Spin Glasses and Other Frustrated Systems

Spin Glasses and Other Frustrated Systems

Debashish Chowdhury
Copyright Date: 1986
Pages: 400
https://www.jstor.org/stable/j.ctt7ztmdr
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    Spin Glasses and Other Frustrated Systems
    Book Description:

    Debashish Chowdhury's critical review of more than a thousand papers not only identifies the complexities involved in the theoretical understanding of the real spin glasses but also explains the physical concepts and the mathematical formalisms that have been used successfully in solving the infiniterange model.

    Originally published in 1987.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5886-6
    Subjects: Physics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. A note for the readers:
    (pp. ix-x)
  3. Table of Contents
    (pp. xi-xiv)
  4. CHAPTER 1 REAL SPIN GLASS (SG) MATERIALS, SG-LIKE MATERIALS AND SG MODELS
    (pp. 1-17)

    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,...

  5. CHAPTER 2 A BRIEF HISTORY OF THE EARLY THEORIES OF SG
    (pp. 18-23)

    It was Blandin (1961) who first exploited the power law behaviour of the exchange interaction to derive some universal properties of canonical SG 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 HamiltonianHand the temperature T in the partition function

    $Z = Tr\exp ( - H/{k_B}T)$

    by the same constant leaves Z, and hence the thermodynamic properties, unaltered. Suppose, this constant is the concentration of the magnetic impurities (e.g., Fe in AuFe), c....

  6. CHAPTER 3 SG “PHASE TRANSITION”: ORDER PARAMETERS AND MEAN-FIELD THEORY
    (pp. 24-37)

    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 Xa.c.at Tgsuggets 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

    ${q_{EA}} = {[ < {s_i}{ > ^2}]_{av}}$(3.1)

    (Our discussion...

  7. CHAPTER 4 SHERRINGTON-KIRKPATRICK (SK) MODEL AND THE SK SOLUTION
    (pp. 38-45)

    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${J_{ij}} = {J_0}/N$. Then the energy can be written as

    $E = - ({J_0}/2N){(\sum {S_i})^2} + {J_0}N/2 = - ({J_0}N/2){M^2} + {J_0}N/2$(4.1)

    which is extensive, as it should be. Similarly, the entropy is given by

    $S = - {K_B}N[(1/2)(1 + M)\ln \{ (1/2)(1 + M)\} + (1/2)(1 - M)\ln \{ (1/2)(1 - M)\} ]$

    (4.2)

    and finally, minimizing the free energy E – TS we get

    $M = \tanh (\beta {J_0}M)$

    which is identical with the standard mean-field result.

    What happens when the latter treatment is extended to EA model of SG? In order to examine whether the EA MFT and the...

  8. CHAPTER 5 INSTABILITY OF THE SK SOLUTION
    (pp. 46-51)

    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

    ${x^\alpha } = x + {e^\alpha }$, and${y^{(\alpha \beta )}} = y + {\eta ^{(\alpha \beta )}}$,

    where

    ${x^\alpha } = {({J_0}/{k_B}T)^{1/2}} < {s^\alpha } > = {({J_0}/{k_B}T)^{1/2}}M$and${y^{(\alpha \beta )}} = (J/{k_B}T) < {s^\alpha }{s^\beta } > = Jq/{k_B}T$, x and y are the equilibrium values given by the SK solution. Substituting for${x^\alpha }$and${y^{(\alpha \beta )}}$in the SK free energy expression and keeping terms up to the second order in${e^\alpha }$and${\eta ^{(\alpha \beta )}}$, the free energy difference was given by$ - (1/2)\Delta $where

    $\Delta = \sum [{\delta _{\alpha \beta }} - ({J_0}/{k_B}T)( < {s^\alpha }{s^\beta } > - < {s^\alpha } > < {s^\beta } > )]{ \in ^\alpha }{ \in ^\beta } + \{ 2J{J^{1/2}}/{({k_B}T)^{3/2}}\} \sum [ < {s^\alpha } > < {s^\alpha }{s^\beta } > - < {s^\alpha }{s^\beta }{s^\delta } > )]{ \in ^\delta }{\eta ^{(\alpha \beta )}} + \sum [{\delta _{(\alpha \beta )(\gamma \delta )}} - {(J/{k_B}T)^2}( < {s^\alpha }{s^\beta }{s^\gamma }{s^\delta } > - < {s^\alpha }{s^\beta } > < {s^\gamma }{s^\delta } > ]{\eta ^{(\alpha \beta )}}{\eta ^{(\gamma \delta )}}$(5.1)

    This quadratic form must be positive definite for a stable solution of the...

