Spin glasses

Recent developments in the theory of spin glasses are discussed. There has been considerable progress, due to Parisi, Sompolinsky, and others, towards understanding the infinite range (mean field) model of Sherrington and Kirkpatrick. Relaxation times diverge in the thermodynamic limit, and this nonergodic behavior is now understood to be the cause of earlier difficulties. There has been less progress in the study of more realistic models with short-range interactions, but numerical studies have shown rather clearly the absence of a finite temperature transition in two dimensions. There is probably no transition ind=3 either, though the evidence is less clearcut, which makes it difficult to understand the sharpness of the freezing observed experimentally. Well below the freezing temperature ESR and torque measurements have been fairly well explained by a theory of Henley, Sompolinsky, and Halperin, in which an important ingredient is anisotropy due to the Dzyloshinsky-Moriya interaction proposed by Fert and Levy.     

Spin glasses and information

After a brief introduction to information theory, we review the close relationship between the theory of spin glasses and information processing, error-correcting codes in particular. An interesting equivalence of the solvability condition of the spin glass problem and the optimal inference condition in information theory is pointed out.     

Spin glasses with long-range interactions

A spin-glass model with a general form of long-range interactions is introduced. The replica method and the renormalization group theory are used and the results are explained from the viewpoint of frustration effects compared with the non- random case. The so-called mean field approximation is shown to be invalid for 3-d RKKY interaction systems.     

Spin glasses on thin graphs

In a recent paper [C. Baillie, D.A. Johnston and J.-P. Kownacki, Nucl. Phys. B 432 (1994) 551] we found strong evidence from simulations that the Ising antiferromagnet on “thin” random graphs — Feynman diagrams — displayed a mean-field spin-glass transition. The intrinsic interest of considering such random graphs is that they give mean-field theory results without long-range interactions or the drawbacks, arising from boundary problems, of the Bethe lattice. In this paper we reprise the saddle-point calculations for the Ising and Potts ferromagnet, antiferromagnet and spin glass on Feynman diagrams. We use standard results from bifurcation theory that enable us to treat an arbitrary number of replicas and any quenched bond distribution. We note the agreement between the ferromagnetic and spin-glass transition temperatures thus calculated and those derived by analogy with the Bethe lattice or in previous replica calculations.

We then investigate numerically spin glasses with a ±J bond distribution for the Ising and Q = 3, 4, 10, 50 state Potts models, paying particular attention to the independence of the spin-glass transition from the fraction of positive and negative bonds in the Ising case and the qualitative form of the overlap distribution P(q) for all of the models. The parallels with infinite-range spin-glass models in both the analytical calculations and simulations are pointed out.     

Spin glasses with short range interaction

The theory is presented of spin glasses for which the condition n _m ^1/3 l « dl is valid, l being the range of interaction forces between spins and n _m the volume concentration of spins. Various examples where such an interaction appears are considered. Thermodynamic and kinetic characteristics of such systems are calculated at high and low temperatures. Particular attention is paid to the transition region; for its analysis percolation theory methods are applied. The theory of one-dimensional spin glasses of several types is constructed. A ‘kinetic’ equation is derived for the cluster distribution function in the transition region and the behaviour of this function is studied.     

Spin glasses models and typical properties

This article presents a critical discussion of some at the recently published theoretical attempts to understand the essential properties of spin glasses. The fundamental difference between phase transition- and cluster-model (with frozen moments) is studied and compared with experiments regarding the critical behaviour of the systems. At present the cluster-model, combined with magnetic freezing mechanism, seems to be more near to the experimental facts, but more experiments near the transition temperature would be very welcome.     

Spin glasses: Some recent experiments

This review of the spin-glass problem will begin by considering the basic ingredients of a spin glass. Then I shall summarize the fundamental experimental properties and the real systems which are being studied. A bit of theory will be presented to show how idealized the models are when compared to the real materials. Finally I shall discuss the dynamics of the spin-glass phase transition and relaxations within the frozen state, drawing upon the muon investigations wherever possible.     

Spin and link overlaps in three-dimensional spin glasses

Excitations of three-dimensional spin glasses are computed numerically. We find that one can flip a finite fraction of an LxLxL lattice with an O(1) energy cost, confirming the mean-field picture of a nontrivial spin overlap distribution P(q). These low energy excitations are not domain-wall-like, rather they are topologically nontrivial and they reach out to the boundaries of the lattice. Their surface to volume ratios decrease as L increases and may asymptotically go to zero. If so, link and window overlaps between the ground state and these excited states become "trivial."     

