Reentrance in an active spin glass model
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Reentrance in an active spin glass model
Kevin R. Pilkiewicz and Joel D. Eaves ∗ Department of Chemistry and Biochemistry, University of Colorado, Boulder (Dated: September 14, 2018)Active matter, whose motion is driven, and glasses, whose dynamics are arrested, seem to lie atopposite ends of the spectrum in nonequilibrium systems. In spite of this, both classes of systemsexhibit a multitude of stable states that are dynamically isolated from one another. While thisdefining characteristic is held in common, its origin is different in each case: for active systems,the irreversible driving forces can produce dynamically frozen states, while glassy systems vitrifywhen they get kinetically trapped on a rugged free energy landscape. In a mixture of active andglassy particles, the interplay between these two tendencies leads to novel phenomenology. Wedemonstrate this with a spin glass model that we generalize to include an active component. In theabsence of a ferromagnetic bias, we find that the spin glass transition temperature depresses withthe active fraction, consistent with what has been observed for fully active glassy systems. When abias does exist, however, a new type of transition becomes possible: the system can be cooled outof the glassy phase. This unusual phenomenon, known as reentrance, has been observed before ina limited number of colloidal and micellar systems, but it has not yet been observed in active glassmixtures. Using low order perturbation theory, we study the origin of this reentrance and, based onthe physical picture that results, suggest how our predictions might be measured experimentally.
I. INTRODUCTION
Active systems, those whose particles exhibit exter-nally driven or self-propelled motion, challenge standarddescriptions of matter. While a growing amount of ev-idence from both simulations and experiments suggeststhat the dynamical structural transitions observed in ac-tive systems bear more than just a superficial resem-blance to the thermodynamic phase transitions of sys-tems at equilibrium,[1–9] the microscopically irreversibledynamics that drive these transitions can lead to a statespace comprised of many similar steady-state configura-tions that are dynamically estranged from one another.This sort of configurational landscape is also observed inglasses, though there it is achieved through a differentmechanism: kinetic trapping on a corrugated free energysurface.In this paper we study a mixture of active and glassyparticles to probe what transpires when these disparatemechanisms compete and interact. Fig. 1 illustrates howsuch a mixture differs from a glassy system in which allthe particles are active. In a fully active system, everyparticle will have its average energy increased by the ex-ternal driving, making it possible, in many cases, to mapthe active system onto its passive counterpart through aneffective temperature (Fig. 1( a )).[10–13] This suggeststhat so long as the driving is not too excessive, the samethermodynamic phases will be observed, only at lowertemperatures (or higher densities), a conclusion that hasbeen borne out in a number of studies.[13–16]When only a fraction of the system is active, however,the uneven distribution of energy will stabilize some con-figurations of the system while destabilizing others (Fig. ∗ [email protected] b )). This has the potential to radically alter the sys-tem’s phase diagram and lead to new physical phenom-ena. Consistent with this expectation, fully active sys-tems in which a fraction of the particles have a highermotility have been observed to exhibit novel patterns ofphase separation.[17, 18]The glass forming system we study in this paper isa generalization of the mean field Ising spin glass, alsoknown as the Sherrington-Kirkpatrick (SK) model. Inthe appropriate limits, fractional activation can be ap-proximated as fractional annealing, and we show howthis annealing modifies quantities like the free energy andthe magnetization. After examining how these modifica-tions alter the familiar SK phase diagram, we demon-strate that while some phase boundaries on the diagrammerely shift or elongate, others change more drasticallyand allow transition pathways between phases that werenot possible in the fully quenched model. Most notableamong these is a reentrant spin glass transition[27] inwhich the spin glass can be cooled into a ferromagnet andthen back to a spin glass as the temperature is loweredat fixed ferromagnetic bias. This behavior is similar incharacter to what has been observed in some colloid andmicellar systems[19–22] as well as numerous simulatedsystems.