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 Effective theory of the glass transition in terms the overlap with an equilibrium configuration p ( x ) • It allows to focus on the relevant field and on the “physical” order 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

Known results: Self-induced disorder and RFIM • 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 β the RFIM Franz, Parisi, Ricci-Tersenghi, Rizzo The terminal critical point in the - phase T ✏ diagram is in the same universality class of T the RFIM Franz & Parisi; Biroli, Cammarota, Tarjus, MT T d The same result holds for the continuous glass transition found at the terminal point T K of the random pinning phase diagram ✏ Cammarota & Biroli; Nandi & Biroli • 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

Intuituve arguments cfr Giulio’s lecture and Silvio’s talk The equilibrium reference configuration acts as a random field 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 overlap magnetization p m configurational entropy magnetic field h s c height of the barrier ferromagnetic coupling J

What do we want/need to compute? Choose an equilibrium reference configuration at random C 0 (according to ) e − β H ( C 0 ) /Z Compute the probability that a copy of the system has an overlap profile with the reference configuration: p ( x ) e − S [ p ( x ) |C 0 ] ∝ X e − β H ( C ) δ [ p ( x ) − Q x ( C , C 0 )] C The cumulants of can be computed through an expansion S [ p ( x ) |C 0 ] in free replica sums: Tarjus & Tissier (cfr Gille’s lecture) a =1 S [ p ( x ) |C 0 ] C 0 e − S rep [ { p a ( x ) } ] = e − P n n n S 1 [ p a ( x )] − 1 X X S rep [ { p a ( x ) } ] = S 2 [ p a ( x ) , p b ( x )] + · · · 2 a =1 a,b =1 c S 1 [ p ( x )] = S [ p ( x ) |C 0 ] ; S 2 [ p 1 ( x ) , p 2 ( x )] = S [ p 1 ( x ) |C 0 ] S [ p 2 ( x ) |C 0 ]

The Kaç version of the Random Energy Model j i Franz, Parisi, Ricci-Tersenghi 2 M configurations on Random energy on each link h ij i each site: C i = { 1 , . . . , 2 M } E h ij i = E ( C i , C j ) iid Gaussian E ( C i , C j ) = 0 X H = E h ij i ( C i , C j ) E ( C i , C j ) E ( C 0 j ) = M δ C i , C 0 i , C 0 i δ C j , C 0 j h ij i Compute the replicated action replicas α = 0 , . . . , n n + 1 ! Y X X e � S rep [ { p a ( i ) } ] ∝ E h ij i ( C α i , C α exp j ) − β δ p a ( i ) ,q ( C 0 i , C a i ) {C α i } h ij i , α a,i ! Y β 2 M X X X = exp δ C α i δ C α δ p a ( i ) ,q ( C 0 i , C β j , C β i , C a i ) 2 j {C α i } h ij i α , β a,i

The overlap matrix Overlap with the p a   1 p 1 p 2 . . . p n reference configuration 1 p 1 q 12 . . . q 1 n     1 p 2 q 21 . . . q 2 n δ C α , C β =     . . . . Overlap among ... . . . . q ab   . . . . the free replicas   1 p n q n 1 q n 2 . . . example ( ): n = 5   1 1 0 0 1 0 i and C a i = C 0 C b C a i = C b i = C 0 i i 1 1 0 0 1 0     i and C b C a i 6 = C b C a i = C 0 i 6 = C 0 0 0 1 ? 0 ?   i i    0 0 ? 1 0 ?  and ??????   C a i 6 = C 0 C b i 6 = C 0 1 1 0 0 1 0   i i   0 0 ? ? 0 1 C a i 6 = C b In the simplest approximation consists in assuming that i in this case (justified in the Kaç limit for large ) M q ab ( i ) = p a ( i ) p b ( i )

The annealed approximation − M β 2 ✓ ◆ X X X S MF rep = p a ( i ) p a ( j ) + M log 2 p a ( i ) 2 a i h ij i − M β 2 X X p a ( i ) p b ( i ) p a ( j ) p b ( j ) ab h ij i Going back to a spin model p ( i ) = (1 + σ i ) / 2 Random field random bond Ising model (with correlated disorder) H MF = − X X ( J + J ij ) σ i σ j + ( H ext − h i ) σ i h ij i i H ext = Md ( β 2 K − β 2 ) / 4 ∝ Ms c p 2 ln 2 /d β K = J = M β 2 / 8 Mean-field ( ) M → ∞ critical temperature ∆ 2 J = M β 2 ∆ 2 h = M β 2 d/ 8 h i h j = h i J ij = M β 2 / 16

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

The periodic pinning Set on the vertices of a -dimensional hyper-cube of size p ( x ) = 1 ` d and elsewhere p ( x ) = 0 Use the variational ansatz ` ξ MF and optimize over m pin m 1 S 1 [ p ( i )] = S MF [ p ( i )] + ∆ S 1 [ p ( i )] 1 1 ✓ β 2 K − β 2 ◆ d ` ∆ S 1 ξ MF pin = β 2 K Additional effective interaction among the p ( i ) ξ pin `

Effective long-range antiferromagnetic interaction Approximate ansatz for possibly long-range pair interaction K ( r ) + external field the effective interaction ∆ H X X X H = − ( J + J ij ) σ i σ j + K ( | i − j | ) σ i σ j + ( H ext + ∆ H + h i ) σ i h ij i i,j i K ( r ) ' Mc r 2 d θ ( ξ MF pin � r ) lowers the transition temperature with respect to mean-field ∆ H (by renormalizing the configurational entropy) H ext ( β ? K ) + ∆ H ( β ? K ) = 0 β ? β K K > β K Study the properties of the critical point using the T T K T ? K effective Hamiltonian

Estimation of the disorder at the critical point p ∆ 2 Numerical simulations in d = 3 h ' 0 . 4 Jd • Neglect the random bonds (ok for large ) M • Neglect correlations between n.n. random fields and random bonds • Use the result of in higher dimensions d = 3 ( Jd ) eff = Jd − 1 Z • r d − 1 K ( r )d r 2 50 48 Ideal Glass 46 Transition 44 M 42 40 38 Normal 36 liquid 34 2 3 4 5 6 7 8 9 10 d

Conclusions & Perspectives • First steps towards the derivation of an Ising-like effective theory for the glass transition close to T K • 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?)