Short-range Spin Glasses Research Articles

Coarse graining of a spin-glass state space

The complex structure of a spin-glass state space can be simplified by a coarse-graining procedure, i.e. microscopic states being assembled into larger clusters. An algorithm for the coarse graining of the state space of a short-range Ising spin glass is provided, which is the basis of a coarse-grained dynamics. Different ways for modelling the transition rates in the coarse-grained state space are discussed. A comparison with the dynamics of the microscopic system shows that the dynamics in the coarse-grained state space gives an appropriate approximation.

Magnetoconductance Noise and Irreversibilities in Submicron Wires of Spin-Glassn+−Cd1−xMnxTe

Signatures of spin-glass freezing such as the appearance of $1/f$ conductance noise, the recovery of universal conductance fluctuations, aging, as well as magnetic and thermal irreversibilities, are detected in mesoscopic wires of ${\mathrm{Cd}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}\mathrm{Te}$:I at millikelvin temperatures. Spectral characteristics of conductance time series are consistent with the droplet model of short-range spin glasses.

The state space of short-range Ising spin glasses: the density of states

The state space of finite square and cubic Ising spin glass models is analysed in terms of the global and the local density of states. Systems with uniform and Gaussian probability distribution of interactions are compared. Different measures for the local state density are presented and discussed. In particular, the question whether the local density of states grows exponentially or not is considered. The direct comparison of global and local densities leads to consequences for the structure of the state space.

Evidence for a genuine ferromagnetic to paramagnetic reentrant phase transition in a Potts spin-glass model

uch experimental and theoretical efforts have been devoted in the past twenty years to search for a genuine thermodynamic reentrant phase transition from a ferromagnetic to either a paramagnetic or spin glass phase in disordered ferromagnets. So far, no real system or theoretical model of a short-range spin glass system has been shown to convincingly display such a reentrant transition. We present here results from Migdal-Kadanoff real-space renormalization-group calculations that provide for the first time strong evidence for ferromagnetic to paramagnetic reentrance in Potts spin glasses on hierarchical lattices. Our results imply that there is no fundamental reason ruling out thermodynamic reentrant phase transitions in all non mean-field randomly frustrated systems, and may open the possibility that true reentrance might occur in some yet to be discovered real randomly frustrated materials.

Long-Time Decay of Thermoremanent Magnetization through Stepwise Changes of the Reentrant Spin-Glass-Like State in the Ni2In-Type Mn7Sn4

The zero-field-cooled magnetization at the magnetic field H = 300 Oe shows anomolous kinks at T A and T e (> T A ) far below the reentrant spin-glass-like transition temperature T f (above which a long-range ferrimagnetic ordering is set up). The irreversibility between the zero-field-cooled and the field-cooled magnetizations, and the time decay of the thermoremanent magnetization (σTRM) are observed below the ferrimagnetic Curie temperature T C . The decay feature can be described by Ogielski's power law σTRM = At -a whose a exhibits peaks at T A , T B , T f and T C . The peak of α at Te can be fitted by the combination of Ising type and Ginzburg-Landau type expressions. Three stepwise changes of the spin-glass-like state below T f and the feature of the coexistence of short-range spin-glass and long-range magnetic ordering components above T f are discussed.

Ordered phase and field-induced domains in a short-range Ising spin glass: Fe0.25Zn0.75F2

Using a local mean-field numerical approach to describe disordered antiferromagnets, with application to FexZn1−xF2, it is shown that the characteristics of the spin glass phase found for x=0.25 is much in agreement with the scenario predicted by the scaling theory of the droplet model. In particular, the spin-reversal domains induced by a magnetic field give rise to a metastable irreversibility line.

Short-range Ising spin glass: Multifractal properties

The multifractal properties of the Edwards-Anderson order parameter of the short-range Ising spin glass model on d=3 diamond hierarchical lattices is studied via an exact recursion procedure. The profiles of the local order parameter are calculated and analysed within a range of temperatures close to the critical point with four symmetric distributions of the coupling constants (Gaussian, Bimodal, Uniform and Exponential). Unlike the pure case, the multifractal analysis of these profiles reveals that a large spectrum of the $\alpha $-H\"older exponent is required to describe the singularities of the measure defined by the normalized local order parameter, at and below the critical point. Minor changes in these spectra are observed for distinct initial distributions of coupling constants, suggesting an universal spectra behavior. For temperatures slightly above T_{c}, a dramatic change in the $F(\alpha)$ function is found, signalizing the transition.

