Transition in coupled replicas may not imply a finite temperature ideal glass transition in glass forming systems
TTransition in coupled replicas may not imply a finite temperature ideal glasstransition in glass forming systems
Juan P. Garrahan
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (Dated: November 9, 2018)A key open question in the glass transition field is whether a finite temperature thermodynamictransition to the glass state exists or not. Recent simulations of coupled replicas in atomisticmodels have found signatures of a static transition as a function of replica coupling. This can beviewed as evidence of an associated thermodynamic glass transition in the uncoupled system. Wedemonstrate here that a different interpretation is possible. We consider the triangular plaquettemodel, an interacting spin system which displays (East model-like) glassy dynamics in the absenceof any static transition. We show that when two replicas are coupled there is a curve of equilibriumphase transitions, between phases of small and large overlap, in the temperature-coupling plane(located on the self-dual line of an exact temperature/coupling duality of the system) which endsat a critical point. Crucially, in the limit of vanishing coupling the finite temperature transitiondisappears, and the uncoupled system is in the disordered phase at all temperatures. We discussinterpretation of atomistic simulations in light of this result.
A series of recent simulation works [1–3] have studiedin some detail the thermodynamic properties of (reason-ably realistic) glass-forming systems when two copies (orreplicas) of the system are coupled . These studies haveproduced evidence for an equilibrium transition betweenstatic phases of small and large overlap between the repli-cas [1–3]. This phase transition seems to be first-orderand the coexistence curve in the temperature (or density,in the case of hard-spheres) and replica coupling planeterminates at a critical point [2, 3]. Such a behaviourof the coupled system is consistent with theoretical pre-dictions [4–7] from the random first-order transition the-ory (RFOT) [8–10]. Furthermore, these numerical re-sults could be interpreted as evidence for the existenceof a thermodynamic transition to the glass state in the uncoupled system. This is an important issue, since acentral unresolved question in the understanding of thekinetic phenomenon we term “glass transition” [11–14]is whether it is caused by an underlying thermodynamicsingularity or not. In particular, it may seem that theobservations of [2, 3] are difficult to reconcile with purelydynamical approaches such as dynamic facilitation [15].The purpose of this paper is to show that this is not thecase, and that it is possible to explain the behaviour ofcoupled replicas without recourse to a thermodynamicsingularity in the uncoupled system.Our results are summarised in Fig. 1. The figure showsthe phase diagram in the temperature, T , and replicacoupling, ε , plane for two coupled copies of a specificglass model, the triangular plaquette model (TPM) [16–20]. The TPM is one of a class of interacting spin modelswhich map to kinetically constrained models [17–19], i.e.that display complex glassy dynamics with simple ther-modynamics. The dynamics of the TPM in particular isthat of the East model [17–19, 21], the facilitated modelwhich captures best the phenomenology of glass formers[22, 23]. Figure 1 shows that the TPM also has a first- ! T high overlaplow overlap critical endpointself-dual line FIG. 1. Exact phase diagram of two coupled triangular pla-quette glass models. The temperature, T , and coupling, (cid:15) , arein units of the energy constant of the model, J . The full lineis a curve of first-order transitions from a phase with low over-lap between the replicas to a phase of high overlap betweenthe replicas. The transition line, ¯ T ( ε ), is on the self-dualcurve of the model (dashed line), and ends at a critical point( ε c /J, T c /J ) = [1 , / log (1 + √ T ( ε →
0) = 0 andthere is no finite T singularity in the uncoupled system. order transition between static phases of low and highoverlap occuring at a temperature ¯ T that depends onthe coupling, ¯ T = ¯ T ( ε ). The curve ¯ T ( ε ) terminates ata critical point ( ε c /J, T c /J ). Crucially, the coexistencecurve meets the T axis at T = 0, that is, ¯ T ( ε →
