# Spin Glass Theory and Further Beyond

David Sherrington  Rudolf Peierls Centre for Theoretical Physics, Oxford OX1
3PU, UK  Jairo R. L. de Almeida  Departamento de Física, Universidade Federal
de Pernambuco, Recife, Brazil

##  Chapter 3 Beyond Ising SK

$m$-vector, Potts, $p$-spin, spherical, induced moment, random graphs

This chapter introduces a variety of spin glass models, beyond the simple
Sherrington-Kirkpatrick (SK) version, that have led to enriched understanding
and application.

###  1 Introduction

When one of us introduced the SK model $\hbox{}{[{\mbox{?}}]}$, with
Kirkpatrick in 1975, following the very innovative paper of Edwards and
Anderson $\hbox{}{[{\mbox{?}}]}$ earlier the same year, he made two choices
that have proven to be influential; (i) an infinite-range intensive exchange
distribution and (ii) Ising spins; the first in the hope of enabling an exact
solution by analogy with and extension of previously known work on pure
magnets and to provide the correct mean-field solution; the second to simplify
the technicalities. In the event, the correct mean-field solution turned out
to be much more subtle and interesting than was anticipated and thus revealing
that mean-field theory need not be trivial – a common perspective at the time.
The use of discrete Ising spins was serendipitous in highlighting a problem
with a ‘natural’ (replica-symmetric) ansatz, recognized through an unphysical
negative entropy feature already in SK. (The problem would not have been
obvious for continuous classical spins where such a feature is a known
pathology111EA $\hbox{}{[{\mbox{?}}]}$ considered $m=3$ spins, presumably
because the stimulating experiments at the time involved Heisenberg spins..)
The probable origin of the unphysical behaviour was later shown by the
demonstration of an instability in an excitation mode in replica-space by the
other of us and Thouless $\hbox{}{[{\mbox{?}}]}$, eventually leading Parisi
between 1979 $\hbox{}{[{\mbox{?}}]}$ and 1983 $\hbox{}{[{\mbox{?}}]}$ to the
correct and highly impactful (full) replica symmetry breaking (RSB)
resolution.

In this chapter we introduce systems with spins and interactions of different
symmetries and indicate some of the qualitatively new consequences that arise.
Mainly, we concentrate on systems with quenched infinite-range interactions,
drawn randomly and independently from identical distributions, in conceptual
analogy with SK, in the expectation of solvability in the thermodynamic limit
$N\rightarrow\infty$ and the definition of the appropriate associated mean-
field theories. We concentrate on equilibrium/Gibbsian statistical mechanics
(statics), leaving dynamics to later chapters.

###  2 Binary interaction models

####  2.1 Local moment spins of $O(m)$ symmetry

In this section we consider systems with classical $m$-vector spins
${\bf{S}}_{i}$ ($m$ integral) beyond Ising ($m=1$) and with SK-like scalar-
product interactions.

In the absence of anisotropy these have the Hamiltonian

$H=-\sum_{(ij)}J_{ij}{\bf{S}}_{i}.{\bf{S}}_{j},$ (1)

with the $J_{ij}$ drawn independently randomly and symmetrically
($J_{ij}=J_{ji}$) à la SK, from a Gaussian distribution of mean222The
inclusion of a finite positive mean, originally by Sherrington and Southern
$\hbox{}{[{\mbox{?}}]}$, aims to emulate the concentration-dependence of
experimental spin glass systems.  $J_{0}/N$ and variance $J^{2}/{N}$. The
subsequent behaviour is essentially as for the $m=1$ case (For RS analysis see
$\hbox{}{[{\mbox{?}}]}$ and $\hbox{}{[{\mbox{?}}]}$). In particular, if the
spin length normalisation is chosen as $|{\bf{S}}|=\sqrt{m}$ the spin glass
transition temperature (in units with Boltzmann’s constant set to unity,
$k_{B}=1$) is $T=J$.

Something new arises when anisotropic terms are included.

