Interface energies in ising spin glasses

The replica method has been used to calculate the interface free energy associated with the change from periodic to antiperiodic boundary conditions in finite-dimensional spin glasses. At mean-field level the interface free energy vanishes, but after allowing for fluctuation effects, a nonzero interface free energy is obtained which is significantly different from numerical expectations.     

Magnetic correlations in three-dimensional ising spin glasses

By a recursive method numerically exact free energies are calculated forL×L×M Ising lattices with random bonds andL=4, 4M10, applying free boundaries in the direction where the lattice is less small and otherwise periodic boundary conditions. Both for the±J model and the gaussian model the specific heat is in fair agreement with Monte Carlo results obtained for much larger lattices. However, the correlation function [S ₀ S _{R T} ² ]_av is found to decay exponentially with distanceR [for 1R9] at temperatures far below the apparent freezing temperatures of the Monte Carlo simulations, implying that there is no nonzero Edwards-Anderson order parameter in equilibrium. This behavior is qualitatively different from Mattis spin glasses (or Ising ferromagnets) where even smaller lattices show pronounced magnetic order at low temperatures. As the Monte Carlo results give evidence for a nonzero Edwards-Anderson order parameter (for not too long observation times), which is fairly independent of lattice size down to sizes of 4³, we suggest that Edwards-Anderson ordering is a nonequilibrium phenomenon visible only in studying dynamic properties.     

Comparative monte Carlo study of ising spin glasses in two to five dimensions

Monte Carlo simulations of the Edwards-Anderson-Ising spin glass with Gaussian distribution of nearest-neighbor exchange forces in four and five dimensions are performed to check the speculation thatd=4 is the lower critical dimensionality. In contrast to this expectation we find no qualitative difference at all to the results in two and three dimensions. We still find that on not too long time-scales there is an apparently rather well defined freezing temperatureT _f , where the susceptibility has a cusp, and belowT _f nonzero order parametersq, can be found as ford=2, 3. But even ford=5 the decay of the Edwards-Anderson order parameter belowT _f is found to be consistent with a logarithmic variation over several decades of observation time. The possible interpretations of this result are discussed. Our data thus suggest that either there is no equilibrium phase transition in all these dimensions, or more likely that a phase transition exists for 2d5 but the properties of the ordered phase may be rather peculiar.Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Nature of ergodicity breaking in ising spin glasses as revealed by correlation function spectral properties

In this Letter we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions C(ij) identical with. We argue that more than one extensive [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking. Monte Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.     

Random anisotropy in ising spin glass

The theory of Ghatak and Sherrington for the Ising spin glass with uniform uniaxial anisotropy has been extended for the random uniaxial anisotropy (RUA). The magnitude of the RUA is assumed to fluctuate randomly with gaussian probability distribution. The susceptibility and the specific heat have been calculated for a system of spin one and zero mean values of the anisotropy and the exchange interaction. It is found that the random fluctuations of the anisotropy produce a cusp in the susceptibility and a broad maximum in the specific heat.     

Dynamics of the infinite-range ising spin glass

The Glauber dynamics of an Ising spin glass with infinite-range interactions and additional static field, h, is investigated near the freezing temperature, T_f. We obtain critical slowing down at and below the de Almeida-Thouless instability line, h_c(T), to order (1?T/T_f)³ with algebraic decay of the spin correlations ～t^?ν, where $\text{ν=}\text{1}\text{2}$ at T_f and $\text{ν≤}\text{1}\text{2}$ for T<T_f.     

GKS inequalities for arbitrary spin ising ferromagnets

Elementary proofs of the first and second Griffiths-Kelly-Sherman (GKS) inequalities are given for higher-spin Ising systems with a Hamiltonian containing only a quadratic form in the spin variables and integer powers of single spin variables. These proofs are obtained using Gaussian random variables. A slight generalization of previous results has been obtained in that the coefficients of the even powers of the spin variables are allowed to be negative.Work supported by NSF Grant GP-36564X.     

Properties of metastable ising models evolving under the Swendsen-Wang dynamics

The behavior of the metastable nearest neighbor Ising model governed by Swendsen-Wang dynamics (SW) is investigated ind=2. The results are compared to those obtained in standard Metropolis dynamics. Both the SW and Metropolis systems are observed to decay from the metastable state via the formation of nucleating droplets. Nucleation rates are measured and found to agree with those predicted by classical nucleation theory. The growth rates of the droplets are observed to differ between the two dynamics. In addition, the dynamic critical exponentz is measured in a mean-field (Curie-Weiss) metastable Ising model at the spinodal. It is found that for SW dynamics,z=2. Since this is the same value as that obtained in the Metropolis case, this result shows that SW does not change the dynamical universality class at the spinodal.     

