Aging, memory, and nonhierarchical energy landscapeof spin jamAnjana Samarakoona, Taku J. Satob, Tianran Chena, Gai-Wei Cherna, Junjie Yanga, Israel Klicha, Ryan Sinclairc,Haidong Zhouc, and Seung-Hun Leea,1

aDepartment of Physics, University of Virginia, Charlottesville, VA 22904; bInstitute of Multidisciplinary Research for Advanced Materials, Tohoku University,Katahira, Sendai 980-8577, Japan; and cDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996

Edited by Zachary Fisk, University of California, Irvine, CA, and approved September 2, 2016 (received for review May 23, 2016)

The notion of complex energy landscape underpins the intriguingdynamical behaviors in many complex systems ranging frompolymers, to brain activity, to social networks and glass transitions.The spin glass state found in dilute magnetic alloys has been anexceptionally convenient laboratory frame for studying complexdynamics resulting from a hierarchical energy landscapewith ruggedfunnels. Here, we show, by a bulk susceptibility and Monte Carlosimulation study, that densely populated frustrated magnets in aspin jam state exhibit much weaker memory effects than spinglasses, and the characteristic properties can be reproduced by anonhierarchical landscape with a wide and nearly flat but roughbottom. Our results illustrate that the memory effects can be used toprobe different slow dynamics of glassy materials, hence opening awindow to explore their distinct energy landscapes.

memory | energy landscape | spin jam | spin glass

If the energy landscape of a system resembles a smooth vasewith a pointy bottom end, upon cooling the system goes quickly

into the lowest energy state, i.e., the global ground state, that isusually associated with crystalline order. If the energy landscapeis more complex with many metastable states, i.e., local minima,then cooling may lead the system into local minima resulting in aglassy order. The concept of such energy landscapes has beeninstrumental in explaining the glassiness that is ubiquitous in awide range of systems, including atomic clusters (1), structuralglasses (2, 3), polymers (4), brain activity (5), and social networks(6). Several different topological types of energy landscapes wereproposed to characterize different glassiness and the associatedslow dynamics (7, 8). For instance, a rugged funnel-shaped land-scape shown in Fig. 1A was proposed to understand the physics ofbiopolymers (9, 10) and dilute magnetic alloys called spin glass (11).Magnetic glass systems (12–16) present a unique opportunity

to microscopically study the relation between the energy land-scape and low temperature properties. The most studied mag-netic glass state is the conventional spin glass realized in dilutemagnetic alloys such as CuMn and AuFe. Here, dilute magneticions (Mn and Fe) in a nonmagnetic metal interact via the long-range Ruderman–Kittel–Kasuya–Yosida (RKKY) interactionwhose magnitude and sign change with distance between therandomly placed magnetic ions (11). The randomness drives thesystem into the spin glass state below a critical temperature, Tf ,that is comparable to the mean-field magnetic energy scale, i.e.,the absolute value of the Curie–Weiss temperature, jΘCW j. Forinstance, for Cu − 2 atomic (at.) % Mn (CuMn2% hereafter),ΘCW =−45 K and Tf = 15.5 K. Another distinct glassy state calleda spin jam has been recently suggested to appear in denselypopulated frustrated magnets (17–21). At the mean-field level,these systems are expected to remain in a classical spin liquiddown to absolute zero temperature, due to macroscopic classicalground state degeneracy. Quantum fluctuations, however, lift thedegeneracy and lead the system to the spin jam state below Tf thatis much lower than jΘCW j. For instance, for SrCr9pGa12-9pO19(SCGO)(p = 0.97), ΘCW =−500 K and Tf = 3.8 K. The starkdifferent ratios of jΘCW j=Tf for the spin jam and spin glass

suggest that the two states might have qualitatively distinct energylandscapes.Aging and memory effects have been key features of glassy

systems due to the intrinsic slow dynamics. The thermo-remanentmagnetization (TRM) method is the most effective way so farto investigate these effects (12–14); for the measurements, thesample is first cooled down from well above Tf to base tem-perature with a single stop for a waiting time, tw, at an in-termediate temperature Tw, under zero field. While waiting atTw, if the system has as many nearly degenerate metastablestates at low energies as spin jam and conventional spin glasseshave, the system will relax to the accessible lower energy statesthan when no waiting is imposed. The longer tw is, the lowerenergy states the system will relax to, which is called “aging.”Once cooled down to base temperature, the TRM is measuredby applying a small field of a few gauss upon heating at aconstant rate. During the measurements, when the temperatureapproaches the temperature of aging, Tw, the system revisits thelower energy states reached during the wait time that are asso-ciated with the energy scale of kBTw, where kB is the Boltzmannconstant. Upon further heating, the system goes to higher energystates allowed within kBT. This is referred to as the aging andmemory effect.We have performed the TRM measurements on two spin

