Dynamical large deviations and glass transitionschimera.roma1.infn.it/SYSBIO/CAPRI2014_SLIDES/Garrahan.pdf · Juan P. Garrahan (University of Nottingham) Dynamical large deviations

Juan P. Garrahan (University of Nottingham)

Dynamical large deviations and glass transitions

D. Chandler, A. Keys (Berkeley) R. Jack (Bath) S. Genway, J. Hickey, R. Turner (Nottingham) £

Dynamics is more than statics !

Canonical example → glass transition problem

Dynamical large-deviations→ s-ensemble method ➙ glasses{Ruelle, Derrida, Lebowitz-Spohn, Gartner-Ellis, Donsker-Varadhan, ...}

Statistical mechanics of trajectories rather than states/configurations

Applications in quantum many-body systems?

Stylised facts about the glass transition

#2:

Dynamical

heterogeneitye.g.

50:50 L-J mixture {Hedges 2009}

t� ⇥� t � ⇥� t� ⇥�

#3: Anomalous response if

driven out-of-equilibrium

This scan was continued beyond the meltingpoint, after which the sample was cooled into theglass and then scanned again to yield the blackcurve. This latter curve represents the behavior ofan ordinary glass of TNB, with Tg = 347 K, asdefined by the onset temperature; it is consistentwith previously reported results for TNB (13–15).

Remarkably, the vapor-deposited samplehas a substantially higher onset temperatureof 363 K. This result indicates that the vapor-deposited material is kinetically much morestable, because higher temperatures are requiredto dislodge the molecules from their glassy con-figurations. For comparison, we isothermally an-nealed the ordinary glass for 6 months at 296 Kand up to 15 days at 328 K (equilibrium wasreached at 328 K). Vapor-deposited samplescreated in only a few hours have much greaterkinetic stability than ordinary glasses aged formany days or months below Tg.

To quantify the thermodynamic stability ofthe vapor-deposited materials, we calculate thefictive temperature (Tf), as defined below. LowerTf values indicate a lower position in the energylandscape. The enthalpy for TNB and IMC sam-ples, obtained by integrating the heat capacity Cp,is plotted in Fig. 1B. The intersection betweenthese data and the extrapolated supercooledliquid enthalpy (red curve) defines Tf for eachsample. For both TNB and IMC, samples pre-pared by vapor deposition have considerablylower enthalpies and Tf values. On the basis ofaging experiments on TNB, we estimate that itwould require at least 40 years of annealing anordinary glass to match Tf for the vapor-depositedsample shown in Fig. 1 (12). The similarity ofthe results for TNB and IMC suggests that vapordeposition can generally produce highly stableglasses.

The thermodynamic stability of these filmscan also be quantified in comparison with theKauzmann temperature (TK), the temperature atwhich the extrapolated entropy of the super-cooled liquid equals that of the crystal (4, 5). Wedefine a figure of merit:

qK ¼Tg − TfTg − TK

ð1Þ

For fragile glassformers such as TNB and IMC,qK is a measure of position on the energylandscape, with a value of 1 (Tf = TK) indicatingthe lowest possible position on the landscape. ForTNB, Magill estimated TK = 270 K (14). Vapordeposition of TNB at Tg – 50 K created filmswith qK = 0.43; by this measure, we have pro-ceeded 43% toward the bottom of the energylandscape for amorphous configurations. In com-parison, annealing the ordinary glass at 296 K(qK = 0.09) or 328 K (qK = 0.22) is relativelyineffective. Similar results were observed forIMC deposited at Tg – 50 K, with qK = 0.23 to0.44, depending on deposition rate. These re-

sults can be put into context by comparisonwith Kovac’s seminal aging experiments onpoly(vinylacetate), where 2 months of anneal-ing achieved qK ≤ 0.17 (16).

Vapor deposition can also create unusuallydense glasses. The ratio of the density of vapor-deposited TNB (rVD) to that of the ordinary glass(ro, prepared by cooling from the liquid) in-creases as the deposition temperature is loweredtoward Tg – 50 K (Fig. 2). Also shown as thesolid line is a prediction of the density if vapordeposition produced an equilibrium supercooledliquid at the deposition temperature (12). For thisrange of deposition temperatures, our samplesnearly achieve this upper bound for the density. Ifwe define a fictive temperature based on density,deposition at 296 K produces Tf ≈ 300 K, slight-ly lower than the fictive temperature based onthe enthalpy (15).

We have used neutron reflectivity to charac-terize diffusion in glasses of TNB. The high spa-tial resolution and large contrast in the scatteringlength of neutrons for hydrogen and deuteriumnuclei make this an excellent technique for quan-tifyingmolecular motion.As schematically shownin the inset of Fig. 3, 300-nm films were preparedby alternately vapor-depositing 30-nm-thick lay-ers of protio TNB (h-TNB) and deuterio TNB(d-TNB) (17). The specular reflectivity R wasmeasured as a function of beam angle relative tothe sample surface. This value, multiplied by q4

for clarity, is plotted as a function of the wavevector q. Reflectivity curves for samples vapor-deposited at different temperatures display dif-fraction peaks; as expected for our symmetricmultilayer samples, only odd diffraction ordersare present. For samples deposited at low tem-perature, diffraction can be observed up to the13th order, indicating very sharp h-TNB/d-TNBinterfaces (15).

