Spin Glass Correlation Research Articles

Multifractality in spin glasses.

We unveil the multifractal behavior of Ising spin glasses in their low-temperature phase. Using the Janus II custom-built supercomputer, the spin-glass correlation function is studied locally. Dramatic fluctuations are found when pairs of sites at the same distance are compared. The scaling of these fluctuations, as the spin-glass coherence length grows with time, is characterized through the computation of the singularity spectrum and its corresponding Legendre transform. A comparatively small number of site pairs controls the average correlation that governs the response to a magnetic field. We explain how this scenario of dramatic fluctuations (at length scales smaller than the coherence length) can be reconciled with the smooth, self-averaging behavior that has long been considered to describe spin-glass dynamics.

Superposition principle and nonlinear response in spin glasses

The extended principle of superposition has been a touchstone of spin-glass dynamics for almost 30 years. The Uppsala group has demonstrated its validity for the metallic spin glass, CuMn, for magnetic fields H up to 10 Oe at the reduced temperature Tr=T/Tg=0.95, where Tg is the spin-glass condensation temperature. For H>10 Oe, they observe a departure from linear response which they ascribe to the development of nonlinear dynamics. The thrust of this paper is to develop a microscopic origin for this behavior by focusing on the time development of the spin-glass correlation length, ξ(t,tw;H). Here, t is the time after H changes, and tw is the time from the quench for T>Tg to the working temperature T until H changes. We connect the growth of ξ(t,tw;H) to the barrier heights Δ(tw) that set the dynamics. The effect of H on the magnitude of Δ(tw) is responsible for affecting differently the two dynamical protocols associated with turning H off (TRM, or thermoremanent magnetization) or on (ZFC, or zero-field-cooled magnetization). This difference is a consequence of nonlinearity based on the effect of H on Δ(tw). Superposition is preserved if Δ(tw) is linear in the Hamming distance Hd (proportional to the difference between the self-overlap qEA and the overlap q[Δ(tw)]). However, superposition is violated if Δ(tw) increases faster than linear in Hd. We have previously shown, through experiment and simulation, that the barriers Δ(tw) do increase more rapidly than linearly with Hd through the observation that the growth of ξ(t,tw;H) slows down as ξ(t,tw;H) increases. In this paper, we display the difference between the zero-field-cooled ξZFC(t,tw;H) and the thermoremanent magnetization ξTRM(t,tw;H) correlation lengths as H increases, both experimentally and through numerical simulations, corresponding to the violation of the extended principle of superposition in line with the finding of the Uppsala Group.11 MoreReceived 22 July 2022Revised 18 May 2023Accepted 31 May 2023DOI:https://doi.org/10.1103/PhysRevB.107.214436©2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMagnetismPhysical SystemsMetallic glassesSpin glassesTechniquesCritical phenomenaCondensed Matter, Materials & Applied Physics

Direct observation of a dynamical glass transition in a nanomagnetic artificial Hopfield network

Spin glasses, generally defined as disordered systems with randomized competing interactions, are a widely investigated complex system. Theoretical models describing spin glasses are broadly used in other complex systems, such as those describing brain function, error-correcting codes, or stock-market dynamics. This wide interest in spin glasses provides strong motivation to generate an artificial spin glass within the framework of artificial spin ice systems. Here, we present the experimental realization of an artificial spin glass consisting of dipolar coupled single-domain Ising-type nanomagnets arranged onto an interaction network that replicates the aspects of a Hopfield neural network. Using cryogenic x-ray photoemission electron microscopy (XPEEM), we performed temperature-dependent imaging of thermally driven moment fluctuations within these networks and observed characteristic features of a two-dimensional Ising spin glass. Specifically, the temperature dependence of the spin glass correlation function follows a power law trend predicted from theoretical models on two-dimensional spin glasses. Furthermore, we observe clear signatures of the hard to observe rugged spin glass free energy in the form of sub-aging, out of equilibrium autocorrelations and a transition from stable to unstable dynamics.

Spin-glass dynamics in the presence of a magnetic field: exploration of microscopic properties

The synergy between experiment, theory, and simulations enables a microscopic analysis of spin-glass dynamics in a magnetic field in the vicinity of and below the spin-glass transition temperature T g. The spin-glass correlation length, ξ(t, t w; T), is analysed both in experiments and in simulations in terms of the waiting time t w after the spin glass has been cooled down to a stabilised measuring temperature T < T g and of the time t after the magnetic field is changed. This correlation length is extracted experimentally for a CuMn 6 at. % single crystal, as well as for simulations on the Janus II special-purpose supercomputer, the latter with time and length scales comparable to experiment. The non-linear magnetic susceptibility is reported from experiment and simulations, using ξ(t, t w; T) as the scaling variable. Previous experiments are reanalysed, and disagreements about the nature of the Zeeman energy are resolved. The growth of the spin-glass magnetisation in zero-field magnetisation experiments, M ZFC(t, t w; T), is measured from simulations, verifying the scaling relationships in the dynamical or non-equilibrium regime. Our preliminary search for the de Almeida–Thouless line in D = 3 is discussed.