Spin glasses with long-range interaction at high temperatures

We study the Ising andN-vector spin glasses with exchange couplings J=(J _ij ;i, jZ ^d ), which are independent random variables with EJ_ij=0 andEJ ⁿ _ij ⁿ n!¦i–j¦ ^–nd , forn, some finite constant >0, and >1/2. For sufficiently small, we show that forE-a.a.J there is a weakly unique, extremal, infinite-volume Gibbs measure _J for which the expectation of a single (component of) spin vanishes and which has the cluster property inL ₂(E) with the same decay as interaction. This work is based on results and methods of Fröhlich and Zegarlinski.     

Spin glasses and nonlinear constraints in portfolio optimization

We discuss the portfolio optimization problem with the obligatory deposits constraint. Recently it has been shown that as a consequence of this nonlinear constraint, the solution consists of an exponentially large number of optimal portfolios, completely different from each other, and extremely sensitive to any changes in the input parameters of the problem, making the concept of rational decision making questionable. Here we reformulate the problem using a quadratic obligatory deposits constraint, and we show that from the physics point of view, finding an optimal portfolio amounts to calculating the mean-field magnetizations of a random Ising model with the constraint of a constant magnetization norm. We show that the model reduces to an eigenproblem, with 2N solutions, where N is the number of assets defining the portfolio. Also, in order to illustrate our results, we present a detailed numerical example of a portfolio of several risky common stocks traded on the Nasdaq Market.     

Spin glasses: Comments on some experimental results

Some of the experimental results on spin glasses commonly interpreted as showing a sharp magnetic phase transition are re-examined. Recent neutron scattering measurements add significantly to our understanding of the phenomena occuring in spin glasses. These results together with those on other physical properties are discussed in terms of a unified picture of freezing of spins in the binary alloys.     

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.     

Spin glasses on Bethe lattices for large coordination number

We study spin glasses on random lattices with finite connectivity. In the infinite connectivity limit they reduce to the Sherrington Kirkpatrick model. In this paper we investigate the expansion around the high connectivity limit. Within the replica symmetry breaking scheme at two steps, we compute the free energy at the first order in the expansion in inverse powers of the average connectivity (z), both for the fixed connectivity and for the fluctuating connectivity random lattices. It is well known that the coefficient of the 1/z correction for the free energy is divergent at low temperatures if computed in the one step approximation. We find that this annoying divergence becomes much smaller if computed in the framework of the more accurate two steps breaking. Comparing the temperature dependance of the coefficients of this divergence in the replica symmetric, one step and two steps replica symmetry breaking, we conclude that this divergence is an artefact due to the use of a finite number of steps of replica symmetry breaking. The 1/z expansion is well defined also in the zero temperature limit. Received 15 July 2002 Published online 31 December 2002     

Spin echo and nuclear orientation study of metallic glasses

Hyperfine fields on Co nuclei in amorphous as-quenched and heat-treated Co₇₅Fe₅B₂₀ samples were studied by conventional NMR and by very low temperature nuclear orientation techniques. The Co spin echo measurement at 1.4 K yielded broad spectra between 130 – 260 MHz, with narrow maxima at 145.5 MHz and 155.1 MHz for as-quenched sample and with a broad maximum at 227 MHz for heat-treated sample well below the recrystallization point. The⁶⁰Co nuclear orientation measurements gave the mean value of hyperfine field B_hf15 T nearly independent of the sample heat-treatment. The spin-lattice relaxation was studied by pulse NMR and also by nuclear orientation thermal cycling technique.     

Spin glasses on the basis of rare earth metal compounds

A new class of materials is constructed of rare earth metals and their compounds by introduction of a large number of defects into the crystal lattice-dislocations and unelastic disclinations. It is shown that superconductive phase state is possible in these materials having spin glass magnetic structure. A technique of such a state observation is suggested.     

Spin glasses and other lattice systems with long range interactions

We study classical lattice systems, in particular real spin glasses with Ruderman-Kittel interactions and dipole gases, with interactions of very long (non-summable) range but variable sign. Using the Kac-Siegert representation of such systems and Brascamp-Lieb inequalities we are able to establish detailed properties of the high-temperature phase, such as decay of connected correlations, for these systems.     

Spin disorder resistivity of spin glasses with application to AgMn

The impurity resistivity due to scattering from disordered spins is calculated for temperatures above the ordering temperature T₀. It varies linearly around T₀, but shows a maximum at higher temperatures, in agreement with experiment.     

Spin domains generate hierarchical ground state structure in J = +/-1 spin glasses

Unbiased samples of ground states were generated for the short-range Ising spin glass with J(ij) = +/-1, in three dimensions. Clustering the ground states revealed their hierarchical structure, which is explained by correlated spin domains, serving as cores for macroscopic zero energy "excitations."