Spin-glass Dynamics Research Articles

Dynamics of mean field spin glasses on short and long timescales

We survey some recent rigorous results regarding the dynamics of spin glasses. We first survey recent results with Gheissari and Ben Arous regarding spectral gaps for these models. In particular, we briefly present the extension of the large deviations based approach of Ben Arous and Jagannath [Commun. Math. Phys. 361, 1–52 (2018)] to the setting of spherical spin glasses, unifying the treatment of the Ising and spherical models. We then turn to the new bounding flows method introduced by Ben Arous et al. [“Bounding flows for spherical spin glass dynamics,” e-print arXiv:1808.00929] regarding the nonactivated dynamics. We end with a report on progress on activated dynamics.

An experiment-oriented analysis of 2D spin-glass dynamics: a twelve time-decades scaling study

Recent high precision experimental results on spin-glass films ask for a detailed understanding of the domain-growth dynamics of two-dimensional spin glasses. To achieve this goal, we numerically simulate the out-equilibrium dynamics of the Ising spin glass for a time that spans close to twelve orders of magnitude (from picoseconds to order of a second), in systems large enough to avoid finite-size effects. We find that the time-growth of the size of the glassy domains is excellently described by a single scaling function. A single time-scale controls the dynamics. diverges upon approaching the T = 0 critical point. The divergence of is Arrhenius-like, with a barrier height that depends very mildly on temperature. The growth of this barrier-height is best described by critical dynamics. As a side product we obtain an impressive confirmation of universality of the equilibrium behavior of two-dimensional spin-glasses.

On the spectral gap of spherical spin glass dynamics

Nous considerons le temps d’atteinte de l’equilibre pour la dynamique de Langevin du modele de verre de $p$-spin spherique de taille $N$. Nous montrons que la constante de log-Sobolev et le trou spectral sont d’ordre $1$ a temperature suffisamment grande, alors que le trou spectral decroit exponentiellement en $N$ a temperature suffisamment basse. Ceci confirme l’existence d’une phase dynamique de haute temperature et d’une phase dynamique verre concernant le trou spectral. Les arguments cles de ces resultats sont la comprehension du processus extremal et de l’energie libre restreinte de Subag–Zeitouni et Subag.

Influence of Concentration Fluctuations on Relaxation Processes in Spin Glasses

Using the unique combination of atomically resolved atom probe tomography (APT) and volume averaged neutron (resonance) spin echo (NRSE and NSE) experiments, the influence of nano-scaled clusters on the spin relaxation in spin glasses was studied. For this purpose, the phase transition from the paramagnetic phase to the spin glass phase in an Fe-Cr spin glass with a composition of Fe 17 . 8 Cr 82 . 2 was studied in detail by means of NRSE. The microstructure was characterised by APT measurements, which show local concentration fluctuations of Fe and Cr on a length scale of 2 to 5 nm, which lead (i) to the coexistence of ferro- and anti-ferromagnetic clusters and (ii) a change of the magnetic properties of the whole sample, even in the spin glass phase, where spins are supposed to be randomly frozen. We show that a generalized spin glass relaxation function, which was successfully used to describe the phase transition in diluted spin glasses, can also be used for fitting the spin dynamics in spin glasses with significant concentration fluctuations.

Spectral Gap Estimates in Mean Field Spin Glasses

We show that mixing for local, reversible dynamics of mean field spin glasses is exponentially slow in the low temperature regime. We introduce a notion of free energy barriers for the overlap, and prove that their existence imply that the spectral gap is exponentially small, and thus that mixing is exponentially slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure which guarantee the existence of these barriers, using the notion of replicon eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient conditions cover large classes of Ising spin models for reversible nearest-neighbor dynamics and spherical models for Langevin dynamics. Finally, in the case of Ising spins, Panchenko's recent rigorous calculation [79] of the free energy for a system of "two real replica" enables us to prove a quenched LDP for the overlap distribution, which gives us a wider criterion for slow mixing directly related to the Franz-Parisi-Virasoro approach [43,60]. This condition holds in a wider range of temperatures.

Ergodic and localized regions in quantum spin glasses on the Bethe lattice.

