12

non-log lsi with a = a′ : deviations ≈ 20% non-log lsi with a = a′ : works up to ≈ 5% log lsi : works better than ≈ 0.1% R a′ A0 A1 A2 φ φ – LSI −0.500 0.662 φ ψ – L1LSI −0.500 0.663 −6 · 10−4 ψ ψ – L2LSI −0.8206 0.7187 0.2424 −0.09087 logarithmic lsi fits data at least down to y ≃ 1.01, with a′ − a ≈ −0.4873 (can we make a conjecture ?)

SLIDE 56

(B) assumption : R(t, s) =

ψ(t)

ψ(s)

2D majority voter/Glauber model

(triangular lattice)

good collapse ⇒ no logarithmic corrections ⇒ x′ = x′ = 0 fR(y) =

1 − 1

y a−a′ h0 − g12,0 ln(1 − 1/y) − 1 2f0 ln2(1 − 1/y)

no logarithmic terms for y ≫ 1

⇒ ξ′ = 0 can normalise ξ′ = 1

F. Sastre (2015)

preliminary

logar. lsi fit data, at least down to y ≃ 1.005.

SLIDE 57

(C) assumption : R(t, s) =

ψ(t)

ψ(s)

1D critical contact process

good collapse ⇒ no logarithmic corrections ⇒ x′ = x′ = 0 fR(y) =

1 − 1

y a−a′ h0 − g12,0 ξ′ ln(1 − 1/y) − g21,0ξ′ ln(y − 1) −1 2f0 ξ′2 ln2(1 − 1/y) + 1 2f0ξ′2 ln2(y − 1)

find empirically :

very small amplitude of ln2-terms ⇒ f0 = 0 require both ξ = 0, ξ′ = 0 BUT : logarithmic factor for y ≫ 1 ?

logar. lsi fit data, at least down to y ≃ 1.002 ; with a′ − a ≃ −0.002.

SLIDE 58

7. Logarithmic conformal & Schr¨
dinger invariance

generalise conformal invariance → doubletts Ψ = ψ φ

scalars : generators : ℓn = −wn+1∂w − (n + 1)wn∆,

∆ : conformal weight commutator : [ℓn, ℓm] = (n − m)ℓn+m

; n, m ∈ Z

invariance : Laplace equation Sψ = ∂w∂ ¯

wψ = 0

is conformally invariant for ∆ = ∆ = 0 since [S, ℓn] = −(n + 1)wnS − (n + 1)nwn−1∆∂ ¯

w

doubletts :

Saleur 92, Gurarie 93

generators ℓn = −wn+1∂w − (n + 1)wn ∆ 1 ∆

‘Laplace’ equation

SΨ = ∂w∂ ¯

w

Ψ = 0

invariance [S, ℓn] = −(n + 1)wnS − (n + 1)nwn−1

∆
∂ ¯

w

SLIDE 59

define two-point correlators :

Gurarie ’93, Rahimi Tabar et al. ’97

F := φ1(w1)φ2(w2) , G := φ1(w1)ψ2(w2) , H := ψ1(w1)ψ2(w2) (a) translation-invariance (ℓ−1) : F = F(w), G = G(w), H = H(w),

w = w1 − w2

(b) dilatation-invariance & special invariance for F(w) ℓ0 : (−w∂w − ∆1 − ∆2) F(w) = 0 ℓ1 :

−w2∂w − 2w∆1
F(w) = 0
⇒ (∆1 − ∆2)wF(w) = 0

if F(w) = 0, then ∆1 = ∆2 . (c) dilatation-invariance & special invariance for G(w) ℓ0 : (−w∂w − ∆1 − ∆2) G(w) = F(w) ℓ1 :

−w2∂w − 2w∆1
G(w) = 0
⇒ (∆1−∆2)G(w) = F(w)

if G(w) = 0, one has : F(w) = 0 and ∆1 = ∆2 .

