Dimensional Ising Spin Glass Research Articles

Numerical study of the dynamics of some long range spin glass models

We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling () gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.

Observation of spin glass behavior in Ba0.8La0.2Ti0.8Co0.2O3

The structural and magnetic properties of Ba0.8La0.2Ti1−xCoxO3 (0 ≤ x ≤ 0.2) have been investigated. All samples show a cubic structure with the space group of Pm3m at room temperature. Ba0.8La0.2TiO3 exhibits a carrier induced Pauli-paramagnetism and Ba0.8La0.2Ti0.9Co0.1O3 shows a Curie-Weiss like paramagnetism. With increasing cobalt doping, Ba0.8La0.2Ti0.8Co0.2O3 seems to undergo a spin glass like transition at Tf = 10.1 K as evidenced by the frequency dependence of ac susceptibility measurements, aging effect measurements, memory effect measurements and magnetic relaxation measurements. The time decay of the thermoremnant magnetization can be well described with the stretched exponential function. It is worthwhile to note that the temperature dependence of relaxation characteristic time τr and the critical exponent n deduced from the function show opposite tendencies below and above Tf, and the possible reason has been discussed. n at Tf close to 2/3 suggests that Ba0.8La0.2Ti0.8Co0.2O3 may be a three dimensional Ising spin glass.

Study of the phase transition in the 3D Ising spin glass from out-of-equilibrium numerical simulations

Using the decay of the out equilibrium spin-spin correlation function we compute the equilibrium Edward-Anderson order parameter in the three dimensional binary Ising spin glass in the spin glass phase. We have checked that the Edward-Anderson order parameter computed from out of equilibrium numerical simulations follows with good precision the critical law as determined in experiments and in numerical studies at equilibrium. We have also studied the dependence of the order parameter with the lattice size. Finally we present a large time study of the scaling of the off-equilibrium fluctuation-dissipation relations.

Generating droplets in two-dimensional Ising spin glasses using matching algorithms

We study the behavior of droplets for two dimensional Ising spin glasses with Gaussian interactions. We use an exact matching algorithm which enables study of systems with linear dimension L up to 240, which is larger than is possible with other approaches. But the method only allows certain classes of droplets to be generated. We study single-bond, cross and a category of fixed volume droplets as well as first excitations. By comparison with similar or equivalent droplets generated in previous works, the advantages but also the limitations of this approach are revealed. In particular we have studied the scaling behavior of the droplet energies and droplet sizes. In most cases, a crossover of the data can be observed such that for large sizes the behavior is compatible with the one-exponent scenario of the droplet theory. Only for the case of first excitations, no clear conclusion can be reached, probably because even with the matching approach the accessible system sizes are still too small.

Scalings of domain wall energies in two dimensional Ising spin glasses.

We study domain wall energies of two dimensional spin glasses. The scaling of these energies depends on the model's distribution of quenched random couplings, falling into three different classes. The first class is associated with the exponent theta approximately -0.28; the other two classes have theta=0, as can be justified theoretically. In contrast to previous claims, we find that theta=0 does not indicate d=d(c)(l) but rather d< or =d(c)(l), where d(c)(l) is the lower critical dimension.

Off-equilibrium fluctuation-dissipation relations in the3dIsing spin glass in a magnetic field

We study the fluctuation-dissipation relations for a three dimensional Ising spin glass in a magnetic field both in the high temperature phase as well as in the low temperature one. In the region of times simulated we have found that our results support a picture of the low temperature phase with broken replica symmetry, but a droplet behavior can not be completely excluded.

State hierarchy induced by correlated spin domains in short-range spin glasses

We generate equilibrium configurations for the three and four dimensional Ising spin glass with Gaussian distributed couplings at temperatures well below the transition temperature T_c. These states are analyzed by a recently proposed method using clustering. The analysis reveals a hierarchical state space structure. At each level of the hierarchy states are labeled by the orientations of a set of correlated macroscopic spin domains. Our picture of the low temperature phase of short range spin glasses is that of a State Hierarchy Induced by Correlated Spin domains (SHICS). The complexity of the low temperature phase is manifest in the fact that the composition of such a spin domain (i.e. its constituent spins), as well as its identifying label, are defined and determined by the ``location'' in the state hierarchy at which it appears. Mapping out the phase space structure by means of the orientations assumed by these domains enhances our ability to investigate the overlap distribution, which we find to be non-trivial. Evidence is also presented that these states may have a non-ultrametric structure.

Relaxation of the field-cooled magnetization of an Ising spin glass

The time and temperature dependence of the field-cooled magnetization of a three dimensional Ising spin glass, Fe_{0.5}Mn_{0.5}TiO_{3}, has been investigated. The temperature and cooling rate dependence is found to exhibit memory phenomena that can be related to the memory behavior of the low frequency ac-susceptibility. The results add some further understanding on how to model the three dimensional Ising spin glass in real space.

Evidence for the Droplet Picture of Spin Glasses

We have studied the Parisi overlap distribution for the three dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures T around 0.7Tc and system sizes upto L=32, we found a P(q) as expected for the full Parisi replica symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behaviour in Monte Carlo simulations is due to the fact that all existing simulations have been done at temperatures too close to the transition temperature so that sytem sizes larger than the correlation length have not been achieved.