By considering the quantum dynamics of a transverse-field Ising spin glass on the Bethe lattice, we find the existence of a many-body localized (MBL) region at small transverse field and low temperature. The region is located within the thermodynamic spin glass phase. Accordingly, we conjecture that quantum dynamics inside the glassy region is split into a small MBL region and a large delocalized (but not necessarily ergodic) region. This has implications for the analysis of the performance of quantum adiabatic algorithms.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

Effect of magnetic fields on spin glass dynamics

The effects of a magnetic field on spin glass dynamics are explored for a ${\mathrm{Cu}}_{0.887}{\mathrm{Mn}}_{0.113}$ thin film of thickness $\mathcal{L}=20$ nm in a multilayer configuration. An experimental protocol removes uncertainties associated with the time dependence of the field-cooled magnetization ${M}_{\text{FC}}(t,T)$. Activated dynamics is exhibited after the spin glass correlation length $\ensuremath{\xi}(t,T)$ has reached $\mathcal{L}$, creating a quasiequilibrium state. The activation energy depends upon the strength of the magnetic field $H$. The magnitude of the activation energy diminishes as ${H}^{2}$, the coefficient of which is proportional to the number of correlated spins. A quantitative fit requires a ``pancakelike'' correlated region, associated with the $T=0$ phase transition for a spin glass in $D=2$ dimensions.

Syntactic Structures and Code Parameters

We assign binary and ternary error-correcting codes to the data of syntactic structures of world languages and we study the distribution of code points in the space of code parameters. We show that, while most codes populate the lower region approximating a superposition of Thomae functions, there is a substantial presence of codes above the Gilbert–Varshamov bound and even above the asymptotic bound and the Plotkin bound. We investigate the dynamics induced on the space of code parameters by spin glass models of language change, and show that, in the presence of entailment relations between syntactic parameters the dynamics can sometimes improve the code. For large sets of languages and syntactic data, one can gain information on the spin glass dynamics from the induced dynamics in the space of code parameters.

A rock-salt-type Li-based oxide, Li3Ni2RuO6, exhibiting a chaotic ferrimagnetism with cluster spin-glass dynamics and thermally frozen charge carriers.

The area of research to discover new Li containing materials and to understand their physical properties has been of constant interest due to applications potential for rechargeable batteries. Here, we present the results of magnetic investigations on a Li compound, Li3Ni2RuO6, which was believed to be a ferrimagnet below 80 K. While our neutron diffraction (ND) and isothermal magnetization (M) data support ferrimagnetism, more detailed magnetic studies establish that this ferrimagnetic phase exhibits some features similar to spin-glasses. In addition, we find another broad magnetic anomaly around 40–55 K in magnetic susceptibility (χ), attributable to cluster spin-glass phenomenon. Gradual dominance of cluster spin-glass dynamics with a decrease of temperature (T) and the apparent spread in freezing temperature suggest that the ferrimagnetism of this compound is a chaotic one. The absence of a unique freezing temperature for a crystalline material is interesting. In addition, pyroelectric current (Ipyro) data reveals a feature in the range 40–50 K, attributable to thermally stimulated depolarization current. We hope this finding motivates future work to explore whether there is any intriguing correlation of such a feature with cluster spin-glass dynamics. We attribute these magnetic and electric dipole anomalies to the crystallographic disorder, intrinsic to this compound.

Evidence of aging and dynamic scaling in the collapse of a polymer.

We investigate a newly framed two-time property for the nonequilibrium evolution dynamics during the collapse of a homopolymer via Monte Carlo simulations of a model polymer. Our results show evidence of aging effects, as observed in the slow dynamics of structural and spin glasses, along with the presence of a dynamic scaling of the autocorrelation function ∼x(-λ(c)) (x being the ratio of the cluster sizes at two different times). We estimate the value of λ(c) unambiguously by applying a finite-size scaling analysis to the numerical data. The value thus obtained obeys a bound which we predict via general theoretical arguments. The results presented should be of general validity and may trigger direct experimental verification in single-polymer dynamics.

Dielectric anomalies and magnetodielectric coupling behavior of single crystalline Ca3Co2O6, a geometrically frustrated magnetic spin-chain system