0) = 0,and the transition vanishes in the absence of coupling.The upshot is two-fold: (i) from a facilitation perspec-tive it is indeed possible to account for the static be-haviour observed in simulations at finite coupling ε ; (ii)the static transition may disappear when the couplingvanishes, and thus one needs to be carful about extrapo-lations of finite ε results for interpretations of what occursin the uncoupled system. a r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov The TPM is defined by the energy function [16–19] E ( s ) ≡ − J (cid:88) (cid:52) s i s j s k , (1)where s i = ± i of a tri-angular lattice ( i = 1 , . . . , N ), and where the sum is overupward pointing triangles (cid:77) , or plaquettes, of the lattice.For simplicity we consider periodic boundary conditions,at least in one direction of the lattice, and such that thelinear size in that direction is a power of two, L = 2 K .In this case the ground state of (1) is unique [16–19],corresponding to all spins pointing up. (Other boundaryconditions lead to the same physics, but the mathematicsis more tedious [18].) One can define plaquette spin vari-ables, τ (cid:77) ≡ − s i s j s k , for all plaquettes (cid:77) of the lattice. Interms of these variables the ground state is τ (cid:77) = − (cid:77) , and the elementary excitation of the system corre-sponds to a defective plaquette, τ (cid:77) = +1. For the chosenboundary conditions, there is a one-to-one mapping be-tween spins s i and plaquettes τ (cid:77) , and the thermodynam-ics is that of a free gas of binary plaquettes [16–19]. Whilethe thermodynamics of the TPM is trivial its dynamicsis not. This is because an elementary dynamical movecorresponds to a spin-flip, s i → − s i , and each spin-flipreverts the three plaquettes τ (cid:77) in which it participates.The plaquette dynamics is kinetically constrained [17–19]leading to complex glassy dynamics at low temperatures.In the case of the TPM the effective dynamics is that ofthe East facilitated model [17–19], and therefore displays“parabolic” super-Arrhenius growth of timescales, tem-perature dependent stretching of correlations, dynamicheterogeneity, and other hallmarks of the glass transition[14].We now consider two TPMs, with a coupling betweenthem that is proportional to their overlap. The totalenergy of the coupled system is defined as, E ( s a , s b ) ≡ − J (cid:88) (cid:52) (cid:0) s ai s aj s ak + s bi s bj s bk (cid:1) − ε (cid:88) i s ai s bi , (2)where s a and s b are the spin configurations in the replicas a and b . The coupling field ε is conjugate to the overlap, q ( s a , s b ) ≡ N − (cid:80) i s ai s bi , which measures the similarityof the spin configurations of the two copies. Equation(2) corresponds to the annealed coupling, in the sensethat both replicas are considered on equal footing. The quenched coupling, in contrast, corresponds to when oneof the replicas is frozen in an equilibrium configuration.We restrict here to the annealed case which is easier totreat analytically. Just like in previous works [1–3], theassumption is that the behaviour of the simpler annealedproblem is a good qualitative indicator of what happensin the more complex quenched problem.The partition sum for the coupled replicas reads, Z ( K , K ) ≡ (cid:88) { s a ,s b } e K (cid:80) (cid:52) ( s ai s aj s ak + s bi s bj s bk ) + K (cid:80) i s ai s bi , with K ≡ J/ k B T and K ≡ ε/k B T . By applying tech-niques similar to those of Refs. [20, 24] it is easy to provethe existence of an exact duality that relates the parti-tion sum at state point ( K , K ) to that at ( K ∗ , K ∗ ) (seeMethods for details). That is, Z ( K , K ) = (sinh 2 K sinh K ) N Z ( K ∗ , K ∗ ) , (3)where the transformation that connects the two statepoints is given by e − K ∗ = tanh K , tanh K ∗ = e − K . (4)The duality relation (3)-(4) for the problem of two cou-pled TPMs is analogous to that of the TPM in a mag-netic field [20], which in turn is the one of the generalizedBaxter-Wu model [24].The self-dual line is defined by the condition that thepartition functions Z ( K , K ) and Z ( K ∗ , K ∗ ) coincide.From (3) we see that this happens when, (cid:18) sinh JT (cid:19) (cid:16) sinh εT (cid:17) = 1 , (5)where we have reverted to the original parameters of theproblem and set k B to unity. Figure 1 shows the self-dual line in the ( ε, T ) plane (for unit J , which just setsthe energy scale of the problem). Note that this lineapproaches the origin as ε ∼ T e − J/T .The self-dual line (5) is the locus of first-order phasetransitions, just like it happens in the generalised Baxter-Wu model [24] or the TPM in a field [20]. These occurat a coupling dependent temperature ¯ T ( ε ), which obeysEq. (5). For a given ε , the transition at ¯ T ( ε ) is between aphase with low overlap between the replicas, at T > ¯ T ( ε ),to one of high overlap, at T < ¯ T ( ε ). The transition lineends at a critical point ( ε c , T c ).The location of the critical point can be obtained fromthe following argument. The duality transformation (4)relates a state point ( ε/J, T /J ) to another state point[( ε/J ) ∗ , ( T /J ) ∗ ], while never connecting the half-planes ε > J and ε < J . That is, if ε/J > <
