Thermodynamic Glass Transition in Finite Dimensions J. Yeo - - PowerPoint PPT Presentation

Thermodynamic Glass Transition in Finite Dimensions

• J. Yeo

Department of Physics, Konkuk University

NSPCS 2008, July 1-4, 2008, KIAS

Collaborator: M. A. Moore, University of Manchester

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 1 / 18

Introduction and motivation

Studies of structural glasses at mean-field level: Connections with infinite-range p-spin glass models A dynamical transition at T = Td below which the ergodicity breaking occurs. Dynamical equations at Td ≈ those of the Mode-Coupling Theory (MCT) of supercooled liquids A discontinuous thermodynamic glass transition at TK (TK < Td) Beyond mean-field theory, no dynamical transition at Td is expected. ⇐ = Activation processes over finite free-energy barriers in finite dimensions What about the thermodynamic transtion?

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 2 / 18

Thermodynamic glass transition in finite dimensions? Vogel-Fulcher law η ∼ exp[A/(T − T0)].

➤ Relaxation time τ ∼ Viscosity η. ➤ But no data near T0 so no evidence of a real transition at T0. ➤ Strong glasses: simple Arrhenius (i.e. T0 = 0): no transition?

Configurational entropy apparently goes to zero at TK (Kauzmann Paradox) sc(T) ∼ (T − TK).

➤ But no data right up to the Kauzmann temperature TK.

The ratio TK/T0 ∼ 0.9 − 1.1 for many glass formers for which TK ∼ 50K − 1000K. Simulations and experiments support exitence of growing length scale L∗(T); increasingly large regions have to move simultaneously for the liquid to move as T → T0.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 3 / 18

All suggest a thermodynamic glass transition as T → T0.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 4 / 18

All suggest a thermodynamic glass transition as T → T0.

Our Work

An effective replica Hamiltonian describing supercooled liquids near their glass transtion is constructed. Suggests there is no actual thermodynamic glass transition for d < 6. Maps the problem onto an Ising spin glass in a field. Uses the droplet picture of spin glasses to understand the phenomena associated with the glass transition. (e. g. Vogel-Fulcher law.)

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 4 / 18

Effective Potential Formalism

Following [Franz and Parisi (1997)] and [Dzero, Schmalian and Wolynes (2005)]: Define the overlap pc(r) = δρ1(r)δρ2(r) between two copies of the liquid. Compute the constrained partition function Z[pc(r), δρ2(r)] = δ(pc(r) − δρ1(r)δρ2(r))ρ1. The effective potential is given by Ω[pc(r)] = −Tln Z[pc, δρ2]ρ2. Use the replica trick to average the logarithm ln Z = lim

n→0(Z n − 1)/n.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 5 / 18

Use an integral representation of the delta function to obtain Ω[pc(r)] = −T

α

Dλα 2π exp

• i
• α
• drλα(r)pc(r)
• ×
• exp
• −i
• α
• drδρα

1 (r)δρ2(r)λα(r)

• ρ2
• ρα

1

. Average over ρα

1 and ρ2 by cumulant expansions.

Ω[pc (r)] ≃ −T Z Y

α

Dλα 2π exp " i X

α

Z drλα(r)pc (r) # × exp " − 1 2 Z d1d2 G2(1, 2) X

α

λα(1)λα(2) + i 6 Z d1d2d3 G2(1, 2, 3) X

α

λα(1)λα(2)λα(3) # × exp h O(λ4

α, λ2 αλ2 β)

i ,

where G(1, 2) = δρ(r1)δρ(r2), G(1, 2, 3) = δρ(r1)δρ(r2)δρ(r3), etc.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 6 / 18

Define qαβ(r) = λα(r)λβ(r) for α = β. Off-diagonal λ vertices

α α β β α α α β β β α α β β γ γ V V

4 6

V

6 (1) (2)

Insert into the expression the identity

1 =

α<β

Dqαβ

α<β

Duαβ 2π × exp  i

• α<β
• dr uαβ(r) (qαβ(r) − λα(r)λβ(r))

  .

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 7 / 18

Trace out λα fields to get upto O(u3)

Ω[pc(r)] ∼ −T

α<β

Dqαβ

α<β

Duαβ 2π exp[i

• α<β
• dr uαβ(r)qαβ(r)]

× exp[ i 2

• α<β
• d1d2d3 A(1, 2, 3)uαβ(1)pc(2)pc(3)]

× exp[−1 2

• α<β
• d1d2 B(1, 2)uαβ(1)uαβ(2)]

× exp[ i 6

• (α,β,γ)
• d1d2d3 W1(1, 2, 3)uαβ(1)uβγ(2)uγα(3)]

× exp[ i 6

• α<β
• d1d2d3 W2(1, 2, 3)uαβ(1)uαβ(2)uαβ(3)],
• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 8 / 18

Examples of W1 and W2 terms u u u u u u u u u

αβ βγ γα αβ γα αβ αβ αβ βγ V V V

4 4 4

6

V

(2) 6

V

(1)

W1 W2 All are functions of G (n)(1, 2, . . . , n) = δρ(1)δρ(2) . . . δρ(n).

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 9 / 18

The Glass Transition Functional

Trace out uαβ fields to get an effective Hamiltonian H[qαβ]: Ω[pc] ∼

α<β Dqαβ exp[−H[q]].

