Aging in the ferromagnetic phase of terbium

We report on aging, rejuvenation and memory effects in the ferromagnetic phase of pure terbium. We have applied an experimental method specifically for investigating slow dynamics of spin glasses, because these effects cannot be interpreted as conventional diffusion after-effects. Results show that relaxation times of the magnetic response are widely distributed, and isothermal aging shifted the distribution towards longer durations. If the sample was heated/cooled after such isothermal aging, the relaxation times shortened as if aging was starting anew; the behavior resembles that in spin glasses. Uniform magnetization experiments indicate that, unlike rejuvenation in spin glasses, ferromagnetic correlations are not returned to disorder by thermal perturbations. In contrast with memory effects in spin glasses, the effects of isothermal aging cannot be recovered once these disappear, even if the system is returned to its initial temperature. The observed results can be explained as collective pinning of the domain walls for which the potential is given by a rugged temperature-sensitive energy landscape.     

Spin glass model based on the Onsager reaction field

Changes in the Onsager reaction field are used to account quantitatively for aging (the decrease in the magnetic susceptibility when cooling in zero field is halted below the glass temperature) and rejuvenation (the disappearance of aging phenomena on further cooling only to reappear at Tw on heating) that characterize spin glasses. These effects must be caused by interactions between the spins since, absent the interactions, the magnetic properties of N spins are just N times the magnetic property of a single spin that cannot display aging. A spin introduced at an empty site with a nonzero field becomes polarized, and the polarized spin in turn polarizes its neighbors, thereby changing the local field. This additional field is the Onsager reaction field. Ma's theory for the reaction field in spin glasses [PRB 22, 4484 (1980)10.1103/PhysRevB.22.4484] has been extended to provide a spin-glass model that can account for the experimental data.     

Aging, rejuvenation and memory phenomena in spin glasses

In this paper, we review several important features of the out-of-equilibrium dynamics of spin glasses. Starting with the simplest experiments, we discuss the scaling laws used to describe the isothermal aging observed in spin glasses after a quench down to the low-temperature phase. We report in particular new results on the sub-aging behaviour of spin glasses. We then discuss the rejuvenation and memory effects observed when a spin glass is submitted to temperature variations during aging, from the point of view of both energy landscape pictures and real-space pictures. We highlight the fact that both approaches point out the necessity of hierarchical processes involved in aging. Finally, we report an investigation of the effect of small temperature variations on aging in spin glass samples with various anisotropies which indicates that this hierarchy depends on the spin anisotropy.     

Memory of multiple aging stages above the freezing temperature in the Relaxor ferroelectric PLZT

The dynamic dielectric susceptibility and the elastic compliance of the relaxor ferroelectric lanthanum lead zirconate titanate (PLZT) 9/65/35 have been measured under different cooling and heating protocols in order to study aging and memory. The memory of multiple aging stages at different temperatures has been found (several dips in the susceptibility curves on heating), as in spin glass systems below the glass transition. Remarkably, in PLZT the memory of several aging stages is retained also above the freezing temperature deduced from the dynamic susceptibilities. The results are discussed in light of the existing models of aging and memory in spin and dipolar glasses.     

Spin anisotropy and slow dynamics in spin glasses

We report on an extensive study of the influence of spin anisotropy on spin glass aging dynamics. New temperature cycle experiments allow us to compare quantitatively the memory effect in four Heisenberg spin glasses with various degrees of random anisotropy and one Ising spin glass. The sharpness of the memory effect appears to decrease continuously with the spin anisotropy. Besides, the spin glass coherence length is determined by magnetic field change experiments for the first time in the Ising sample. For three representative samples, from Heisenberg to Ising spin glasses, we can consistently account for both sets of experiments (temperature cycle and magnetic field change) using a single expression for the growth of the coherence length with time.     

New developments in the Nonequilibrium Dynamics of spin glasses

We present an overview of recent developments in the nonequilibrium dynamics of spin glasses. The prevailing theoretical pictures are described and discussed in light of numerical results and experimental observations. We focus on aging phenomena like waiting-time-dependent length scales, scaling of two-time quantities, more complex temperature protocols during aging, and the violation of the fluctuation–dissipation theorem. Finally, aging in quantum spin glass models coupled to a dissipative bath is discussed.     

