Ising-like effective theory for the glass transition Marco Tarzia - - PowerPoint PPT Presentation

Ising-like effective theory for the glass transition

Marco Tarzia LPTMC, Université Paris VI Spin glasses: an old tool for new problems

Cargèse, August 25 - September 6 2014

with G. Biroli, C. Cammarota, and G. Tarjus

Introduction & motivation

• It allows to focus on the relevant field and on the “physical”
• rder parameter and to obtain a more intuitive description of the

glass transition

• It leads naturally to a scalar field theory in presence of quenched

disorder, which is easier to handle than the original replica field

• theory. It can be studied using standard tools of statistical physics

(nonperturbative RG, numerical simulations, ...) cfr Gille’s lecture

• It allows to go beyond mean-field theory and RFOT, and to study

the nature and the critical properties of the critical points. It naturally allows to identify the possible mechanisms that could destroy the glass transition in finite dimensions

Effective theory of the glass transition in terms the

• verlap with an equilibrium configuration

p(x)

Known results: Self-induced disorder and RFIM

T Td TK

✏

• The distribution of the overlap fluctuations has been computed in

numerical simulations of glass forming systems and have been interpreted in terms of an effective RFIM Stevenson & al

• Analysis of perturbation theory of the Replica Field Theory

Critical overlap fluctuations close to the dynamical transition in the

• regime are in the same universality class of the spinodal point of

the RFIM Franz, Parisi, Ricci-Tersenghi, Rizzo β The terminal critical point in the - phase diagram is in the same universality class of the RFIM

Franz & Parisi; Biroli, Cammarota, Tarjus, MT

The same result holds for the continuous glass transition found at the terminal point

• f the random pinning phase diagram

Cammarota & Biroli; Nandi & Biroli

✏

T

Intuituve arguments cfr Giulio’s lecture and Silvio’s talk

Local fluctuations of the Franz-Parisi potential due to the density fluctuations of the reference configuration Local fluctuations of the configurational entropy and of the surface tension Random field random bond Ising model

The equilibrium reference configuration acts as a random field

• verlap

configurational entropy height of the barrier p sc

h

m magnetization magnetic field ferromagnetic coupling J

What do we want/need to compute?

e−Srep[{pa(x)}] = e− Pn

a=1 S[p(x)|C0] C0

Srep[{pa(x)}] =

n

X

a=1

S1[pa(x)] − 1 2

n

X

a,b=1

S2[pa(x), pb(x)] + · · · The cumulants of can be computed through an expansion in free replica sums: Tarjus & Tissier (cfr Gille’s lecture) S[p(x)|C0]

S1[p(x)] = S[p(x)|C0] ; S2[p1(x), p2(x)] = S[p1(x)|C0]S[p2(x)|C0]

c

e−S[p(x)|C0] ∝ X

C

e−βH(C)δ[p(x) − Qx(C, C0)] Choose an equilibrium reference configuration at random (according to ) Compute the probability that a copy of the system has an overlap profile with the reference configuration: p(x) e−βH(C0)/Z C0

The Kaç version of the Random Energy Model

2M configurations on each site: Ci = {1, . . . , 2M}

i

j

Franz, Parisi, Ricci-Tersenghi

Compute the replicated action H = X

hiji

Ehiji(Ci, Cj) E(Ci, Cj) = 0 E(Ci, Cj)E(C0

i, C0 j) = MδCi,C0

iδCj,C0 j

Random energy on each link hiji

eSrep[{pa(i)}] ∝ X

{Cα

i }

exp − β X

hiji,α

Ehiji(Cα

i , Cα j )

! Y

a,i

δpa(i),q(C0

i ,Ca i )

= X

{Cα

i }

exp β2M 2 X

hiji

X

α,β

δCα

i ,Cβ i δCα j ,Cβ j

! Y

a,i

δpa(i),q(C0

i ,Ca i )