SLIDE 60

(d) dilatation-invariance & special invariance for H(w)

with ∆ := ∆1 = ∆2

ℓ0 : (−w∂w − 2∆) H(w) = G(w) + G(−w) ℓ1 :

−w2∂w − 2w∆
H(w) = 2wG(w)
⇒ G(w) = G(−w)

Consequences : G(w) = G(−w) = G0|w|−2∆ = w−2∆

2

G0|y − 1|−2∆ w dH(w) dw + 2∆H(w) + 2G0|w|−2∆ = 0 and finally H(w) = (H0 − 2G0 ln |w|) |w|−2∆ = w−2∆

2

(H0 − 2G0 ln |y − 1| − 2G0ln |w2|) |y − 1|−2∆

with w = w1 − w2 and y = w1/w2.

Simultaneous log corrections to scaling and modified scaling function

SLIDE 61

Logarithmic Schr¨

dinger-invariance : construct doubletts Ψ =

ψ φ

Formally, scaling dimensions x become a Jordan matrix :

x → x 1 x

repeat same calculation to find co-variant two-point (reponse) functions :

F := φ1(t1, r)φ∗

2(t2, 0) , G := φ1(t1, r)ψ∗ 2(t2, 0) ,

H := ψ1(t1, r)ψ∗

2(t2, 0)

and one obtains, with t = t1 − t2 and y = t1/t2

Hosseiny & Rouhani ’10

F =

0 , G = δx1,x2δM1,M∗

2 |t2|−x1G0|y − 1|−x1exp

−M1

2 r2 t

,

H =

δx1,x2δM1,M∗

2 |t2|−x1(H0 − G0(ln |y − 1|− ln |t2|)) |y − 1|−x1 exp

−M1

2 r2 t

Simultaneous log corrections to scaling and modified scaling function

SLIDE 62

8. Logarithmic ageing-invariance

Schr¨

dinger-invariance cannot be a dynamical symmetry for

ageing, since it contains time-translations X−1 ! Go to ageing algebra age(d) :=

X1,0, Y (j)

±1/2, M0, D(jk) j,k=1,...d

Need generalised form of generator Xn = −tn+1∂t − n + 1 2 tnr · ∇r − M 2 (n + 1)ntn−1r2 − n + 1 2 xtn − nξtn with the two independent scaling dimensions x and ξ construct logarithmic ageing-invariance by the formal changes :

mh 13

x → x x′ x

, ξ →

ξ ξ′ ξ′′ ξ

SLIDE 63

concentrate on time-dependence X0 = −t∂t − 1 2 x x′ x

, X1 = −t2∂t − t

x + ξ x′ + ξ′ ξ′′ x + ξ

compute the commutator

[X1, X0] = X1 + 1

2t x′ξ′′

−1 1

!

= X1 = ⇒ x′ξ′′

!

= 0 x′ = 0 : either, ξ ξ′ ξ′′ ξ

→

ξ+ ξ−

is diagonalisable

⇒ non-logarithmic case. Or else, it reduces to a Jordan form ⇒ 2nd case. ξ′′ = 0 : simultaneous Jordan forms ⇒ generic case. (one can arrange for x′ = 0 or x′ = 1). we can always arrange for ξ′′ = 0.

SLIDE 64

co-variance conditions (with ∂i = ∂/∂ti) :

nly r = 0

F = φ1φ∗

2= F(t1, t2), G12 = φ1ψ∗ 2, G21 = ψ1φ∗ 2, H = ψ1ψ∗ 2

t1∂1 + t2∂2 +

1 2 (x1 + x2)

F(t1, t2)

=

t2

1 ∂1 + t2 2 ∂2 + (x1 + ξ1)t1 + (x2 + ξ2)t2

F(t1, t2)

=

t1∂1 + t2∂2 +

1 2 (x1 + x2)