1) then( ε/J ) ∗ > < ε = J is aspecial point: for this value of the coupling the problemdefined by Eq. (2) becomes equivalent to the Baxter-Wumodel [25]. This model has a critical transition at sometemperature T c , which we can obtain from the self-dualcondition (4) when ε c = J , (cid:18) sinh JT c (cid:19) = 1 ⇒ ( ε c , T c ) = (cid:18) J, J log (1 + √ (cid:19) . (6)The first-order line of (2)-(3) will end at this criticalpoint, see Fig. 1. Note that while the critical temper-ature of the Baxter-Wu model is the same as that of thetwo-dimensional Ising model, its critical properties arethose of the two-dimensional four-state Potts model [25].The structure of the phase diagram of Fig. 1 is interest-ing as it offers an alternative interpretation on the recentnumerical results on replicated systems [1–3]. The super-cooled regime of the TPM is that of T /J <
1. Withinthis regime, Fig. 1 shows that the coexistence coupling ¯ ε changes rapidly with temperature. If one were to simu-late the coupled TPMs for a range of temperatures sat-isfying 1 (cid:38) T /J (cid:29) T axis at some finite temperature.The results presented here highlight some of the differ-ences between theoretical approaches to the glass transi-tion [13, 14], and in particular RFOT [8–10] versus dy-namical facilitation [15]. In order to have a thermody-namic transition to an ideal glass state at non-zero tem-perature, as RFOT advocates [8–10], it is necessary tohave excitations to low energy states which are extended[26] rather than pointlike. Extended excitations wouldbe needed, on one hand to support interfaces, and on theother to prevent their proliferation due to entropy, whichwould destroy the transition.In contrast, the central tenet of the facilitation ap-proach is that excitations are localised [15, 27]. The en-tropy of mixing of such excitations destabilises any or-dered (or randomly frozen) state [28]. Plaquette modelslike the TPM are a clear example of this mechanism. Theplaquette interactions of the energy function (1) can beminimised by a multiplicity of local spin arrangements,four per plaquette in the case of the TPM (while thereis one, or few depending on boundaries [18], global en-ergy minima). Whenever there is a mismatch betweensuch local arrangements the energy cost is localised inan excited plaquette. Defects are thus pointlike ratherthan extended. The “deconfinement” of defects is alwaysentropically favoured, and the TPM and similar plaque-tte models are disordered at all temperatures [16–19].Structural rearrangement is concentrated around defects[17, 18] and these systems show all the features of facili-tated dynamics [15, 29]. (These mechanisms are not dif-ferent from those of other systems with local constraints,such as dimer coverings [30] or spin-ice [31].)The introduction of a magnetic field [20], or other inter-actions [32], can lead to a confinement of defects whichthen allows for a transition to an ordered state. Thisis essentially what happens for two coupled TPMs: thecoupling between replicas can induce a confinement ofthe difference between the defect arrangements in thetwo replicas, leading not to an ordered state in eitherreplica, as in the single TPM in a field, but to a phaseof large overlap between the two copies. A finite cou-pling, however, is required at all non-zero temperaturesto keep this confinement and prevent the two copies frombecoming independent. This contrasts to what RFOTwould predict occurs for glass formers. It is also note- worthy, that the TPM is one of several simple systems,with local and disorder-free interactions (the East modelis another example), which can be solved in finite dimen-sions and be shown to display the ideal behaviour of thedynamical facilitation approach. This is a further con-trast with RFOT where solvable low-dimensional modelsthat furnish that perspective are yet to be found [33, 34].In summary, we have shown that when two replicas ofa triangular plaquette model are coupled there can bea first-order transition, at finite coupling strength, be-tween a phase of low overlap where the two copies areidependent, to one of high overlap where the two copiesare correlated. At high enough temperature this transi-tion ends at a critical point. These are the qualitativefeatures revealed by numerical simulations of replicatedglass forming models [1–3]. A key observation here is thatwhen the coupling is removed the phase transition disap-pears. There are two messages to take from these results:first, the numerically observed behaviour of replicatedliquids is not inconsistent with dynamic facilitation, asan ideal model from this approach displays essentiallythe same behaviour; and second, it may not be safe toassume that a transition at finite coupling implies a finitetemperature/density thermodynamic glass transition inthe uncoupled system. Methods.
