Thermal Memory in Glassy Systems

Nicholas M. Boffi and Po-Yi Ho

Recent experiments have demonstrated that disordered, glassy systems can exhibit relaxation dynamics thatdepend on the system’s thermal excitation history: that is, they retain a thermal memory. We investigate thermalmemory by studying two disordered systems. We first study a model of a random walk in a random environment.We review a real space renormalization group method to obtain exact solutions for the effective dynamics of thesystem. In particular, we obtain the diffusion front and the two-time diffusion front. The RG technique togetherwith simple numerical simulations is applied to investigate two-temperature effective dynamics. Subsequently,we study a two dimensional generalization of this model: diffusion in the square lattice Ising spin glass. Diffusionin this system is simulated using kinetic Monte Carlo techniques and the Python code used to do so is provided.Two simulation models were considered: one in which the distribution of the bond energies is taken to havean explicit dependence on the maximum temperature the glass has ever been exposed to, and one in which thedistribution of bond energies is held fixed, but the system is allowed to relax at a higher temperature before thediffusive dynamics are studied. Slower relaxations are found for higher values of Tmax in both cases.

1. INTRODUCTION

Glassy systems often exhibit slow, “aging” dynamicsthat depend on the system’s excitation history, such asthe waiting time of a perturbation. A recent experiment,summarized in Fig. 1, demonstrates that such systemscan also exhibit thermal memory, in which the relaxationdynamics depend on the maximum temperature to whichthe system was exposed. The essence of the experimentrelies on a new cryogenic fabrication method, which en-ables the creation of an electron glass at extremely lowtemperatures. This allows the experimentalists to con-trol the highest temperature to which the glass has everbeen exposed. Notably, the rate of the logarithmic relax-ation depends on the ratio of the current and maximumtemperatures T/Tmax of the system [1].

The hypothesized explanation from Ref. [1] for thisdependence is that the glass, when fabricated at cryogenictemperatures, is “born” in a highly energetically unfa-vorable state. Unlike glassy systems created through aquenching procedure, the glasses in this experiment donot start in a high temperature state above the glass transi-tion, and hence the level of equilibration of the final statehas no dependence on an experimental quench rate. Bytuning Tmax, one is able to directly control the amount ofequilibration the system has ever been able to undergo,and hence control the depth of the local minimum in theenergy landscape defining the current state of the glass.Higher values of Tmax lead to more equilibration, a morestable state, and slower relaxation dynamics.

Here, we study thermal memory in disordered systemsin an attempt to further understand this result outside ofa qualitative explanation. We first investigate an analyti-cally tractable model of a particle diffusing on a randomlandscape. The many minima of the random landscapespan all scales, and the diffusive behavior of the particlenaturally exhibits slow dynamics. We review a real spacerenormalization group method to obtain exactly the dif-fusion and two-time diffusion fronts. This model offers

a rare example of a disordered system with analyticallysolvable nonequilibrium dynamics. Simple numericalsimulations of the effective dynamics confirm that theparticle exhibits both aging and thermal memory.

We then study a more realistic two-dimensional modelof diffusion on the square lattice Ising spin glass via ki-netic Monte Carlo methods. We consider two versionsof this model. The first has an explicit Tmax dependencein the distribution of bond energies, and diffusion is stud-ied at fixed current temperature T for several values ofTmax. The second is an “atomistic” approach in whichthe distribution of bond energies has no Tmax dependence,but systems are allowed to equilibrate for a short time atTmax > T before the diffusion at T is simulated. Data isonly recorded from the diffusion simulation at temper-ature T . The first of these models essentially tests thehypothesis that higher Tmax leads to deeper minima andhence slower relaxation, while the second simulates theexperiment directly to see if we can capture the emer-gence of a Tmax dependence from the dynamics. We findslower diffusion for higher values of Tmax in both cases.The second approach is particularly interesting, as thethermal memory occurs organically from the simulation,directly in line with the recent experiment [1].

I. RSRG OF RWRE

We study a particle in a disordered 1D landscape. Theparticle performs Arrhenius diffusion across randomlydrawn energy barriers. Denote the potential of site i byUi. Without loss of generality, the barrier heights fi canbe made alternating, such that fi =Ui−Ui+1 = (−1)i Fi,where Fi = |Ui−Ui+1|. Sites i and i+ 1 are connectedby bonds of random length li. The bond variables arechosen independently from bond to bond. The model istherefore specified by the distribution of barrier heightsand lengths P(F, l). We refer to this model as randomwalks in random environments (RWRE, also known asthe Sinai model), see Fig. 2. We focus on the symmetric

2

case where the ascending and descending barriers havethe same statistics, but the techniques discussed can beapplied to asymmetric cases and finite landscapes.

We review a real space renormalization group (RSRG)analysis of RWRE following Ref. [2]. All results repro-duced below can be found in Ref. [2], but we more clearlycomment on various parts of the derivation. The RSRGtechnique allows us to extract effective dynamics thatare exact at long times. Deviations from the effective

FIG. 1. (Top) A quench-condensed Au film on Si/SiO substrateforms a disordered electronic system. The quenched, disorderedgeometry and long-range Coulomb interactions lead to frus-trated, glassy dynamics. (Bottom) (a) Change in conductance asa function of time for one sample following the heating protocolin (b). (b) The sample was exposed to a maximum tempera-ture Tmax = 170K. (c) The slope of the logarithmic relaxationdepends on T/Tmax. From Ref. [1].

FIG. 2. (a) Effective dynamics of a particle in a random land-scape. If the particle is adjacent to a renormalized bond, theparticle moves to the bottom of the renormalized valley. (b) Thedecimation of the smallest barrier, illustrated to be F2. FromRef. [2]. (Bottom) A numerical demonstration of the RSRGprocedure described in the text. Blue is the original landscape.Black is the renormalized landscape. Red traces the effective dy-namics of a particle that begins near the middle of the landscapebut arrives at the second-from-left renormalized valley.

dynamics can be accounted for by rare events in the RGprocedure. Below, we review the RSRG analysis and ob-tain the diffusion front and the two-time diffusion front.The analytical results are compared to stochastic simula-tions. We then extend the analysis to investigate the two-temperature diffusion front, and compare the observedaging and thermal memory phenomena to the experimentdiscussed in the introduction.