The dielectric behavior of the single crystals of the spin-chain system Ca3Co2O6, undergoing geometrically frustrated antiferromagnetic ordering below 25 K, has been investigated as a function of temperature and magnetic field (H) and compared with magnetization (M) behavior. The results provide evidence for anisotropic magnetodielectric (MDE) coupling in this compound. Ac susceptibility exhibits a strong frequency dependence for H//c with changes of this feature with the application of external dc magnetic field. No feature in ac susceptibility could be observed for H⊥c, thereby providing evidence for strong anisotropic spin-glass behavior. Interestingly, the strong frequency dependence in dielectric is present for both the crystallographic directions (E//c and E⊥c where E is the electric field) with a negligible influence of H, despite the existence of MDE coupling. This result appears to suggest different dynamics of electric dipole and spin-glass, although they are coupled with each other. In addition, interestingly, there is also a step at one-third of high-(magnetic) field value of MDE at some temperatures tracking a similar step in M(H). This work also confirms following unusual features reported on polycrystals, when measured along c-axis. The dielectric constant exhibits a broad peak around 50–130 K when the electric field is applied along the spin-chain (crystallographic ‘c’ direction); however this feature is essentially absent for the perpendicular orientation. This finding supports the role of incipient spin-chain ordering to induce magnetodielectric coupling. There is a signature of ‘magnetodielectric phase coexistence’ when the magnetic field is applied along the spin-chain irrespective of the direction of applied electric field.

The connection between statics and dynamics of spin glasses

We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z (T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.

Temperature chaos in a Ge:Mn thin-film spin glass

Temperature changes in thin-film Ge:Mn spin-glass dynamics are presented that exhibit temperature chaos (TC) when the spin-glass correlation length $\ensuremath{\xi}(t,T)$ grows to its thickness $\mathcal{L}$. For small $\mathcal{L}\ensuremath{\approx}15.5$ nm, the transition to chaos takes place over a temperature range $\mathrm{\ensuremath{\Delta}}T$ sufficiently large to exhibit both reversible and chaotic behavior. The value of $\mathrm{\ensuremath{\Delta}}T$ can be related to the critical exponent for TC, $\ensuremath{\zeta}$. Experimentally, $\ensuremath{\zeta}$ is found to be $\ensuremath{\approx}1.06$, in the range of recent simulations. The presence of a specific length scale $\mathcal{L}$ allows the transition to chaos to be examined over measurable laboratory temperature changes. The transition is found to be abrupt. Bulk materials, with a distribution of crystallite sizes, will smear out the transition, resulting in a very slow crossover. The abruptness of the transition and its nature are compared with recent theoretical calculations.

Phase transition in random adaptive walks on correlated fitness landscapes

We study biological evolution on a random fitness landscape where correlations are introduced through a linear fitness gradient of strength c. When selection is strong and mutations rare the dynamics is a directed uphill walk that terminates at a local fitness maximum. We analytically calculate the dependence of the walk length on the genome size L. When the distribution of the random fitness component has an exponential tail, we find a phase transition of the walk length D between a phase at small c, where walks are short (D∼lnL), and a phase at large c, where walks are long (D∼L). For all other distributions only a single phase exists for any c>0. The considered process is equivalent to a zero temperature Metropolis dynamics for the random energy model in an external magnetic field, thus also providing insight into the aging dynamics of spin glasses.

Spin glass dynamics at the mesoscale

The mesoscale allows a new probe of spin glass dynamics. Because the spin glass lower critical dimension ${d}_{l}g2$, the growth of the correlation length $\ensuremath{\xi}(t,T)$ can change the nature of the spin glass state at a crossover time ${t}_{\text{co}}$ when $\ensuremath{\xi}({t}_{\text{co}},T)=\ensuremath{\ell}$, a minimum characteristic sample length (e.g., film thickness for thin films and crystallite size for bulk samples). For thin films, and times $tl{t}_{\text{co}}$ such that $\ensuremath{\xi}(t,T)l\ensuremath{\ell}$, conventional three-dimensional dynamics is observed. When $tg{t}_{\text{co}}$, a crossover to $d=2$ behavior takes place. The parallel correlation length, associated with a ${T}_{g}=0$ transition, increases in time from the saturated value of the perpendicular correlation length $\ensuremath{\ell}$ to an equilibrium value of the parallel correlation length proportional to ${T}^{\ensuremath{-}\ensuremath{\nu}}$. This results in a pancakelike correlated state, with a thickness $\ensuremath{\ell}$ and a temperature-dependent in-plane radius that increases with decreasing temperature. Activated dynamics is associated with these states. Measurements on Cu:Mn thin films are analyzed quantitatively within this framework. We extract a temperature-dependent activation energy from a fit to the frequency dependence of the dynamic susceptibility. The extrapolated temperature at which the activation energy would become large is close to the extrapolated glass transition temperature from ac susceptibility measurements. All known relevant experimental data are consistent with this approach. For polycrystalline materials, there is a distribution of length scales $\mathcal{P}(\ensuremath{\ell})$. For sufficiently broad distributions, a logarithmic time dependence is derived for the time decay of the thermoremanent magnetization ${M}_{\text{TRM}}(t,T)$ using an approach originally derived by Ma. Properties dependent upon an effective waiting time ${t}_{w}^{\mathrm{eff}}$ are derived that are consistent with experiment, and further measurements are suggested.

