Spin Glasses and Other Frustrated Systems

The following sections are included:

• Acknowledgements

• A note for the readers

• CONTENTS

Magnetic systems exhibit various different types of ordering depending on the temperature T, external magnetic field H, etc. (see Hurd 1982 for an elementary introduction). An experimentalist usually identifies a magnetic material as a spin glass (SG) if it exhibits the following characteristic properties: (i) the low-field, low-frequency a.c. susceptibility χ_a.c.. (T) exhibits a cusp at a temperature T_g, the cusp gets flattened in as small a field (H) as 50 Gauss, (a better criterion is the divergence of the nonlinear susceptibility, as we shall see in chapter 16), (ii) no sharp anomaly appears in the specific heat, (iii) below T_g, the magnetic response is history -dependent; viz. the susceptibility measured in a field-cooled sample is higher than that when cooled in zero -field, (iv) below T_g, the remanent magnetization decays very slowly with time, (v) below T_g, a hysteresis curve, laterally shifted from the origin, appears, (vi) below T_g, no magnetic Bragg scattering, chracteristic of long-range order (LRO), is observed in neutron scattering experiments, thereby demonstrating the absence of LRO (vii) susceptibility begins to deviate from the Curie law at temperatures T >> T_g…

It was Blandin (1961) who first exploited the power law behaviour of the exchange interaction to derive some universal properties of canonical SO alloys. The dominant exchange interaction in these alloys is the RKKY Interaction (1.7a). Suppose, the cosine factor in (1.7a) can be dropped and its effect can be simulated by supplying +1 and −1 randomly. Note that dividing the Hamiltonian ℋ and the temperature T in the partition function

The spontaneous magnetization is a measure of the long-range orientational (spin) order in space, and is the order parameter that distinguishes ferromagnetic phase from the paramagnetic phase (for an excellent introduction to the concepts of order parameter and broken symmetry see Anderson 1984). There is no long-ranged order (LRO) in SG, as demonstrated by neutron scattering experiments (see appendix B). However, the cusp in X_a.c. at T_g suggets the onset of some kind of orientational ordering. In order to describe this new kind of ordering, Edwards and Anderson (EA) (1975) introduced an order parameter, defined as

We know that the MFT for nn ferromagnets becomes exact in the infinite-range limit. As an example, consider an N-spin system where every-spin interacts with all the others with the same exchange J_ij = J_O/N. Then the energy can be written as

The unphysical low temperature behaviour of the SK solution is a consequence of its instability, as was shown subsequently (de Almeida and Thouless 1978). In order to test the stability of the SK solution, let us introduce the variables

Since the replica-symmetric solution of the SK model turned out to be unstable, Thouless, Anderson and Palmer (TAP) (1977) developed a solution of the latter model in terms of the local magnetization without using the replica trick…

Almeida and Thouless (1978) (AT) suggested that breaking the permutation symmetry between the replicas might yield stable solution (s) of the SK model. Suppose P_n is the group of permutation of n elements. At first it might appear that a pattern i.e…

Sompolinsky (1981a) proposed a dynamical approach to the statics of the SK model based on the Sompolinsky-Zippelius (1981, 1982) dynamical theory (to be discussed in chapter 19). As mentioned in chapter 3, Sompolinsky (1981a) introduced an order parameter (function) q(x) which is defined as…

In the formalism of equilibrium thermodynamics, a system is called ergodic if its properties observed experimentally are equal to the corresponding ensemble average. To put the ideas on a more precise footing, let us describe the dynamical evolution by T_t, so that if w is a point in the phase space Ω at t = 0 then T_tw is the position in the phase space at time t. Suppose μ(w) is t the probability measure in the phase space. The phase space average of a quantity f is given by