Time series of neutron reflectivity curveswere obtained for two vapor-deposited samplesduring annealing at 342 K for samples depositedat 330 K (Fig. 4A) or 296 K (Fig. 4B). During 8hours of annealing, all diffraction peaks (exceptthe first-order peak) for sample A decayed tozero, indicating that substantial interfacial broad-ening had occurred because of interdiffusion ofh-TNB/d-TNB. During the 16 hours of anneal-ing at 342 K for sample B, no detectable inter-diffusion occurred, even on the single-nanometerlength scale. We emphasize that the only differ-ence between these two samples was the temper-ature at which the substrate was held duringdeposition.

Figure 4A illustrates the behavior of an or-dinary glass annealed near Tg; as shown else-where (17), interdiffusion in this sample ischaracteristic of the equilibrium liquid. In con-trast, the sample deposited near Tg – 50 K (Fig.4B) is kineticallymuchmore stable, in qualitativeagreement with the high onset temperature shownfor the vapor-deposited sample in Fig. 1A. Wecan quantify the magnitude of this stability interms of the equilibrium structural relaxation time

Fig. 1. (A) Heat capacity, Cp, of TNB samples: vapor-deposited directly into a DSC pan at 296 K at a rateof ~5 nm/s (blue); ordinary glass produced by coolingthe liquid at 40 K/min (black); ordinary glass annealed at 296 K for 174 days (violet), 328 K for 9 days(gold), and 328 K for 15 days (green). (Inset) Structure of TNB. (B) Enthalpy of TNB and IMC samples.Heat capacities of the samples shown in (A) are integrated to obtain the curves shown for TNB. Similarexperimental conditions were used for IMC (15).

Fig. 2. Density of vapor-deposited TNB films(rVD) normalized to the density of the ordinaryglass (ro), with both measured at room temper-ature. Experimental density ratios (filled squares)were calculated from x-ray reflectivity measure-ments on 100- to 300-nm films by measuring filmthickness before and after annealing above Tg(15). The solid line indicates the expected densityif the samples were prepared in thermal equilib-rium with Tf = Tdeposit.

19 JANUARY 2007 VOL 315 SCIENCE www.sciencemag.org354

REPORTS

on F

ebru

ary

29, 2008

ww

w.s

cie

ncem

ag.o

rgD

ow

nlo

aded fro

m

{Swallen+ 2007}

#1:Slowdown w/o

structural change

distance

stat

ic S

F

log time

dyna

mic

SF

density

super-cooled

normal

relaxation rate

dynamic correlation range

staticrange

66 A. Cavagna / Physics Reports 476 (2009) 51–124

1000K / T

log

(vis

cosi

ty /P

)

13

11

9

7

5

3

1

–1

–3

15

–52 4 6 8 100

Fig. 6. The growth of the viscosity close to the glass transition—Logarithmof the viscosity of several substances as a function of the inverse temperature.The horizontal line marks the value � = 1013 Poise, which conventionally defines the dynamic glass transition. (Reprinted with permission from [39];copyright of Societa’ Italiana di Fisica.)

Tg

spec

ific

hea

t

∆Cp

glass

liquid

crystal

temperature

Fig. 7. Specific heat vs. temperature at the dynamic glass transition — The specific heat drops at the dynamic glass transition to approximately thesame value it has in the crystal phase. This is because below Tg we are not giving the system enough time to be ergodic. Roughly speaking, a glass is stuckin a single potential energy minimum for a long time, so that it looses all the configurational degrees of freedom.

we sharply cut the number of degrees of freedom accessible to the system. This causes a sharp drop (up to a factor 2) ofthe constant pressure specific heat cp at Tg [1]. A schematic view of the typical behaviour of cp(T ) is reported in Fig. 7. Theexperimental time is smaller than the ergodicity time, i.e. the time needed by the system to explore a representative fractionof the phase space. In this dynamic sense, we can say that the system is no longer ergodic.

This phenomenon becomes all the more clear when we notice that the specific heat below Tg drops to a value veryclose to that of the crystalline phase. In a crystal the motion of all particles consists of vibrations around their (ordered)equilibrium positions, without any kind of rearrangement. Ergodicity is broken and the system is confined to one (absolute)energy minimum in the phase space. The behaviour of the specific heat thus suggests that also in a glass at low temperatureparticles vibrate around their (disordered) equilibrium positions, with almost no structural rearrangement. Ergodicity isdynamically broken and the glass is confined to one (local) energy minimum in the phase space. For this reason, the specificheat is approximately the same in the crystal as in the low temperature glass.

Even though the view of a glass stuck in a local minimum is good enough to understand the behaviour of static quantitiesas the specific heat, it is unfortunately far too simplistic if we want to understand the dynamic properties of the off-equilibrium phase. This is not our focus, but we nevertheless must be a bit more precise here. A glass is something morecomplicated than a system vibrating around an amorphous minimum of the energy. Were this simple picture true, theglass would be at a (broken-ergodicity) equilibrium within this minimum, as it happens to the crystal. However, a glass isdrastically out of equilibrium. Even though one-time quantities (as the volume or the energy) may look almost constant inthe long time limit, two-time quantities (as the dynamic correlation function) showa stark off-equilibriumbehaviour, in thatthey depend explicitly on both times, rather than on their difference. In other words, the properties of the system depend onthe time elapsed from the instant the systemwas cooled below Tg . This is aging. The reasons for this behaviour are complex