  9. CHAPTER 6 THOULESS-ANDERSON-PALMER (TAP) SOLUTION OF THE SK MODEL
    (pp. 52-61)

    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.

    We begin with the following identity for the Ising spins$\exp ( - \beta {J_{ij}}{S_i}{S_j}) = 2\cosh (\beta {J_{ij}})[1 - {S_i}{S_j}{B_{ij}}]$(6.1)

    where${B_{ij}} = \tanh (\beta {J_{ij}})$. Now, the free energy can be expanded as a series of diagrams in which the lines represent${B_{ij}}$’s and the vertices${S_1}$’s. Thus, the free energy$F = {F_{pairs}} + {F_{Loops}}$where

    ${F_{pairs}} = - {k_B}T\prod \ln \cosh (\beta {J_{ij}})$(6.2)

    and

    $\begin{array}{l}{F_{Loops}} = - {k_B}T{(\ln T{r_s}\prod (1 - {B_{ij}}{S_i}{S_j})]_{av}}\\ {B_{ij}} \\\end{array}$(6.3)

    It is easy to show that${F_{pairs}}$...

  10. CHAPTER 7 PARISI SOLUTION OF THE SK MODEL AND ITS STABILITY
    (pp. 62-67)

    Almeida and Thouless (1978) (AT) suggested that breaking the permutation symmetry between the replicas might yield stable solutionis) of the SK model. Suppose Pnis the group of permutation of n elements. At first it might appear that a pattern

    ${P_n} \to {P_{n/m}} \otimes {({P_m})^{n/m}}$

    i.e.,

    ${P_0} \to {P_0} \otimes {({P_n})^0}$in the limit$n \to 0$

    would be sufficient (Blandin et al. 1980). Unfortunately, such a weak symmetry breaking is not sufficient and Parisi (1979a,b, 1980a) suggested the scheme

    ${P_0} \to {({P_{m1}})^0} \otimes {({P_{mr}})^0} \otimes \cdots {P_0}$

    so that one can have infinitely broken symmetry. However, parametrizing an n X n matrix in the limit n=0 is very difficult. Fortunately, the following physical requirements provide some...

  11. CHAPTER 8 SOMPOLINSKY’S DYNAMICAL SOLUTION OF THE SK MODEL AND ITS STABILITY
    (pp. 68-72)

    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

    $q(x) = {[ < {s_i}(0){s_i}({t_x}) > ]_{av}}$(8.1)

    where x varies between 0 and 1. q(x) is a measure of the correlation that has not decayed at the time scale tx. Sompolinsky also introduced another order parameter

    $\Delta (x) = {T_x}(x) - (1 - {q_{EA}})$(8.2)

    where x(x) is the local susceptibility measured at the frequency${\omega _x} = t_x^{ - 1}$. Both q(x) and Δ(x) are assumed to be continuous functions of x....

  12. CHAPTER 9 ERGOOICITY, PURE STATES, ULTRAMETRICITY, AND FLUCTUATION-DISSIPATION THEOREM
    (pp. 73-88)

    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 Tt, so that if w is a point in the phase space Ω at t = 0 then Ttw is the position in the phase space at time t. Suppose μ(w) is the probability measure in the phase space. The phase space average of a quantity f is given by

    $ < f > = \int {d\mu (w)f(w)} $

    A system is called ergodic with respect to Ttif...

  13. CHAPTER 10 p-SPIN INTERACTION AND THE RANDOM ENERGY MODEL
    (pp. 89-95)

    The p-spln 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 \to \infty $(Derrida 1980a, b, 1981). The REM is defined as follows:

    (i) the system has 2Nenergy levels Ei(i=1, . . . , 2N),

    (ii) the energy levels Eiare independent random variables,

    (iii) these energy levels have a Gaussian distribution

    $p(E) = A\exp ( - {E^2}/N{J^2})$(10.1)

    where

    $A = {(N\Pi {J^2})^{ - 1/2}}$,

    The latter model can be solved exactly without using replicas. Let n(E) be the number of...