The RSRG procedure for RWRE is conceptually sim-ple, but offers a rare example of a disordered systemwhose long-time dynamics are exactly solvable. At longtimes, the diffusive behavior of the particle is dominatedby large barriers. Hence, the RG procedure decimatesthe bonds with the smallest barriers Γ = Fmin in a givenlandscape. Suppose Fi = Γ is such a bond. At time scalesmuch longer than the Arrhenius time scale t0 exp(Fi/T ),the rate for the particle to hop between sites i− 1 andi+ 2 is approximately the same as it would be if sites iand i+1 were decimated, and sites i−1 and i+2 wereconnected by a bond with barrier F ′ and length l′,

F ′ = Fi−1−Fi +Fi+1 (1)l′ = li−1 + li + li+1 (2)

3

see Fig. 2. Since this procedure preserves the modelstructure of alternating, independent bonds, this is ourcoarse-graining step.

The RG flows are as follows. Since bonds remainindependent, it is useful to define renormalized bonds atscale Γ as ξi = Fi−Γ and the associated distribution asPΓ (ξ = F−Γ, l). Following a decimation at scale Γ, thedistribution PΓ (ξ , l) is renormalized as

∂ΓPΓ (ξ , l)= ∂ξ PΓ (ξ , l)+PΓ (0, ·)∗l PΓ (·, ·)∗ξ ,l PΓ (·, ·)

−2PΓ (ξ , l)∫ ∞

0dl′PΓ

(0, l′)+2PΓ (ξ , l)

∫ ∞0

dl′PΓ(0, l′),

(3)

where ∗ denotes convolution with respect to the sub-script and dotted variables, i.e. PΓ (·, ·) ∗ξ ,l PΓ (·, ·) =∫ ∞

0 dl′ ∫ ∞

0 dξ ′PΓ (ξ −ξ ′, l− l′)PΓ (ξ ′, l′). The first termon the RHS is due to the change in the definition of ξ asΓ increases. The second term accounts for the new bondscreated by decimation. A new bond is formed at level ξ ′ ifa decimated bond (at level ξ = 0) has neighboring bondsthat add to ξ ′, and similarly for its length. Since the bondsare independent, the probability of the renormalized bondcan be obtained as a convolution of the probability of thecurrent bonds. The third term accounts for the decimationof the two bonds adjacent to the smallest barriers. Thelast term rescales the distribution to keep it normalized byaccounting for the overall net loss of a fraction of bonds.This is the logic that we will use to write down other RGflows below. This net loss also gives the evolution of thetotal number nΓ of bonds as

∂ΓnΓ =−nΓ2∫ ∞

0dl′PΓ

(0, l′). (4)

The average bond length l̄Γ =∫

dξ dllPΓ similarly evolvesas

∂Γ l̄Γ = l̄Γ2∫ ∞

0dl′PΓ

(0, l′). (5)

In the symmetric case considered here, the RG flowssimplify to(

∂Γ−∂ξ)

PΓ (ξ , l) = PΓ (0, ·)∗l PΓ (·, ·)∗ξ ,l PΓ (·, ·) . (6)

The RSRG procedure allows us to extract an effec-tive dynamics that approximate the Arrhenius dynamicsat long times. In the effective dynamics, the particle attime t is at the bottom of the renormalized valley at scaleΓ = T ln(t/t0) which contains the initial point, see Fig. 2.We set the microscopic timescale t0 to unity. The effec-tive dynamics become the exact dynamics as Γ = T ln tapproaches ∞, which was shown rigorously in Ref. [3].This can be seen by considering the broadening of PΓunder RSRG. At large Γ, the renormalized landscape con-sists entirely of deep valleys with high barriers. Therefore,the particle with high probability will be at the bottom of

the valley which contains the initial point. Corrections tothe effective dynamics come from rare events when twoadjacent bonds have barriers within order T of each other.

The RG flow given by Eq. 6 can be solved exactlyas follows. First, we nondimensionalize the barrier andlength variables. Length variables can be normalized by2σ =

∫dFF2P(F) where P(F) is the initial unrenormal-

ized distribution. The dimensionless length variable canthen be written as λ = l/Γ2. The dimensionless barriervariable is η = ξ/Γ. The rescaled probability distributionis PΓ (η ,λ ) = Γ3PΓ (ξ , l). Convolutions become multi-plications when Laplace transformed. Hence, Laplacetransforming Eq. 6 from λ → p,

[Γ∂Γ− (1+η)∂η +2p∂p−1]PΓ (η , p) =PΓ (0, p)PΓ (·, p)∗η PΓ (·, p) . (7)

We look for solutions of the form P̃(η , p) =a(p)exp(−ηb(p)). At p = 0, P̃(η , p = 0) =∫ ∞

0 dλ ′P(η ,λ ′) = P(η) must be normalized so thata(0) = b(0) = PΓ (0). Substituting this requirement intothe RG flow, we find that for each p, the following non-linear ordinary differential equations must be satisfied∂b/∂Γ = −a2, and ∂a/∂Γ = −ba. These can be inte-grated explicitly to show that Eq. 7 has the fixed point

P̃(η , p) =

( √p

sinh(√

p))exp(−η√pcoth(√p)) . (8)

At p = 0, we find that P(η) has the form

PΓ (F) = θ (F−Γ)exp(−(F−Γ)/Γ)

Γ, (9)

where θ is the step function. This is a natural result, sincethe decimation procedure will lead to long regions of as-cending or descending bonds. In other words, the coarse-grained probability distribution of barriers is exponentialwith width 〈F〉 = Γ = T ln t. What is the convergencetowards this fixed point P∗ (η) = exp(−η)? We defineP = P∗+q for small q. The linearized RG flows are then

[Γ∂Γ− (η +1)−1]q = 2qP∗+q(0)ηe−η . (10)

The analysis of the linearized RG flow is too long tobe included for our purposes here. Unlike usual RG ap-proaches, the RSRG method applied to RWRE is usefulnot because it characterizes phase transitions, but becauseit characterizes effective dynamics that are otherwise dif-ficult to obtain. Nevertheless, it has been shown that per-turbations converge as Γ−1 towards the fixed point witheigenvector (η−1)exp(−η), corresponding to a shift inΓ. The explicit form of the bond length distribution canbe found by integrating Eq. 8.

