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Effective theories of the glass transition: thermodynamic versus dynamic approaches Gilles Tarjus (LPTMC, CNRS/Univ. Paris 6) + G. Biroli (IPhT, Saclay), C. Cammarota (Rome), M. Tarzia (LPTMC) Warwick 2014 1 Outline • In search of an effective theory and an appropriate order parameter for the glass transition: dynamic or thermodynamic? • Contrasting the dynamical facilitation and the landscape approaches. Enlarging the phase diagram. • Mean-field dynamical transitions. • Beyond mean-field: Constraints and pinning fields. 2 The glass transition: What makes the problem interesting... There are hints that glass formation involves ✴ some form of universality ✴ some form of collective/cooperative behavior Yet, of an unusual kind... ➡Search for effective theories of glass-forming liquids ➡What is the appropriate local order parameter? A thermodynamic/structural or a purely kinetic approach? Some theoretical ingredients •Frustration: The energy of a system cannot be minimized by simultaneously minimizing all the local interactions. => Multiplicity of low-energy (‶metastable″) states. •Thermal activation in a rugged (free) energy landscape: Presence of an exponentially large number of metastable states that may trap the system. => Relaxation slowdown is associated with thermally activated escape from metastable states. •Dynamical facilitation: Mobility triggers mobility in nearby regions. => Spatial correlations in the dynamics. Different ways to incorporate the ingredients in a general theory! Local order parameter(s) •Local structural order: Observables characterizing the locally preferred molecular arrangement in the liquid, if present (e.g., related to bond-orientational order). => e.g., poly-tetrahedral/icosahedral in metallic glasses. • Similarity or ‶overlap″ between configurations: Measures of the similarity between two equilibrium configurations of the liquid. High overlap => in the same state (‶localized″) Low overlap => in different states (‶delocalized″) •Local mobility field: Mobility or activity defined by following the dynamics in small volumes of space over short periods of time. => Easier at low T where mobility is localized and scarce. Which is the best starting point to describe glassy liquids? Weak constraints from comparison to experimental data... With the help of (unavoidable ?) adjustable parameters, several theories fit the same data equally well Configurational entropy/RFOT (Wolynes et al.) Facilitation (Garrahan-Chandler) log(viscosity or time) vs Tg/T Frustration (Kivelson-GT) Focus on: Dynamical facilitation & KCM’s (rarefaction of dynamical paths) vs free-energy landscape approach (rarefaction of metastable states) Contrasting the predictions by enlarging the parameter space The configurational entropy/landscape scenario as the mean-field theory of glass-forming liquids [Wolynes, Kirkpatrick, Thirumalai, 80’s + Parisi-Mezard-Franz +many!!!] large number of metastable states • Exponentially (1) (2) (3) s = effective potential = F[ !(r)] sc the liquid s c(configurational c entropy) that may trap between two temperatures, a dynamical transition N1 N2 N3 at Td and a “random first-order transition” at TK. states states states • Order parameter = overlap between equilibrium configurations => sbuild a Landau functional (i) kB Log(Ni )/ Vdrop ~ c through the replica formalism. • • cut in !( r ) space FIG. 1: Appearance of an exponentially large number of Found in mean-field spin glasses, in mean-field-like FIG. 2: The typical energy lands meta-stable configurations and the possibility for a system to phase as a function of density ( realize these configurations gives(HNC, rise to the configurational approximations of liquids DFT), in the hard-sphere fluid in to des potential, which is used entropy. The latter serves as the driving force for the structhe nonergodic2013-14] glassy states and theture limit of in infinite dimension change the glassyspatial phase. The energy cost to[Kurchan-Zamponi-Parisi, match the panel). boundaries between the different density configurations gives riseof to fluctuations the energy barriers. fluctuations of the configuraRole in The finite dimensions ???????????? tional entropy give rise to the energy barrier