{Angell 1995}

220 240 260T (K)

0.8

1

1.2

1.4

1.6

1.8

2

Cp (J

/gK

)

cool -20 K/mheat +20 K/m

{Biroli-JPG, JCP Pespective: The Glass Transition 2013}

Statics ⇒ Dynamics

eg. RFOT {Parisi+Wolynes+many others}

ideal models e.g. p-spin spin glass

Thermodynamic

Perspectives on glass transition

Statics does not ⇒ Dynamics

metric → Dynamic facilitation

ideal models KCMs {Anderson+Andersen+Jackle+many others}

Dynamic

#2: Dynamical heterogeneity

Cold/dense Lennard-Jones mixture{L. Hedges}

NA = NB = 104 (1 : 1.4)T = 1.1 < Tonset

Motion begets motion ➙ dynamical facilitation

Effective excitations are localised

Interesting structure in trajectories not in configurations/states

{Keys-et-al, PRX 2011}

Dynamic facilitation → kinetically constrained models

�t |P⌅ =W|P⌅ ⇥ W =�

⇧(n⇧�1 + ⇥)⇥⇤⌅+⇧ + ⌅�⇧ � ⇤(1� n⇧)� n⇧

⇤+ (⇧⇤ ⇧� 1)

East model{Jackle 1991}

FA model{Fredrickson-Andersen 1984}

constraint = operator valued rate

Trivial statics but heterogeneous & hierarchical dynamics

{Sollich-Evans 1999}�rel�x ⇡ �0 expÇA

T2+B

T

åEast model

Dynamic facilitation → kinetically constrained models

�t |P⌅ =W|P⌅ ⇥ W =�

⇧(n⇧�1 + ⇥)⇥⇤⌅+⇧ + ⌅�⇧ � ⇤(1� n⇧)� n⇧

⇤+ (⇧⇤ ⇧� 1)

constraint = operator valued rate

East model{Jackle 1991}

FA model{Fredrickson-Andersen 1984}

Trivial statics but heterogeneous & hierarchical dynamics

{Sollich-Evans 1999}�rel�x ⇡ �0 expÇA

T2+B

T

åEast model

(b)

-15

-10

-5

0

0.5 0.75 1

(a) 0

4

8

12

16

-1 0 1 2 3 4

3BRP3Sty

5-PPEAFEHB2O3

BNBP2IB

BPCBSC

BePhCAKNO3

CN60.0CN60.2CN60.4Cum-1Cum-2dBAFDBP-1DBP-2DC704

DCHMMSDEP

DHIQ

dIBPDMPDOP

DPGDMEDPG

EHER

FANGly

KDEmTCP

MTHF-1MTHF-2

mTolNBB

NBS710NBS

nProp-1nProp-2

NS66NS80

OTP-1

OTP-2OTP-3

PDEPG

PHIQPPGPS1PS2PS3PTSB

Sal-1Sal-2Sal-3Sqa

TANAB-1TANAB-2

TCPtNB

TPGTPPXyl 0

4

8

12

16

0 5 10 15

(b)

-15

-10

-5

0

0.5 0.75 1

(a) 0

4

8

12

16

-1 0 1 2 3 4

3BRP3Sty

5-PPEAFEHB2O3

BNBP2IB

BPCBSC

BePhCAKNO3

CN60.0CN60.2CN60.4Cum-1Cum-2dBAFDBP-1DBP-2DC704

DCHMMSDEP

DHIQ

dIBPDMPDOP

DPGDMEDPG

EHER

FANGly

KDEmTCP

MTHF-1MTHF-2

mTolNBB

NBS710NBS

nProp-1nProp-2

NS66NS80

OTP-1

OTP-2OTP-3

PDEPG

PHIQPPGPS1PS2PS3PTSB

Sal-1Sal-2Sal-3Sqa

TANAB-1TANAB-2

TCPtNB

TPGTPPXyl 0

4

8

12

16

0 5 10 15

{Elmatad-Chandler-JPG,

JPCB 2009/2010}

{cf. Bassler 1987, Rossler+

1998, Hecksher+ 2007,

McKenna+ 2013}

��⇥⇥⇥⇥⇥⇥⇥⇥exp�

C

T � T0

⇥NB: no VFT

{Angell 1995}

66 A. Cavagna / Physics Reports 476 (2009) 51–124

1000K / T

log

(vis

cosi

ty /P

)

13

11

9

7

5

3

1

–1

–3

15

–52 4 6 8 100

Fig. 6. The growth of the viscosity close to the glass transition—Logarithmof the viscosity of several substances as a function of the inverse temperature.The horizontal line marks the value � = 1013 Poise, which conventionally defines the dynamic glass transition. (Reprinted with permission from [39];copyright of Societa’ Italiana di Fisica.)

Tg

spec

ific

hea

t

∆Cp

glass

liquid

crystal

temperature

Fig. 7. Specific heat vs. temperature at the dynamic glass transition — The specific heat drops at the dynamic glass transition to approximately thesame value it has in the crystal phase. This is because below Tg we are not giving the system enough time to be ergodic. Roughly speaking, a glass is stuckin a single potential energy minimum for a long time, so that it looses all the configurational degrees of freedom.

we sharply cut the number of degrees of freedom accessible to the system. This causes a sharp drop (up to a factor 2) ofthe constant pressure specific heat cp at Tg [1]. A schematic view of the typical behaviour of cp(T ) is reported in Fig. 7. Theexperimental time is smaller than the ergodicity time, i.e. the time needed by the system to explore a representative fractionof the phase space. In this dynamic sense, we can say that the system is no longer ergodic.

