# Aging, memory, and nonhierarchical energy landscape of spin jam

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

## Significance

Our bulk susceptibility and Monte Carlo simulation study of aging and memory effects in densely populated frustrated magnets (spin jam) and in a dilute magnetic alloy (spin glass) indicates a nonhierarchical landscape with a wide and nearly flat but rough bottom for the spin jam and a hierarchical rugged funnel-type landscape for the spin glass.

## Abstract

The notion of complex energy landscape underpins the intriguing dynamical behaviors in many complex systems ranging from polymers, to brain activity, to social networks and glass transitions. The spin glass state found in dilute magnetic alloys has been an exceptionally convenient laboratory frame for studying complex dynamics resulting from a hierarchical energy landscape with rugged funnels. Here, we show, by a bulk susceptibility and Monte Carlo simulation study, that densely populated frustrated magnets in a spin jam state exhibit much weaker memory effects than spin glasses, and the characteristic properties can be reproduced by a nonhierarchical landscape with a wide and nearly flat but rough bottom. Our results illustrate that the memory effects can be used to probe different slow dynamics of glassy materials, hence opening a window to explore their distinct energy landscapes.

If the energy landscape of a system resembles a smooth vase with a pointy bottom end, upon cooling the system goes quickly into the lowest energy state, i.e., the global ground state, that is usually associated with crystalline order. If the energy landscape is more complex with many metastable states, i.e., local minima, then cooling may lead the system into local minima resulting in a glassy order. The concept of such energy landscapes has been instrumental in explaining the glassiness that is ubiquitous in a wide range of systems, including atomic clusters (1), structural glasses (2, 3), polymers (4), brain activity (5), and social networks (6). Several different topological types of energy landscapes were proposed to characterize different glassiness and the associated slow dynamics (7, 8). For instance, a rugged funnel-shaped landscape shown in Fig. 1*A* was proposed to understand the physics of biopolymers (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 landscape and low temperature properties. The most studied magnetic glass state is the conventional spin glass realized in dilute magnetic alloys such as *Cu*Mn and *Au*Fe. Here, dilute magnetic ions (Mn and Fe) in a nonmagnetic metal interact via the long-range Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction whose magnitude and sign change with distance between the randomly placed magnetic ions (11). The randomness drives the system into the spin glass state below a critical temperature, *Cu* − 2 atomic (at.) % Mn (*Cu*Mn2% hereafter), _{9p}Ga_{12-9p}O_{19} (SCGO)(*p* = 0.97),

Aging and memory effects have been key features of glassy systems due to the intrinsic slow dynamics. The thermo-remanent magnetization (TRM) method is the most effective way so far to investigate these effects (12⇓–14); for the measurements, the sample is first cooled down from well above *T*_{f} to base temperature with a single stop for a waiting time, *t*_{w}, at an intermediate temperature *T*_{w}, under zero field. While waiting at *T*_{w}, if the system has as many nearly degenerate metastable states at low energies as spin jam and conventional spin glasses have, the system will relax to the accessible lower energy states than when no waiting is imposed. The longer *t*_{w} is, the lower energy states the system will relax to, which is called “aging.” Once cooled down to base temperature, the TRM is measured by applying a small field of a few gauss upon heating at a constant rate. During the measurements, when the temperature approaches the temperature of aging, *T*_{w}, the system revisits the lower energy states reached during the wait time that are associated with the energy scale of *k*_{B}*T*_{w}, where *k*_{B} is the Boltzmann constant. Upon further heating, the system goes to higher energy states allowed within *k*_{B}*T*. This is referred to as the aging and memory effect.