First, consider the application of a uniform uni-directional magnetic field,
in a direction denoted below by 1,

$H=-\sum_{(ij)}{J_{ij}{\bf{S}}_{i}.{\bf{S}}_{j}}-\sum_{i}hS^{1}_{i}.$ (2)

In 1981, Gabay and Toulouse $\hbox{}{[{\mbox{?}}]}$ pointed out that this
system would now have a new transition to a spin glass phase transverse to the
field direction (i.e. symmetric in the $m\neq 1$ hyper-plane) beneath a
characteristic (GT) line in $(h,T)$, starting from $T=1$ for $h=0$ and with
$h_{GT}$ going asymptotically to $\infty$ as $T\rightarrow 0$. Replica
symmetry breaks beneath this line, gradually and strongly in the transverse
direction (as for an ($m-1$)-dimensional spin glass) but also weakly in the
1-direction, until a cross-over to stronger RSB around an AT-like $(h,T)$ line
$\hbox{}{[{\mbox{?}},{\mbox{?}},{\mbox{?}}]}$; see Figs 2 and 2.

(i)
(ii) .

Figure 1: $m$-vector phase diagrams: (i) $J_{0}=0$ in an applied field H: (a)
GT transverse spin-glass line, (b) AT crossover (figure from
$\hbox{}{[{\mbox{?}}]}$,with permission. ©American Physical Society(1981),
(ii) with also a positive exchange distribution mean $J_{0}$; solid lines
denote phase transitions, dashed lines denote quasi-AT crossovers (figure
adapted from $\hbox{}{[{\mbox{?}}]}$, with permission ©Physical Society of
Japan (1983)).

(a)
(b)

Figure 2: Schematic longitudinal spin glass susceptibilities in field $H$: (a)
Ising, (b) $m$-vector. The upper curves are field-cooled, the lower zero-
field-cooled, reflecting the difference between $[1-\int{dxq_{||}(x)}]/T$ and
$[1-q_{||max}]/T$. Figures from $\hbox{}{[{\mbox{?}}]}$, with permission
©Physical Society of Japan(1983).

Second, for a $J_{0}=0$ $m>2$-vector system with external fields randomly and
independently distributed over the full $m$-hypersphere there is no GT
analogue, only AT lines which are $m$-dependent $\hbox{}{[{\mbox{?}}]}$; see
Fig.(4).

Third, consider the case with also quadratic single-site anisotropy

$H=-\sum_{(ij)}{J_{ij}{\bf{S}}_{i}.{\bf{S}}_{j}}-\sum_{i}hS^{1}_{i}-\sum_{i}D(S^{1}_{i})^{2}.$
(3)

It exhibits three different spin glass phases already for $J_{0}=0,h=0$, as
shown in Fig.(4). All three phases exhibit full RSB. For a more general
situation, including $J_{0}\neq 0$, $h\neq 0$, RSB and Parisi solutions, see
$\hbox{}{[{\mbox{?}}]}$.

Figure 3: AT line for $m$-vector spin glasses in quenched $m$-isotropic local
random fields $\hbox{}{[{\mbox{?}}]}$, with permission. ©Amerlcan Physical
Society (2010).

Figure 4:  Phase diagram of anisotropic spin glass. Figure from
$\hbox{}{[{\mbox{?}}]}$ ©IOP Publshing; see also $\hbox{}{[{\mbox{?}}]}$.

####  2.2 Potts spins

All $m$-vector spins, including Ising ($m=1$), have definiteness symmetry with
pairwise interactions, $J_{ij}$ and $-J_{ij}$ each leading to a unique
relative ordering of the pair of spins $i$ and $j$ to minimize their energy.
This is not the case for $p$-state Potts spins $\hbox{}{[{\mbox{?}}]}$ with
$p>2$, where a ferromagnetic $J$ prefers identical Potts states on i and j,
whereas an antiferromagnetic $J$ leads to $(p-1)$ degenerate optima. To
investigate possible consequences of the lack of this symmetry, Elderfield and
Sherrington (1983) $\hbox{}{[{\mbox{?}}]}$ considered the $p$-state Potts
extension of SK333 Erzan and Lage $\hbox{}{[{\mbox{?}}]}$ also introduced and
studied the model, contemporaneously but independently.