[23–26] A perturbation theory argument revealsthe physical origin of this phenomenon, and we show,to leading order, that the effect of the fractional activa-tion on the free energy landscape is consistent with thephysical picture in Fig. 1( b ). The fundamental mecha-nisms uncovered by this analysis extend beyond the spe-cific magnetic interactions studied here, and we concludewith a discussion of how our results can be generalizedto more complex systems and how such a system mightbe studied experimentally. k B T F ree E n er gy Configurational Coordinate ( a )( b ) F ree E n er gy Configurational Coordinate k B T eff FIG. 1. Full versus fractional activation. ( a ) A schematic rep-resentation of the free energy landscape for a glassy system.The black dot denotes a reference low energy configuration ofthe system, and the black dashed lines delineate the amountof thermal energy ( k B T ) available to the system before ac-tivation and the larger amount of effective thermal energy( k B T eff ) available afterwards due to the driving of the sys-tem. Prior to activation, the system is kinetically trapped inthe indicated minimum, but activation postpones this trap-ping to lower temperatures. ( b ) The same free energy land-scape before (solid black curve) and after (dashed red curve)fractional activation of the system. The uneven distributionof energy in the system stabilizes some states while destabi-lizing others. II. THE MODEL
We start with the standard Sherrington-Kirkpatrickmodel, which consists of N Ising spins interacting ac-cording to the following Hamiltonian. H = − X ( ij ) J ij S i S j − h X i S i (1)In the above, the Ising spin variables S i can only takevalues of +1 or − h is an external magnetic field, andthe first sum is over all N ( N − / J ij are chosen from a Gaussiandistribution. P ( J ij ) = (cid:18) N πJ (cid:19) / exp (cid:20) − N ( J ij − J /N ) J (cid:21) (2)The usual motivation for these random couplings isthat the strength and sign of the exchange interaction, J ij , varies as a function of the distance between eachpair of spins. In a disordered material, these distances will be stochastic, so, for a sufficiently large system, onecan approximately select the coupling constants for theseinteractions from the distribution in equation (2). For agiven sample, the J ij do not change and thus are consid-ered “quenched” interactions, but when the free energyof the whole system is computed, it must be averagedover all realizations of the coupling constants.We generalize this model to allow for a fixed fraction, µ , of the spins to become active. We imagine there is anexternal driving force coupled to these spins, as well as africtive force that keeps the system in a steady state. Inthe limit of strong activation, the exchange couplings ofthe active spins will fluctuate on time scales that are shortcompared to the spin relaxation time of the quenched de-grees of freedom, so the steady state of this fractionallyactive system may be approximated by the thermal equi-librium of a fractionally annealed system. We term theresulting model the “fractionally annealed Sherrington-Kirkpatrick” (FASK) model. A pictorial representationof this model is shown in Fig. 2( a ). Fig. 2( b ) empha-sizes the basic similarities between our model and a morerealistic fractionally active glass former, discussed at theend of the paper. ( a )( b ) FIG. 2. Depiction of the model. ( a ) A simple pictorial rep-resentation of the FASK model, with the quenched, inactivespins drawn as blue circles and the active spins drawn as redcircles. The arrows inside each circle indicate the particle’sspin state, and the motion of the active spins is depicted asmotion blur. ( b ) A pictorial representation of a potential ex-perimental system that would behave as a fractionally activeglass former. The blue spheres represent silica beads, the halfred, half white spheres represent silica beads that are halfcoated in platinum, and the light blue background representshydrogen peroxide solvent. Arrows indicate the direction ofself-propulsion for the active colloid particles. To construct the partition function for this system, wefirst divide the pairs of spins into two non-intersectingsets: a set with annealed interactions