We have performed the TRM measurements on two spin jam prototypes, SCGO(*p* = 0.97) and BaCr_{9p}Ga_{12-9p}O_{19} (BCGO)(*p* = 0.96) in which the magnetic Cr^{3+} (3d^{3}) ions form a highly frustrating quasi-2D triangular network of bipyramids (17⇓⇓⇓–21) and a spin glass prototype *Cu*Mn2% in which the 2% low concentration of the magnetic Mn atoms is embedded in the nonmagnetic Cu metal. Strong aging and memory effects have been observed in *Cu*Mn2%, whereas the effects are much weaker in SCGO and BCGO. Fig. 2 *A–C* shows the TRM data obtained from SCGO(*p* = 0.97), BCGO(*p* = 0.96), and *Cu*Mn2%, respectively, with several different values of *t*_{w} ranging from 6 min to 100 h, at *t*_{w}. These indicate the existence of numerous metastable states and slow dynamics in all systems. Despite the similarity, there is a clear difference: For the *Cu*Mn2% magnetic alloy, considerable aging occurs at *T*_{w} even for a short *t*_{w} of 6 min (data in violet in Fig. 2*C*), whereas for the spin jam SCGO(*p* = 0.97) and BCGO(*p* = 0.96), there is very small aging for *t*_{w} = 6 min (data in violet in Fig. 2 *A* and *B*). Furthermore, in the case of *Cu*Mn2%, as *t*_{w} increases, the memory effect increases to develop a dip at *T*_{w} for *p* = 0.97) such a dip never appears even for *t*_{w} = 100 h; instead only a weak memory shoulder appears.

The memory effect can be quantified by the aging-induced relative change in the magnetization, *D* shows *p* = 0.97) (solid symbols), BCGO(*p* = 0.96) (symbols with a line), and *Cu*Mn2% (open symbols), as a function of *t*_{w}. These data are consistent with a previous study on SCGO(*p* = 0.956) with *Cu*Mn2%, as *t*_{w} increases from 6 min to 100 h, *p* = 0.97), *p* = 0.96), from 0.7% to 3.1%, as *t*_{w} increases from 6 min to 10 h. The increase rate of *p* = 0.97). We emphasize that over this wide range of *t*_{w} up to 100 h the susceptibility curve is always monotonically dependent on temperature up to the freezing point (Fig. 2 *A* and *B*), in sharp contrast to *Cu*Mn2%.

Fig. 3 shows the memory effect for various values of *Cu*Mn2% than in SCGO and BCGO; for *Cu*Mn2%, *p* = 0.97) [BCGO(*p* = 0.96)], *Cu*Mn2% may hint at an energy landscape with a more hierarchical structure. On the other hand, the weak memory effect, observed in SCGO and BCGO, which is uniform over a wide range of

Rejuvenation and memory effects have proved difficult to reproduce in standard simulations of supercooled liquids or spin glasses, due to the large phase space to be covered and large spread of time scales involved (22, 23). Several successful attempts were made, such as a multilayer random energy model (24) and a model of thermally activated number sorting (23). None of the studies, however, investigated how different topologies of the energy landscape will impact the memory effects. Here we have done so by taking a phenomenological approach based on a multilayer energy model. As shown later, this approach reproduces qualitatively the differences between memory effects associated with 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 by numerous local minima, the distribution and connectivity of these minima are very different. Here we adopt the so-called barrier tree representation (8, 25, 26) in which the local minima correspond to leaves of the tree, whereas the branching points denote the barriers separating disconnected valleys and/or minima. Details can be found in Fig. S1 and discussion in *Supporting Information*.

Fig. 1*A* shows a funnel-type barrier tree that is characteristic of the conventional spin glass. A rugged funnel here corresponds to a single long branch (the global minimum) with many dead branches splitting from it (8, 25). The experimentally observed memory effect is intimately related to a multitude of energy and time scales in the low-energy configurational space. For the funnel-type landscape, a hierarchical structure of energy scales is encoded in the different levels of the barrier tree. The energy barriers *A–C* for three different

In contrast to ordinary spin glass, the energy structure in a spin jam results from quantum and classical fluctuations breaking an exactly flat landscape (20). Importantly, the energy scale for glass transition *B*. As the local minima in spin jam result from the original zero energy mode of the classical spin Hamiltonian, it is plausible that the energy minima here are clustered into different branches (labeled by *m*) each characterized by a different energy scale *D–F*, shows a memory effect that depends on both *A–C*), the susceptibility here exhibits a wide shoulder-like feature over a much wider range of *A*, for *A* and *B*) vs. the substantial dip for the ordinary spin glass (Fig. 2*C*). Furthermore the functional dependence of the memory effect on waiting time is nicely reproduced for both systems in Fig. 2*D* over three orders of magnitude.