More importantly, Eq. 4 implies that the number ofbonds nΓ decays asymptotically as Γ−2. Since the averagebond length l̄Γ ∼ 1/nΓ scales inversely with the number

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of bonds, l̄Γ ∼ Γ2. In other words, the position of theparticle following the effective dynamics is

x∼ T 2 ln2 t. (11)

This reproduces the classical result obtained by Ref. [3],which introduced the RWRE model. Already, we see thatthe RWRE model reproduces the slow dynamics aspectof our experimental motivation.

The full expression for the single time diffusion front isobtained as follows. We wish to calculate the probabilityP(x, t|0,0) that a particle starting at x = 0 at time t = 0 isat x at time t under the effective dynamics in our RSRGprocedure. Since the particle with high probability will beat the bottom of the renormalized valley containing x = 0at t = 0, the distribution is δ function shaped. To obtainthe diffusion front, we average over the quenched disorder(i.e. the various random landscapes) or equivalently theinitial conditions. The probability that x = 0 belongs toa renormalized bond of length l at scale Γ is the lengthof a given bond divided by the average length of a bondtimes the probability that a given bond has length l, orlPΓ (l)/

∫dllPΓ (l), where PΓ (l) =

∫dηP(η , l). Further-

more, the distance |x| between the starting point and thebottom of the bond is uniformly distributed over [0, l].Hence, averaging over l, we find that

P(x, t|0,0) =∫ ∞|x| dlPΓ (l)

2∫

dllPΓ (l). (12)

The substitution of the fixed point can be carried outexplicitly. We quote the result from Ref. [2], which is

P(x, t|0,0) = 1T 2 ln2 t

q(

xT 2 ln2 t

), (13)

where

q(X) =4π ∑

(−1)n

2n+1exp(−π2 (2n+1)2 |X |/4

). (14)

This agrees with previous rigorous results, validating theRSRG method [4].

II. AGING IN RWRE

We now calculate the two-time diffusion front. Thetwo-time diffusion front is the probability that a particlestarting at x = 0 at t = 0 first arrives at x′ at t = t ′, then atx at t = t,

P(x, t,x′, t ′

)= P(x, t,x′, t ′|0,0). (15)

This quantity is important in aging protocols that apply aperturbation for a waiting time tw ≡ t ′. We anticipate thatsince in the effective dynamics, the barrier scales withboth temperature and time as Γ∼ T ln t, this quantity will

also be important in thermal memory protocols that applya thermal excitation Tmax. We focus on large t and t ′, forwhich there are several regimes.

We calculate first the probability D(t ′, t) that a particleremains at the bottom of a renormalized valley and doesnot move between t ′ and t. This will be used later inthe full expression for the two-time diffusion front. Wewrite the RG flow of D in a similar manner as the RGflow of P(ξ , l). Denote by DΓ,Γ′ (ξ ) the probability thata particle on a bond F = Γ+ξ does not move between Γ′and Γ. In other words, the bond has not been decimatedand the same-valley neighbor has also not been decimated.Therefore,(

∂Γ−∂ξ)

DΓ,Γ′ (ξ ) =−2PΓ (0)DΓ,Γ′ (ξ )+PΓ (0)PΓ (·)∗ξ DΓ,Γ′ (·) . (16)

The first term on the RHS accounts for forbidden dec-imations, while the second term accounts for alloweddecimations. At large Γ, PΓ has reached the fixed pointfound above. Defining α = Γ/Γ′, we find after substitut-ing the fixed point and rescaling,

[α∂α − (1+η)∂η +1]Dα (η) =∫ η1

0e−η1Dα (η−η1) .

(17)The RG flow is accompanied by the initial conditionDα=1 (η), which is the probability that a particle is ona bond with F = Γ′ (1+η) at scale Γ′. This probabil-ity is just the length of the bond divided by the aver-age length of bonds, or Dα=1 (η) =

∫dλλP(η ,λ )/λ̄ =

e−η [(1+2η)/3]. The solution to Eq. 17 is of the formAe−aη +Be−bη . The exact expression turns out to be,

Dα (η) = e−η[

5+ 2α−1 e1−α

3α2

]+ e−αη

[2e1−α

3(a−1)

].

(18)Eq. 18 is the combination of two exponential terms. Thefirst term is exp(−F/Γ) and is expected from the effectivedynamics, whereas the second term is exp(−F/Γ′) anddemonstrates that the system retains memory of the initialphase at scale Γ′. This form demonstrates aging propertiesin the variable Γ, which leads to both aging properties int and thermal memory in T . The probability D(t, t ′) isfinally obtained as

∫dηDα (η), which simplifies to

D(t, t ′)=

(ln t ′

ln t

)2(53− 2

3e−(ln t/ ln t

′−1)). (19)

Eq. 19 shows that when α ≈ 1 or when t ′ ≈ t, D(t, t ′)∼7/3−4/3α . In other words, the number of jumps acrossthe origin grows as 4/3ln t. This is consistent with pastrigorous results [4].

