PHYSICAL REVIEW E 91, 032148 (2015)
Glassy slowdown and replica-symmetry-breaking instantons A llan A dam s,1’* Tarek A n o u s,1’' Jaehoon L ee ,1’* and Sho Y aida1’2' 1’ 'Center fo r Theoretical Physics, Massachusetts Institute o f Technology, Cambridge, Massachusetts 02139, USA 2Department o f Chemistry, Duke University, Durham, North Carolina 27708, USA (Received 17 July 2014; revised manuscript received 26 February 2015; published 30 March 2015) Glass-forming liquids exhibit a dramatic dynamical slowdown as the temperature is lowered. This can be attributed to relaxation proceeding via large structural rearrangements whose characteristic size increases as the system cools. These cooperative rearrangements are well modeled by instantons in a replica effective field theory, with the size of the dominant instanton encoding the liquid's cavity point-to-set correlation length. Varying the parameters of the effective theory corresponds to varying the statistics of the underlying free-energy landscape. We demonstrate that, for a wide range of parameters, replica-symmetry-breaking instantons dominate. The detailed structure of the dominant instanton provides a rich window into point-to-set correlations and glassy dynamics. DOI: 10.1103/PhysRevE.91.032148
PACS number(s): 64.60.De, 64.70.Q -
W hen glass-form ing liquids are cooled over a m odest range o f tem peratures, their dynam ics slows down by many orders of m agnitude. T his slowdown m anifests itself in a rapid increase o f shear viscosity and structural relaxation time. Despite decades o f w ork, the physics o f the slow dow n— also observed in a w ide variety o f system s including granular and biological system s— has yet to be properly described [1,2]. We develop a method w ith w hich to fill this gap, and, in so doing, we find nonperturbative effects triggered by replica-sym m etrybreaking (RSB) instantons, m aking predictions for correlations in glassy system s and opening up additional research directions for understanding glassy dynam ics. G lassy slowdown is believed to be controlled by growing cavity point-to-set (PTS) correlations [3], w hose correlation length diverges together w ith the relaxation time: this has been rigorously established for a large class o f graphical m odels [4], and there are num erical sim ulations that support this hypothesis for glass-form ing liquids [5,6]. To define cavity PTS correlations, consider a m any-body system at equilibrium . First, specify a cavity, say a spherical ball o f size R , and pin everything outside o f it— thus fixing a set. Now random ize the particles inside the cavity and allow them to reequilibrate under the influence o f the force exerted by the external pinned particles. The cavity PTS correlator m easures the overlap betw een the new configuration and the original at a p o in t inside the cavity. W hen the cavity is sufficiently small, the interior configuration will be strongly constrained by the fixed configuration outside the cavity, so the cavity PTS correlation should be large. In contrast, w hen the cavity is sufficiently large, the tw o configurations becom e statistically independent deep in their interiors. T his crossover from high correlation to low defines the cavity PTS correlation length, £ ptS. The physics o f cavity PTS correlations can be captured by an effective theory of a replica field qab(r), where qab(r) = q b d r) and c/„„(r) = 0, w ith a ,b = 1 , 2 , . . . ,N r [7], The original equilibrium configuration singles out a replica
‘[email protected] ' [email protected] [email protected] §[email protected] 1539-3755/2015/91(3)7032148(6)
index, say a = 1, w hich acts as a fictitious disorder for reequi librated configurations inside the cavity. T he field com ponents q ia ir) w ith a ^ 1 then characterize the position-dependent overlap betw een the original and the reequilibrated configura tions, w hile the others characterize the overlap betw een two independent reequilibrated sam ples im m ersed in the fictitious disorder. Pinning the external particles m eans that the overlap c/ifi(r) m ust be large outside the cavity. Inside small cavities, the pinned boundary conditions keep the field in the high-overlap m etastable state throughout. Inside large cavities, in contrast, the boundary conditions cannot prevent the field from finding a low -overlap m inim um near the core. T his crossover, taking place at £pts , is precisely w hat an instanton captures, in the lim it Afr -> 1. The dom inance o f large replica-field instantons indicates the need for large cooperative rearrangem ents in order for the system to continue sam pling its phase space, resulting in sluggish dynam ics. The effective-field-theoretic approach allows us to explore cavity PTS correlations in generic glassy system s by w riting down a generic effective action incorporating all interactions in the qab that are sym m etric under perm utations o f the N r indices [7]. We start w ith the action [8,9] l S[qab( r)]