Here we prove the duality relations quotedin the text above. We follow closely the notation of Ref.[20]. The spins s i live on the sites i ∈ Λ of the triangularlattice Λ. The plaquettes in turn are located on the sites (cid:77) ∈ Λ ∗ of the dual lattice Λ ∗ . The plaquette variable τ (cid:77) is a function of the spins immediately to the left, rightand top of (cid:77) . If we denote these sites in Λ by l (cid:77) , r (cid:77) , t (cid:77) ,respectively, then we have τ (cid:77) = − s l (cid:77) s r (cid:77) s t (cid:77) . Similarly,the spin s i participates in three plaquettes immediatelyto the left, right and bottom of i . We denote these siteson Λ ∗ by l i , r i , b i , respectively.Just like in Refs. [20, 24] for the cases of the generalisedBaxter-Wu model or the TPM in a field, duality is provedby introducing extra degrees of freedom and tracing outthe original ones. The partition sum (3) can be written Z ( K , K ) = (cosh K ) N (cosh K ) N (cid:88) { s a ,s b } (cid:89) (cid:77) ∈ Λ ∗ (cid:0) K s al (cid:77) s ar (cid:77) s at (cid:77) (cid:1) (cid:0) K s bl (cid:77) s br (cid:77) s bt (cid:77) (cid:1)(cid:89) i ∈ Λ (cid:0) K s ai s bi (cid:1) . If we introduce auxiliary binary variables { ˜ σ a (cid:77) , ˜ σ b (cid:77) = 0 , } on the dual lattice, we can express the partition sum as Z ( K , K ) = (cosh K ) N (cosh K ) N (cid:88) { s a ,s b } (cid:88) { ˜ σ a , ˜ σ b } (cid:89) (cid:77) ∈ Λ ∗ (tanh K ) ˜ σ a (cid:77) +˜ σ b (cid:77) (cid:0) s al (cid:77) s ar (cid:77) s at (cid:77) (cid:1) ˜ σ a (cid:77) (cid:0) s bl (cid:77) s br (cid:77) s bt (cid:77) (cid:1) ˜ σ b (cid:77) (cid:89) i ∈ Λ (cid:0) K s ai s bi (cid:1) , and by rearranging the factors, Z ( K , K ) = (cosh K ) N (cosh K ) N (cid:88) { s a ,s b } (cid:88) { ˜ s a , ˜ s b } (cid:89) (cid:77) ∈ Λ ∗ (tanh K ) (˜ s a (cid:77) +˜ s b (cid:77) )+1 (cid:89) i ∈ Λ ( s ai ) (˜ s ali ˜ s ari ˜ s abi +1) × (cid:0) s bi (cid:1) (˜ s bli ˜ s bri ˜ s bbi +1) (cid:0) K s ai s bi (cid:1) , where we have defined the auxiliary spins, ˜ s (cid:77) ≡ σ (cid:77) − ±
1. In the above expression spins on different sites i are uncoupled and can be traced out. If we sum overthe four possible values of the pair of spins ( s ai , s bi ) theterms with ˜ s al i ˜ s ar i ˜ s ab i (cid:54) = ˜ s bl i ˜ s br i ˜ s bb i vanish. This means thatall the auxiliary a and b plaquettes have to be same.But for the chosen boundary conditions (see above) thespin-plaquette mapping is one-to-one, which means that˜ s a (cid:77) = ˜ s b (cid:77) for all (cid:77) . After summing over all the spins( s a , s b ) we therefore obtain, Z ( K , K ) = 2 N/ (sinh 2 K ) N (sinh 2 K ) N/ (cid:88) { ˜ s } (cid:89) (cid:77) ∈ Λ ∗ (tanh K ) ˜ s (cid:77) (cid:89) i ∈ Λ (tanh K ) ˜ s li ˜ s ri ˜ s bi , where we have renamed ˜ s a → ˜ s . Now we have a singleset of spins ˜ s , and the partition sum of the two coupledTPMs can be written as, Z ( K , K ) = 2 N/ (sinh 2 K ) N (sinh 2 K ) N/ Y ( ˜ K , ˜ K ) , (7)where Y ( ˜ K , ˜ K ) is the partition sum of a single TPM ina field [20], with e − K = tanh K , e − ˜ K = tanh K . (8)From Ref. [20] we know that the TPM in a field has anexact duality, namely, Y ( ˜ K , ˜ K ) = (cid:16) sinh 2 ˜ K sinh 2 ˜ K (cid:17) N/ Y ( ˜ K ∗ , ˜ K ∗ ) , (9)with e − K ∗ = tanh ˜ K , e − K ∗ = tanh ˜ K . (10)The partition sum of the dual TPM in a field, withparemeters ˜ K ∗ , ˜ K ∗ , in turn is equivalent to that of apair of replicated TPMs, with parameters K ∗ , K ∗ , viaEqs. (7)-(10). From this we recover the duality of thecoupled TMPs given in Eqs. (3)-(4). Acknowledgements.
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