ABOUT THE OVERLAP DISTRIBUTION IN MEAN FIELD SPIN GLASS MODELS

We continue our presentation of mathematically rigorous results about the Sherrington-Kirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica symmetry breaking. As a byproduct, we show that the self-averaging of the Edwards-Anderson fluctuating order parameter, with respect to the external quenched noise, implies that the overlap distribution is given by the Sherrington-Kirkpatrick replica symmetric Ansatz. This extends previous results of Pastur and Scherbina. Finally, we show how to generalize our results to realistic short range spin glass models.

Dynamic Fluctuations in a Short-range Spin Glass Model

We study the dynamic fluctuations of the soft-spin version of the Edwards-Anderson model in the critical region for $T\rightarrow T_{c}^{+}$. First we solve the infinite-range limit of the model using the random matrix method. We define the static and dynamic 2-point and 4-point correlation functions at the order $O(1/N)$ and we verify that the static limit obtained from the dynamic expressions is correct. In a second part we use the functional integral formalism to define an effective short-range Lagrangian $L$ for the fields $\delta Q^{\alpha\beta}_{i}(t_{1},t_{2})$ up to the cubic order in the series expansion around the dynamic Mean-Field value $\overline{{Q}^{\alpha\beta}}(t_{1},t_{2})$. We find the more general expression for the time depending non-local fluctuations, the propagators $[\langle\delta Q^{\alpha\beta}_{i}(t_{1},t_{2}) \delta Q^{\alpha\beta}_{j}(t_{3},t_{4})\rangle_{\xi}]_{J}$, in the quadratic approximation. Finally we compare the long-range limit of the correlations, derived in this formalism, with the correlations of the infinite-range model studied with the previous approach (random matrices).

Short-range Potts spin-glass model: Renormalization-group method.

We study the phase diagram of the short-range Potts spin glass within the Migdal-Kadanoff renormalization group of the hypercubic lattices. Using some analytical approaches and performing the real-space renormalization group of the problem we show that the model, for q\ensuremath{\gtrsim}2, freezes into a Potts spin-glass phase at finite temperature, ${\mathit{T}}_{\mathrm{SG}}$\ensuremath{\ne}0, in d=4. This indicates an algebraic power-law decay of the correlation functions in this low-temperature phase. We show that the Potts spin-glass phase does not occur in d=3 for q\ensuremath{\gtrsim}2 and calculate the lower critical dimension for all continuous values of q. The specific-heat critical exponent is found to be large and negative. \textcopyright{} 1996 The American Physical Society.

A Puzzle on Fluctuations of Weights in Spin Glasses

In certain mean field models for spin glasses there occurs a one step replica symmetry breaking pattern. As an example of general $1/N$-corrections in such systems, the fluctuations in the internal energy are calculated. For this specific quantity the outcome is known from the specific heat. It is shown that the correct result can be derived by assuming that the both the height and the location of the breakpoint fluctuate. This effect enlarges the commonly considered space of allowed fluctuations of the order parameter in loop calculations for short range spin glasses of this type. The phenomenon does not occur in spin glasses with infinite order replica symmetry breaking.

Magnetic phase diagram, static properties, and relaxation of the insulating spin glass CoCl2