The picture that emerges from the bulk susceptibility and Monte Carlo simulations is that the energy landscape of a spin jam is qualitatively different from the rugged funnel-type landscape of a spin glass. The hierarchical structure allows a natural realization of multiple energy scales (e.g., ref. 27) that is crucial to the memory effect. On the other hand, the aging dynamics in the spin jam are well described by an essentially nonhierarchical barrier tree with more uniform branching. This result is consistent with the fact that the rough energy landscape in spin jam results from quantum fluctuations that lift the otherwise degenerate classical ground states. In particular, the weak memory effect at short times found in a spin jam may be interpreted as a result of the large time it takes the system to wander among the numerous roughly 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 an effective reduction of degrees of freedom. In SCGO (20, 21) and kagome antiferromagnet (28) the origin of the reduction can be induced by quantum fluctuations. We remark that the transition bears some analogy to the transition from a structural (mechanical) liquid state to a mechanical jam by increasing the concentration of the atoms, i.e., pressure (29). Both frustrated magnets and the mechanical jam have a large number of metastable states, other than their ground states, in the vicinity of their liquid states. The configurational entropy of these states ranges from extensive, as in mechanical jams and coplanar states of the kagome antiferromagnet (28, 30), to subextensive as in the locally collinear states of an ideal SCGO (20). Both types of systems are expected to feature a relatively shallow energy landscape of accessible states due to their proximity to a uniform liquid state. It is interesting to note a possibly related observation that the ensemble of metastable states in self-generated Coulomb glasses is shallow compared with more ordinary 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’s hierarchical energy landscape (even with a fractal structure) resembles the so-called scale-free network (33), proposed to explain internet connections and ecological and neural networks (34). In this network, there are highly connected dominating nodes, each of which corresponds to the global minimum of a rugged funnel. On the other hand, the spin jam’s nonhierarchical landscape corresponds to a network consisting of weakly connected clusters that are homogenous on a larger scale.

## Model and Monte Carlo Simulations

Understanding the dynamics of a glassy system via the energy landscape approach has led to promising insights into puzzling phenomena, such as temperature-dependent aging and memory effect. In this framework, the energy landscape is seen as a set of basins of attraction, and the system evolves through a succession of jumps between their local minima (35, 36). This approach focuses on the interbasin transitions without treating explicitly the fast (high-frequency) intrabasin dynamics. Over the past decades, much effort has been devoted to characterizing the structure and topology of the energy landscape for various glass-forming systems (8, 25, 37). In particular, the so-called disconnectivity graph (25) has become a widely used approach for visualizing and representing the multidimensional potential energy surface. The disconnectivity graph summarizes the local minima and saddle points of an energy landscape into a tree. Each leaf in this tree representation corresponds to a local minimum, whereas the branching point is a transition state (saddle point) connecting different local minima. Another approach to describe the energy landscape is to use the language of complex networks (33, 36).

The disconnectivity graph (also referred to as a barrier tree representation) can be constructed numerically from the database of local minima and the kinetic pathways for small molecules or lattices (37). Monte Carlo sampling is often required to construct the representative barrier tree for a larger system. These approaches require microscopic details of the system at hand, which are hard to process for complicated physical systems. An alternative, phenomenological, approach for complex systems uses a statistically based characterization of the barrier tree, which is the method adopted in our work. For a given energy landscape representation, either through a barrier tree or through a complex network, the dynamics of the system can be simulated as a random walk on the tree or the network. A master equation is often used to study the resultant dynamics (25, 35, 36). Here we use the Markov chain Monte Carlo simulations coupled with a dynamical tree method to study the memory effect of spin glass and spin jam states. Our Monte Carlo approach offers the advantage of being applicable to barrier trees with complex structures without the need of introducing further approximations as in the master equation method.

In our simulations, the relaxation dynamics of the system are modeled as a random walk on the barrier tree. Each node of the tree, corresponding to either a local minima or a saddle point, represents a specific microscopic spin configuration. Transition between two nodes corresponds to modifying a small number of spins. In our simulations, the structure and properties of the barrier trees are characterized statistically. Specifically, the tree is described by a set of random numbers satisfying certain probability distributions.