The RSRG procedure can again be applied to calculatethe full two-time diffusion front. To do so, we considerquantities that track the end points of bonds, and write the

5

corresponding RG flow equations. Define

Ω++Γ,Γ′(ξ ,xL,xR,x′L,x

′R)

as the probability that the initial point belongs to a de-scending bond with ends [−x′L,x′R] at scale Γ′ and to adescending bond with barrier height ξ = F−Γ with ends[−xL,xR] at scale Γ. There are four such quantities, for thefour combinations of descending and ascending bonds atthe two scales Γ′ and Γ′. From Ω, the two-time diffusionfront can be obtained as

P(x, t,x′ > 0, t ′

)= θ (x)

∫dξ dxLdx′LΩ

++Γ,Γ′(xR = x,x′R = x

′)+θ (−x)

∫dξ dxRdx′LΩ

−+Γ,Γ′(xL =−x,x′R = x′

), (20)

and similarly for x′ < 0. By the usual logic of keepingtrack of decimated bonds, but now also keeping track ofthe direction and the ends of the bonds, the RG flows forthe four Ω quantities can be written as(

∂Γ−∂ξ)

Ω±ε′

Γ,Γ′ =−2PΓ (0)Ω±ε ′Γ,Γ′

+∫

dydl1dl2PΓ (0, l2)PΓ (·, l1)∗ξ Ω±ε′

Γ,Γ′ (·,xL = y)δ (xL− (y+ l1 + l2))

+∫

dydl2dl3PΓ (0, l2)PΓ (·, l3)∗ξ Ω±ε′

Γ,Γ′ (·,xR = y)δ (xR− (y+ l2 + l3))

+∫

dl1dl3dy1dy2PΓ (·, l1)∗ξ PΓ (·, l3)Ω±ε′

Γ,Γ′ (xL = y1,xR = y2)×

δ (xL− (y1 + l1))δ (xR− (y2 + l3)) , (21)

where ε ′=±1. The first term on the RHS accounts for theforbidden decimations, while the other terms account forcontributions from decimated bonds while keeping trackof the ends and directions of the renormalized bonds. TheRG flows in the asymmetric case must also carefully dis-tinguish between ascending and descending bonds. TheRG flows in Eq. 21 are supplemented by the initial condi-tion at Γ = Γ′,

Ωεε′

Γ,Γ′ = δεε ′δxLx′L δxRx′R ωε ′Γ′(ξ ,x′L,x

′R),

where ω is the probability that the initial position be-longs to a bond with barrier ξ and ends [−x′L,x′R]at scale Γ′. Like before, this probability is ω =∫

dl′PΓ′ (ξ , l′)/l̄Γ′δ (l′− (x′L + x′R)). Since the RG flowsinvolve convolutions of probability distributions, they canbe solved by taking the Laplace transform to xL → µ ,xR → ν , x′L → µ ′, x′R → ν ′, and using the fixed pointsolution for the distribution of barrier heights. The equa-tions involve only exponential terms, so we look for solu-tions that are also combinations of exponential terms,Ω̃ = Ae−ξ uεΓ(µ) + Be−ξ uεΓ(ν) +Ce−ξ u

εΓ(µ

′) + De−ξ uεΓ(ν

′)

where the coefficients A through D depend on ε , ε ′, andall four Laplace variables. Furthermore, we can use theproperty that the RG flows are decoupled except for the

ξ = 0 factors to simplify the RG flows into homogeneousordinary differential equations. The usual techniques forsolving ODEs can then be applied. The linearly indepen-dent solutions can be found exactly after tedious algebrathat span several pages in the appendix of Ref. [2], whichwe will not reproduce. The reason that these calculationscan be carried out exactly is because the involved inte-grals are only of exponential terms. The final results canbe checked to reduce to the results already derived in theappropriate limits.

There are several regimes. First, t− t ′ ∼ t ′α with α > 1.Since the second phase is long, the bond containing theinitial position is typically decimated between the twoscales Γ′ and Γ and the particle moves. If α → ∞, thetime evolution of x at t and of x′ at t ′ are decoupled. Ifα > 1, the two-time diffusion front can be scaled as

P(x, t,x′, t ′

)∼ 1

Γ4Pα

(X =

x

(T ln t)2, X̃ ′ =

x′

(T ln t)2

),

(22)where Pα

(X , X̃ ′

)= Dα

(X̃ ′)

δ(X− X̃ ′

)+ P̃α

(X , X̃

). The

first contribution comes from particles that do not movebetween scales Γ and Γ′, and the second contributioncan be calculated as detailed above. An explicit expres-sion is not provided because the inverse Laplace trans-form is very complicated. When α → ∞, we recover cor-rectly that Dα→∞

(X̃ ′)= 0 and P̃α→∞ =α2q

(α2X̃ ′

)q(X),

where the second contribution has decoupled, implyingthat the system has lost memory of the first phase [2].The second regime has t− t ′ ∼ t ′α with α < 1. In thisregime, the particles typically do not move between val-leys, but only between equilibrated minima at the bottomof a valley. Motion is therefore the result of rare eventssuch as two neighboring minima separated by a to-be-decimated barrier. These rare events contribute sublead-ing corrections to the effective dynamics of the particles.In obtaining the two-time diffusion front, we have alsoobtained the trajectory of two independent particles in thesame quenched landscape, each diffusing with a differenttemperature T . If one particle has trajectory x(t) undertemperature T , and another has trajectory x′ (t) at tem-perature T ′ < T , then the effective dynamics gives thatP(x,x′, t)∼ Pα=Γ′/Γ=T ′/T

(x

ln2 t, x′

ln2 t

). Lastly, the proba-

bility distribution can be used to calculate the momentsof the displacement x(t)− x(t ′). For example, we quotefrom Ref. [2] the mean of the displacement,

〈x(t)− x(t ′)〉 ∼(T ln t ′

)2 [a(α)+ e1−α b(α)] , (23)where a and b are polynomials of α .

III. THERMAL MEMORY IN RWRE

We now use these results to investigate two-temperatureeffective dynamics. First, we corroborate the scaling

6

found above by carrying out the RSRG procedure numeri-cally. We find that indeed, x∼ Γ2, see Fig. 3. This scalingimplies that the RWRE model has thermal memory. Ifthe system evolves for a time t ′ at temperature Tmax, theparticle will be found at position x∼ Γ2 ∼ T 2max ln2 t ′. Ifthe system then evolves to time t, where t− t ′ ∼ t ′α withα > 1, at temperature T , what is the position of the parti-cle at time T ? At large t, the particle position must againscale as Γ2 ∼ (Γ′+∆Γ) ∼ T 2 ln2 t. Since the particle isalready in a valley with barrier Γ′ ∼ Tmax ln t ′, the effec-tive dynamics will slow down to approach this scaling atlarge times. Unfortunately, we were not able to obtain anexact expression for the particle position at short timesfollowing the temperature shift. Since the RG flows inEq. 21 also applies for a two-temperature diffusion front,given more time, we would derive an expression for thedisplacement in terms of α ≡ (T ln t)/(Tmax ln t ′). Theexpression should resemble Eq. 23, and we expect thedynamics to show dependence on the ratio T/Tmax, asobserved in the experiments discussed in the introduction.