jam prototypes, SCGO(p = 0.97) and BaCr9pGa12-9pO19(BCGO)(p = 0.96) in which the magnetic Cr3+ (3d3) ions form ahighly frustrating quasi-2D triangular network of bipyramids(17–21) and a spin glass prototype CuMn2% in which the 2% lowconcentration of the magnetic Mn atoms is embedded in thenonmagnetic Cu metal. Strong aging and memory effects havebeen observed in CuMn2%, whereas the effects are much weakerin SCGO and BCGO. Fig. 2 A–C shows the TRM data obtainedfrom SCGO(p = 0.97), BCGO(p = 0.96), and CuMn2%, re-spectively, with several different values of tw ranging from 6 minto 100 h, at Tw=Tf ∼ 0.7. All samples exhibit similar aging andmemory effects that increase with increasing tw. These indicate

Significance

Our bulk susceptibility and Monte Carlo simulation study ofaging and memory effects in densely populated frustratedmagnets (spin jam) and in a dilute magnetic alloy (spin glass)indicates a nonhierarchical landscape with a wide and nearlyflat but rough bottom for the spin jam and a hierarchical ruggedfunnel-type landscape for the spin glass.

Author contributions: S.-H.L. designed research; A.S. and T.J.S. designed and performedexperiment; T.C. and G.-W.C. designed and performed MC simulations; A.S., J.Y., R.S., andH.Z. provided samples; and S.-H.L., G.-W.C., and I.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: shlee@virginia.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1608057113/-/DCSupplemental.

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the existence of numerous metastable states and slow dynamicsin all systems. Despite the similarity, there is a clear difference:For the CuMn2% magnetic alloy, considerable aging occurs atTw even for a short tw of 6 min (data in violet in Fig. 2C), whereasfor the spin jam SCGO(p = 0.97) and BCGO(p = 0.96),there is very small aging for tw = 6 min (data in violet in

Fig. 2 A and B). Furthermore, in the case of CuMn2%, as twincreases, the memory effect increases to develop a dip at Tw fortw J 3 h. On the other hand, for SCGO(p = 0.97) such a dipnever appears even for tw = 100 h; instead only a weak memoryshoulder appears.The memory effect can be quantified by the aging-induced

relative change in the magnetization, ðMaging −Mref Þ=Mref , whereMaging and Mref are the magnetization with and without aging,respectively. Fig. 2D shows ðMaging −Mref Þ=Mref , measured atTw=Tf ∼ 0.7 for SCGO(p = 0.97) (solid symbols), BCGO(p =0.96) (symbols with a line), and CuMn2% (open symbols),as a function of tw. These data are consistent with a previousstudy on SCGO(p = 0.956) with tw up to 5.83 h (13). Inthe case of CuMn2%, as tw increases from 6 min to 100 h,ðMaging −Mref Þ=Mref continues to gradually increase from3.4% to 8.2%. On the other hand, for SCGO(p = 0.97),ðMaging −Mref Þ=Mref increases gradually from 0.6% to 2.4%, andfor BCGO(p = 0.96), from 0.7% to 3.1%, as tw increases from6 min to 10 h. The increase rate of ðMaging −Mref Þ=Mref seems todecrease for tw > 10 h, reaching 2.7% at tw = 100 h for SCGO(p =0.97). We emphasize that over this wide range of tw up to 100 hthe susceptibility curve is always monotonically dependent ontemperature up to the freezing point (Fig. 2 A and B), in sharpcontrast to CuMn2%.Fig. 3 shows the memory effect for various values of

0.4K   Tw=Tf K 1 measured with tw = 10 h. All systems exhibitmaximal memory effect for Tw=Tf ∼ 0.7. When Tw=Tf increasesor decreases from the maximal value, then the memory ef-fect becomes weaker. The weakening, however, is more rapidin CuMn2% than in SCGO and BCGO; for CuMn2%,ðMaging −Mref Þ=Mref decreases from 7.2% for Tw=Tf ∼ 0.7 to 2.7%for Tw=Tf ∼ 0.9, whereas for SCGO(p = 0.97) [BCGO(p = 0.96)],ðMaging −Mref Þ=Mref decreases from 2.4% (3.1%) for Tw=Tf ∼ 0.7 to1.7% (2.5%) for Tw=Tf ∼ 0.9. The pronounced memory effectsfound in CuMn2% may hint at an energy landscape with a morehierarchical structure. On the other hand, the weak memory effect,observed in SCGO and BCGO, which is uniform over a wide range

A B

Fig. 1. (A) (Upper) Schematic energy landscape of a conventional spin glassthat consists of many hierarchical rugged funnels and (Lower) the corre-sponding hierarchical tree representation. (B) (Upper) Schematic diagram ofnonhierarchical energy landscape of the spin jam that has wide nearly flatrough bottom and (Lower) the corresponding nonhierarchical barrier treerepresentation.