3D Co(II) coordination polymer with ferrimagnetic-like layers based on azide and tetrazolate bridges showing slow magnetic dynamics.

A novel 3D Co(II) coordination polymer, [Co6(4-ptz)6(N3)6]n (1) (4-ptz = 4-(pyridinyl)tetrazole), was synthesized by in situ hydrothermal reaction and characterized magnetically. Compound 1 consists of ferrimagnetic-like Co(II) layers with the title heterobridges and exhibits the spin-glass slow dynamics. The study on spin-glass behavior in mixed azide and tetrazolate bridged compounds is the first example.

Low-temperature dynamics of long-ranged spin-glasses: full hierarchy of relaxation times via real-space renormalization

We consider the long-ranged Ising spin-glass with random couplings decaying as a power-law of the distance, in the region of parameters where the spin-glass phase exists with a positive droplet exponent. For the Metropolis single-spin-flip dynamics near zero temperature, we construct via real-space renormalization the full hierarchy of relaxation times of the master equation for any given realization of the random couplings. We then analyze the probability distribution of dynamical barriers as a function of the spatial scale. This real-space renormalization procedure represents a simple explicit example of the droplet scaling theory, where the convergence towards local equilibrium on larger and larger scales is governed by a strong hierarchy of activated dynamical processes, with valleys within valleys.

Dynamical transition in the D=3 Edwards-Anderson spin glass in an external magnetic field.

We study the off-equilibrium dynamics of the three-dimensional Ising spin glass in the presence of an external magnetic field. We have performed simulations both at fixed temperature and with an annealing protocol. Thanks to the Janus special-purpose computer, based on field-programmable gate array (FPGAs), we have been able to reach times equivalent to 0.01 s in experiments. We have studied the system relaxation both for high and for low temperatures, clearly identifying a dynamical transition point. This dynamical temperature is strictly positive and depends on the external applied magnetic field. We discuss different possibilities for the underlying physics, which include a thermodynamical spin-glass transition, a mode-coupling crossover, or an interpretation reminiscent of the random first-order picture of structural glasses.

Direct dynamical evidence for the spin glass lower critical dimension 2

A dynamical method is introduced to study the effect of dimensionality on phase transitions. Direct experimental measurements for the lower critical dimension for spin glasses is provided as an example. The method makes use of the spin glass correlation length ξ(t,T). Once nucleated, it can become comparable to sample dimensions in convenient time and temperature ranges. Thin films of amorphous Ge:Mn alloys were prepared with thickness L≈15.5 nm. Conventional behavior is observed as long as ξ(t,T)<L. At the measurement time tco, when ξ(tco,T)≈L, the time dependence is observed to cross over to exponential. These results are interpreted using spin glass dynamics, and are consistent with a lower critical dimension for spin glasses, dℓ, between 2<dℓ<3.

Assembly-mediated interplay of dipolar interactions and surface spin disorder in colloidal maghemite nanoclusters

Controlled assembly of single-crystal, colloidal maghemite nanoparticles is facilitated via a high-temperature polyol-based pathway. Structural characterization shows that size-tunable nanoclusters of 50 and 86 nm diameters (D), with high dispersibility in aqueous media, are composed of ∼13 nm (d) crystallographically oriented nanoparticles. The interaction effects are examined against the increasing volume fraction, φ, of the inorganic magnetic phase that goes from individual colloidal nanoparticles (φ = 0.47) to clusters (φ = 0.72). The frozen-liquid dispersions of the latter exhibit weak ferrimagnetic behaviour at 300 K. Comparative Mössbauer spectroscopic studies imply that intra-cluster interactions come into play. New insight emerges from the clusters' temperature-dependent ac susceptibility that displays two maxima in χ''(T), with strong frequency dispersion. Scaling-law analysis together with the observed memory effects suggests that a superspin-glass state settles-in at TB ∼ 160-200 K, while at lower-temperatures, surface spin-glass freezing is established at Tf ∼ 40-70 K. In such nanoparticle-assembled systems, with increased φ, Monte Carlo simulations corroborate the role of the inter-particle dipolar interactions and that of the constituent nanoparticles' surface spin disorder in the emerging spin-glass dynamics.