CHAPTER 1
REAL SPIN GLASS (SG) MATERIALS, SG-LIKE MATERIALS AND SG MODELS
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 xa.c. (T) exhibits a cusp at a temperature Tg, 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 Tg, 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 Tg, the remanent magnetization decays very slowly with time,
(v) below Tg, a hysteresis curve, laterally shifted from the origin, appears,
(vi) below Tg, 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 >> Tg.
(see appendix A for a list of the SG materials and appendix B for a summary of the general features of the experimental results). There are several other systems (not necessarily magnetic) which share at least some of the features of SG and hence we shall call these materials SG-like systems (see chapter 27).
From the theoretical point of view, the SG materials have two common features: (a) the disorder is "quenched" and (b) the interaction between the spins are in conflict with each other leading to "frustration" (don't worry if you are not familiar with the two terns within quotes, we shall explain these terns in sections 1.2 and 1.3, respectively.
1.1 SG Models:
The simplest model for a d-dimensional magnetic system is the Heisenberg Hamiltonian (see Mattis 1981, 1985)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
where [??]i, [??]j, etc. are the m-component spin vectors (the models corresponding to m=1 and m=2 are called the Ising model and the XY model, respectively) at the positions [??]i, [??]j, etc. in a d-dimensional space interacting with one another, the strength of the interaction being Jij. In the general case Jij depends on [??]i-[??]j. (The nature of Jij in some of the real SG materials has been listed in appendix A.) Most of the theoretical works so far have focussed attention only on classical spins. Such a description, in terms of "spin vectors" is valid provided there exist "good" local moments in the system. There exists a class of random magnetic alloys, e.g., AuCo, RhFe, RhCo, etc. (Murani and Coles 1970), whose SG transition temperature Tg is much lower than the Kondo temperature Tk. Such systems, called Stoner glass (Hertz 1979, 1980), are described better by the Stoner model (Shimizu 1981, Gautier 1982) rather than by the Heisenberg Hamiltonian (1.1). However, we shall not discuss Stoner glasses further in this book.
The exchange interaction Jij on the right hand side of (1.1) is called short-ranged or long-ranged depending on (Mattis 1988) whether [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Disorder can be Introduced Into magnetic models in two different ways - either through "bond disorder" or "site disorder". In the bond-random models of SG one assumes the exchange bonds Jij to be independent random variables distributed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
where Pc is some suitable continuous or discrete distribution and the presence of the second term incorporates dilution. In other words, only a finite fraction cb of the exchange bonds are nonzero, the strength of the latter are determined by the distribution Pc. In the Gaussian model
Pc(Jij) = (2πj2)-1/2 exp[- (Jij - J0)2/2J2) (1.3)
In the nondilute Gaussian model, most often used in the literature (Edwards and Anderson 1975), cb = 1. In the Sherrington-Kirkpatrick (SK) model each spin is assumed to Interact with every other spin in the system (i.e., the lattice coordination number z -> ∞) with an exchange interaction distributed as in (1.3). However, the requirement that the free energy is an extensive thermodynamic quantity is fulfilled provided
J0 = [??]0/N and J = [??]/N1/2
where [??]0 and [??] are both intensive. Thus, SK model is truly long-ranged.
In the generalized ±J model, P0(Jij) is assumed to be discrete, having only two values +J and -J with probabilities cbf and cbm respectively, i.e.,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
where cbf + cba = cb. Notice that for cba = 0 (1.4) represents a dilute ferromagnet and for cb = cbf + cba = 1 (1.4) reduces to the nondilute [+ or -]J model (Toulouse 1977). Another possible generalization of the nondilute [+ or -]J model is to have a variable strength of one of the interactions (say, the antiferromagnetic) by replacing J by aJ where a is a variable parameter (Wolff and Zittartz 1986). So far most of the theoretical attention has been focussed on the Gaussian model and the [+ or -]J model.
Now let us consider the site-disorder models where only a finite nonzero fraction c of the lattice sites are occupied (randomly) by the spins; the remaining sites are occupied by nonmagnetic atoms or molecules. One such model is defined by the Hamiltonian (Binder et al. 1979)
X = -J1 [summation] c1cjsisj - J2 [summation] cicjsisj (1.5)
where J1 > 0 and J2< 0, the first summation is carried over nearest neighbor pairs and the second summation is carried over next-nearest-neighbor pairs; ci is the occupation probability of the i-th site (ci = 1 if the i-th site is occupied by a spin, and ci = 0 otherwise, so that (1/X)[summation]ci = c, where X is the total number of lattice sites). In another site-disordered model the interaction Jij between any two sites [??]i and [??]j (not necessarily nearest-neighbors) is given by
Jij = JRKKY([??]i - [??]j) ci cj (1.6)
where JRKKY is Ruderman-Kittel-Kasuya-Yoshida (RKKY) exchange interaction (see Mattis 1981, 1985). The latter interaction arises from the indirect exchange interaction between the localized moments mediated via the conduction electrons, the conduction electron-spin interacts with a localized impurity spin through the s-d exchange interaction Jsd. The expression for JRKKY in a d-dimensional nonrandom system is given by (Larsen 1981)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
where C is a constant depending on the characteristic parameters of the system, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Fermi wave vector, Jsd is the strength of the s-d exchange interaction and Jv and Yv are the Bessel and the Neumann functions, respectively. The expression (1.7) reduces to the appropriate forms for JRKKY in d=2 (Fischer and Klein 1975) and in d=3 (Ruderman and Kittel 1954, Kasuya 1956, Yoshida 1957); the latter is given by
Jd=3RKKY = V0 (cos (2kFrij + φ))/r3ij (1.7a)
where Vo determines the "amplitude" of the oscillating factor of the interaction and φ is a phase factor. Is the RKKY Interaction long-ranged or short-ranged? The answer to this question will be presented in chapter 17. In random magnetic systems the RKKY interaction gets damped by the disorder (deGennes 1962, Kaneyoshi 1975, 1979, Poon 1978, deChatel 1981); for weak disorder the latter interaction in d=3 (see Larsen 1985 for the general expression in d dimension) is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)
where δ is the mean free path of the conduction electrons.