102 103 104 105 1060

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Fig. 2. (A–C) Bulk susceptibility, χDC = M/H, where M and H are magnetization and applied magnetic field, respectively, obtained from (A) SCGO(p = 0.97),(B) BCGO(p = 0.96), and (C) a spin glass CuMn2%, with H = 3 G. Symbols and lines with different colors indicate the data taken with different waiting times, tw,ranging from 0 h to 100 h, at Tw/Tf ∼ 0.7, where Tw and Tf are the waiting and the freezing temperature, respectively. (D) From the data shown in A–C, the agingeffect was quantified for the three systems by (Mref – M)/Mref, where Mref is the magnetization without waiting, and it was plotted as a function of tw in a logscale. The “+” symbols mark the results of our MC simulations. Details of the simulations can be found in Supporting Information.

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of 0.4K   Tw=Tf K 1, suggests an energy landscape with a lesshierarchical structure.Rejuvenation and memory effects have proved difficult to re-

produce in standard simulations of supercooled liquids or spinglasses, due to the large phase space to be covered and largespread of time scales involved (22, 23). Several successful attemptswere made, such as a multilayer random energy model (24) and amodel of thermally activated number sorting (23). None of thestudies, however, investigated how different topologies of theenergy landscape will impact the memory effects. Here we havedone so by taking a phenomenological approach based on a mul-tilayer energy model. As shown later, this approach reproduces

qualitatively the differences between memory effects associatedwith different landscapes.We performed Monte Carlo simulations on two types of

energy landscapes suggested for the spin glass and spin jam.Although the energy surface in both cases is characterized bynumerous local minima, the distribution and connectivity ofthese minima are very different. Here we adopt the so-calledbarrier tree representation (8, 25, 26) in which the local min-ima correspond to leaves of the tree, whereas the branchingpoints denote the barriers separating disconnected valleys and/or minima. Details can be found in Fig. S1 and discussion inSupporting Information.

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Fig. 3. (A–C) χDC and (D–F) (Mref – M)/Mref measured for (A and D) SCGO(p = 0.97), (B and E) BCGO(p = 0.96), and (C and F) CuMn2%, with tw = 10 h, atvarious waiting temperatures.

A B C

D E F

Fig. 4. A–C and D–F show the simulated DC susceptibility during the reheating process for the hierarchical and nonhierarchical trees, respectively, at threedifferent waiting temperatures Tw = 0.2Tf (A and D), 0.4Tf (B and E), and 0.6Tf (C and F). Different curves in each panel correspond to varying tw measured inunits of the total cooling time.

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Fig. 1A shows a funnel-type barrier tree that is characteristic ofthe conventional spin glass. A rugged funnel here corresponds toa single long branch (the global minimum) with many deadbranches splitting from it (8, 25). The experimentally observedmemory effect is intimately related to a multitude of energy andtime scales in the low-energy configurational space. For thefunnel-type landscape, a hierarchical structure of energy scales isencoded in the different levels of the barrier tree. The energybarriers «l at level l are characterized by a temperature Tl suchthat T1 >T2 >⋯>TL, where L is the number of levels of thetree (24). The freezing temperature is Tf ≈T1. The relaxation ofthe system in this hierarchical structure exhibits complex tem-perature-dependent dynamics. Typically, because the relaxationtime at level l scales as τl ∼ τ0   e«l=T , where τ0 is a microscopictime scale, the relaxation dynamics start to show exponentialslowing down at level l when T <Tl. Depending on the pop-ulation of dead-end local minima at each level, the system fluc-tuates over a small window of levels determined by TW in theexperiments. A longer tw at this temperature allows the system torelax to a deeper and larger (entropically) valley of the energy sur-face. The memory effect observed during the reheating process re-sults from the fact that the system is trapped in this special landscapebasin. The susceptibility, χDC, as a function of temperature is shownin Fig. 4 A–C for three different Tw. The DC susceptibility com-puted using a random magnetization model (24) shows a clear dipthat depends on Tw as well as tw. In particular, a longer tW gives riseto a larger susceptibility reduction. It should be noted that othercontributions to χDC such as continuous spin fluctuations are notincluded in the landscape tree dynamics simulations.In contrast to ordinary spin glass, the energy structure in a spin