A ≡ { ( ij ) | i =1 , ..., µN, j > i } and a set with quenched interactions Q ≡ { ( ij ) | i = µN + 1 , ..., N, j > i } , where we havenumbered the active spins with labels 1 through µN , andthe passive spins with labels µN + 1 through N . Thisdivision allows us to rewrite the first sum on the right ofequation (1) as X ( ij ) J ij S i S j = X ( ij ) ∈A J ij S i S j + X ( ij ) ∈Q J ij S i S j A similar factorization of the product over ( ij ) allows usto write down the desired partition function. Z µ = tr S Z Y ( ij ) ∈A (cid:18) N / dJ ij (2 πJ ) / (cid:19) exp X ( ij ) ∈A (cid:18) βJ ij S i S j − N ( J ij − J /N ) J (cid:19) × exp X ( ij ) ∈Q βJ ij S i S j + βh X i S i The trace in this expression is over the 2 N distinct spinconfigurations of the system. After performing the Gaus-sian integrals over the annealed interactions, this parti-tion function can be reduced to the following form. Z µ = exp "(cid:18) βJ (cid:19) µ (2 − µ ) N × tr S exp β X ( ij ) ∈Q ( J ij − J /N ) S i S j + β ( J /N ) X ( ij ) S i S j + βh X i S i It is important to note that although we are treating thisactive system as if it were at thermal equilibrium, for µ > f , in the FASKmodel, averaged over the quenched interactions, can becomputed using the usual replica trick.[30, 31] Since wewill primarily be concerned with phase boundaries, it issufficient to evaluate the free energy within the assump-tion of replica symmetry. The derivation proceeds simi-larly to that of the standard SK model free energy,[31, 32]so only the final result will be shown here. − βf = (cid:18) βJ (cid:19) (cid:2) (1 − q µ ) + 2 µq µ (cid:3) − βJ M + 1 − µ (2 π ) / Z ∞−∞ dz e − z ln [2 cosh η ( z )]+ µ ln [2 cosh β ( J M + h )] (3) In the above, η ( z ) = β (cid:16) Jq / µ z + J M + h (cid:17) and the or-der parameters q µ and M are defined through the follow-ing self-consistency relations. q µ = 1 − µ (2 π ) / Z ∞−∞ dz e − z tanh η ( z ) M = 1 − µ (2 π ) / Z ∞−∞ dz e − z tanh η ( z )+ µ tanh β ( J M + h ) (4)In the extreme cases of µ = 0 and µ = 1, equation (3)reduces, as required, to the familiar results of the fullyquenched SK model and the fully annealed mean fieldIsing model, respectively.Differentiating equation (4) with respect to the field h and taking the limit h → χ M = 1 − qk B T − J (1 − q ) (5)This expression is identical to that obtained for the usualSK model, except that now the overlap order parameter q is defined as follows. q = q µ + µ tanh β ( J M )These results are all for the replica symmetric solu-tion of the free energy. The validity of this solution isdetermined by the following stability condition, which isanalogous to that found by de Almeida and Thouless[33]for the SK model.( βJ ) (1 − µ )(2 π ) / Z ∞−∞ dz e − z sech η ( z ) < III. RESULTS
For convenience, we will use reduced units for the re-mainder of the paper where temperature is scaled by k B /J and all energies are given in units of J .In the T - h plane, there is a single phase transition oc-curring at h = 0 between a paramagnetic phase ( q = 0, M = 0) and a spin glass phase ( q = 0, M = 0). Thespin glass transition temperature, T f , can be computedas a function of the active fraction µ by finding the tem-perature at which equation (6) becomes an equality for q µ , J , and h all set to zero. The result is plotted in Fig.3( a ). T f = p − µ In their treatment of a fully active spin glass system,Berthier and Kurchan[13] found a roughly linear rela-tionship between the magnitude of their driving forceand the depression of their glass transition temperature,and though our result becomes highly nonlinear as µ ap-proaches unity, for µ less than roughly 0 .
5, a linear fitis very good (see Fig. 3( a )). For small to moderateamounts of activation, the shift of the paramagnetic tospin glass transition temperature is qualitatively similarregardless of whether the whole system gets partially an-nealed or one fraction of it gets fully annealed. Active Fraction ( (cid:1) ) T r a n s i t i o n T e m p er a t u re ( T f ) Temperature (T) E x t er n a l F i e l d ( h ) ( a ) ( b ) FIG. 3. Phase diagram in the T − h plane. ( a ) The spin glasstransition temperature T f plotted versus the active fraction µ . The black line represents a best fit for the curve up to µ = 0 .
5. The slope of this line is roughly 0 .
58, a little largerthan what one would get from a linear Taylor expansion about µ = 0. ( b ) The Almeida-Thouless stability line in the T - h plane, plotted for active fractions µ = 0 (red), µ = 0 . µ = 0 .
50 (blue), and µ = 0 .