The magnetic phase transition in amorphous ferromagnets and in spin glasses

The magnetic phase transition in materials with exchange disorder (amorphous ferromagnets, spin glasses) is discussed. In the critical temperature range the behavior of amorphous ferromagnetic transition metal-metalloid glasses is found to be similar to the one derived for a three-dimensional homogeneous Heisenberg ferromagnet. The most prominent difference between disordered and homogeneous materials is manifested in a large temperature range of deviations from the mean field behavior beyond the critical region, as observed experimentally for the temperature dependence of the linear susceptibility of amorphous ferromagnets and of the nonlinear susceptibility of spin glasses. A molecular field theory with correlations in space and time is developed, which relates the deviations from the mean field behavior to the interplay between the temperature dependent thermal correlations in the spin system and the spatial fluctuations of the material. Application to dynamical processes (kinetic critical slowing down) is discussed.     

Aging effects in an anderson insulator

Aging, commonly observed in glasses, is a manifestation of breakdown of time-translational invariance. Here we demonstrate experimentally aging effects in the electronic system of an Anderson insulator. The aging phenomenon in the electron glass appears to be much less sensitive to temperature than in other systems. The differences in the behavior of the electron glass and a spin glass system are discussed in terms of some microscopic differences between the two systems.     

Corrections to scaling in disordered systems with competing interactions

Using a mean-field effective-field model for disordered systems with competing interactions we have (a) obtained explicit expressions for the corrections to scaling for both ferromagnets and spin glasses up to order h³ in the magnetization, and (b) performed numerical calculations of the non-linear susceptibility of both ferromagnets and spin glasses as a function of reduced field and temperature. Scaling plots constructed from the numerical data show that the departures from universality are much more significant for spin glasses than they are for ferromagnets and that, while the discrepancy can be reduced through a redefinition of the scaling variables, a knowledge of the proper non-universal scaling correction is essential for spin glasses if a reasonable degree of universality is to be achieved.     

End of aging in a complex system

Aging phenomena in complex systems have been used as an important tool to investigate the physics of complexity. In particular, aging effects in spin glasses, measured using the thermoremanent magnetization decays, have been instrumental as a probe of complex equilibrium and nonequilibrium dynamics. In this Letter, we show that aging found in spin glass materials has a finite lifetime. After the aging part of the decay has ended, we find a post-aging decay which is apparently logarithmic in nature. This decay is independent of the waiting time and part of the same mechanism that produces aging.     

The spin wave excitations and the temperature dependence of the magnetization in iron,cobalt, nickel and their alloys

A critical review is given on the literature concerning spin waves in the ferromagnetic transition metals and their alloys. It is shown that spin waves can account for the low temperature behaviour of the magnetization in metallic glasses as well as in nickel(-alloys). A systematic correlation between Curie-temperature and spin wave stiffness is worked out, holding for crystalline and amorphous alloys of iron, cobalt and nickel. The reduced magnetization versus temperature curves of (amorphous) alloys are interpreted without referring to disorder effects.     

Exchange-mediated spin-lattice relaxation of Fe3+ ions in borate glasses

Spin-lattice relaxation times (T1) of two borate glasses doped with different concentrations of Fe2O3 were measured using the Electron Spin-Echo (ESE) technique at X-band (9.630 GHz) in the temperature range 2-6K. In comparison with a previous investigation of Fe3+-doped silicate glasses, the relaxation rates were comparable and differed by no more than a factor of two. The data presented here extend those previously reported for borate glasses in the 10-250K range but measured using the amplitude-modulation technique. The T1 values were found to depend on temperature (T) as T(n) with n approximately 1 for the 1% and 0.1% Fe2O3-doped glass samples. These results are consistent with spin-lattice relaxation as effected by exchange interaction of a Fe3+ spin exchange-coupled to another Fe3+ spin in an amorphous material.     

On the connection between spin glasses and gauge field theories

A simple connection between Ising spin glasses and the Z₂ lattice gauge theory, at negative plaquette temperatures, is presented. It is first shown that annealed models give useful lower bounds on the free energy and ground-state energy of spin glasses. However, they have unphysical low temperature properties (e.g. a negative entropy), which are related to a temperature dependence of the frustration. A restricted annealing scheme is presented which remedies this deficiency through the introduction of a pure gauge coupling counterterm. The possible phase diagrams of the lattice gauge system and their relevance to spin glass transitions are discussed.     

Aging and memory effects in a clathrate

The out-of-equilibrium low-frequency complex susceptibility of the orientational glass methanol(73%)-beta-hydroquinone-clathrate is studied using temperature-stop protocols in aging experiments. Although the material does not have a sharp glass transition aging effects including rejuvenation and memory similar to the effects in spin glasses are found at low temperatures.     