α = 0, . . . , n n + 1 replicas

Ehiji = E(Ci, Cj) iid Gaussian

The overlap matrix

Ca

i = C0 i and Cb i 6= C0 i

Ca

i 6= Cb i

Cb

i = C0 i

Ca

i = C0 i and

Ca

i = Cb i

and ?????? Ca

i 6= C0 i

Cb

i 6= C0 i

qab(i) = pa(i)pb(i) Ca

i 6= Cb i

In the simplest approximation consists in assuming that in this case (justified in the Kaç limit for large ) M δCα,Cβ =        1 p1 p2 . . . pn p1 1 q12 . . . q1n p2 q21 1 . . . q2n . . . . . . . . . ... . . . pn qn1 qn2 . . . 1        Overlap with the reference configuration pa Overlap among the free replicas qab

         1 1 1 1 1 1 1 ? ? ? 1 ? 1 1 1 ? ? 1         

example ( ):

n = 5

The annealed approximation

Random field random bond Ising model (with correlated disorder) Going back to a spin model p(i) = (1 + σi)/2 HMF = − X

hiji

(J + Jij)σiσj + X

i

(Hext − hi)σi ∆2

J = Mβ2

∆2

h = Mβ2d/8

J = Mβ2/8 hihj = hiJij = Mβ2/16 Hext = Md(β2

K − β2)/4 ∝ Msc

Mean-field ( ) critical temperature M → ∞ βK = p 2 ln 2/d SMF

rep =

X

a

✓ − Mβ2 2 X

hiji

pa(i)pa(j) + M log 2 X

i

pa(i) ◆ − Mβ2 X

ab

X

hiji

pa(i)pb(i)pa(j)pb(j)

Beyond the annealed computation

A variational Ground-State approximation ξP S p = 1 p = 0 p = 1 Regions with closer than the point-to-set length p = 1 Locally decrease sc Induce a spontaneous RSB among the replicas forced to have p = 0 n Consider (first cumulant of the effective action) pa(i) = p(i) ∀a 1 1 1 m qab(i) =         1 · · · · · · . . . . . .         p(i) = 0 ✓2M − 1 n/m ◆ combinatorial factor

The periodic pinning

Set on the vertices of a -dimensional hyper-cube of size and elsewhere p(x) = 1 p(x) = 0

d `

m 1 ∆S1

` ` ξpin ` S1[p(i)] = SMF

1

[p(i)] + ∆S1[p(i)] Use the variational ansatz and optimize over m Additional effective interaction among the p(i) ξMF

pin

ξMF

pin =

✓β2

K − β2

β2

K

◆

1 d

Effective long-range antiferromagnetic interaction

Approximate ansatz for the effective interaction possibly long-range pair interaction + external field K(r) ∆H

H = − X

hiji

(J + Jij)σiσj + X

i,j

K(|i − j|) σiσj + X

i

(Hext + ∆H + hi)σi

T Study the properties of the critical point using the effective Hamiltonian lowers the transition temperature with respect to mean-field (by renormalizing the configurational entropy) ∆H K(r) ' Mc r2d θ(ξMF

pin r)

βK Hext(β?

K) + ∆H(β? K) = 0

β?

K > βK

TK T ?

K

Estimation of the disorder at the critical point

p ∆2

h

Jd ' 0.4 Numerical simulations in d = 3

• Neglect the random bonds (ok for large )
• Neglect correlations between n.n. random fields and random bonds
• Use the result of in higher dimensions
• d = 3

M (Jd)eff = Jd − 1 2 Z rd−1K(r)dr Ideal Glass Transition

34 36 38 40 42 44 46 48 50 2 3 4 5 6 7 8 9 10

d

M Normal liquid

Conclusions & Perspectives

• First steps towards the derivation of an Ising-like effective theory

for the glass transition close to

• Random field random bond Ising model (with correlated disorder)

+ long-range antiferromagnetic interaction

• Go beyond RFOT. Use the effective Hamiltonian to study the

properties of the original model (length scales, critical points, ...)

• Identify mechanisms which may destroy the Ideal glass transition
• Check the robustness of the results with respect to other

geometry of the overlap profile

• Extend the computation to the second cumulant of the effective

action (possibly long-range correlated disorder?)

• Extend the calculation to other models (similar results are found

for a generic effective Ginzburg-Landau replicated action)

• Improve the variational ground state approximation (low

temperature expansion?) TK