The p-spin interaction defined by (1.21) is a generalization of the SK model, the latter corresponds to p=2. It can be shown that the p-spin interaction model is identical with the random energy model (REM) in the limit p → ∞ (Derrida 1980a,b, 1981). The REM is defined as follows: (i) the system has 2^N energy levels E_i (i=1,…,2^N), (ii) the energy levels E_i are independent random variables, (iii) these energy levels have a Gaussian distribution

The following sections are included:

• The Mattis Model

• The van Hemmen Model

• The Generalized van Hemmen model (Provost-Vallee model)

The condition

The Ising model (m = 1) and the sperical model (m = ∞) represent two extreme situations. Intuitively, the Heisenberg model with finite m seems to be the most realistic and therefore deserves special attention. For the economy of words, we shall call the SG models with finite m as “vector SG”

We have been exploring the possibility of a finite temperature phase transition in the SG models. We shall see in chapter 24 that the SG transition is possible in short-ranged SG models at T ≠ 0 only in d ≥ 2. Therefore, at first sight the study of onedimensional SG models might seem an irrelevant exercise. However, the long-ranged interaction leads to highly nontrivial behavior of SG models! Kotliar, Anderson and Stein (KAS) (1983) introduced the one-dimensional model (1.17). The range of the interaction is determined by the parameter σ. The distribution of ∈_ij is given by

Anisotropy of a SG Hamiltonian can arise in two different ways: (a) suppose the exchange Hamiltonian is given by

As is well known, the ferromagnetic transition in magnetic systems is signalled by the divergence of the linear susceptibility. The field conjugate to the linear susceptibility is the uniform external field. Although the cusp in the linear susceptibility has been used extensively as one of the main criteria for the SG ordering, the corrcet criterion would be the divergence of the EA-order parameter susceptibility χ_EA (5.17) because the mean square field is naturally the field conjugate to the EA order parameter. But, it is difficult to “measure” the latter susceptibility in laboratory experiments. However, since near T_g (Chalupa 1977)…

The technique of high-temperature expansion was quite successful in the study of the phenomenon of phase transition before the powerful technique of renormalization group (RG) was applied. The basic philosophy of the high temperature expansion (1/T) is quite simple. One can expand the relevant physical quantity (say, the susceptibility) in a power series in 1/T, retain manageably large number of terms and infer the behaviour near the critical temperature by indirect methods. The natural dimensionless expansion parameter is K = J/k_BT but it is often more convenient to expand in terms of tanh(J/k_BT) (see Mattis 1988 for an elementary introduction). Then one estimates the critical point and the corresponding critical exponent analyzing the series either by the ratio method or by the Pade approximants or by some other efficient technique (see Gaunt and Guttmann 1974 for a detailed review of the series expansion technique)…

The nonrelativistic version of the Goldstone’s theorem (Lange 1965, 1966) states that if the ground state of the system has a lower continuos symmetry than that of the Hamiltonian, there must be an excitation mode whose spectrum in the long-wavelength limit (k → 0) extends to zero without any gap. However, certain extra conditions must also be satisfied, for example, the range of the interaction must be finite. Such Goldstone modes in ferromagnets with nearest-neighbour (nn) exchange interaction are very well known, these are the spin waves (magnons). We also know that the 0(3) symmetry of the state (assuming the spin dimension m = 3) is completely broken at the SG transition temperature. Can one expect spin wave excitations in the low temperature phase of the m-vector SG models? We shall not discuss the theories of spin Naves in the Mattis model (see Sherrington 1977, 1978, 1979, van Hemmen 1980, Canisius and van Hemmen 1981). We shall focus our attention to the Halperin-Saslow (HS) (1977) theory and its various extensions. These theories have been developed following closely the hydrodynamic theory of excitations in systems with long-ranged order (LRO) (Halperin and Hohenberg 1969). (Note that there are subtle differences between the hydrodynamic modes and the so-called “collisionless” modes (see Anderson 1984). But for our purposes in this book such distinctions will not be essential.) We shall first explain the regimes of the validity of the hydrodynamic theory…