  14. CHAPTER 11 SEPARABLE SPIN GLASS MODELS
    (pp. 96-99)

    The simplest among all the separable SG models is the Mattis model where the randomness is incorporated into Jijthrough the relation (1.9). The classical Mattis model is equivalent to a model for pure system because of the reason explained in chapter 1. However, the properties of the quantum Heisenberg Mattis model is nontrivial because the transformation${S_i} \to {\zeta _i}{S_i}$is not necessarily canonical (Sherrington 1979, Johnston and Sherrington 1982, Nishimori 1981). By the latter statement we mean that a transformation

    ${\tau _i} = {\zeta _i}{S_i}$

    would change the hamiltonian (1.1) into the hamiltonian

    ${\rm H} = - \sum {J_{ij}}{\tau _i}{\tau _j}$

    corresponding to a nonrandom system but the commutation relation for the...

  15. CHAPTER 12 THE SPHERICAL MODEL OF SG
    (pp. 100-102)

    The condition

    $\sum {S_i}^2 = N$(12.1)

    where N is the number of spins in the system, is called the spherical constraint. Therefore, the Hamiltonian for the spherical model can be written as

    $H = - (1/2)J{\Sigma _i}{\Sigma _{j \ne 1}}{S_i}{S_j} + \mu (\Sigma {S_i}^2 - N)$(12.2)

    where the spherical constraint has been incorporated through the Lagrange multiplier (see Mattis 1985 for an introduction to the spherical model). is determined from the requirement that the condition (12.1) be satisfied on the average, i.e.,$ < \Sigma {S_i}^2 > = N$. Thus, the free energy for the spherical model is evaluated on the surface of an N-sphere. Let us evaluate the free energy without using the replica trick. Suppose,...

  16. CHAPTER 13 MFT OF VECTOR SG: MIXED PHASE
    (pp. 103-108)

    The Ising model (m = 1) and the eperical model ($m = \infty $) 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 SQ models with finite m as “vector SG”

    ${\rm H} = - \sum {J_{ij}}\Sigma {S_{i\mu }}{S_{j\mu }} - {\rm H}\Sigma {S_{i1}}$

    where μ (μ = l, 2, . . , m) denotes spin components and the magnetic field is applied along the direction 1. Moreover, the normalization of the spin magnitude implies

    $\Sigma S_{i\mu }^2 = m$.

    Most of our attention will be focussed on the Gaussian model where the exchange interaction${J_{ij}}$are...

  17. CHAPTER 14 OTHER LONG-RANGED MODELS
    (pp. 109-110)

    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$T \ne 0$only in$d \ge 2$. 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$\sigma $. The distribution of is given by${\varepsilon _{ij}}$is given by

    The high temperature expansion...

  18. CHAPTER 15 ANISOTROPIC EXCHANGE INTERACTIONS AND SG
    (pp. 111-120)

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

    $H = - \Sigma ({J_{XX}}S_i^XS_j^X + {J_{YY}}S_i^YS_j^Y + {J_{ZZ}}S_i^ZS_j^Z$(15.1)

    where Jxx, Jyyand Jzzare the diagonal elements of the exchange matrix J (the off-diagonal elements are zero). If${J_{XX}} = {J_{YY}} = 0$and${J_{ZZ}} \ne 0$the model is called the Ising model, whereas${J_{XX}} = {J_{YY}} \ne 0$and Jzz= 0 corresponds to the isotropic XY model, and finally,${J_{XX}} = {J_{YY}} = {J_{ZZ}}$in the isotropic Heisenberg model. The model (15.1) would be anisotropic if the strength of at least one of the exchange constants Jxx, Jyyand Jzzis different from that of the other two....

  19. CHAPTER 16 NONLINEAR SUSCEPTIBILITIES, AT AND GT LINES AND SCALING THEORIES
    (pp. 121-127)

    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${\chi _{EA}}$(5.17) because the mean square field${{\tilde h}^2}$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 Tg(Chalupa 1977)...

  20. CHAPTER 17 HIGH-TEMPERATURE EXPANSION, RENORMALIZATION GROUP; UPPER AND LOWER CRITICAL DIMENSIONS
    (pp. 128-136)

    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 expamsion parameter is K = J/kBT but it is often more convenient to expand in terms of tanh(J/kBT) (see Mattis 1985 for an elementary introduction)....

  21. CHAPTER 18 SPIN DYNAMICS IN VECTOR SG: PROPAGATING MODES
    (pp. 137-148)

    The nonrelativistic version of the Goldstone’s theorem (Lange 1965, 1966) states that if the ground state of the system has a lower continues symmetry than that of the Hamiltonian, there must be an excitation mode whose spectrum in the long-wavelength limit ($\begin{array}{l} A = {(N\Pi {J^2})^{ - 1/2}} \\ k \to 0 \\ \end{array}$) 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...