We take pc(r) = 0, which is always a solution to δΩ/δpc = 0 ⇒ liquid phase. We obtain H[q] =

• dr

c 2

• α<β

(∇qαβ(r))2 + t 2

• α<β

q2

αβ(r)

−w1 6 Tr q3(r) − w2 3

• α<β

q3

αβ(r)

• .

The coefficients c, t, w1 and w2 will be functions of the temperature and density of the liquid, with smooth dependence on them. The same replica functional arises in studies of p-spin model and Potts models.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 10 / 18

Properties of the Functional

w2/w1 > 1: Two transitions at mean-field level: a dynamic transition at Td and a first-order thermodynamic glass transition at Tc (below which pc = 0). The dynamics near Td ∼ Mode-coupling Theory (MCT) There is no simple connection to MCT in our formalism, since the density fluctuations are integrated out. Beyond mean-field theory: No true metastable states in finite dimensions ⇒ No dynamical transition.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 11 / 18

w2/w1 < 1: A transition to the glass state (if any) is continuous. A continuous transition allows a growing lengthscale. A short-range p-spin model studied by Parisi, Picco and Ritort (1999)

H = −

• ij
• J(1)

ij σiτiσj + J(2) ij σiτiτj + J(3) ij σiσjτj + J(4) ij τiσiτj

• .

J(1)

ij

• etc. are independent quenched random couplings between n.n

sites on which two kinds of Ising spin σi and τi sit. This model has the same effective functional as the one derived here with w2/w1 < 1.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 12 / 18

Mapping onto Ising Spin Glass in a Field

Moore and Drossel (2002) showed that the transition (if any) in the model for w2/w1 < 1 was in the same universality class as that of Ising spin glass in a field. H = −

• <ij>

JijSiSj − h

• i

Si. For h = 0, there is a transition at Tc at least for d ≥ 3. de Almeida-Thouless (AT) line in h − T plane where a continuous ‘replica symmetry breaking’ occurs: Exists at mean-field level and possibly for d ≥ 6. Bray and Roberts (1986) RG study: No fixed point in an ǫ expansion in 6 − ǫ dimensions. Moore’s argument (2005) using perturbative analysis of diagrams: No AT transition for d < 6.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 13 / 18

Replicated spin glass functional in a field

H[q] =

• dr

c 2

• α<β

(∇˜ qαβ(r))2+ t 2

• α<β

˜ q2

αβ(r)−w1

6 Tr ˜ q3(r)−h2

α<β

˜ qαβ(r)

• .

Edwards-Anderson order parameter ˜ q = 1

N

• i < Si >2= 0 when h = 0.

We take T ≈ T0 ≪ Tc, so ˜ q ≈ 1. w2 3

• α<β

q3

αβ ≈ w2

3

• α<β

q2

αβqαβ ⇐

⇒ h2

α<β

qαβ

w2 ≈ h2

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 14 / 18

Use of Droplet Scaling Ideas

For applications to d = 3 structural glass systems, assume there is no AT line → no genuine thermodynamic glass transition. Long lengthscales ⇐ Large droplets of size L∗(T) exist in the field. Droplets: low energy excitations of reversed spins, (compact with fractal-like surfaces) Imry-Ma argument: (L∗/ξ(T))θ ∼

• ˜

q(L∗)d/2h ⇒ L∗ ξ(T) ∼ 1 h

2 d−2θ

➤ Droplets contain O(Ld) spins with typical magnetic moment O(√˜ qLd/2). ➤ Free energy cost of a droplet: (L/ξ(T))θ. (ξ(T) correlation length for zero-field transition, θ ∼ 0.2 at d = 3)

Time scale to create a droplet of size L∗ is τ ∼ τ0 exp[B(L∗)/T].

➤ To create a droplet one needs to pass over a free energy barrier B(L) ∼ (L/ξ(T))ψ, (θ < ψ < d).

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 15 / 18

Fragile glasses

➤ We expect, provided T is not too close to T0, w2 ≈ h2 ≈ T − T0 = ⇒ L∗ ξ(T) ∼ 1 (T − T0)

1 d−2θ .

➤ Generalized Vogel-Fulcher law: τ ∼ τ0 exp

• C

(T − T0)

ψ d−2θ

• Strong Glasses

➤ w2 does not get small. ➤ L∗ is nearly temperature independent. ➤ Simple Arrehnius timescales.

It is only fragile glasses which can be associated with a lengthscale L∗(T) which grows with decreasing temperature.

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 16 / 18

The Configurational Entropy

The droplets of size L∗ are largely uncorrelated with each other and can be re-orientated = ⇒ Entropy of order ln 2. The entropy per spin or particle, sc(T) is sc(T) ∼ ln 2 (L∗)d ∼ w

d d−2θ

2

. As θ ∼ 0.2 and w2 ∼ (T − T0), sc(T) vanishes nearly linearly. (Again this is not expected to hold too close to T0.)

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 17 / 18

Conclusions

We have constructed an effective Hamiltonian for supercooled liquids near the putative thermodynamic glass transition. It is equivalent to that of an Ising spin glass in a field. Using droplet scaling arguments and simple assumptions, we were able to explain phenomenologically main features of fragile glasses. (e. g. Vogel-Fulcher law)

• J. Yeo (Konkuk)

Thermodynamic Glass Transition NSPCS 2008 18 / 18