This phenomenon becomes all the more clear when we notice that the specific heat below Tg drops to a value veryclose to that of the crystalline phase. In a crystal the motion of all particles consists of vibrations around their (ordered)equilibrium positions, without any kind of rearrangement. Ergodicity is broken and the system is confined to one (absolute)energy minimum in the phase space. The behaviour of the specific heat thus suggests that also in a glass at low temperatureparticles vibrate around their (disordered) equilibrium positions, with almost no structural rearrangement. Ergodicity isdynamically broken and the glass is confined to one (local) energy minimum in the phase space. For this reason, the specificheat is approximately the same in the crystal as in the low temperature glass.

Even though the view of a glass stuck in a local minimum is good enough to understand the behaviour of static quantitiesas the specific heat, it is unfortunately far too simplistic if we want to understand the dynamic properties of the off-equilibrium phase. This is not our focus, but we nevertheless must be a bit more precise here. A glass is something morecomplicated than a system vibrating around an amorphous minimum of the energy. Were this simple picture true, theglass would be at a (broken-ergodicity) equilibrium within this minimum, as it happens to the crystal. However, a glass isdrastically out of equilibrium. Even though one-time quantities (as the volume or the energy) may look almost constant inthe long time limit, two-time quantities (as the dynamic correlation function) showa stark off-equilibriumbehaviour, in thatthey depend explicitly on both times, rather than on their difference. In other words, the properties of the system depend onthe time elapsed from the instant the systemwas cooled below Tg . This is aging. The reasons for this behaviour are complex

#1

“Thermodynamics” of trajectories: s-ensemble

spac

e

timeBMLJ East

time-integrated order parameter: K = activity

active K � 0

inactive K � 0

spac

e

timeBMLJ East


active K � 0

inactive K � 0

P(K)

K

large-deviations of time-integrated observables


spac

e

timeBMLJ East


active K � 0

inactive K � 0

{e.g. Ruelle, Lebowitz-Spohn, Gartner-Ellis, Donsker-Varadhan,

Lecomte+, many otherscf. Full Counting Statisitcs}

s ↔ K t = “volume”= “free-energy” �= “entropy” �

large deviationsProb(K) ⇡ e�t �(K)

Zt(s) ⌘ he�sK i ⇡ et �(s)


largest eigenvalue ⇥(s)W ! Ws =X

�n��1î Ä

��+� + ��ä� �(1� n�)� n�

óe�s

Zt(s)is “transfer matrix” of “partition sum” Ws


spac

e

timeBMLJ East


active K � 0

inactive K � 0






0.00

0.04

0.08

θ(s)

-0.2 -0.1 0 0.1 0.2s

0

0.2

0.4

0.6

K(s)

East model

{JPG-Jack-Lecomte-Pitard-van Duijvendijk-van Wijland, PRL 2007, JPA 2009}

first-order dynamical transition

at sc = 0

when tobs !�


�n��1î Ä

��+� + ��ä� �(1� n�)� n�

óe�s



�n��1î Ä

��+� + ��ä� �(1� n�)� n�

óe�s


spac

e

timeBMLJ East


active K � 0

inactive K � 0






BMLJ

{Hedges-Jack-JPG-Chandler, Science 2009}

0 0.02 0.04 0.06s

0.03

0.04

0.05

0.06

Ks /

Nt ob

s

T = 0.7

0 0.02 0.04 0.06s

0.02

0.03

0.04

T = 0.6

0.5 1 1.5K / K*

0

1

2

3

P(K

/ K

*)

0.5 1 1.5K / K*

0

1

2

3

tobs=33τ

33τ

tobs=40τ

40τ

tobs=17τ

17τ

20τ27τ

tobs=10τ

10τ

(f)

(e)

(d)

27τ20τ

0 0.02 0.04 0.06s

0.03

0.04

0.05

0.06

Ks /

Nt ob

sT = 0.7

0 0.02 0.04 0.06s

0.02

0.03

0.04

T = 0.6

0.5 1 1.5K / K*

0

1

2

3

P(K

/ K

*)

0.5 1 1.5K / K*

0

1

2

3

tobs=33τ

33τ

tobs=40τ

40τ

tobs=17τ

17τ

20τ27τ

tobs=10τ

10τ

(f)

(e)

(d)

27τ20τ

0 0.02 0.04 0.06s

0.03

0.04

0.05

0.06

Ks /

Nt ob

s

T = 0.7

0 0.02 0.04 0.06s

0.02

0.03

0.04

T = 0.6

0.5 1 1.5K / K*

0

1

2

3P(

K /

K*)

0.5 1 1.5K / K*

0

1

2

3

tobs=33τ

33τ

tobs=40τ

40τ

tobs=17τ

17τ

20τ27τ

tobs=10τ

10τ

(f)

(e)

(d)

27τ20τ

0 0.02 0.04 0.06s

0.03

0.04

0.05

0.06

Ks /

Nt ob

sT = 0.7

0 0.02 0.04 0.06s

0.02

0.03

0.04

T = 0.6

0.5 1 1.5K / K*

0

1

2

3

P(K

/ K

*)