Importantly however, numerics of the actual Arrheniusdynamics on disordered landscapes do not show the pre-dicted T 2 ln2 t scaling, see Fig. 3. This is not becausethe simulations have not reached large enough times, be-cause the dynamics are consistent between short timesand long times. Nevertheless, the actual dynamics agreequalitatively with the effective dynamics of the RSRGprocedure: 1) At large t < t ′, the dynamics scale univer-sally as x∼ (ln t)y with y≈ 4.5. The differences in Tmaxcontribute only an offset. 2) After t ′, the dynamics aredifferent for the systems that have experienced differentTmax. Systems that have experienced higher Tmax showslower dynamics. For example, in systems that experi-enced T/Tmax = 0.375, the particle is trapped for a longtime after cooling, whereas in systems that experiencedT/Tmax = 0.75, the particle begins to diffuse shortly aftercooling. The dynamics after cooling do not scale with ln tor t. Given more time, it would be useful to investigatethe convergence time for the system to resume scalingwith ln t. We expect this convergence time to depend onT/Tmax.

IV. DISCUSSION

The RSRG procedure applied to RWRE allowed usto extract not only system behavior around some fixedpoint, but also effective dynamics for the particle that isexact at long times. The procedure can be applied to morecomplicated models, including random walks on finiteand biased landscapes. Other than the diffusion and two-time diffusion fronts, the procedure can also be applied tocalculate “recurrence” quantities such as the number ofreturns to the origin or first passage times that are oftenof interest for disordered systems.

The effective dynamics obtained show both aging and

thermal memory. However, numerics show discrepanciesbetween the actual dynamics and the effective dynam-ics. We therefore could not corroborate the theory withour numerical results. Further works should begin byclarifying the discrepancies, for example by investigatingwhether the found scaling depends on the underlying dis-tributions used to generate the disordered landscapes or

FIG. 3. (Top) Effective dynamics in the RSRG procedure. Nu-merics of the RSRG procedure show that particle position x,averaged over quenched disorder, indeed scales as Γ2. (Mid-dle) Simulations of actual Arrhenius dynamics (implementedvia the Gillespie algorithm which produces exact trajectoriesfor stochastic processes) on disordered landscapes show slow,logarithmic dynamics that depend on the system temperature.Before t ′ = 106, the system is at various values of Tmax. Aftert ′, the system is at T = 0.75. (Bottom) Log-log plot of x as afunction of ln t. The actual dynamics exhibit thermal memory,since the dynamics after t ′ depend not only on T , but also onTmax.

7

the underlying Arrhenius dynamics. The existence of aconceptually appealing and analytically solvable RSRGmethod nevertheless encourages investigating aging andthermal memory phenomena along this direction.

V. A SPIN GLASS MODEL

In tandem with the theoretical effort to predict thermalmemory of the glass presented in the previous sections, weconsider diffusion on a two-dimensional Ising spin glass.This is the natural generalization of the one dimensionalSinai model, and is justified as the recent experiment wasconducted on thin films [1]. In particular, we consider thespin glass Hamiltonian

−βH = ∑〈i, j〉

ai jT

σiσ j (24)

where the sum runs over all nearest neighbor bonds on thelattice and the bonds are drawn from a uniform distribu-tion. σi = 1 is to be interpreted as a particle and σi =−1as a vacancy. As a first pass, we encode the dependenceon Tmax directly in the distribution of bond energies, andassume the width is proportional to Tmax. Our goal is tounderstand the manifestation of this dependence of thebond energy distrbution on Tmax in the quantitity

〈R2(t)

〉,

where R2(t) is the squared distance from the starting po-sition of a tracer particle, and the average is computedover many tracer particles and many realizations of thedisorder.

An interesting “atomistic” approach involves fixing thedistribution of energy barriers (i.e. no dependence onTmax) and simulating the experiment directly. One wayto do this is to choose several values {T (1)max,T (2)max, . . .}.Diffusive dynamics can be simulated at a given T (i)max fora short time, and the state of the system can be storedand then used for a longer dynamic simulation at a fixedT < T (i)max for all i. Tracer particles can be tracked inthis portion of the simulation where T is identical acrosssimulations.

〈R2(t,T (i)max)

〉can be studied identically as

in the description in the previous paragraph, but P(∆E),the distribution of energy barriers, could also be computeddirectly.

This section of the paper is organized as follows. Wefirst provide a brief review of methods for simulatingglassy systems, and the inherent challenges therein. Wethen discuss our chosen method and describe its imple-mentation, commenting on its efficiency and room forextensions and optimizations. We provide results fromlonger 50×50 simulations for the model in which we as-sume the distribution of energy barriers has width given byTmax for several values of Tmax, and demonstrate that dif-fusion is slower for higher values of Tmax as is physicallyexpected. We then provide preliminary results on smaller

25×25 simulations using the “atomistic” approach. In-terestingly, we demonstrate that a dependence on Tmaxemerges naturally from the simulations.

VI. NUMERICAL BACKGROUND

Spin glasses are notoriously difficult to simulate, bothin the kinetic Monte Carlo setting and the equilibriumsetting [5–7]. The hallmark of a glassy system is a com-plex energy landscape with numerous metastable stateswith long escape times. Computationally, this implies thatstandard Monte Carlo procedures will require intractablesimulation length to capture the dynamics [5, 6]. Thecause of this phenomenon is simple: the energy barri-ers to escape a metastable well are in general very high,and so the acceptance probability in a basic Monte Carloalgorithm is disastrously low.