■
r
N'
1
U— 1 a,b= 1_
J
,
Nr ,
+
y 4
4
‘la b -
« 3
t
,
2 (V ? o t)- + 2 Qab
w -
\
^ v,qabqiK-qca c=l
,
~ J Cfa b
(i)
/
and we see how varying the param eter w, w hich couples different com ponents o f the replica field, changes the character of the dom inant instanton. W hile this action is not com pletely general, it is sufficient for dem onstrating that RSB in PTS correlations is generic, in the follow ing sense: turning on more general couplings such as J 2 a ’.b.c=\ Oabqbc may trigger higher-step RSB [10], but it does not change the fact that replica sym m etry is generically broken. Since the effective action treats the replicas sym m etrically, one might naively expect that the dom inant instantons do not distinguish betw een replica indices. Solutions of this type are called replica-sym m etric (RS), and w e begin by analyzing
032148-1
©2015 American Physical Society
ALLAN ADAMS, TAREK ANOUS, JAEHOON LEE, AND SHO YAIDA
them. Varying the action (1) gives the saddle-point equations for qab(r), N, - v 2qab + tqab - m l h + yqlb = u rib + X !
(2)
QcicQcb
c= 1
fora ^ b. Inserting the RS ansatz qab(r) = (1 - 8ab)Q{r) into this saddle-point equation and taking Nr l gives - V 2Q + t Q - w Q 2 + y Q 3 = 0 .
To test the stability of the RS instanton, we consider infinitesimal perturbations around the RS solution, qab (r) = (1 - $ab)Q * (r) + Sqab (r), and we compute the eigenvalues of the resulting Hessian. The eigenperturbations 8qab (r) of the Hessian can be sorted into three categories [13,14]. The first consists of perturbations that are symmetric among the replicas, Sqab (r) = (1 - 8ab)8i ( r ) .
(3)
This equation can be derived from a reduced action,
A W 1, RS
[ - V 2 + 1 - 2wQ* + 3yQ*2] 8u (r)
Sqs i (r) = (1 —
Glassy slowdown and replica-symmetry-breaking instantons A llan A dam s,1’* Tarek A n o u s,1’' Jaehoon L ee ,1’* and Sho Y aida1’2' 1’ 'Center fo r Theoretical Physics, Massachusetts Institute o f Technology, Cambridge, Massachusetts 02139, USA 2Department o f Chemistry, Duke University, Durham, North Carolina 27708, USA (Received 17 July 2014; revised manuscript received 26 February 2015; published 30 March 2015) Glass-forming liquids exhibit a dramatic dynamical slowdown as the temperature is lowered. This can be attributed to relaxation proceeding via large structural rearrangements whose characteristic size increases as the system cools. These cooperative rearrangements are well modeled by instantons in a replica effective field theory, with the size of the dominant instanton encoding the liquid's cavity point-to-set correlation length. Varying the parameters of the effective theory corresponds to varying the statistics of the underlying free-energy landscape. We demonstrate that, for a wide range of parameters, replica-symmetry-breaking instantons dominate. The detailed structure of the dominant instanton provides a rich window into point-to-set correlations and glassy dynamics. DOI: 10.1103/PhysRevE.91.032148
PACS number(s): 64.60.De, 64.70.Q -
W hen glass-form ing liquids are cooled over a m odest range o f tem peratures, their dynam ics slows down by many orders of m agnitude. T his slowdown m anifests itself in a rapid increase o f shear viscosity and structural relaxation time. Despite decades o f w ork, the physics o f the slow dow n— also observed in a w ide variety o f system s including granular and biological system s— has yet to be properly described [1,2]. We develop a method w ith w hich to fill this gap, and, in so doing, we find nonperturbative effects triggered by replica-sym m etrybreaking (RSB) instantons, m aking predictions for correlations in glassy system s and opening up additional research directions for understanding glassy dynam ics. G lassy slowdown is believed to be controlled by growing cavity point-to-set (PTS) correlations [3], w hose correlation length diverges together w ith the relaxation time: this has been rigorously established for a large class