Aging phenomena in spin glasses and frustrated ferromagnets

An experimental protocol developed for spin glasses has been applied to a reentrant system. We compare the main features of the aging phenomena in both spin glass and ferromagnetic phases.     

Ultrametricity in three-dimensional edwards-anderson spin glasses

We perform an accurate test of ultrametricity in the aging dynamics of the three-dimensional Edwards-Anderson spin glass. Our method consists in considering the evolution in parallel of two identical systems constrained to have fixed overlap. This turns out to be a particularly efficient way to study the geometrical relations between configurations at distant large times. Our findings strongly hint towards dynamical ultrametricity in spin glasses, while this is absent in simpler aging systems with domain growth dynamics. A recently developed theory of linear response in glassy systems allows us to infer that dynamical ultrametricity implies the same property at the level of equilibrium states.     

Complexity of vector spin glasses

We study the annealed complexity of the m-vector spin glasses in the Sherrington-Kirkpatrick limit. The eigenvalue spectrum of the Hessian matrix of the Thouless-Anderson-Palmer free energy is found to consist of a continuous band of positive eigenvalues in addition to an isolated eigenvalue and (m-1) null eigenvalues due to rotational invariance. Rather surprisingly, the band does not extend to zero at any finite temperature. The isolated eigenvalue becomes zero in the thermodynamic limit, as in the Ising case (m=1), indicating that the same supersymmetry breaking recently found in Ising spin glasses occurs in vector spin glasses.     

Field-induced parameters of re-entrant magnets and concentrated spin glasses

We have used electron spin resonance measurements to derive the temperature and frequency dependences of the field-induced magnetization [M(T, f)] and anisotropy field [H_an (T)] in a number of amorphous alloys belonging to the series (Fe_pNi_1−p)₇₅P₁₆B₆Al₃. In re-entrant (p > p_c, the critical concentration for ferromagnetism) alloys at hi gh frequencies (f = 35 GHz, field ≈ 12 kOe) M reduces as T^3/2 at high T and as T below ≈ 40 K, the deviation from T^3/2 becoming more marked as p → p⁺_c. For p close to p_c, lowering the frequency first causes the T term to increase and ultimately ( ≈ 4 GHz) changes the variation of M with T to that discovered previously for concentrated spin glasses, namely M is constant at low T and drops linearly at high temperatures. For the re-entrants, the results are interpreted on the basis of a model which invokes an energy gap in the spin-wave spectrum, introduces a non-zero density of states of the gap energy and takes into consideration a low-q cut-off in the spin-wave integral in thelow-T (T) regime.In the concentrated spin glasses [M (0) - M (T)]/ M (0) is well represented by the function [exp (Δ / T) - 1]^-1, where Δ has values close to the corresponding Curie-Weiss temperatures θ_p but much larger than the respective spin glass transition temperatures T_SG. The temperature dependence of H_an is largely given by the function (1 - T/T^*), where T^* is equal to the zero-field freezing temperature for the re-entrants and T_SG for the spin glasses, respectively.     

Spectral distribution of relaxation times in spin glasses

Recent neutron scattering measurements on spin glasses show that the dynamics of the spin systems can be best described in terms of wide spectral distribution of relaxation times evolving continuously with decreasing temperature but which is devoid of any critical behaviour, either speeding up or slowing down, at any finite temperature including the spin glass “freezing temperature T_sg”. It is argued that the latter temperature itself is dependent on the time constant of measurement for all spin glasses in general; the observed variation with frequency being less pronounced in some systems than others owing to some special characteristics of their spin dynamics such as, for example, the presence of parallel channels of rapid relaxation provided by the Korringa coupling in metallic spin glasses. The neutron scattering measurements presented here enable us to propose plausible forms for the density of relaxation times of the spin system and to show that the logarithmic frequency dependence of the freezing temperature observed in low frequency ac susceptibility measurements follows naturally from a uniform density of relaxation times at these frequencies.     

The temperature dependence of the magnetic susceptibility of spin glasses

The temperature dependence of the magnetic susceptibility of spin glasses is calculated exactly on the basis of a magnetic cluster model, following Néel. The susceptibility of a single cluster below its blocking temperature is taken to be negligibly small compared to its value above. A distribution of blocking temperatures may be obtained from observations on spin glasses. Conditions for the sharpness of the susceptibility peak are discussed and the low-temperature variation of the susceptibility is also obtained.