Ultrasound in spin glasses

The change of the sound velocity v(,T) and the damping of sound waves (,T) in spin glasses are calculated in the frame-work of an Ising model with a random distribution of exchange interactions. The calculation is based on linearized equations of motion for the spins and on an improved mean field approximation which includes the Onsager reaction field. Near to the freezing temperatureT _f and at high temperatures v(,T) and (,T) turn out to be approximately proportional to the real and the imaginary parts of the dynamical susceptibility. For the special case of infinite range interactions atT=T _f one has v(, T_f) ( )^1/2 and (, T_f) (/)^1/2 where is the relaxation time of independent spins. However, already slightly aboveT _f the frequency dependence of both quantities becomes rather small for RKKY spin glasses. At high temperatures both, v(,T) and (,T) vary asT ^–1.SFB 125 Aachen-Jülich-Köln     

Numerical results for ground states of spin glasses on Bethe lattices

The average ground state energy and entropy for ±J spin glasses on Bethe lattices of connectivities k + 1 = 3..., 26 at T = 0 are approximated numerically. To obtain sufficient accuracy for large system sizes (up to n = 2¹²), the Extremal Optimization heuristic is employed which provides high-quality results not only for the ground state energies per spin e_k+1 but also for their entropies s_k+1. The results indicate sizable differences between lattices of even and odd connectivities. The extrapolated ground state energies compare very well with recent one-step replica symmetry breaking calculations. These energies can be scaled for all even connectivities k + 1 to within a fraction of a percent onto a simple functional form, e _{k + 1} = E _SK - (2E _SK + )/, where E _SK = - 0.7633 is the ground state energy for the broken replica symmetry in the Sherrington-Kirkpatrick model. But this form is in conflict with perturbative calculations at large k + 1, which do not distinguish between even and odd connectivities. We also find non-zero entropies per spin s_k+1 at small connectivities. While s_k+1 seems to vanish asymptotically with 1/(k + 1) for even connectivities, it is numerically indistinguishable from zero already for odd k + 1 ≥ 9. Received 9 August 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: sboettc@emory.edu www.physics.emory.edu/faculty/boettcher     

Block spin techniques applied to the ising model

A detailed study of the critical behaviour of the Ising model in 1+1 and 2+1 dimensions is made using an approximate real space renormalization transformation which involves block spins. The critical indices α, β, η, and ν are calculated and compared with previous results, as is the critical couplingy _c. The method is shown to respect one of the scaling relations, and in 1+1D some exact results are reproduced (y _c=1, ν=1).     

Insulating spin glasses

The possibility of obtaining spin glasses by addition of impurities in an antiferromagnetic insulator is examined. Dipolar interactions are briefly considered but the attention is focussed on Heisenberg systems. Equivalence with the Edwards-Anderson model is derived in a theoretical case. Experimental realisations, such as quasi-one dimensional systems, and spinels, are reviewed. A weak concentration of non-magnetic impurities can give rise to a new state that we call semi spin glass, in which a ferromagnetic component coexists with a transverse, spin glass component. An important case is when the pure system has a high ground state degeneracy (cooperative paramagnet). Non-magnetic impurities or other forms of disorder can transform it into a spin glass.     

Thermopower of spin glasses

We calculate the thermopowerS _d(T) of spin glasses on the basis of thesd exchange model with additional interactions between the impurity spins by means of time dependent perturbation theory. Similarly to the Kondo effect, the observed giant thermopower can only be explained if one also takes into account the spin independent interactionV between the magnetic impurities and the conduction electrons. We obtain a Kondo termS _d ⁽¹⁾ (T) (which reduces for vanishing spin interactions to the thermopower of Kondo systems) and in addition a resonance termS _d ⁽²⁾ (T) of opposite sign which vanishes for vanishing interactions. The superposition of both terms leads to an additional extremum of the total thermopower and to a change of sign at a temperatureT ₀ which depends on the Kondo temperature and on the magnetic excitations of the spin glass and therewith on the freezing temperatureT _f. We obtain at low temperatures for oscillating and relaxing spin glass modesS _d ⁽¹⁾ T² andS _d ⁽²⁾ T. At high temperaturesS _d ⁽¹⁾ (T) decreases as |lnT|^–3 andS _d ⁽²⁾ asT ^–1. A close relationship between the temperature dependent resistivity in Born approximation andS _d ⁽¹⁾ (T) is pointed out.SFB 125 Aachen-Jülich-Köln     

Superconductivity of spin glasses

The coexistence of superconducting and spin glass phases is investigated. The dependence of the phase transition temperature in the spin glass state both on the impurity concentration and the superconductivity state is given. The influence of the spin glass phase on the renormalized frequency and the order parameter of the conduction electrons is determined.