0.5 1 1.5K / K*

0

1

2

3

tobs=33τ

33τ

tobs=40τ

40τ

tobs=17τ

17τ

20τ27τ

tobs=10τ

10τ

(f)

(e)

(d)

27τ20τ

BMLJ (MD/TPS, N=150)

first-order dynamical transition

when tobs � �

at sc ¶ 0

activity vs. counting field

{see also, Lecomte-Pitard-van Wijland 2011,

Speck-Chandler 2012 Speck-Malins-Royall 2012}


dynamical phase-diagram (tobs !�, N!�)

Active phase

Inactive phase

T

s

Tg

To

{Hedges-Jack-JPG-Chandler, Science 2009, Elmatad-Jack-Chandler-JPG, PNAS 2010,

Elmatad-Jack 2013}

non-equilibrium

“glass”

equilibrium liquid

real dynamics s = 0➙ can we access ?sc ¶ 0Accessible from normal dynamics via cumulants and Lee-Yang zeros

{Flindt-JPG, PRL 2013; Hickey-Flindt-JPG 2014}


{Keys-JPG-Chandler, PNAS 2013 and arXiv:1401.7206}

Preparing glasses with s-ensemble

Ti =

spac

e

time

spac

e

time

spac

e

time

a

b

c

0.5 0.4 0.3 0.2 0.3 0.4 0.5

East Model, T =

f

pi =10

0.50.4

0.30.2

si =10

East model

T =

{cf. Sollich-Evans 2003}

space-time bubbles (active & equil.)

space-time stripes (inactive & noneq.)

} � =✓ tobs

�o

◆T} � =✓�g�o

◆Tg

non-equilibrium characteristic length in glassy state

Ti =

spac

e

time

spac

e

time

spac

e

time

a

b

c

0.5 0.4 0.3 0.2 0.3 0.4 0.5

East Model, T =

f

pi =10

0.50.4

0.30.2

si =10

East model

T =

{cf. Sollich-Evans 2003}

space-time bubbles (active & equil.)

space-time stripes (inactive & noneq.)

} � =✓ tobs

�o

◆T} � =✓�g�o

◆Tg

1 101001000100001e+051e+061e+071e+080

5

10

15

20

25

T=0.25T=0.2

P(�)

0 25 50 750 5 10 15 20

0.9

1

1001000100001e+051e+061e+071e+080

5

10

15

20

25

1 10 100 1000100000

5

10

15

20

25aging

tage100 102 104 106 108

ecooling s-ensemble

tobsν -1 104 106 108 101 102 103 104

T=0.72T=0.5

dc

` ne

coolings-ensembleaging

b

equilibriumcooling

a

`

Z(�

)

`5 10 15 20

0.8

0.9

1.0

1.1

1.2

1.3

average low 0.1%

spac

e

time 10τ

a b c∆ ri / σ

≥ 0.60.50.40.30.20.10.0

d=2, N=5x104 d=3, N=106

r / σ

F(r)

r / σZ(r) 5 10 15 20

1.0

1.1

-

AA

non-equilibrium characteristic length in glassy state

{Keys-JPG-Chandler, PNAS 2013 and arXiv:1401.7206}

Preparing glasses with s-ensemble

Statics ⇒ Dynamics

eg. RFOT {Parisi+Wolynes+many others}

ideal models e.g. p-spin spin glass

Thermodynamic

Statics does not ⇒ Dynamics

metric → Dynamic facilitation

ideal models KCMs {Anderson+Andersen+Jackle+many others}

Dynamic

low overlap (liquid) ➞ high overlap (glass){Franz-Parisi}

numerical evidence {Berthier 2013, Parisi-Seoane 2013}

transition in space of trajectories

active (liquid) ➞ inactive (glass)

Perspectives on glass transition

Triangular plaquette model (TPM):{Newman-Moore 1999, JPG-Newman 2000}

E = �J

2

�

⇥s�sjsk

Thermodynamics: !

1-1 mapping spins-plaquettes

free plaquettes → free localised defects ⇒ disordered ∀ T

Overlap transitions and facilitation {JPG, PRE 2014; Turner-Jack-JPG 2014}

Dynamics: (effectively) kinetically constrained

Statics trivial, dynamics complex & glassy, but singular only at T=0 (cf. dynamic facilitation)

Relaxation is hierarchical �! � = e1/T2

cf. East facilitated model {Sollich-Evans 1999}

Two coupled TPMs (annealed): E = �J

2

X

4

⇣s�� s

�j s

�k + sb� s

bj s

bk

⌘� �X

�s�� s

b�

Exact duality:

a

b

a’

b’e�2K

01

e�2K03

= t�nhK3

= t�nhK1

(2KJ = �J, K� = ��)Z(KJ, K�) =�sinh2KJ sinhK�

�N Z(K�J , K

�� )

e�K�� = t�nhKJ, t�nhK�

J = e�K�


{cf. Franz-Parisi}

Two coupled TPMs (annealed): E = �J

2

X

4

⇣s�� s

�j s

�k + sb� s

bj s

bk

⌘� �X

�s�� s

b�

Self-dual:

Çsinh

J

T

å✓sinh

�

T

◆= 1

Cf. TMP in field {Sasa 2010} & generalised Baxter-Wu {Nienhuis 2010}

1st order static transition at finite coupling

ending at CP (*Ising*) {Turner-Jack-JPG 2014}

transition vanishes at ε → 0


0 0.1 0.2ε

0

0.5T

high overlap

low overlap

critical endpoint

self-dual line

Classify trajectories of TPM according to dynamical activity → s-ensemble


1st order transition in activity & overlap

-0.04 -0.02 0 0.02 0.04s

0

0.5

1

1.5

2

K/<

T>

K/<T>

L=4, T=0.5

0

0.2

0.4

0.6

0.8

Over

lap

Overlap

L=8 Excitation Density

T

S-0.05 0 0.05 0.1 0.15 0.2

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Tem

pera

ture Active

Inactive

S

TPM in D=3 → SPyM E = �JX

pys�sjsks�sm

spins-pyramids 1-1 same dualities as TPM

A typical area of the TPTC network adsorbedon the HOPG surface is shown in Fig. 2A. Theterphenyl backbones of the molecules appear asbright rodlike features, and the molecular arrange-ment is unusual because it exhibits hexagonalorientational order but no translational symmetry.The hexagonal order may be discerned from thearray of blue dots overlaid on dark contrast fea-tures (corresponding to depressions or pores) inFig. 2A and, using calibration scans of the graph-ite substrate, we found that the hexagonal arrayhas a period of 16.6 T 0.8 Å oriented at an angleof T6° to the HOPG substrate (16). Although thepores are regularly arranged, the molecular net-work enclosing them is not translationally ordered.Figure 2, B to F, shows that the molecular ar-rangements enclosing different pores (highlightedareas in Fig. 1A) are hexagons formed by a vary-ing number of molecules.

For example, Fig. 2B shows a hexagon formedby three molecules with edges that alternate be-tween a terphenyl backbone and a carboxylic acid–carboxylic acid junction. Figure 2, C and D, showstwo alternative hexagonal arrangements formedby the junction of four molecules with two edgesformed by the terphenyl backbone. Similarly, thejunction in Fig. 2E is formed by five moleculeswith one terphenyl edge, and the junction in Fig.2F is formed by six molecules with no terphenyledges. The lengths of the hydrogen-bonded andterphenyl edges (equivalent to d1 and d2 as de-fined in Fig. 1B) are calculated to be 9.6 and 8.7 Å[the intermolecular binding energyEHB = 0.80 eVis calculated to be the same for the parallel andarrowhead arrangements (16)], giving estimatedwidths of the hexagons in Fig. 2 ranging from15.8 Å (Fig. 2B) to 16.6 Å (Fig. 2F), which is ingood agreement with the measured periodicity.The molecular array shown in Fig. 2A may bebuilt by combining these five structural units inan arrangement that exhibits orientational sym-metry but no translational order.

The network may be mapped onto a tiling byreplacing each molecule with a rhombus [see (25)

for another example linking molecular arrays totiling problems]. Each molecule in the networkpoints along one of three high-symmetry directions,and we have chosen, for clarity, to represent thesethree molecular orientations as rhombi with differ-ent colors. To illustrate the tiling, we have convertedeach of the hexagonal structural units discussedabove into rhombi (Fig. 2). The representations ofthe junctions in Fig. 2, B to F, correspond to verti-ces where three, four, five, or six rhombi meet.These diagrams also show that, at a molecularlevel, the mapping is possible because the inter-molecular bonds between neighboring moleculesare located at the midpoint of the rhombus edges(Fig. 2G). We suggest that this symmetry is key toidentifying other candidate molecules that mightform similar networks.

The molecular network displayed in Fig. 2Acan be mapped into rhombi, and the resultanttiling is shown in Fig. 2H. The mapping directlyaccounts for the presence of orientational sym-metry combined with an absence of translationalorder because the rhombus vertices (pores in theSTM images) fall on a hexagonal lattice, eventhough the arrangement of rhombi is not ordered.Thus, we demonstrate that the molecular array isequivalent to a rhombus tiling.

We also observed tiling defects in the form oftriangular voids enclosed completely by rhombi(Fig. 3). These voids are topological defects thatoccur in two states of effective “charge” corre-sponding to triangles pointing either “up” or“down” and have been considered theoreticallybut have not previously been observed (26–28).We observed ~3 × 10−3 defects per adsorbed mol-ecule and may unequivocally distinguish thesevoids from other less intrinsically interesting de-fects, such as vacancies. The triangular defectshave been observed to propagate through the net-work, as shown in Fig. 3, C to H. This movementresults in a rearrangement of a single molecule(or tile) within the network. Figure 3, C and F,shows a comparison of images before and aftersuch a transition, in which, as expected, effective

charge is conserved. The triangular defect under-goes a second movement between Fig. 3, E andG. In our images, this transition appears to bemediated by the temporary presence of an addi-tional species at the defect site, as highlighted inFig. 3, C and E, possibly an additional TPTCmolecule temporarily bound by hydrogen bond-ing. Although it is difficult to determine the exactdetails of the atomistic mechanism for defect move-ment, this sequence of images shows that defectpropagation through the network gives rise to areordering of molecular tiles and facilitates a tran-sition between different local energy minima.