  22. CHAPTER 19 SPIN DYNAMICS IN SG: RELAXATIONAL MODES AND CRITICAL DYNAMICS
    (pp. 149-161)

    In the proceeding chapter we have discussed the dynamics of SG at$T < < {T_g}$, mostly using hydrodynamic approach. The latter probe essentially 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$q\xi < < 1$breaks down in this regime. Besides, the hydrodynamic theory does not describe the decay of a nonconserved 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...

  23. CHAPTER 20 FRUSTRATION, GAUGE INVARIANCE, DEFECTS AND SG
    (pp. 162-173)

    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).

    The geometry of the spin-ordering in a SG has similarity with the “parallel transport” of a tangent vector on a curved surface (e.g., in general relativity). The misfit between the various lines of transport is expressed by the “frustration” and “curvature”, respectively, in the two cases. In this sense, frustrated plaquettes are “curved” whereas unfrustrated ones are “flat”; all the plaquettes in...

  24. CHAPTER 21 IS THE SG TRANSITION ANALOGOUS TO THE BLOCKING OF SUPERPARAMAGNETIC CLUSTERS?
    (pp. 174-177)

    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 Morrlsh 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...

  25. CHAPTER 22 IS THE SG TRANSITION ANALOGOUS TO PERCOLATION?
    (pp. 178-180)

    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...

  26. CHAPTER 23 IS THE SG TRANSITION ANALOGOUS TO THE LOCALIZATION-DELOCALIZATION TRANSITION?
    (pp. 181-185)

    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$\exp (i\overrightarrow k \cdot \overrightarrow r )$characterized by the corresponding k values. The susceptibility is given by$x(\overrightarrow k ) = 1/\{ T - J(\overrightarrow k )\} $. As the temperature is lowered starting from a high value, a magnetic ordering takes place at${T_c} = {J_{\max }}(\overrightarrow k ),{J_{\max }}(\overrightarrow k )$being the largest eigenvalue of the matrix J. If$J(\overrightarrow k )$is maximum for k = 0, the ordered phase is ferromagnetic, for$k \ne 0$is antiferromagnetic. Below Tcthe corresponding eigenstate is macroscopically magnetized. This type of macroscopic condensation into one particular a...

  27. CHAPTER 24 COMPUTER SIMULATION STUDIES AND ‘NUMERICALLY EXACT’ TREATMENT OF SG MODELS
    (pp. 186-202)

    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...

  28. CHAPTER 25 TRANSPORT PROPERTIES OF SG AND SOUND ATTENUATION IN SG
    (pp. 203-210)

    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...

  29. CHAPTER 26 MISCELLANEOUS ASPECTS OF SG
    (pp. 211-230)

    We have mentioned the features of the local field distribution P(h) in SG in various different contexts in the earlier chapters in this book. In this chapter we shall summarize the works of different authors on P(h) and compare with one another. Almost all the analytical works have been carried out within the MFA. As mentioned in chapter 2. Marshall (1960) and Klein and Brout (1963) derived P(h) for the Ising SG with RKKY exchange interaction and argued that taking proper care of the lower cutoff on the inter-spin separation imposed by the lattice P(h) turns out to be Gaussian....

  30. CHAPTER 27 SG-LIKE SYSTEMS
    (pp. 231-276)

    Several systems with electric moments exhibit glassy behavior over a certain range of composition. Most of the physical properties of such glassy phases are qualitatively similar to the corresponding properties of SG materials. Therefore, in recent years efforts have been made to understand the orientational orderings in these systems in the light of our present knowledge of the ordering of the magnetic moments in SG materials. Examples of materials exhibiting dipolar and quadrupolar glass phases are (KCN)x(KBr)1-x(see section 27.1.1), solid hydrogen (see section 27.1.2), mixed crystals of (Rb)1-x(NH₄)xH₂PO₄ (see section 27.1.3), (KCN) x (NaCN) (Luty and Ortiz-Lopez 1983, Loidi...

  31. CHAPTER 28 CONCLUSION
    (pp. 277-277)

    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,...

  32. APPENDIX A: SG SYSTEMS AND THE NATURE OF THE INTERACTIONS
    (pp. 278-282)
  33. APPENDIX B: GENERAL FEATURES OF THE EXPERIMENTAL RESULTS
    (pp. 283-324)
  34. REFERENCES:
    (pp. 325-374)
  35. ADDENDUM
    (pp. 375-385)