75 (purple). The insetshows that these curves all collapse onto the µ = 0 mastercurve when the temperature and field are both scaled by afactor of (1 − µ ) − / . We can go further and use the stability condition ofequation (6) to plot the entire Almeida-Thouless (AT)stability line for different values of µ . The results areshown in Fig. 3( b ). While the entire curve is shifted tolower temperatures with increasing active fraction, theamount each point gets shifted decreases with increasingfield due to all the curves converging towards infinite fieldas T →
0. If one scales the temperature by a factor ofone over T f , it is clear that each of these curves willcross the temperature axis at T = 1, but, interestingly, ifthe external field is also scaled by that same factor, thecurves for different µ all collapse onto the fully quenchedcurve (see the inset of Fig. 3( b )).The FASK model phase diagram is much richer in the J - T plane, because in addition to a paramagnetic phaseand a spin glass phase with M = 0, there is also a ferro-magnetic phase and a spin glass phase with M = 0, oftenreferred to as a mixed phase.[35] The boundary betweenthe region of the phase diagram with M = 0 and thatwith M = 0 can be determined by finding where the sus-ceptibility diverges. Using equation (5), one finds thatthe Curie temperature T c is given as a function of J bythe following relation. T c = J (1 − q µ ( T c ))Note that when q µ = 0 (in the paramagnetic phase),the above simplifies to T c = J . The remaining phase boundaries can be found by using the stability conditionof equation (6). An example of the phase diagram thatresults from these considerations is shown in Fig. 4( a ),for µ = 0 .
50. The replica symmetric phase diagram ofthe fully quenched model is plotted in light gray for com-parison. ( a )( b ) PM FMFSGSG Reentrant
FIG. 4. Phase diagram in the J - T plane. ( a ) The FASKmodel phase diagram plotted in the J - T plane for µ = 0 . µ = 0). The labels PM, FM, SG, andFSG refer to the paramagnetic, ferromagnetic, spin glass, andferromagnetic spin glass phases respectively. The shaded redregion gives the range of J for which reentrance is possible.( b ) Plots of the FASK model phase diagram for µ = 0 . µ = 0 .
50 (blue), and µ = 0 .
75 (purple). In eachphase diagram, the horizontal line is given by T f = √ − µ (see Fig. 3( a )). As µ approaches one, the curve separatingthe two spin glass phases approaches the line T = J , and theregion where reentrance can occur increases in size. A non-zero active fraction causes the paramag-netic/spin glass transition line to shift to a lower temper-ature T = √ − µ and terminate at a lower value of J ,also equal to √ − µ . The paramagnetic/ferromagnetictransition line is still the curve T = J , but now it ter-minates at the point (cid:0) √ − µ, √ − µ (cid:1) instead of (1 , µ <
1, whosearea grows linearly with µ . Specifically, for an increasein active fraction equal to ∆ µ , this area changes by(1 / µ .The impact of a non-zero active fraction on the lowtemperature region of the phase diagram is more dra-matic. For µ >
0, the spin glass/ferromagnetic spin glassphase boundary bends in the opposite direction, connect-ing the points (cid:0) √ − µ, √ − µ (cid:1) and (0 , J >
0. Though the replicasymmetric solution is not valid in this region of the phasediagram, the phase boundary it predicts does approachthe line T = J as µ →
1, as physically required, soit is likely to be at least qualitatively correct. The ATline separating the ferromagnetic and ferromagnetic spinglass phases also changes shape for µ >
0, bending in onitself to create a reentrant region where it is possible, justby lowering the temperature, for the system to transitionfrom a spin glass to a ferromagnet back to a spin glass.In most systems with a reentrant glass transition,repulsive interactions dominate the higher temperatureglass phase while attractive interactions dominate in thelower temperature glass.[19–22] If, as in a lattice gas,[36]one views antiferromagnetism and ferromagnetism as re-pulsion and attraction, respectively, then the same basicphenomenology holds in the FASK model. The initialspin glass formation is driven by antiferromagnetic in-teractions that compete with the ferromagnetic bias tocause frustration, while the reentrant spin glass is char-acterized by some degree of ferromagnetic order, a resultof the more prevalent interactions winning out at lowtemperature. The range of J over which reentrance canoccur is shown as a shaded region in Fig. 4( a ), and, inFig. 