The magnetic properties of ${\mathrm{CoCl}}_{2}$\ensuremath{\cdot}${\mathrm{H}}_{2}$O are examined by dc magnetization measurements as a function of temperature and field. This insulating system has both ferromagnetic and antiferromagnetic interactions, the distribution of which may be randomized to some degree by removal of one water from the parent system ${\mathrm{CoCl}}_{2}$\ensuremath{\cdot}${2\mathrm{H}}_{2}$O. It behaves as a reentrant spin glass. A field-induced transition is observed in the 4--6 kG range, a much smaller field than the 31.3-kG metamagnetic transition in ${\mathrm{CoCl}}_{2}$\ensuremath{\cdot}${2\mathrm{H}}_{2}$O. The nonlinear susceptibility displays an unusual variation with temperature, but does not scale in the conventional way. The thermoremanent magnetization (TRM) is examined as a function of cooling field and temperature. Its time development approximates a stretched exponential decay and the parameters characterizing this are analyzed. It is also found that the TRM scales well according to a T ${\mathrm{log}}_{10}$(t/${\mathrm{\ensuremath{\tau}}}_{0}$) form, for temperatures not much above \ensuremath{\sim}0.7${\mathit{T}}_{\mathit{g}}$, with ${\mathrm{\ensuremath{\tau}}}_{0}$\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}10--}$${10}^{\mathrm{\ensuremath{-}}9}$ s. Irreversibility lines are derived from measurements of field-cooled and zero-field-cooled magnetizations. A weak irreversibility sets in at ${\mathit{T}}_{\mathit{g}}$(H=0)=9.0 K and displays no detectable field dependence up to 1 kG. A strong irreversibility sets in at ${\mathit{T}}_{\mathit{g}}$(H=0)=8.4 K and displays a strong field dependence. The results are analyzed using theories which consider anisotropy. The strong irreversibility line conforms much better to a (reduced) form ${\mathrm{\ensuremath{\tau}}}_{\mathit{g}}$\ensuremath{\propto}${\mathit{h}}^{0.53}$, recently proposed for the 3D Ising short-range spin glass, than to the usual De Almeida--Thouless mean-field form ${\mathrm{\ensuremath{\tau}}}_{\mathit{g}}$\ensuremath{\propto}${\mathit{h}}^{2/3}$.

Equilibrium behaviour of quantum Ising spin glass

A phenomenological theory of the ordered phase of short-range quantum Ising spin glass is developed in terms of droplet excitations, and presented in detail. These excitations have free energies that provene from an interplay between the classical excitation energies ϵ L , with a broad distribution whose characteristic magnitude grows with length scale L as L θ , and quantum tunneling rates Γ L , decreasing faster than exponentially with L. At temperature T = 0, the equal-time spatial correlations due to quantum fluctuations do not show the power-law decay that occurs for T > 0 due to thermal fluctuations, but ather an exponential decay of Ornstein-Zernicke type. At finite, but still very small T, there exists a crossover length scale L ∗ (T), set by ħΓ L ∗(T) ⋍ k B T , below which the droplets behave quantum-mechanically and above which the behaviour is essentially classical. As for the classical spin glass, only a small fraction of droplets are thermally active at low T; it is shown that many of the low- T static properties are dominated by the thermally active droplets at the crossover length L ∗(T) . The uniform static linear susceptibility is found to diverge at T = 0 below the lower critical dimension, d l , and to be finite above d l . The static nonlinear susceptibility diverges in all dimensions, d. The zero temperature linear ac susceptibility χ(ω) is dominated by droplets at a length scale such that ω is of order the characteristic frequency of the quantum system. The behaviour near to the quantum critical point is discussed within a conventional scaling framework if it is approached at strictly zero temperature as well as from finite T. Implications of the existence of Griffiths singularities at the critical point and the disordered phase are pointed out: In the disordered phase, Griffiths singularities dominate the low- T specific heat and the long-time correlations.

Local phase-space structure and low-temperature dynamics of short-range Ising spin glasses.

We study the phase-space geometry and the low-temperature relaxation of short-range Gaussian Ising spin-glasses, by finding all the configurations and their interconnection within a ``pocket'' of states surrounding a low-lying energy minimum. Thermalization within the pocket is modeled by a master equation, which is solved numerically, yielding the time-dependent propagator. The findings show that the relaxation proceeds by a series of local equilibrations in nested ``valleys,'' and support the dynamical predictions of hierarchical relaxation models.

"Valley structures" in the phase space of a finite 3D Ising spin glass with +or-I interactions

Exact results for ground states and low-lying excitations of short-range Ising spin glass systems in three dimensions are presented. Using an exact method of nonlinear discrete optimization valley structures in the phase space can be analysed for finite systems. The existence of non-trivial breaking of ergodicity at zero temperature is shown by arranging the highly degenerate ground states in several valleys, which are connected only by the excited states.