Scaling Law Describes the Spin-Glass Response in Theory, Experiments, and Simulations.

The correlation length ξ, a key quantity in glassy dynamics, can now be precisely measured for spin glasses both in experiments and in simulations. However, known analysis methods lead to discrepancies either for large external fields or close to the glass temperature. We solve this problem by introducing a scaling law that takes into account both the magnetic field and the time-dependent spin-glass correlation length. The scaling law is successfully tested against experimental measurements in a CuMn single crystal and against large-scale simulations on the Janus II dedicated computer.

Confinement, reduced entanglement, and spin-glass order in a random quantum spin-ice model

We study an effective spin model derived perturbatively from random transverse-field Ising model on the pyrochlore lattice. The model consists of spin-configurations on the pyrochlore lattice, restricted to the spin-ice subspace, with spins interacting with random Ising exchange couplings as well as ring exchanges along the hexagons of the lattice. This model is studied by exact diagonalization upto N=64 site systems. We calculate spin-glass correlation functions and local entanglement entropy $S_T$ between spins in a single tetrahedron and the rest of the system. We find that the model undergoes two phase transitions. At weak randomness the model is in a quantum spin-ice phase where $S_T=\ln{6}$. Increasing randomness first leads to a frozen phase, with long-range spin-glass order and $S_T=\ln{2}$ corresponding to the Cat states associated with Ising order. Further increase in randomness leads to a random resonating-hexagon phase with a frozen backbone of spins and a broad distribution of entanglement entropies. The implications of these studies for non-Kramers rare-earth pyrochlores are discussed.

Slowing down of spin glass correlation length growth: Simulations meet experiments

The growth of the spin-glass correlation length has been measured as a function of the waiting time $t_{\mathrm{w}}$ on a single crystal of CuMn (6 at.\%), reaching values $\xi\sim 150$ nm, larger than any other glassy correlation-length measured to date. We find an aging rate $\mathrm{d}\ln\,t_{\mathrm{w}}/\mathrm{d}\ln\,\xi$ larger than found in previous measurements, which evinces a dynamic slowing-down as $\xi$ grows. Our measured aging rate is compared with simulation results by the Janus collaboration. After critical effects are taken into account, we find excellent agreement with the Janus data.

Bimodal Ising spin glass in two dimensions: The anomalous dimension η

Direct measurements of the spin glass correlation function $G(R)$ for Gaussian and bimodal Ising spin glasses in dimension two have been carried out in the temperature region $T \sim 1$. In the Gaussian case the data are consistent with the known anomalous dimension value $\eta \equiv 0$. For the bimodal spin glass in this temperature region $T > T^{*}(L)$, well above the crossover $T^{*}(L)$ to the ground state dominated regime, the effective exponent $\eta$ is clearly non-zero and the data are consistent with the estimate $\eta \sim 0.28(4)$ given by McMillan in 1983 from similar measurements. Measurements of the temperature dependence of the Binder cumulant $U_{4}(T,L)$ and the normalized correlation length $\xi(T,L)/L$ for the two models confirms the conclusion that the 2D bimodal model has a non-zero effective $\eta$ both below and above $T^{*}(L)$. The 2D bimodal and Gaussian interaction distribution Ising spin glasses are not in the same Universality class.

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.

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.

Aging at the spin-glass/ferromagnet transition: Monte Carlo simulations using graphics processing units

We study the the non-equilibrium ageing behaviour of the +/-J Edwards-Anderson model in three dimensions for samples of size up to N=128^3 and for up to 10^8 Monte Carlo sweeps. In particular we are interested in the change of the ageing when crossing from the spin-glass phase to the ferromagnetic phase. The necessary long simulation times are reached by employing a CUDA-based GPU implementation, which allows for single-spin flip times as small as 8ps. We measure typical spin glass correlation functions in space and time to determine the growing length scale and extract the constituting exponents. We observe a clear signature of the disorder-driven equilibrium transition in the non-equilibrium behavior.

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.

Low-temperature onset of the spin glass correlations in the ensemble of oriented Stoner–Wohlfarth nanoparticles

We reveal the low-temperature anomaly in the temperature and angular dependencies of the coercivity in granular (CoFeB)x–(SiO2)1−x films with oriented in-plane anisotropy. Namely, at T&amp;lt;100 K the in-plane angular dependence of coercive field acquires two maxima corresponding to easy and hard (in a film plane) directions. This signifies the emergence of coercivity for hard direction in a film plane. The experimental results are explained in terms of a random field model, which describes the onset of spin glass-like correlations in the ensemble of oriented weakly interacting Stoner–Wohlfarth particles.