In all the models (1.2)-(1.8) P(Jij) is independent of temperature and hence the quenched nature of the disorder. As we shall see in the next section, the random sign of the interaction leads to frustration. However, randomness in the sign of the Interaction does not necessarily guarantee that it is frustrated. For example, let us consider the Mattis model (1976) which is defined by (1.1) together with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.9)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are independent random variables assuming the values +1 and -1 with equal probability. Mattis model is a special case (J1 = J3 = 0) of the Luttinger model (1976) which is defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)
where J1, J2 and J3 are constants. However, as we shall in section 1.3, the Mattis model as well as the Luttinger model do not contain frustration. Therefore, these two models fail to capture one of the essential features of SG.
The van Hemmen model (1982) is defined by
X = - (J0/N) [summation] sisj - [summation] Jij sisj (1.11)
where J0 is the nonrandom ferromagnetic exchange and randomness is incorporated through Jij which is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [??]i and ηi are identically distributed independent random variables with zero mean.
Provost and Vallee (1983) generalized the van Hemmen model as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)
where [??]i are N independent identically distributed random p-vectors (μ = i, ..., p) with mean zero. The special case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is identical with the van Hemmen model.
The models described so far can be divided into two classes: separable and non-separable models. The Mattis model, the van Hemmen model and the generalized van Hemmen model (Provost-Vallee model) belong to the former class whereas the SK model belongs to the latter. An elegant way of writing all these separable models of random magnetic systems is as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where J is a p X p symmetric matrix and [??]i are N independent identically distributed Gaussian p-vectors with zero mean. For example, the Mattis model corresponds to a scalar J, the van Hemmen model corresponds to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and so on. All the separable models discussed so far have a common characteristic, viz., the dimension of the matrix J is finite. The close correspondence between the SK model and the separable model with infinite value of p (Benamira et al. 1985) will be examined in chapter 14.
A model with only a long-ranged interaction or with only a short-ranged interaction is an ideal situation whereas the Hamiltonian of a real SG material is expected to have both long-ranged and short-ranged parts. Bowman and Halley (1982) studied a model
H = HS + HL (1.13)
where the short-ranged part HS is nearest-neighbor ferromagnetic or antiferromagnetic exchange Hamiltonian whereas the long-ranged part HL is the SK Hamiltonian. Notice that the short-ranged parts In the van Hemmen as well as in the Bowman-Halley model are nonrandom. Morgenstern and van Hemmen (1984) generalized the model (1.11) so as to incorporate randomness also In the short-ranged part of the Hamiltonian, the latter was assumed to be Gaussian-distributed.
All the Hamiltonians (1.1)-(1.13) assume the spins to be "hard", i.e., the spins can have only fixed finite values. For example, an Ising spin is "hard" because it can take only two values +1 and -1. In other words, the spin weighting function for Ising spins consists of two delta functions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i
Now, one can generalize this idea to introduce "soft" spins where each component of the spin is allowed to vary between -∞ and +∞ and the distribution exp[-Ws (si) is usually assumed to be
exp[-Ws(S)] = exp[-ri !S!2 - ui !S!4] (u>0)
in the so-called Landau-Ginzburg-Wilson model. The partition function is given by (see, for example, Fisher 1982, Sherrington 1981)
Z = Tr exp(-Heff)
where
Heff = -[summation]Kij sisj - [summation] Ws(si) (1.14)
with Kij = Jij/kBT, kB is the Boltzmann constant. One must remember that ri and ui are quenched variables in SG. One recovers the hard spin model (1.1) in the limit ri -> -∞, ui -> +∞, so that (-2ri/ui) -> s2i. The competing nature of the random interaction in SG is taken into account by assuming a distribution of Kij in (1.14) with, say, zero mean (Hertz and Klemm 1979).
All the models of random magnets listed above, including (1.14) are defined on a discrete lattice. However, continuum version of (1.1), viz.,
H = (1/2) ∫ ddx [r0 !φ!2 + (u/4) !φ!4 + | ([bar.V]-iQ(x))φ(x)|2] (1.15)
with a gauge-invariant derivative of the field φ, has also been studied (Hertz 1978). For a given Q, (1.15) describes a spin density wave with the wave vector Q. The effective Hamiltonian (1.15) for SG differs from the corresponding expressions for random ferromagnets (RF) and random antiferromagnets (RAF) by the fact that Q is x-dependent in SG in contrast to x-independent Q for RF and RAF. However, we shall see in chapter 20 that just arbitrary x-dependence of Q does not necessarily guarantee that (1.15) is Indeed a good model for SG. The continuum analogue of the discrete Mattis model (1.9) will be identified in chapter 20.