To determine whether the observed rhombustilings are ordered or random, we followed pre-vious theoretical studies (10, 12) and introducedan effective height h(x,y) at each vertex (x,y). Theheight was calculated with the scheme shown inFig. 4A, in which a displacement along a rhombusedge leads to a change in height of T1. By ar-bitrarily choosing an origin with zero height, it ispossible to define h(x,y) for all vertices of a perfect(defect-free) tiling. Within this scheme, a tiling maybe visually considered as a perspective of the sur-face of a simple cubic lattice when viewed along a(111) direction. More formally, the rhombus tilingis equivalent to the projection of an irregular sur-face of a three-dimensional simple cubic crystalonto a (111) plane of the cubic lattice. A map ofeffective height of the STM image (Fig. 2A) isshown in Fig. 4C.

Within the random tiling hypothesis (11), thetilings may be analyzed by introducing an effec-tive free energyG, which, assuming that all vertextypes (shown in Fig. 2) are degenerate, is deter-mined entirely by an entropic contribution and isgiven by G ¼ ðKo=2Þ∫j∇hj2 dxdy. This contribu-tion is equivalent to the energy of a deformedsurface with elastic constantKo. The gradient ∇hcorresponds to the projection in the (x,y) planeof the normal to the representative surface. Thetilings that are generated by this free energy havea height representation for which ⟨∇h⟩ ¼ 0, thatis, a surface which on average is flat and par-

Fig. 3. (A, C, E, and G) STMimages showing two separatemovements of a single defectthrough the network structure.(B, D, F, and H) Tiling repre-sentation of the network struc-ture during the defect motion.The effective rearrangements ofrhombi in the tiling are markedby the black arrows in (D) and(F). Transient image artifacts ob-served within the defect sitebefore defect motion are high-lighted by blue dashed squares[(C) and (E)]. Scanning condi-tions for all images were It =0.021 nA and Vt = 1200 mV.

A C E G

B D F H

www.sciencemag.org SCIENCE VOL 322 14 NOVEMBER 2008 1079

REPORTS

on

No

ve

mb

er

14

, 2

00

8

ww

w.s

cie

nce

ma

g.o

rgD

ow

nlo

ad

ed

fro

m

dynamics facilitated by localised free defects

Similar features in tilings / dimer coverings / “spin-ice”-like systems

appears in Ising models of magnetism and leads to ordered phaseswith broken symmetry for |J|! kBT, where kB is the Boltzmann con-stant and T the temperature; these phases may be either ferromagneticor antiferromagnetic depending on the sign of J, although entropicterms in the free energy dominate for |J|≪ kBT, in which case aparamagnetic phase occurs. Our experiments and simulations showthat this rich phase behaviour may be investigated by preparing mol-ecular arrays with differing values of D.

Results and discussionFigure 1c–h shows images of molecular networks prepared underseveral different conditions (see Methods), referred to as Experiments1–6, respectively, and the corresponding representations of rhombustiling, in which each molecule is represented by a rhombus colouredaccording to its orientation. To characterize our experimentaltilings we defined an order parameter, C¼ (n0p – p0n)/(n0pþ p0n),where n and p represent the fraction of rhombus tile junctions in

non-parallel and parallel orientations, respectively, and n0¼ 0.608and p0¼ 0.392 are the equivalent values for a defect-free, ideal,random tiling and were estimated numerically (see Methods). Assuch, C equals 1 in a fully parallel phase, 21 in a fully non-parallelphase and 0 for an ideal random tiling. C was calculated for each ofExperiments 1–6; the images in Fig. 1c–h are placed in order ofdecreasing C, from C¼ 0.22 to C¼ –0.68.

Figure 1b is a schematic of the expected equilibrium phasediagram13–16. The two relevant thermodynamic parameters aretemperature, T (in units of 1, the characteristic hydrogen bondenergy) and energetic bias, D. Ideal random tilings are observedfor D¼ 0. For D . 0, non-parallel bonding is favoured, whichresults in increasingly negative values of C. For sufficiently largeD the system orders into a crystalline phase dominated by non-parallel bonds. This transition occurs at D/kBT¼ 0.454(3), a valuecalculated in the limit of zero temperature and the transition is ofthe Kosterlitz–Thouless kind16. For non-zero temperatures of

d1

d2Non-parallel

orderParallelorder

Randomtiling

1 2 3 4 5 6

Exp. 1: ψ = 0.22DPBDTC

Nonanoic acid

Exp. 2: ψ = –0.08TPTC

Heptanoic acid

DPBDTC

i j k l

a b

c d e f g h

TPTCCoronene

Exp. 3: ψ = –0.25TPTC

Octanoic acid

Exp. 4: ψ = –0.35TPTC/60 °C

Nonanoic acid

Exp. 5: ψ = –0.43TPTC

Nonanoic acid

Exp. 6: ψ = –0.68TPTC and coronene

Nonanoic acid

kBT/ε

Δ/kBT

Figure 1 | Tetracarboxylic acid supramolecular assemblies and rhombus tilings. a, Parallel and non-parallel intermolecular bonding orientations, withbackbone (d1) and bond (d2) lengths indicated, overlaid onto the corresponding representations of the rhombus tiles. b, Schematic phase diagram of theinteracting rhombus tiling model. Three phases are expected depending on the interaction energy D: a random-tiling phase, an ordered phase dominated bynon-parallel bonding at large positive D and a second ordered phase dominated by parallel bonding at large negative D. c–h, STM images (sections of largerarea scans) of tetracarboxylic acid supramolecular networks at alkanoic acid–HOPG interfaces and the corresponding rhombus tilings, in a sequence ofdecreasing C. Experiments are labelled 1–6 and were performed at room temperature (apart from Experiment 4, which was at 60 8C) using differentcombinations of TPTC, DPBDTC, coronene and solvents, as specified in the text boxes. STM image contrast originates from molecular backbones and, in h,coronene (see Methods for imaging parameters; all scale bars¼ 50 Å). i–k, Molecular ball-and-stick diagrams of DPBDTC (i), TPTC (j) and coronene (k).l, Diagram of coronene adsorbed at the vertex of six TPTC molecules.