G12(t1, t2) +

x′

2

2 F(t1, t2) =

t2

1 ∂1 + t2 2 ∂2 + (x1 + ξ1)t1 + (x2 + ξ2)t2

G12(t1, t2) + (x′

2 + ξ′ 2)t2F(t1, t2)

=

t1∂1 + t2∂2 +

1 2 (x1 + x2)

G21(t1, t2) +

x′

1

2 F(t1, t2) =

t2

1 ∂1 + t2 2 ∂2 + (x1 + ξ1)t1 + (x2 + ξ2)t2

G21(t1, t2) + (x′

1 + ξ′ 1)t1F(t1, t2)

=

t1∂1 + t2∂2 +

1 2 (x1 + x2)

H(t1, t2) +

x′

1

2 G12(t1, t2) + x′

2

2 G21(t1, t2) =

t2

1 ∂1 + t2 2 ∂2 + (x1 + ξ1)t1 + (x2 + ξ2)t2

H(t1, t2)

+(x′

1 + ξ′ 1)t1G12(t1, t2) + (x′ 2 + ξ′ 2)t2G21(t1, t2)

=

8 linear eqs. for 4 functions in 2 variables ⇒ expect unique solution, up to normalisations.

SLIDE 65

Solve these via the following ansatz, with y := t1/t2 > 1 . Set F(y) := yξ2+(x2−x1)/2(y − 1)−(x1+x2)/2−ξ1−ξ2. Then F(t1, t2) = t−(x1+x2)/2

2

F(y) f (y) G12(t1, t2) = t−(x1+x2)/2

2

F(y)

j∈Z

lnj t2 · g12,j(y) G21(t1, t2) = t−(x1+x2)/2

2

F(y)

j∈Z

lnj t2 · g21,j(y) H(t1, t2) = t−(x1+x2)/2

2

F(y)

j∈Z

lnj t2 · hj(y) must find the functions f , g12,j, g21,j, hj ; where j ∈ Z Solving these equations gives the following explicit scaling forms :

SLIDE 66

F(t1, t2) = t−(x1+x2)/2

2

F(y) f0 G12(t1, t2) = t−(x1+x2)/2

2

F(y)

g12(y) − ln t2 · x′

2

2 f0

G21(t1, t2)

= t−(x1+x2)/2

2

F(y)

g21(y) − ln t2 · x′

1

2 f0

H(t1, t2)

= t−(x1+x2)/2

2

F(y)

h0(y) − ln t2 · 1

2(x′

1g12(y) + x′ 2g21(y))

+ln2 t2 · x′

1x′ 2

4 f0

with the explicit functions, and the normalisations f0, g12,0, g21,0, h0

g12(y) = g12,0 +

x′

2

2 + ξ′

2

f0 ln
y

y − 1

g21(y)

= g21,0 −

x′

1

2 + ξ′

1

f0 ln |y − 1| −

x′

1

2 f0 ln |y| h0(y) = h0 −

x′

1

2 + ξ′

1

g21,0 +
x′

2

2 + ξ′

2

g12,0
ln |y − 1| −
x′

1

2 g21,0 −

x′

2

2 + ξ′

2

g12,0
ln |y|

+ 1 2 f0

x′

1

2 + ξ′

1

ln |y − 1| +

x′

1

2 ln |y| 2 −

x′

2

2 + ξ′

2

2 ln2

y

y − 1

Separate log corrections to scaling and modified scaling functions

add time-translations ⇒ logarithmic Schr¨

dinger-invariance

Hosseiny & Rouhani 10

SLIDE 67

9. Conclusions

physical ageing occurs naturally in many (ir)reversible systems relaxing towards (non-)equilibrium stationary states considered here : simple magnets & interface growth equilibrium properties (FDT) are strongly broken ⇒ equilibrium never reached ! various algebras : sch, age, cga, ecga, . . . several surprises in logarithmic representations large-distance properties from analyticity in dual space try to find shape of scaling functions from local scale-invariance have explored logarithmic local scale-invariance ; causality numerical experiments on auto-response function