Triangular Ising antiferromagnets with quenched nonmagnetic impurities

In a random spin system, the cooperation of randomness and frustration will lead to a spin-glass phase. However, in geometrically frustrated spin systems, quenched nonmagnetic impurities lift frustration locally. This makes randomness and frustration in these systems as competitors rather than cooperators. By mapping the dilute triangular Ising antiferromagnetic system to elastic array of noncrossing strings, we find that the nonmagnetic impurities in the spin system play roles of pinning centers in the string system. Calculation shows that in the ground state of this system, the spin-glass correlation is power-law decayed, quite different from the standard behavior of spin glass in which spin-glass correlation between two spins at infinite distance tends to a finite value. This indicates that triangular Ising antiferromagnets with quenched nonmagnetic impurities cannot be a spin glass. Instead, in the ground states, they present properties of vortex glass.

The spin glass correlation length and the crossover from three to two dimensions

The identification of a time and temperature dependent spin glass correlation length, ξ ( t , T ) , has consequences for samples of finite size. Previous experiments on layered samples show a suppression of the freezing temperature with decreasing sample thickness. The correlation length exceeding the finite sample thickness can explain these results. This novel analysis lends further credence to the correlation length paradigm of understanding spin glasses.

Gabay–Toulouse phase transition in Heisenberg spin glasses in three dimensions

We examine three-dimensional ±J Heisenberg models with and without random anisotropies in a magnetic field. We calculate both the stiffness exponent θs at zero temperature and spin-glass correlation lengths for the longitudinal and transverse spin components at finite temperatures. We suggest that, contrary to a chirality scenario predicted by Kawamura and his co-workers (Kawamura 1992 Phys. Rev. Lett. 68 3785, 1998 Phys. Rev. Lett. 80 5421, Hukushima and Kawamura 2000 Phys. Rev. E 61 R1008), a Gabay–Toulouse phase transition might occur when the anisotropies are absent, although the results, except for the correlation length of the chirality for larger sizes, suggest no phase transition when they are present.

Lack of monotonicity in spin glass correlation functions

We study the response of a spin glass system with respect to the rescaling of its interaction random variables and investigate numerically the behaviour of the correlation functions with respect to the volume. While for a ferromagnet the local energy correlation functions increase monotonically with the scale and, by consequence, with respect to the volume of the system we find that in a general spin glass model those monotonicities are violated.

Logarithmic growth law in the two-dimensional Ising spin glass state resulting from the electron doping in single-layered manganites

The ac susceptibility of the electron-doped single-layered manganite ${\mathrm{La}}_{1.1}{\mathrm{Sr}}_{0.9}\mathrm{Mn}{\mathrm{O}}_{4}$ is analyzed in detail. A quasi-two-dimensional antiferromagnetic (AFM) order with Ising anisotropy is stabilized below ${T}_{N}\ensuremath{\sim}80\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. We show that below ${T}_{N}$, a rare two-dimensional (2D) spin-glass (SG) correlation develops with the same Ising anisotropy as the AFM state. Using simple scaling arguments of the droplet model, we derive a scaling form for the ac susceptibility data of a 2D SG, which our experimental data follow fairly well. Due to simplifications in this 2D case, the proposed scaling form only contains two unknown variables, $\ensuremath{\psi}\ensuremath{\nu}$ and ${\ensuremath{\tau}}_{0}$. Hence, the logarithmic growth law of the SG correlation predicted by the droplet model is convincingly evidenced by the scaling of our experimental data. The origin and nature of this 2D SG state are also discussed.

Quenched disorder and spin-glass correlations inXYnematics

We present a theoretical study of the equilibrium ordering in a 3D XY nematic system with quenched random disorder. Within this model, treated with the replica trick and Gaussian variational method, the correlation length is obtained as a function of the local nematic order parameter and the effective disorder strength. These results clarify what happens in the limiting cases of diminishing order parameter and disorder strength, that is near a phase transition of a pure system. In particular, it is found that quenched disorder is irrelevant as the order parameter tends to zero and hence does not change the character of the continuous XY nematic to isotropic phase transition. We discuss how these results compare with experiments and simulations

Nonequilibrium Relaxation Analysis of the Spin-Glass Correlation Length

Relaxation functions of the spin-glass correlation length are investigated in the three-dimensional ±J XY model. The algebraic divergence is observed at the spin-glass transition temperature, T sg ≃ 0.45. The finite-time scaling analysis provides results consistent with our previous scaling analysis of the spin-glass susceptibility. The relaxation functions in the low-temperature phase exhibit the same behavior with the Ising spin glass: the exponent is linearly dependent on the temperature as 1/z(T) ≃ 0.21T/T sg . Relaxation functions of the chiral-glass correlation length do not exhibit such behavior as the spin-glass ones do. All the findings suggest that the transition is driven by the spin degrees of freedom.