In the preceeding chapter we have discussed the dynamics of SG at T ≪ T_g, mostly using hydrodynamic approach. The latter probe esentially the small amplitude long-wavelength oscillations of the systems about equilibrium. However, such an approach is not applicable near a critical point because the hydrodynamic condition qζ ≪ 1 breaks down in this regime. Besides, the hydrodynamic theory does not describe the decay of a non-conserved order parameter to equilibrium. In the SG terminology, the basic harmonic approximation strictly restricts the spin wave excitation to the small energy excitations within a given free) energy minimum whereas near the critical temperature sufficient thermal energy is available for flipping of large clusters so that the system can hop from a given minimum to another…

We have defined the concept of frustration in chapter 1 at an elementary level. In this chapter we shall examine its meaning and the deeper implications in a more general context (see also Erzan 1984 for a review)…

Tholence and Tournier (1974) and Wohlfarth (1977a) propsed that the SG transition is not a true thermodynamic phase transition but is very similar to the phenomenon of blocking of superparamagnetic single domain particles in rock materials (see Morrish 1965 for an introduction). Let us briefly review the latter phenomenon. The magnetic particles in rock materials are small enough to consist of a single domain and large enough to consist of a large number of moment-carrying atoms (or molecules). At high temperatures an assembly of such particles behave paramagnetically where each particle possesses a large magnetic moment and hence the name superpara-magnetism. In nature such particles get an extra contribution to their energy from anisotropy effects - anisotropy induced either by shape anisotropy, or crystalline anisotropy or by external stress. Since no domain wall motion is possible in such single domain particles they acquire magnetization by coherent rotation of magnetic moments against anisotropies present in them. Neel (1947) indicated the possibility of freezing of the moments of these single domain particles. The remanent magnetization relaxes as M_r = M_s exp(−t/τ) where M_s is the full magnetization when the field is switched off and the relaxation time τ is given by (1/τ) = (1/τ₀) exp(−KV/k_BT) where K the anisotropy constant and V the volume of the particle, with τ₀ ≃ 10⁻⁹ sec. For a particular measurement, characterized by a typical measurement time τ_m, particles with a volume larger than a critical size appear frozen because their relaxation time will be longer than τ_m (compare with the theory of the two-level systems introduced in chapter 2)…

Smith (1975) propsed a percolation (see Stauffer 1985 for an introduction to the concept of percolation) model of the SG transition. The basic idea behind this theory goes as follows: as the system is cooled from a high temperature a fraction of the spins ‘lock’ together to form clusters within which spins are very strongly correlated. The clusters keep growing bigger and bigger at the expense of the ‘loose’ spins as the temperature is lowered more and more. This cluster evolution consists of two processes- more and more ‘loose’ spins lock together to form clusters and clusters formed at a higher temperature coalesce to form bigger clusters. At a temperature T_g an infinite cluster of the locked spins forms ( i.e., percolation takes place) and the corresponding temperature is identified as the SG transition temperature. Abrikosov (1978a,b, 1980), Cyrot (1981) and Mookerjee and Chowdhury (1983b) improved the theory by taking the effect of the finite mean-free path of the RKKY interaction and frustration into account…

Let us first briefly review the eigenstate-space technique of studying the phase transition in nonrandom systems. In such systems the eigenstate of the exchange matrix are plane waves exp characterized by the corresponding k values. The susceptibility is given by . As the temperature is lowered starting from a high value, a magnetic ordering takes place at being the largest eigenvalue of the matrix J. If is maximum for k = 0, the ordered phase is ferromagnetic, for k ≠ 0 it is antiferromagnetic. Below T_c the corresponding eigenstate is macroscopically magnetized. This type of macroscopic condensation into one particular mode is reminiscent of the Bose condensation…

