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 = T d below which the ergodicity breaking occurs. Dynamical equations at T d ≈ those of the Mode-Coupling Theory (MCT) of supercooled liquids A discontinuous thermodynamic glass transition at T K ( T K < T d ) Beyond mean-field theory, no dynamical transition at T d 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 − T 0 )] . ➤ Relaxation time τ ∼ Viscosity η . ➤ But no data near T 0 so no evidence of a real transition at T 0 . ➤ Strong glasses: simple Arrhenius (i.e. T 0 = 0): no transition? Configurational entropy apparently goes to zero at T K (Kauzmann Paradox) s c ( T ) ∼ ( T − T K ) . ➤ But no data right up to the Kauzmann temperature T K . The ratio T K / T 0 ∼ 0 . 9 − 1 . 1 for many glass formers for which T K ∼ 50 K − 1000 K . 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 → T 0 . J. Yeo (Konkuk) Thermodynamic Glass Transition NSPCS 2008 3 / 18
All suggest a thermodynamic glass transition as T → T 0 . J. Yeo (Konkuk) Thermodynamic Glass Transition NSPCS 2008 4 / 18
All suggest a thermodynamic glass transition as T → T 0 . 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 p c ( r ) = δρ 1 ( r ) δρ 2 ( r ) between two copies of the liquid. Compute the constrained partition function Z [ p c ( r ) , δρ 2 ( r )] = � δ ( p c ( r ) − δρ 1 ( r ) δρ 2 ( r )) � ρ 1 . The effective potential is given by Ω[ p c ( r )] = − T � ln Z [ p c , δρ 2 ] � ρ 2 . Use the replica trick to average the logarithm n → 0 ( Z n − 1) / n . ln Z = lim J. Yeo (Konkuk) Thermodynamic Glass Transition NSPCS 2008 5 / 18
Use an integral representation of the delta function to obtain � � � � D λ α � � Ω[ p c ( r )] = − T exp i d r λ α ( r ) p c ( r ) 2 π α α �� � �� � � � d r δρ α × exp − i 1 ( r ) δρ 2 ( r ) λ α ( r ) . α ρ 2 ρ α 1 Average over ρ α 1 and ρ 2 by cumulant expansions. " # Z Y D λ α Z X Ω[ p c ( r )] ≃ − T exp d r λ α ( r ) p c ( r ) i 2 π α α " # 1 i Z Z d 1 d 2 G 2 (1 , 2) d 1 d 2 d 3 G 2 (1 , 2 , 3) X X × exp − λ α (1) λ α (2) + λ α (1) λ α (2) λ α (3) 2 6 α α h O ( λ 4 α , λ 2 α λ 2 i × exp β ) , where G (1 , 2) = � δρ ( r 1 ) δρ ( r 2 ) � , G (1 , 2 , 3) = � δρ ( r 1 ) δρ ( r 2 ) δρ ( r 3 ) � , etc. J. Yeo (Konkuk) Thermodynamic Glass Transition NSPCS 2008 6 / 18
Define q αβ ( r ) = λ α ( r ) λ β ( r ) for α � = β . Off-diagonal λ vertices γ α β α β α α γ β α α α β β β β (2) (1) V V V 4 6 6 Insert into the expression the identity D u αβ � � � � 1 = D q αβ 2 π α<β α<β � � . × exp i d r u αβ ( r ) ( q αβ ( r ) − λ α ( r ) λ β ( r )) α<β J. Yeo (Konkuk) Thermodynamic Glass Transition NSPCS 2008 7 / 18
Trace out λ α fields to get upto O ( u 3 ) � � � � D u αβ � � Ω[ p c ( r )] ∼ − T D q αβ exp[ i d r u αβ ( r ) q αβ ( r )] 2 π α<β α<β α<β × exp[ i � � d 1 d 2 d 3 A (1 , 2 , 3) u αβ (1) p c (2) p c (3)] 2 α<β × exp[ − 1 � � d 1 d 2 B (1 , 2) u αβ (1) u αβ (2)] 2 α<β × exp[ i � � d 1 d 2 d 3 W 1 (1 , 2 , 3) u αβ (1) u βγ (2) u γα (3)] 6 ( α,β,γ ) × exp[ i � � d 1 d 2 d 3 W 2 (1 , 2 , 3) u αβ (1) u αβ (2) u αβ (3)] , 6 α<β J. Yeo (Konkuk) Thermodynamic Glass Transition NSPCS 2008 8 / 18
Examples of W 1 and W 2 terms u u u αβ αβ αβ V 4 V V 4 4 (1) (2) V V 6 6 u u u u u u γα γα βγ αβ αβ βγ W 1 W 2 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 u αβ fields to get an effective Hamiltonian H [ q αβ ]: � � Ω[ p c ] ∼ α<β D q αβ exp[ − H [ q ]] . We take p c ( r ) = 0, which is always a solution to δ Ω /δ p c = 0 ⇒ liquid phase. We obtain � � c ( ∇ q αβ ( r )) 2 + t � � q 2 H [ q ] = d r αβ ( r ) 2 2 α<β α<β − w 1 6 Tr q 3 ( r ) − w 2 � � q 3 αβ ( r ) . 3 α<β The coefficients c , t , w 1 and w 2 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 w 2 / w 1 > 1: Two transitions at mean-field level: a dynamic transition at T d and a first-order thermodynamic glass transition at T c (below which p c � = 0). The dynamics near T d ∼ 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
w 2 / w 1 < 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) � � J (1) ij σ i τ i σ j + J (2) ij σ i τ i τ j + J (3) ij σ i σ j τ j + J (4) � H = − ij τ i σ i τ j . ij J (1) etc. are independent quenched random couplings between n.n ij sites on which two kinds of Ising spin σ i and τ i sit. This model has the same effective functional as the one derived here with w 2 / w 1 < 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 w 2 / w 1 < 1 was in the same universality class as that of Ising spin glass in a field. � � H = − J ij S i S j − h S i . < ij > i For h = 0, there is a transition at T c 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 � � c q αβ ( r )) 2 + t αβ ( r ) − w 1 � � � q 3 ( r ) − h 2 � q 2 H [ q ] = d r ( ∇ ˜ ˜ 6 Tr ˜ ˜ q αβ ( r ) . 2 2 α<β α<β α<β q = 1 � i < S i > 2 � = 0 when h � = 0. Edwards-Anderson order parameter ˜ N We take T ≈ T 0 ≪ T c , so ˜ q ≈ 1. w 2 αβ ≈ w 2 � � ⇒ h 2 � q 3 � q 2 αβ � q αβ ⇐ q αβ 3 3 α<β α<β α<β w 2 ≈ h 2 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: 1 L ∗ ( L ∗ /ξ ( T )) θ ∼ � q ( L ∗ ) d / 2 h ˜ ⇒ ξ ( T ) ∼ 2 h d − 2 θ ➤ Droplets contain O ( L d ) spins with typical magnetic moment O ( √ ˜ qL d / 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 ∼ τ 0 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
Recommend
More recommend