Significantly more progress has been made in spin glassthermalization than kinetics [5]. Algorithms that improveupon equilibration rates seek to avoid the problem of“critical slowing down” which arises in typical single spinflip Monte Carlo sampling. Some methods, such as theSwendson-Wang or Wolff algorithms, build and flip entireclusters of spins at a time, but are not immediately appli-cable to spin glasses, and in particular not to kinetic spinglass simulations [5]. More complex and recent methodssuch as parallel tempering and replica-exchange algo-rithms involve simulating and exchanging multiple copiesof the system at different temperatures or configurations.These classes of algorithms have recently enjoyed sig-nificant success in spin glass thermalization, but too areinapplicable to kinetic simulations, as they exploit cleverbut nonphysical dynamics to arrive at a thermalized finalstate [5, 8, 9].

Indeed, in some sense, the difficulty of kinetic simu-lations of glasses is unavoidable: we are interested inthe dynamics, which by definition are slow. Thermaliza-tion, on the other hand, seeks to arrive at a Boltzmann-distributed state by any means necessary, and can do so inany non-physical way so long as the result is sound. Anolder rejection-free method sometimes known as the N-fold way is essentially the only method to conduct kineticsimulations of a glassy system the authors were able tofind to date [10].

VII. NUMERICAL METHOD ANDIMPLEMENTATION

We now turn to describe the kinetic Monte Carlo ap-proach we use to simulate the diffusive dynamics, as wellas comment on its implementation, which is nontrivialand comprises roughly 950 lines of Python code (not all ofwhich is needed for the generation of the featured results).Our approach is based on the N-fold way [10].

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In the context of statistical physics, this method wasoriginally developed by Bortz, Kalos, and Lebowitz asa way to improve the speed of Monte Carlo algorithmsfor thermalization [10, 11]. The fundamental idea is toemploy a rejection-free approach so as to avoid discardedattempts that often take place for low-probability transi-tions. In switching to a rejection-free approach, particu-larly for kinetic phenomena, it is important to properlydefine time. In our case, we anticipate anomalous dif-fusion due to the uniform distribution of bond strengths.However, because spin exchanges are accepted at everyiteration of the algorithm, it is clear that in “simulationtime” where one iteration corresponds to ∆t = 1, we willsimply observe standard diffusion. Hence, it is paramountto increment time appropriately based on the availabletransition rates.

This is formalized as follows. We use a process-list ap-proach in which we enumerate all possible transitions thesystem can undergo. For an N×N grid, there are 2N×Nsuch transitions when considering spin exchanges, wherefor every particle or vacancy we need only consider thepossibility that it exchanges with the particle or vacancyabove it or to the right of it. We compute the rate for agiven transition as follows:

k = e−∆E/T (25)

i.e., we set units of time such that the jump rate for alltransitions is unity, and we consider units such that kB =1. For the case of the spin glass Hamiltonian, ∆E iscomputed simply. If we define:

envi j = ai+1, jσi+1, j +ai−1, jσi−1, j+ai, j+1σi, j+1 +ai, j−1σi, j−1 (26)

then σi j×envi j/T is the local exponent of the Boltzmannweight ascribed to σi j. We can then easily compute

∆E = σi j(envi j−envi′ j′)+σi′ j′(envi′ j′ −envi j) (27)

for a swap between σi j and σi′ j′ . Note that swaps can onlyoccur between σi j and σi+1, j,σi−1, j,σi, j+1 or σi, j−1 triv-ially because we are on a square lattice. Computationally,the shared bond will cancel in Eq. 27 and can be droppedfor roughly a factor of 14 speedup (19 floating-point oper-ations are reduced to 15).

The code is organized into three files: diffuse.py,ising.py, and union find.py. diffuse.py con-tains the necessary functions for the kinetic simulationsand ising.py contains relevant functions for settingup the simulation as well as some basic thermaliza-tion techniques using standard Monte Carlo procedures.union find.py is an efficient implementation of theUnionFind data structure (downloaded, not written by theauthors - see the comments in the code) used for imple-mentation of an interesting cluster identification algorithmultimately unneeded for the generated results.

We start with an initially random configuration ofparticles and vacancies with a specifiable concentra-tion of particles. This is created in the functioninstantiate grid() in the file ising.py. We sim-ply begin with a grid of vacancies and repeatedly convertrandom vacancies into particles until we reach the desiredparticle density. We also draw a symmetric matrix ofbond energies ai j from a uniform distribution with thebasic function draw bonds().

A linear array of transition rates conveniently namedrates is computed in the function compute rates()by looping over the grid and computing the up-switch and right-switch transition rate with the functioncompute ur rate() for every spin on the grid usingEqs. 25 and 27. For every spin (i, j), we first storethe up transition rate and then the right transition rate.Hence, the two rates for spin (i, j) can be obtained asrates[2∗ (i+N ∗ j)+{0,1}] where the curly braces in-dicate the two possibilities.

To select what transition actually occurs at a giventimestep, we proceed as follows. We create an array ofcumulative sums of the available rates. We then drawa random number uniformly from the interval (0,∑i ki)where the sum runs over all possible transition rates. Weselect the event corresponding to the first partial sumlarger than the random number so that events are cho-sen with probability proportional to the magnitude oftheir rate. To determine the grid elements involved in theswitch and the direction of switch, we use the formulal = 2∗ (i+N ∗ j)+{0,1} and a combination of modularand integer arithmetic for l the index of the partial sumchosen. The time increment ∆t is determined stochasti-cally by drawing ∆t ∼ Exp(∑i ki), i.e., we draw from anexponential distribution with rate parameter given by thesum of all transition rates for all possible transitions onthe lattice [5, 6, 11].

After each event, the available transition rates changelocally. Some thought demonstrates that the only rateswhich change are those of the next nearest neighbors ofthe particle and vacancy involved in the swap. Giventhe indices and hop direction of the particle involved,the function get nnn() returns a list of the grid indiceswhose rates need to be recomputed. This is used byupdate rates() to update only the rates which havechanged from the previous timestep. This procedure canthen be iterated: at every timestep, we pick a transitionusing the random procedure previously described, updatethe rates, and continue.