fluctuations. dom hard sphere packing29 , in e constraint forces are finite, but the transition e at non-zero s. This result means that the Thewith importance of adding fields/constraints... system finite short-ranged forces of interat exhibits long relaxation times and dynamic This is a way to enlarge the phase diagram, possibly generating without equilibrium thermodynamic transitions bona fide transitions. ng theory random first-order theory [3] => [5] Oneand can check qualitative behavior and universality classes ical non-equilibrium glass transition. The sowithout recourse to adjustable parameters. For instance: In KCM’s (dynamical facilitation), add a source s conjugate to the mobility field T (b) T Criticality Active Inactive Inactive Criticality s s neric space-time phase diagram KCMs [9].constraint, There isElmatad a firstet al., PNAS, 2010] Non-equilibrium phase diagramfor T versus s [soft y that occupies the s = 0 axis, separating an active fluid phase The importance of adding fields/constraints... Similar spirit yet quite different: Adding equilibrium/ thermodynamic constraints induces new phase transitions and critical points in the mean-field theory of glass-forming liquids and structural glasses (random-first order transition scenario) 0.09 Field ε conjugate to the overlap versus T [Franz-Parisi, PRL 1997] εc,Tc T versus the concentration of randomly pinned particles [Cammarota-Biroli, PNAS 2012] 0.75 0.08 0.07 (co,To) 0.7 0.05 T Coupling ! 0.06 0.04 0.65 0.03 Td 0.01 ‘ f cK(T) 0.02 0 cd (T) cd(T) 0.6 0.6 0.62 0.64 0.66 0.68 0.7 Temperature T 0.72 0.74 0.76 0.78 TK 0 0.05 c 0.1 0.15 FIG. 1: Phase diagram for the spherical 3-spin model with spins pinned from an equilibrium con Contrasting dynamical-facilitation and configurational-entropy scenarios Diagram T vs field ε conjugate to the overlap (sketch from mean-field theory): 3 regions of interest I, II, III T Tc, εc LOW Td I : Spinodal(s) III MCT TK RFOT 0 : First-order transition II Tc, εc : Terminal critical point Td HIGH OVERLAP : Dynamical (MCT) singularity TK : Thermodynamical RFOT ε Contrasting dynamical-facilitation and landscape scenarios Obstacles for a direct comparison: • Dynamical facilitation and KCM’s have trivial thermodynamics. • The landscape/configurational-entropy approach is essentially formulated at the mean-field level. => Introduce a mean-field limit in the dynamical facilitation models. => Map back KCM’s on models without kinetic constraints and with nontrivial static (multi-spin) interactions. => Include fluctuations in the landscape approach. Contrasting ...: Region I near the (meanfield) dynamical glass transition • Quasi-equilibrium description of the beta-regime near the MCT transition (Td): Critical phenomenon can be described as a cubic field theory in a random source (RFIM spinodal, dsup=8) for the fluctuations of the overlap between configurations [Franz, Parisi and coll., 2011-2013] • KCM’s on Bethe lattice and regular random graphs [a mean-field limit for dynamical facilitation]: The transition has the same character as MCT (discontinuous, relations between exponents, etc). • Yet, the order parameters for the quasi-equilibrium description are very different: ‶nonlocal mapping″ [Foini et al, J. Stat. Mech. 2012, Franz-Sellitto, J. Stat. Mech., 2013] • Beyond mean-field: seemingly very different behaviors.... + Digression on KCM’s and bootstrap percolation on hyperbolic lattices [Sausset et al., J. Stat. Phys., 2010] Contrasting ...: Region II near the terminal critical point in a nonzero applied source coupled to the overlap Mean-field RFOT-like phase diagram T Td approach: Add fluctuations to the mean-field description -> Biroli, Cammarota, Tarzia, GT, PRL, 2014 Tc, εc II • For the dynamical-facilitation TK 0 • For the landscape/thermodynamic ε approach: Consider spin models with plaquette interactions (duality with KCM’s) -> Garrahan, PRE 2014 The overlap between configurations as an order parameter: Sketch of formalism (I) • Liquid (N,V,T): Hamiltonian H[rN], with rN = atomic configuration. Pick an equilibrium reference configuration r0N and define the microscopic overlap between rN and r0N as N j i � r + r 0 N N j i q̂x [r , r0 ] = f (|r − r0 |)δ(x − ) 2 i,j=1 where f (r) = 0 for r > a with a << atomic