Violation of the fluctuation-dissipation theorem in finite-dimensional spin glasses

We study the violation of the fluctuation-dissipation theorem in the three and four dimensional Gaussian Ising spin glasses using on and off equilibrium simulations. We have characterized numerically the function X(C) that determine the violation and we have studied its scaling properties. Moreover we have computed the function x(C) which characterize the breaking of the replica symmetry directly from equilibrium simulations. The two functions are numerically equal and in this way we have established that the conjectured connection between the violation of fluctuation dissipation theorem in the off-equilibrium dynamics and the replica symmetry breaking at equilibrium holds for finite dimensional spin glasses. These results point to a spin glass phase with spontaneously broken replica symmetry in finite dimensional spin glasses.

Chaos and universality in a four-dimensional spin glass

We present a finite size scaling analysis of Monte Carlo simulation results on a four dimensional Ising spin glass. We study chaos with both coupling and temperature perturbations, and find the same chaos exponent in each case. Chaos is investigated both at the critical temperature and below where it seems to be more efficient (larger exponent). Dimension four seems to be above the critical dimension where chaos with temperature is no more present in the critical region. Our results are consistent with the Gaussian and bimodal coupling distributions being in the same universality class.

Mean field dynamical exponents in finite-dimensional Ising spin glass

We report the value of the dynamical critical exponent z for the six dimensional Ising spin glass, measured in three different ways: from the behavior of the energy and the susceptibility with the Monte Carlo time and by studying the overlap-overlap correlation function as a function of the space and time. All three results are in a very good agreement with the Mean Field prediction z=4. Finally we have studied numerically the remanent magnetization in 6 and 8 dimensions and we have compared it with the behavior observed in the SK model, that we have computed analytically.

On the approximability of the `ground-state' problem for Ising spin glasses by quantum devices

A lower bound result about the approximability of the energy function of Ising spin glasses in the quantum context is determined. The ground-state problem for the three- dimensional Ising spin glasses represented on two-level grids such that the vertical interactions are at most p n (where n is the number of spins set on the grid), is considered. We prove that, unless NP BQP, there is a constant >0 such that every quantum approximate polynomial- time algorithm finds a solution with absolute error greater than p n, with high probability, infinitely often.

Chaos in a two-dimensional Ising spin glass

We study chaos in a two dimensional Ising spin glass by finite temperature Monte Carlo simulations. We are able to detect chaos with respect to temperature changes as well as chaos with respect to changing the bonds, and find that the chaos exponents for these two cases are equal. Our value for the exponent appears to be consistent with that obtained in studies at zero temperature.

Ordering Temperatures and Critical Exponents in Ising Spin Glasses.

We propose a numerical criterion which can be used to obtain accurate and reliable values of the ordering temperatures and critical exponents of spin glasses. Using this method we find an value of the ordering temperature for the $\pm J$ Ising spin glass in three dimensions which is definitely non-zero and in good agreement with previous estimates. We show that the critical exponents of three dimensional Ising spin glasses do not appear to obey the usual universality rules.

Theory of the Critical State of Low-Dimensional Spin Glass

We analyse the critical region of finite-($d$)-dimensional Ising spin glass, in particular the limit of $d$ closely above the lower critical dimension $d_\ell$. At criticality the thermally active degrees of freedom are surfaces (of width zero) enclosing clusters of spins that may reverse with respect to their environment. The surfaces are organised in finite interacting structures. These may be called {\em protodroplets}\/, since in the off-critical limit they reduce to the Fisher and Huse droplets. This picture provides an explanation for the phenomenon of critical chaos discovered earlier. It also implies that the spin-spin and energy-energy correlation functions are multifractal and we present scaling laws that describe them. Several of our results should be verifiable in Monte Carlo studies at finite temperature in $d=3$.

Thermo-cycling experiments with the three-dimensional Ising spin glass model

A characteristic feature of the non--equilibrium dynamics of real spin glasses at low temperatures are strong aging effects. These phenomena can be manipulated by changing the external parameters in various ways: a thermo-cycling experiment consists for instance of a short heat pulse during the waiting time, by which the relaxation might be strongly affected. Results of numerical experiments of this kind, performed via Monte Carlo simulations of the three dimensional Ising spin glass model, are presented. The theoretical implications are discussed and the scenario found is compared with the experimental situation.

Nonequilibrium dynamics and aging in the three-dimensional Ising spin-glass model

The low temperature dynamics of the three dimensional Ising spin glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function $C(t,t_w)$ shows a crossover at the waiting (or {\em aging}) time $t_w$ from algebraic {\em quasi-equilibrium} decay for times $t$$\ll$$t_w$ to another, faster algebraic decay for $t$$\gg$$t_w$ with an exponent similar to one for the remanent magnetization.

The four dimensional Ising spin glass: A Monte Carlo study (invited)

We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.

On the magnetisation of the ground states in two dimensional Ising spin glasses

We study the configurations of minimum energy among those with zero magnetisation in Ising spin glasses. We formulate that problem as a graph partitioning problem and we give an algorithm for it. We determine the concentration of negative interactions such that those configurations give a ground state.