Recent advances in the computer technology have provided powerful tools for the physicists, viz., high-speed computers with large core memories (e.g., Cray 2, Cray XMP, Cray 1, Cyber 205, etc.), special-purpose computing machines (e.g., the special purpose machine of Condon and Ogielski at the Bell Labs.) and multiprocessor machines (e.g., the distributed array processor (DAP) at the Queen Mary College London, etc.). Computational physics has become the third branch of physics, bridging the gap between theoretical physics and experimental physics (see, for example, Kalos 1985, Binder 1986 for the future prospects of computational physics). The introduction to the computer simulation technique in this chapter is quite long. This is not so because of any personal bias of the author but because of the fact that so far results of computer simulation of SG models have played the most decisive role in estimating the lower critical dimension (LCD) and the physical properties of realistic SG models…

So far in this book we have not explicitly treated the interaction between the ‘localized’ d-electrons of the transition metals and the s-electrons in the transition metal-noble metal alloys. In chapter 1 we absorbed the effects of the s-d interaction into the RKKY interaction which is mediated via the s-d exchange interaction. In chapter 19 we have assumed that the s-electrons form a part of the heat bath that causes the spinflip. In this chapter we shall investigate explicitly the effects of the s-d interaction on the transport properties of the metallic SG. Moreover, so far we have always assumed the spins to be rigidly frozen so that the interaction between the spins and the lattice vibrations were also not taken into account. In this chapter we shall review the theories of the interaction between the spin and the phonon degrees of freedom so as to analyze the velocity and the attenuation of sound waves in SG.

The following sections are included:

• Local Field Distribution in SG

• Yang-Lee Method for SG and Other Frustrated Models

• SG on Hierarchical Lattices and Fractals

• Transverse Ising SG Model

• SG in The Clock Model

• Potts Glass

• SG On The Bethe Lattice

• Sensitivity of the Free Energy to The Boundary Conditions; Exchange Stiffness versus Scaling Stiffness

• Induced Moment SG

• SG Freezing and Superconductivity in SG Materials

The following sections are included:

• Electric Dipolar and Ouadrupolar Glasses
  • (KCN)_x(KBr)_1-x: A Director Glass?
  • Solid Hydrogen- A Quadrupolar Glass?
  • Rb_1-x(NH₄)_xH₂PO₄ -Hydrogen-bonded Mixed Crystal of Ferroelectric and Antiferroelectric Materials

• Electron Glass in Doped Compensated Semiconductors?

• Superconducting Glass Phase in Granular Superconductors and Josephson Junction Arrays

• TRAVELLING SALESMAN PROBLEM, COMPUTER DESIGN AND SG

• SG MODEL OF CRYSTAL SURFACES

• ULTRADIFFUSION: DYNAMICS ON ULTRAMETRIC SPACE

• NEURAL NETWORKS, CONTENT ADDRESSABLE MEMORIES, PATTERN RECOGNITION AND SG

• Modelling of Prebiotic Evolution and The Origin of Life: Analogy with SG?

Finally, what is the present status of our understanding of the physics of SG? The answer is, of course, subjective. An optimist would say, we have not only understood the qualitative nature of the order in a large class of magnetic materials we have also developed mathematically elegant formalism of replica symmetry breaking, the first successful utilization of the special purpose machine for simulating short-ranged SG has opened up a new era in computational science, we have developed formalisms which may, in near future, lead to breakthroughs in computer science and biophysics. On the other hand, a pessimist would, perhaps, say that our present knowledge about SG is no better than the knowledge of the n blind persons about an elephant (Fig. 28.1).

The following sections are included:

• APPENDIX A SG SYSTEMS AND THE NATURE OF THE INTERACTIONS

• APPENDIX B GENERAL FEATURES OF THE EXPERIMENTAL RESULTS
  • Magnetization and susceptibility measurements
  • Specific heat measurements
  • Neutron scattering
  • Electron spin resonance (ESR)
  • Nuclear magnetic resonance (NMR)
  • Muon spin relaxation (μSR)
  • Mossbauer spectroscopy
  • Resistivity and Magnetoresistance

• REFERENCES

• ADDENDUM