o f graphical m odels [4], and there are num erical sim ulations that support this hypothesis for glass-form ing liquids [5,6]. To define cavity PTS correlations, consider a m any-body system at equilibrium . First, specify a cavity, say a spherical ball o f size R , and pin everything outside o f it— thus fixing a set. Now random ize the particles inside the cavity and allow them to reequilibrate under the influence o f the force exerted by the external pinned particles. The cavity PTS correlator m easures the overlap betw een the new configuration and the original at a p o in t inside the cavity. W hen the cavity is sufficiently small, the interior configuration will be strongly constrained by the fixed configuration outside the cavity, so the cavity PTS correlation should be large. In contrast, w hen the cavity is sufficiently large, the tw o configurations becom e statistically independent deep in their interiors. T his crossover from high correlation to low defines the cavity PTS correlation length, £ ptS. The physics o f cavity PTS correlations can be captured by an effective theory of a replica field qab(r), where qab(r) = q b d r) and c/„„(r) = 0, w ith a ,b = 1 , 2 , . . . ,N r [7], The original equilibrium configuration singles out a replica
‘[email protected] ' [email protected] [email protected] §[email protected] 1539-3755/2015/91(3)7032148(6)
index, say a = 1, w hich acts as a fictitious disorder for reequi librated configurations inside the cavity. T he field com ponents q ia ir) w ith a ^ 1 then characterize the position-dependent overlap betw een the original and the reequilibrated configura tions, w hile the others characterize the overlap betw een two independent reequilibrated sam ples im m ersed in the fictitious disorder. Pinning the external particles m eans that the overlap c/ifi(r) m ust be large outside the cavity. Inside small cavities, the pinned boundary conditions keep the field in the high-overlap m etastable state throughout. Inside large cavities, in contrast, the boundary conditions cannot prevent the field from finding a low -overlap m inim um near the core. T his crossover, taking place at £pts , is precisely w hat an instanton captures, in the lim it Afr -> 1. The dom inance o f large replica-field instantons indicates the need for large cooperative rearrangem ents in order for the system to continue sam pling its phase space, resulting in sluggish dynam ics. The effective-field-theoretic approach allows us to explore cavity PTS correlations in generic glassy system s by w riting down a generic effective action incorporating all interactions in the qab that are sym m etric under perm utations o f the N r indices [7]. We start w ith the action [8,9] l S[qab( r)]
■
r
N'
1
U— 1 a,b= 1_
J
,
Nr ,
+
y 4
4
‘la b -
« 3
t
,
2 (V ? o t)- + 2 Qab
w -
\
^ v,qabqiK-qca c=l
,
~ J Cfa b
(i)
/
and we see how varying the param eter w, w hich couples different com ponents o f the replica field, changes the character of the dom inant instanton. W hile this action is not com pletely general, it is sufficient for dem onstrating that RSB in PTS correlations is generic, in the follow ing sense: turning on more general couplings such as J 2 a ’.b.c=\ Oabqbc may trigger higher-step RSB [10], but it does not change the fact that replica sym m etry is generically broken. Since the effective action treats the replicas sym m etrically, one might naively expect that the dom inant instantons do not distinguish betw een replica indices. Solutions of this type are called replica-sym m etric (RS), and w e begin by analyzing
032148-1
©2015 American Physical Society
ALLAN ADAMS, TAREK ANOUS, JAEHOON LEE, AND SHO YAIDA
them. Varying the action (1) gives the saddle-point equations for qab(r), N, - v 2qab + tqab - m l h + yqlb = u rib + X !
(2)
QcicQcb
c= 1
fora ^ b. Inserting the RS ansatz qab(r) = (1 - 8ab)Q{r) into this saddle-point equation and taking Nr l gives - V 2Q + t Q - w Q 2 + y Q 3 = 0 .
To test the stability of the RS instanton, we consider infinitesimal perturbations around the RS solution, qab (r) = (1 - $ab)Q * (r) + Sqab (r), and we compute the eigenvalues of the resulting Hessian. The eigenperturbations 8qab (r) of the Hessian can be sorted into three categories [13,14]. The first consists of perturbations that are symmetric among the replicas, Sqab (r) = (1 - 8ab)8i ( r ) .
(3)
This equation can be derived from a reduced action,
A W 1, RS
[ - V 2 + 1 - 2wQ* + 3yQ*2] 8u (r)
Sqs i (r) = (1 —