4( b ), a side-by-side plot of the FASK model phasediagram for several values of µ reveals that this rangegrows with increasing active fraction.We can better understand the origin of reentrance inthis model by performing a perturbative analysis of themagnetization, similar to that used to derive the Bornapproximation in quantum mechanics. Equation (4) canbe rewritten as M = (1 − µ ) M q + µM a , where M q isthe expression for the magnetization of a fully quenchedsystem ( µ = 0) and M a is the corresponding expressionfor a fully annealed system ( µ = 1). The mobile andimmobile spins both contribute to the total magnetiza-tion proportionally to their fractional composition of thesystem, though these contributions are coupled by theirmutual dependence on the same total magnetization M .If we were to ignore this coupling, a zeroth order approx-imation to the total magnetization would be M ( J , T ) ≈ (1 − µ ) M ∗ q ( J , T ) + µM ∗ a ( J , T ) , (7)where M ∗ q and M ∗ a are the magnetizations that a purequenched and pure annealed system would have, respec-tively, at the given values of J and T .Inserting the zeroth order solution back into the righthand side of equation (4) for h = 0, one obtains thefollowing result. M ≈ − µ (2 π ) / Z ∞−∞ dz e − z tanh " q / µ z + J M ∗ q + µh eff T + µ tanh (cid:20) J M ∗ a − (1 − µ ) h eff T (cid:21) (8)In the above, we have defined an effective magnetic field as follows. h eff ≡ J ( M ∗ a − M ∗ q ) (9)The interpretation of this result is as follows. Reentranceis only observed when J < T < J , in which case M ∗ q = 0 and M ∗ a = 0. The inactive component of thesystem thus feels, to leading order, an effective magneticfield from the active component that can cause it to alignout of the spin glass phase into a ferromagnet. The factthat h eff is proportional to µ in the first term on the righthand side of equation (8) also explains why increasingthe active fraction broadens the range of J over whichreentrance occurs.If the system is fully annealed, we can recast the Hamil-tonian using the Weiss form of mean field theory.[37] H = − J N X i =1 M ∗ a S i In the above, we have neglected the term that comes fromintegrating over the annealed degrees of freedom, since atfixed T and µ it is just a constant. Expanding about thissolution by replacing M ∗ a with the zeroth order approxi-mation in equation (7), we get the following approximateresult. H ≈ − X ( ij ) J ij S i S j − µh eff N X i =1 S i (10)In the above, h eff is the same as in equation (9), and allcoupling constants are quenched. Equation (10) suggeststhat for µ close to unity, the system looks, to leadingorder, like a fully quenched system in the presence ofan effective magnetic field. This field selectively stabi-lizes configurations of the system that have a net align-ment with it and destabilizes those that align againstit, consistent with the physical picture depicted in Fig.1( b ). The difference M ∗ a − M ∗ q is largest for T < J and J <
1, which is precisely where the phase diagram ofthe FASK model differs most strikingly from that of thefully quenched system.
IV. DISCUSSION
In the mean field Ising spin glass, activating a fractionof the system gives rise to new physical phenomena–mostnotably a reentrant transition from the spin glass phaseto the ferromagnetic phase. The origin of this reentrantbehavior lies in the fact that the active component willstart to magnetize at low temperatures, generating a lo-cal magnetic field that can, for a certain range of J ,overpower the frustrated interactions of the passive spinsand induce a net magnetization in the entire system.One can imagine reentrance occurring in other systemsthrough a parallel mechanism. In a fractionally acti-vated glass forming liquid, for example, the active parti-cles would be harder to vitrify than the passive particles,leading to a glass phase with pockets of active particlesin a liquid-like state. It is conceivable that for a limitedrange of densities, these pockets could transfer enough oftheir driven energy to the surrounding passive particlesto break them out of their cages and cause reentranceto the liquid phase. This effect will be enhanced if analigning mechanism is present, in which case the activeparticles will tend to exhibit cooperative motion.While it is easy enough in theoretical treatments toleave the task of selectively activating a fraction ofthe system to some Maxwellian mephisto, designing apractical experimental method for accomplishing thistask is more difficult. Recent experimental work has shown that silica particles, a well-known colloidal glassformer,[38, 39] half coated in platinum undergo self-propelled motion in hydrogen peroxide.[40] A dense col-loidal suspension of silica particles in which only a frac-tion were so coated might therefore be viable as a frac-tionally active glass forming system (see Fig. 2( b )).Simple spin glass models have led to many insightsinto the nature of the glassy state, and these conceptsand tools have had applications in fields as distinct asprotein folding and neurosicence. In the nascent field ofactive glass formers, spin glass models will likely continueto play a key role. [1] H.J. Bussemaker, A. Deutsch, and E. Geigant, Phys. Rev.Lett. L99 (1999).[3] A. Czir´ok, A-L. Barab´asi, and T. Vicsek, Phys. Rev. Lett.
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