diameter • Define an overlap field p(x) and its associated effective hamiltonian: � N dr N N N −βH[rN ] S[p; r0 ] = − log δ[p − q̂[r , r0 ]] e N! • The correlation functions are generated by the functional W[ε; r0N]: W [�; r0 N ] = − log � Dp e−S[p;r0 N ]+ � x �(x)p(x) Sketch of formalism (II) • One is interested in the cumulants of W[ε; r0N], S[p; r0N], etc, when averaged over r0N => The reference configuration r0N plays the role of a ‶quenched disorder″. • Goal: To derive an effective theory for the overlap field p(x) => approximate form for S[p; r0N] or its cumulants by integrating out short-distance fluctuations (through liquid-state theory, general Landau-like arguments, etc). • To generate the cumulants: Introduce replicas raN, a=1,...,n, each coupled to a distinct field εa => Now, one can define replicated overlap fields pa(x) between raN and r0N and overlap fields qab(x) between raN and rbN. Sketch of formalism (III) • Generically, the effective hamiltonian for the overlap fields pa(x) is obtained from e−Srep [{pa }] = exp(− n � a=1 S[pa ; r0 N ]) ∝ � � ab�= Dqab e−S[{pa ,qab }] 0.2 where S[{pa,qab}] is obtained by integrating out short-distance fluctuations (through liquid-state theory, general Landau-like arguments, etc) and at the Landau level for pa=qab =Q has the typical form illustrated here: 0.15 0.1 Td>T> TK 0.05 0 -0.05 0 Q • Crucial: The fields pa can be critical (near the terminal critical point) but never the fields qab due to the external field provided by the coupling with the pa’s !!!!!! 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 FIG. 7. The annealed potential VA (q) as a function of q for p = 4 and various values of T , in o from top to bottom. For p = 4 one has Tc = 0.503 and Ts = 0.544. The curves are normalized IV. REAL GLASSES A. The Model We have tested the ideas presented in this note for binary fluids. Although we did no of the phase transition line, we found evidence for a first order transition in presence of in the annealed and in the quenched case. The model we consider is the following. We of different sizes. Half of the particles are of type A, half of type B and the interaction a ! " σ(i) + σ(k) #12 , H(x) = rthier Relevance of the 2-state, first-order-like, effective description after integrating out short-distance fluctuations Université Montpellier 2, Montpellier, France 1, 2013) rational entropy in supercooled liquids, based ize a system in a single metastable state. We show that the temperature dependence of the broad temperature range. This allows a new time to configurational entropy. Finally we onfigurational entropy. Evidence from finite-size computer simulations => Landau-like potential with cubic term ε 8 1.6 procedure in a similar problem). Indeed, in the umbrella sam1 7 pling approach, one needs to compute p(q) to obtain the over1.4 1 500 lap potential [W (q) = − log P(q)], which is much costlier in 6 0.8 N = 64N 250 time and less precise than only recording the central point of 1.2 124 5 the distribution and performing a line integral with it. 62 1 4 0.6 We run simulations at 0.3 ≤ ! ≤ 0.57 at 21 values of q 0.555 0.56 0.565 0.57 T =between 13 φ evenly spaced 0 and 1. In addition, we consider five 3 (100 0.4 for N = 500 7and 50 for N = 250 for 0.54 ≤ ! ≤ 0.57) re2 alizations of each experiment, and results presented here are 1 averaged over all these samples. The set up is the follow0.2 0 ing. We start with a thermalization of each ofΣthe two repli0.3 0.5 0.55 cas. We consider thermalizations of "0 elementary MC steps -1 0.4 0.53 0.57 0 (EMCSs)[39], defining EMCS as N attempts at ordinary individual random particle moves. Only once the initial config0 0.2 0.4 0.6 0.8 1 q 0 are thermalized, 0.2 0.4we run0.6 0.8 simulations 1 at urations the tethered QIn order to ensure the thermalfixed q using the weight (11). FIG. 1: (Color online) Derivative of the replica potential, Ŵ % (q) ob0.8 ization, we consider two alternative experiments. On the one Effective ‶annealed″ potential (left, Berthier,tained PREas2013) and its derivative (right, (9) at different packing fractions. Inset: Phase diagram hand, we move sequentially from q = 1 to 0 in steps of 0.05, (# , ! ) extracted fromplotted the Maxwell construction different system PRE 2014) forprocedure. a glass-forming liquid model versus the for overlap and Parisi-Soane, on the other, we consider the reverse At each sizes. The errors are computed using the jackknife method. 