jam results from quantum and classical fluctuations breaking anexactly flat landscape (20). Importantly, the energy scale for glasstransition Tf is determined by the fluctuations and is two ordersof magnitude smaller than the Curie–Weiss temperature (20).We expect the resulting landscape to feature broad basins andwithin each basin numerous microstates, as shown in Fig. 1B. Asthe local minima in spin jam result from the original zero energymode of the classical spin Hamiltonian, it is plausible that theenergy minima here are clustered into different branches (labeledby m) each characterized by a different energy scale Tm. For aparticular cluster or branch of minima, the temperature Tm un-derscores the energy barrier due to quantum fluctuations. Tm is arandom variable and is uniformly distributed in the interval of½0,Tf �. Tw sets a threshold such that clusters with Tm >Tw exhibitslow relaxation dynamics, whereas a longer tW helps the system findthe cluster with a lower overall energy and larger entropy. Thisproperty underscores the weak memory effect observed in spinjam. Again, the fact that the system is trapped in this special clustermanifests itself as the memory effect during rewarming. The sim-ulated susceptibility of the nonhierarchical tree, shown in Fig. 4D–F, shows a memory effect that depends on both Tw and tw,similar to the spin glass. However, the salient feature is ratherdifferent: Contrary to the narrow dip in the hierarchical tree thatappears even for short waiting time tw > 1 (Fig. 4 A–C), the sus-ceptibility here exhibits a wide shoulder-like feature over a muchwider range of tw for each Tw. As shown in more detail in Fig. S2A,for Tw = 0.6  Tf the nonhierarchical landscape fails to yield a

narrow dip over six orders of magnitude of the Monte Carlo (MC)steps. Remarkably, this finding is consistent with the experimentaldata revealing a shoulder-like feature for spin jam (Fig. 2 A and B)vs. the substantial dip for the ordinary spin glass (Fig. 2C). Fur-thermore the functional dependence of the memory effect onwaiting time is nicely reproduced for both systems in Fig. 2D overthree orders of magnitude.The picture that emerges from the bulk susceptibility and

Monte Carlo simulations is that the energy landscape of a spinjam is qualitatively different from the rugged funnel-type land-scape of a spin glass. The hierarchical structure allows a naturalrealization of multiple energy scales (e.g., ref. 27) that is crucialto the memory effect. On the other hand, the aging dynamics inthe spin jam are well described by an essentially nonhierarchicalbarrier tree with more uniform branching. This result is consistentwith the fact that the rough energy landscape in spin jam resultsfrom quantum fluctuations that lift the otherwise degenerateclassical ground states. In particular, the weak memory effect atshort times found in a spin jam may be interpreted as a result of thelarge time it takes the system to wander among the numerousroughly equivalent minima at a given energy scale.The transition from a spin liquid to a spin jam in densely

populated frustrated magnets upon cooling may be viewed as aneffective reduction of degrees of freedom. In SCGO (20, 21) andkagome antiferromagnet (28) the origin of the reduction can beinduced by quantum fluctuations. We remark that the transitionbears some analogy to the transition from a structural (mechanical)liquid state to a mechanical jam by increasing the concentration ofthe atoms, i.e., pressure (29). Both frustrated magnets and themechanical jam have a large number of metastable states, otherthan their ground states, in the vicinity of their liquid states. Theconfigurational entropy of these states ranges from extensive, as inmechanical jams and coplanar states of the kagome antiferro-magnet (28, 30), to subextensive as in the locally collinear states ofan ideal SCGO (20). Both types of systems are expected to featurea relatively shallow energy landscape of accessible states due totheir proximity to a uniform liquid state. It is interesting to note apossibly related observation that the ensemble of metastable statesin self-generated Coulomb glasses is shallow compared with moreordinary electron glasses relying on quench disorder (31).The two fundamentally different trees studied here can be cast

in the framework of complex networks (32, 33). The spin glass’shierarchical energy landscape (even with a fractal structure) re-sembles the so-called scale-free network (33), proposed to explaininternet connections and ecological and neural networks (34). Inthis network, there are highly connected dominating nodes, each ofwhich corresponds to the global minimum of a rugged funnel. Onthe other hand, the spin jam’s nonhierarchical landscape corre-sponds to a network consisting of weakly connected clusters thatare homogenous on a larger scale.

ACKNOWLEDGMENTS. Work at the University of Virginia by S.H.L. and I.K.was supported in part by US National Science Foundation (NSF) Grants DMR-1404994 and DMR-1508245, respectively. A.S. is supported by Oak RidgeNational Laboratory. Work at Tohoku University was partly supported byGrants-in-Aid for Scientific Research (24224009, 23244068, and 15H05883) fromMinistry of Education, Culture, Sports, Science and Technology of Japan. Workat the University of Tennessee was supported by US NSF Grant DMR-1350002.

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