Dynamical phase diagram of the van Hemmen model of spin glass

The dynamical behaviour of the van Hemmen model of spin glass is studied by comparing the time evolution of two configurations subjected to the same thermal noise. It was found that the system undergoes several dynamical phase transitions among the paramagnetic, ferromagnetic, spin-glass and mixed phases, The dynamical phase diagram seems to be in good agreement with the static one. In the spin glass phase, the system presents a quite different dynamical behaviour from that found in short-range spin glass models as well as in the Sherrington-Kirkpatrick model.

Aging in two-dimensional Ising spin glasses.

Aging of two-dimensional (2D) Ising spin glasses with short-range interactions ${\mathrm{Rb}}_{2}$${\mathrm{Cu}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Co}}_{\mathit{x}}$${\mathrm{F}}_{4}$ (${\mathit{T}}_{\mathit{c}}$=0) is studied in the critical regime via the low-frequency ac susceptibility and via the dc magnetization for x=0.22 and 0.33. After quenching to low temperatures, slow logarithmic relaxation is observed for both the in-phase and out-of-phase ac susceptibilities. Using the droplet scaling model, the spin-glass correlation length R is concluded to increase after a temperature quench as R\ensuremath{\propto}(lnt${)}^{1/\mathrm{\ensuremath{\psi}}}$, with t in units of some microscopic time. The exponent \ensuremath{\psi}=1.0\ifmmode\pm\else\textpm\fi{}0.1 compares with recent theoretical results and numerical simulations for 2D short-range Ising spin glasses. The waiting-time dependence found for the field-cooled and zero-field-cooled dc magnetizations points to a markedly slower crossover from the quasiequilibrium to the nonequilibrium regime than in 3D. From experiments with different cooling histories, finally, the equilibrium spin-glass state is inferred to be strongly dependent on the temperature.

Dynamic scaling in Cd1-xMnxTe spin glasses: A probe of the spin autocorrelation function.

The dynamic scaling relations for the frequency-dependent magnetic susceptibility are derived, using the spin autocorrelation function q(t)\ensuremath{\propto}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathit{a}}$ exp[-(t/\ensuremath{\tau}${)}^{\mathit{b}}$], which successfully describes results from Monte Carlo simulations on a three-dimensional Ising spin-glass model. This functional form accounts very well for the observed frequency and temperature dependences of the imaginary component of the ac magnetic susceptibility in ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Te, above the spin-glass transition temperature ${\mathit{T}}_{\mathit{c}}$. The parameters a and b obtained from the scaling analysis are very comparable to those found for ${\mathrm{Fe}}_{0.5}$${\mathrm{Mn}}_{0.5}$${\mathrm{TiO}}_{3}$, close to ${\mathit{T}}_{\mathit{c}}$, emphasizing the similar critical behavior of short-range insulating spin glasses, of Heisenberg and Ising type.

Multiple states and thermodynamic limits in short-ranged Ising spin-glass models

We propose a test to distinguish, both numerically and theoretically, between the two competing pictures of short-ranged Ising spin glasses at low temperature: ``chaotic'' size dependence. Scaling theories in which at most two pure states (related by a global spin flip) occur require that finite-volume correlations (with, say, periodic boundary conditions) have a well-defined thermodynamic limit. We argue, however, that the picture based on the infinite-ranged Sherrington-Kirkpatrick model, with many noncongruent pure states, leads to a breakdown of the thermodynamic limit. The argument combines rigorous and heuristic elements; one of the fomer is a proof that in the infinite-ranged model itself, non-self-averaging implies chaotic size dependence. Numerical tests, based on chaotic size dependence, could provide a more sensitive measure than the usual overlap distribution P(q) in determining the number of pure states.

New critical-point exponent and new scaling laws for short-range Ising spin glasses.

We investigate the chaotic temperature dependence of the renormalized couplings in a short-range Ising spin glass. Some of our analytic and numerical results, obtained by the Migdal-Kadanoff renormalization scheme but argued to have general validity, differ from those of the scaling and droplet theory. First, chaos is present also at and above the critical temperature. Second, between ${\mathit{d}}_{\mathit{l}}$=2.46 (the lower critical dimension) and ${\mathit{d}}_{+}$=3.4, the chaos in the critical region is characterized by a new critical exponent ${\mathrm{\ensuremath{\zeta}}}_{\mathit{c}}$ that has hitherto escaped attention.