NATURE CHEMISTRY DOI: 10.1038/NCHEM.1199 ARTICLES

NATURE CHEMISTRY | VOL 4 | FEBRUARY 2012 | www.nature.com/naturechemistry 113

random tiled

ordered

T

field

free → confined defects

Eg. molecular random tilings {Beton+Champness+Blunt+Whitelam+Stannard+...+JPG}

“mosaics” for real

KCMs and many-body localisation in closed quantum systems{Hickey-Genway-JPG, arXiv:1405.5780}

Many-body localisation (MBL) transition: {Basko-Aleiner-Altshuler 2006, Huse+, many others}

‣ Cf. Anderson localisation but for interacting system

‣ Singular change throughout spectrum

‣ Eigenstates change from “thermal” (ETH {Deutsch, Sdrenicki}) to MBL

‣ Observables do not relax in MBL phase

‣ Often thought of as “glass transition” but modelled with disorder

Can KCMs (as models for classical glasses) say anything about quantum MBL?


Recap: active-inactive “space-time” transitions in KCMs (eg. East/FA)

W ! Ws =X

�n��1îe�sÄ��+� + ��ä� �(1� n�)� n�

ó+ (�$ �� 1)

Largest e/value = cumulant G.F. for activity → 1st order phase transition 0.00

0.04

0.08

θ(s)

-0.2 -0.1 0 0.1 0.2s

0

0.2

0.4

0.6

K(s)

0.00

0.04

0.08

θ(s)

-0.2 -0.1 0 0.1 0.2s

0

0.2

0.4

0.6

K(s)

0.00

0.04

0.08

θ(s)

-0.2 -0.1 0 0.1 0.2s

0

0.2

0.4

0.6

K(s)

Can transform into Hermitian operator through equilibrium distribution

Consider as Hamiltonian and corresponding quantum unitary dynamics |�ti = e��tHs |�0i

Hs ⌘ �P�1WsP = �X

�n��1îe�sp�� (1� n�)� n�

ó+ (�$ �� 1)

{Merolle-Chandler-JPG, 2005, JPG+Jack+Lecomte+van Wijland+, PRL 2007, JPA 2009, + others}


Signatures of MBL transition: (i) relaxation / non-relaxation of observables

|�ti = e��tHs |�0iHs = �X

�n��1îe�sp�� (1� n�)� n�

ó+ (�$ �� 1)

does not relax on inactive side s>0

relaxes on active side s<0

hMit =X

�h�t |�z� |�titime evolution of magnetisation

-2

0

2

4

6

8

10

0 20 40 60 80

⟨M ⟩

t

(a)

8 10 12 14 16-2

-1.5

-1

-0.5

Log ⟨Q

⟩

N

(c)

s = -2-1.5

-1-0.5

0

0.5

Log ⟨Q

⟩

(b)

s = 0.51

1.52


Signatures of MBL transition: (ii) transitions throughout spectrum

N = 9

N = 16

ETH on active side s<0

no ETH on inactive side s>0


�n��1îe�sp�� (1� n�)� n�

ó+ (�$ �� 1)


Signatures of MBL transition: (iii) level spacing statistics


�n��1îe�sp�� (1� n�)� n�

ó+ (�$ �� 1)

0.3

0.35

0.4

0.45

0.5

0.55

-0.5 0 0.5 1 1.5 2 2.5

⟨ r

⟩

s

GOE

Poiss.

N = 68

1012

GOE 0.53

Poisson 0.39

Many symmetries in clean system !Can remove with site disorder: !!!KCM dynamics unchanged

�! |�� g��| (g⌧ �)


Signatures of MBL transition: (iv) localisation onto classical basis

widely spread active

s<0

e/states concentrated

inactive s>0


�n��1îe�sp�� (1� n�)� n�

ó+ (�$ �� 1)

⇒ active-inactive transition ➝ 1st order MBL transition in whole spectrum

MBL transition without disorder

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-4 -2 0 2 4

IPR

�

(a)

� = 6� = 8� = 10� = 12� = 14

-100 -50 0 50 1000

0.05

LDO

S

� - �� ⇥

(c) � = 20

0.01

LDO

S

� = -2(b)

0

0.1

0.2

0.3

0.4

-4 -2 0 2 4

dIP

R/d

s

�

N = 14N = 12

SUMMARY

“Thermodynamics of trajectories” based on LD theory

Glass transition as a active/inactive transition in trajectory space

KCM glass models as models for MBL without disorder

Documents

Dynamical large deviations and glass transitionschimera.roma1.infn.it/SYSBIO/CAPRI2014_SLIDES/Garrahan.pdf · Juan P. Garrahan (University of Nottingham) Dynamical large deviations