There are a few caveats that must be addressed withthe rate calculation. First, vacancy-vacancy exchangesinvolve an energy change of zero but are non-physical;hence we set such rates to zero.

Particle-particle exchanges, on the other hand, arehighly physical. In fact, in a ferromagnetic model, weexpect the formation of clusters as the diffusive dynam-ics proceeds, and within a cluster there will be many

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particle-particle exchanges. It is clear from Eq. 27 thatparticle-particle exchanges incur an energy cost of zeroand hence occur with a rate of 1. Thus, within a cluster,there will be standard diffusion even in the correctly incre-mented time. This is problematic from a simulation pointof view, as we are interested in extracting 〈R2(t)〉 for anumber of tracer particles. If we allow particle-particleexchanges, the interesting glassy dynamics which occurdue to low-probability transitions will essentially neveroccur, and our simulation (unless run for an extraordinaryamount of time) will simply yield a slight modification tostandard diffusion within the clusters.

A simple approach to correct this is to set particle-particle rates to zero, as we did for vacancy-vacancy rates.This leads to a more subtle issue: in all likelihood, a tracerparticle will end up in a cluster at some point, or perhapseven start within a cluster. If we neglect particle-particletransitions, the only way for a tracer particle to escapethe interior of a cluster is for the cluster to shatter; thisis nonphysical, and can take an extreme amount of time.In reality, the particle will diffuse throughout the clusterrapidly on the timescale of the glassy transitions. Hence,it will spend time on the boundary, and can easily escapeby hopping to a vacancy while it is on the boundary.

We handle this using what we deem a “well-mixed”approximation. We set all particle-particle transition ratesto be equal to zero. However, for every switch, we searchfor and identify the cluster to which the hopping particlebelongs. We uniformly select from the cluster the particlethat made this transition, and record the modified vectordistance update if this uniformly selected particle is one ofthe tracer particles, as well as update its current position.The tracking of and updates to the tracer particles canbe found in the function diffuse lt samp(). Note thatthis procedure is an approximation; in reality, it wouldbe necessary to solve a diffusion equation on the domainof the cluster. We expect that this approximation is un-faithful to the true system dynamics but will not destroya dependence on Tmax, and in future work it would beinteresting to identify the effect of this approximation andunderstand how it can be improved.

The cluster identification is implemented using a basicbreadth-first search on the lattice computed by the func-tion identify cluster(). An alternative method, theHoshen-Kopelman algorithm, is implemented in the func-tion hoshen kopelman() using the attached UnionFinddata structure implementation in union find.py. Whilethe Hoshen-Kopelman algorithm is a fast way to identifyall clusters on the grid, it is clear that two or three clusterscan be shattered or joined with a single exchange. Be-cause of this fact, there is no simple way to precomputeall clusters on the grid and modify them on the fly; toidentify the cluster to which a particle belongs, the clustermust be be recomputed on each iteration. In the worstcase, if there were a system-spanning cluster, the breadth-first search would take asymptotically as much time as

the Hoshen-Kopelman algorithm, and hence BFS is moreefficient on average. The Hoshen-Kopelman algorithm isleft included for possible extensions to this work.

Even accounting for particle-particle and vacancy-vacancy swapping as described above, we still encounterthe so-called “low-barrier problem” [6, 11]. At everytimestep, there are consistently several transitions withrates orders of magnitude higher than all others rates thatappear in the problem. Left alone, the presence of thesehighly probable transitions leads to repeated switchingback and forth between several states of the system. Inother words, the system becomes trapped in a “superbasin”[7, 11]. We take the simplistic approach of artificially rais-ing these lower barriers (lowering the rates) according toa certain specifiable rate threshold. This approximationshould be valid, as we expect that these fast transitionswill equilibrate on the timescale of interest in the problem[11]. Nevertheless, it would be interesting to probe theeffects of thresholding by using more advanced methods,such as those based on hashing the system state to preventrepeatedly revisiting the same states, or those based ontransition matrix diagonalization to escape superbasins[11].

Finally, we discuss the method used to track tracers.We randomly select n tracers tracers whose trajectorieswe track over the course of a simulation. To do so, wekeep track of R2i (t) where the index i runs over the tracers.We store only the current vector (xi,yi) distance from thetracer’s starting location but the entire R2i (t) trajectory.This distance is incremented carefully, accounting forperiodic boundary conditions and the larger jumps due tohopping out of a cluster. There is a subtle issue that ariseshere when considering our goal of averaging over thedisorder across simulations. Because the time incrementis stochastic, the time points at which the trajectory isstored will not be consistent across simulations. Thismakes averaging over the disorder challenging a-priori.

This is alleviated as follows. We discretize the loga-rithm of time into points log(t) j (note the index is out-side the argument of the logarithm). Say tracer parti-cle l’s location was last stored at time ti and say thatlog(t)k = max j (log(t) j) such that log(t) j < log(ti). Wethen set R2l (log(t)k) = R

2l (ti). We ensure the discretiza-

tion is fine enough such that log(t)k+1 < log(ti+1) whereti+1 is the next time any tracer particle moves (i.e., thereare many discretization points between stochastic timeincrements at which we record data). If tracer parti-cle l′ is the particle that moves at ti+1, we find the k′

such that log(t)k′ < log(ti+1) < log(t)k′+1. We fill inthe trajectories of all other tracer particles m 6= l′ suchthat R2m(log(t)q) = R

2m(log(t)k) for all k < q ≤ k′. For

tracer particle l′, we do the same for all q such thatk < q < k′, and set R2l′(log(t)k′) = R

2l′(ti+1). In this

way, we ensure that all trajectories across all simula-tions are sampled at identical time points, and hencecan average directly. This is all handled in the function

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diffuse tracers lt samp().

VIII. RESULTS AND DISCUSSION

The model in Eq. 24 was simulated using the codeand methods described in the previous section to under-stand the dependence of the disorder and tracer parti-cle averaged 〈R2(t)〉 on Tmax. Individual realizationsof the disorder were simulated in parallel using themap and Pool functionalities provided by the Pythonmultiprocessing library. Simulations were run onThinkMate workstations with 2016 Intel Xeon proces-sors. Each workstation was able to simulate forty realiza-tions of the disorder simultaneously and simulations werespread over three workstations.