0.6 value q we remaintemperatures using thesystem latter size "int = 0.1"0 EMCS, atofdifferent forand a small 0.05"0 EMCS in the analysis. We completely avoid hystereW’(q) βV (Q) 3 Results: Universality class • If present, the terminal critical point is in the universality class of the RFIM (random field Ising model). Effective (disordered) hamiltonian for the overlap fluctuations ϕ(x)=p(x)-pc (up to ϕ4): � � � r3 r4 r2 2 3 4 S[φ] = c[∂φ(x)] + φ(x) + φ(x) + φ(x) 2 3! 4! x � � � δr3 (x) δr2 (x) 2 3 + − h(x)φ(x) + φ(x) + φ(x) 2 3! x 2 h(x) random field (most relevant).Expressions for the parameters • When one proceeds instead to an ‶annealed calculation″ (just two coupled replicas): universality class of pure Ising model (no quenched disorder) [Confirmed via a different method by Franz-Parisi: J. Stat. Mech.,2014] Results: Existence of a critical point in D=3? One can evaluate the effective parameters of the RFIM description for specific models: The transition exists if the ratio of the effective disorder strength to the effective interaction (surface tension) is small enough => accounts for long-distance fluctuations. From simulations of the 3D RFIM, the critical ratio ≃ 2.3. => The Wolynes et al Landau theory for the fragile glass-forming liquid o-TP: Ratio ≃ 0.94, the transition seems to survive. => The disordered p-spin model with p=3, Ratio ≃ 4.48, no transition [cf. also Cammarota et al, PRB 2013] ated thermodynamic glass transition in the uncoupled system. We nt interpretation is possible. We consider the triangular plaquette m which displays (East model-like) glassy dynamics in the absence w that when two replicas are coupled there is a curve of equilibrium ses of small and large overlap, in the temperature-coupling plane an exact temperature/coupling duality of the system) which ends n the limit of vanishing coupling the finite temperature transition system is in the disordered phase at all temperatures. We discuss Annealed computation (2 coupled replicas) lations in light of this result. The terminal critical point in plaquette spin models for the 2D triangular plaquette spin model (Garrahan, 2014) 2 3] have studied ties of (reasonself-dual line two copies (or 1.5 e studies have low sition between overlap critical T 1 ween the repliendpoint be first-order high ure (or density, 0.5 overlap coupling plane h a behaviour 0 0 0.5 1 1.5 2 heoretical pre! transition thenumerical Critical reendpoint universality of thetriangular 2D 4-state FIG. 1. Exact = phase diagram of class two coupled plar the existence models. The(ν=2/3, temperature, T , and �, are modelquette ≠ 2Dglass Ising model α=2/3 vscoupling, ν=1, α=0). ss state in the in units of the energy constant of the model, J. The full line No finite-T transition when ε=0. issue, since a is a curve of first-order transitions from a phase with low over- Potts Contrasting ...: Region III near the meanfield static transition (RFOT) • Interesting per se and to compare with the dynamic facilitation picture: in the latter, the presence of localized defects destroys any finite-T transition in the absence of constraints. • Difficulty for deriving an effective theory for the overlap with a reference configuration: All overlap fields are now affected by the presence of static point-to-set correlations (that precisely diverge in mean-field at the RFOT TK) => more tricky to integrate out the qab’s for fixed pa’s. • Work in progress: indication for a RFIM with a renormalized effective external field (configurational entropy) and antiferromagnetic interactions on top of the ferromagnetic ones. Conclusion • Contrasting the dynamical facilitation and the landscape approaches => Enlarging the phase diagram by applying fields/constraints. • Transitions in the enlarged diagram give info on the structure of the configurations and metastable states. Describable by a 2-state Ising theory or not? Consequences for the nature of the defects? • Need more systematic investigation in the dynamical- facilitation approach: quenched case, 3D, more models... • More easily testable in simulations of liquid models. 24