We first discuss the statistical description of the barrier tree for a conventional spin glass. Here, the tree has a hierarchical structure with many levels. A node at a lower level (larger *l*) corresponds to a lower energy state (Fig. 1*A*). The barrier energy *l* is an independent random variable with an exponential distribution

The characteristic temperature *l* scales as *l* when

At each level, whether the node is a local minimum (and thus a dead end) or a saddle point is specified by a constant *l* is a local minimum. Another random number *l* is a random number uniformly distributed in the interval

We use the standard Metropolis dynamics in our Monte Carlo simulations. Because the barrier tree is specified only statistically, there is no need to create a tree at the beginning of the dynamical simulations. Instead, we generate the barrier tree dynamically according to the desired statistical properties as discussed above. However, additional bookkeeping is required to describe a system currently at level *i*) Determine whether this node is a local minimum (a dead end) or a saddle point. This can be done by generating a random number *r* uniformly distributed between 0 and 1. If *ii*) If the current node is a saddle point, then we generate another uniformly distributed random number *iii*) For a downhill update, we first increase the level by one. Next we generate new random numbers *iv*) Finally, for an uphill update we first compute the energy cost

We note that the dynamical tree simulation is valid as long as the number of branchings

The spin jam glassy state, on the other hand, is characterized by a very different energy landscape. This is because the numerous minima in the spin jam originate from quantum fluctuations that lift the otherwise flat energy surface at the classical level; we expect a uniform, nonhierarchical barrier tree structure, shown in Fig. 1*B* in the main text, for spin jam. The lack of hierarchical structure in this type of tree indicates that the weaker memory effect of spin jam results from a different mechanism. As the local minima in a spin jam result from the original zero energy mode of the classical spin Hamiltonian, it is plausible that the energy minima in the spin jam are grouped into clusters with different average barrier heights. This nonhierarchical tree resembles the so-called “banyan tree” pattern (8, 36). In this tree structure, different clusters are separated by a large barrier energy *m*) are random numbers generated from an exponential distribution *m* uniformly distributed in the interval

We performed our Monte Carlo simulations following the same protocol as the experiments. The temperature decreases linearly during the cooling process, except the waiting-time period. Numerically, we start at an initial temperature of 1.5 *A* and *B*, *Insets*. During the reheating part of the simulations, a small magnetic field *H* is included to generate a finite magnetization. The dc susceptibility is simply *A* and *B* shows the results obtained with different waiting times at *T*_{w} = 0.6*T*_{f} for the spin jam and the spin glass model, respectively. Different curves in each panel correspond to varying numbers of MC steps that waited at *T*_{w}, which are proportional to the real waiting time *t*_{w}. Fig. S2 *C* (spin jam model) and *D* (spin glass model) shows *t*_{w} (in seconds) in the experiments, as shown in Fig. 2*D* in the main text. In Fig. S2 *C* and *D*, *C* and *D*, *Insets* at *A–C* in the main text). Furthermore, the overall dependence of *t*_{w} reproduces our experimental data when scaled to the maximum value of *D*). We note that our MC calculations based on the multilayer random energy model do not take into account other possible sources of magnetization. As a result, different scaling factors for

The memory effect in the hierarchical tree arises from the temperature-dependent relaxation dynamics. For a given waiting temperature

## 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 Ridge National Laboratory. Work at Tohoku University was partly supported by Grants-in-Aid for Scientific Research (24224009, 23244068, and 15H05883) from Ministry of Education, Culture, Sports, Science and Technology of Japan. Work at the University of Tennessee was supported by US NSF Grant DMR-1350002.

## Footnotes

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^{1}To whom correspondence should be addressed. Email: shlee{at}virginia.edu.

Author contributions: S.-H.L. designed research; A.S. and T.J.S. designed and performed experiment; T.C. and G.-W.C. designed and performed MC simulations; A.S., J.Y., R.S., and H.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.

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

Freely available online through the PNAS open access option.

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