Due to the cluster search, scaling with system size canbe very poor; we hence restrict ourselves to an antiferro-magnetic model to minimize the size and formation ofclusters. We also expect the “well mixed” approximationto be more valid in the antiferromagnetic case for thesereasons.

Simulations in the Tmax-dependent model (ai j drawnuniformly over the interval (−Tmax,0)) were conductedon a 50× 50 grid of spins with a particle density ofρ = .75 and periodic boundary conditions in both dimen-sions. Diffusive dynamics were simulated at a similarlyarbitrary T = .75 in each case. Each simulation was al-lowed to run for 50 Monte Carlo steps (MCS), or stepsper particle, totalling 503 = 125,000 steps per simulation.Optimizations that enable scaling up to larger system sizesto understand finite size effects are of significant interestto the authors for future work.

1625 tracer particles were tracked per realization ofthe disorder. Tracer particle trajectories were averagedin addition to the disorder average, leading to a total of65,000 trajectories per value of Tmax. Each realizationof the disorder took roughly 12 hours of wall time tocomplete. Results for five values of Tmax = .75,1,2,3,7.5are included in Fig. 4. Note that higher values of Tmax leadto slower diffusion (smaller 〈R2(t)〉). Assuming slowerdiffusion would lead to slower relaxation rates after aperturbation, this precisely corroborates the hypothesis ofthe experiment.

The trajectory differences apparent in Fig. 4 are mostpronounced at long times. This makes intuitive sense;ultimately, the larger barriers for higher Tmax prevent dif-fusion of the tracer particles to far distances even for longtimes, while the lower barriers for lower values of Tmaxeventually enable escape.

We present some preliminary results from the “atom-istic” approach in Fig. 5. Due to time constraints, sim-ulations were conducted on a 25× 25 grid. 437 tracerparticles were followed per simulation with again 40 real-izations of the disorder simulated in parallel per value ofTmax, corresponding to 17480 tracer particles per value of

FIG. 4. Depiction of the dependence of 〈R2(t)〉 on Tmax inthe model where Tmax is treated as the width of the uniformdistribution from which the bond energies are drawn. Resultswere computed on a 50×50 grid. There is a clear qualitativedifference in the trajectories for different values of Tmax, in thatdiffusion is slower for higher Tmax.

Tmax. Simulations were allowed to proceed for 20 MCSat Tmax. The state of the system was then saved and sim-ulated for 100 MCS at T = .5 where diffusive data wasrecorded. Each realization of the disorder took roughlyone hour of wall time on the same 2016 Intel Xeon CPU.The width of the uniform distribution was taken to be 5 inall cases (ai j ∈ (0,5)). The particle density is ρ = .75. In-terestingly, a Tmax dependence emerges organically fromthe simulation. Note that the dependence of 〈R2(t)〉 looksmore like a power law of log(t) in these smaller simula-tions than in the 50×50 simulations. It is unclear if thisis a finite size effect or if the “curling away” present inFig. 4 at long times would look similar if carried out foreven longer times. This needs to be investigated further infuture work with optimizations that enable longer simula-tions and larger systems. More discussion can be foundin the caption of Fig. 5.

IX. CONCLUSIONS AND FUTURE DIRECTIONS

Via kinetic Monte Carlo methods, we demonstrateda dependence of diffusion in the two dimensional Isingspin glass on the maximal temperature the glass has beenexposed to. In the first case, we showed that strongerbond strengths on average, as expected, lead to verifiablydifferent trajectories of 〈R2(t)〉. We then showed that onecan produce a Tmax-dependent 〈R2(t)〉 trajectory directlyfrom the kinetics of the problem by allowing the systemto dynamically evolve at a fixed Tmax and then computingthe trajectory over some T < Tmax in analogy to the ex-periment of Ref. [1]. Together, these simulations provide

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FIG. 5. Depiction of the qualitative dependence of 〈R2(t)〉 onTmax in the “atomistic” approach where the distribution of bondenergies is held fixed, but the system is allowed to equilibrateat a higher temperature before the dynamics are tracked at aconsistent lower temperature. These are preliminary results froma small 25×25 grid. Trajectories scale as a power law in log(t).Curve fitting provides 〈R2(t)〉 ∼ log(t)8, which is interesting tocompare to the theoretical one dimensional result of x∼ log(t)2.However, due to the small system size, it is unclear if this lawshould be trusted. Note that there is a clear difference in thetrajectories for different values of Tmax, and that this emergesnaturally from the simulation.

further support for the hypothesis that exposing the glass

to higher values of Tmax allows for the control of the levelof equilibration of the glass, and hence the strength of thebonds on average.

The majority of this work involved developing the codeto be able to conduct these simulations. Now that thecode has been developed and the expected dependence onTmax realized, many possibilities are open. Future workinvolves optimizations for scaling to larger system sizes,categorizing the dependence on Tmax when it enters ex-plicitly as a parameter, understanding the effects of ratethresholding and exploring more advanced methods toovercome the low barrier problem, considering the effectsof the choice of distribution of energy barriers, comparingantiferromagnetic and ferromagnetic models, and under-standing the effects of ρ and T on the results. Finally,it is very interesting to explore the “atomistic” approachfurther, particularly on larger grids, and see if a quanti-tative dependence on Tmax can be extracted from somerelevant quantity computable from the simulation. In theideal case, the atomistic approach can be combined witha method to introduce a perturbation in the system, fromwhich relaxation can be measured as in the experiment.

X. AUTHOR CONTRIBUTIONS ANDACKNOWLEDGMENTS

NB focused on the 2D Ising spin glass model andMonte Carlo simulation methods. PH focused on RWRE.Both authors discussed both aspects of the project to-gether and wrote the paper equally. The authors thankMehran Kardar, Ariel Amir, and Chris Rycroft for impor-tant discussions.

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