$H={\sum_{(ij)}J_{ij}(p{\delta_{\pi(i)\pi(j)}}-1)};\pi=1,..p$ (4)

where the $J_{ij}$ are distributed as in eqn.(1).

This system turned out to have unusual RSB behaviour compared with the Ising
higher $m$ vector systems $\hbox{}{[{\mbox{?}},{\mbox{?}},{\mbox{?}}]}$, as
illustrated in Fig (5) for the Parisi order function $q(x);x\in(0,1)$ for
increasing $p$, based on formulae interpolated to real $p$.

Figure 5: Potts Glasses: Parisi order functions for small
$\tau=(T_{c}-T)/T_{c}$;
(i) $p=2\equiv$ Ising (FRSB), (ii) $p_{c1}>p>2$, (iii) $p_{c2}>p>p_{c1}$
(C1RSB),
(iv) $p>p_{c2}$ (D1RSB). Figures from $\hbox{}{[{\mbox{?}}]}$. ©Physical
Society of Japan (1986)

Figure 6: Types of RSB $\overline{P(q)}$: (i) FRB ($p_{c1}>p>2$); (ii) C1RSB
(continuous onset) ($p_{c2}>p>p_{c2}$) (iii) D1RSB (discontinuous onset)
($p>p_{c2}$ )

There are two critical Potts dimensions characterising significant changes in
the overlap distribution; $p_{c1}=2.82$ which separates a range of overlaps
continuous between peaks at $q=0$ and a finite $q_{EA}$, i.e. full RSB, and
another with 1RSB, itself split into two parts, the first part onsetting
continuously up to the second critical value $p_{c2}=4$, the second part for
$p>p_{c2}$, for which the 1RSB onset is discontinuous one-step
$\hbox{}{[{\mbox{?}}]}$ but without latent heat, now known as a ‘random first
order transition’ (RFOT)$\hbox{}{[{\mbox{?}}]}$ in recognition of the
discontinuity in state-overlap as opposed to normal Ehrenfest first order; see
Fig. (5). Corresponding overlap distributions, $\overline{P(q)}$, are
illustrated in Fig (6).

Qualitatively similar behaviour is found in other systems lacking symmetry of
definiteness; e.g. $\hbox{}{[{\mbox{?}}]}$

###  3 p-spin interactions

Another interesting and influential extension of the original binary-
interaction SK model was introduced by Derrida in 1980
$\hbox{}{[{\mbox{?}},{\mbox{?}}]}$, with the binary interaction term in the SK
Hamiltonian replaced by one involving $p$ interacting spins.

####  3.1 Ising spins

Derrida considered p interacting Ising spins,

$H_{p}=-\sum_{(i_{1}i_{2}....i_{p})}J_{i_{1}....i_{p}}{\sigma}_{i_{1}}{\sigma}_{i_{2}}.....{\sigma}_{i_{p}};{\hskip
5.69046pt}\sigma=\pm 1$ (5)

with the $J_{i_{1}....i_{p}}$ randomly chosen from a Gaussian probability
distribution of variance $J^{2}/N^{(p-1)}$. For $p\to\infty$ he showed that
the problem simplifies into a random energy model (REM) which he solved
exactly, demonstrating freezing into the lowest energy state below a critical
temperature, corresponding to 1RSB but without the need for a replica
formulation.

Subsequently, in 1984, Gross and Mézard $\hbox{}{[{\mbox{?}}]}$ further
considered the $p\to\infty$ model with several of the different methods used
for the SK model – including the replica method – to explicitly exhibit the
structure of the spin glass phase as 1RSB with a discontinuous transition.

In a further important study in 1985 $\hbox{}{[{\mbox{?}}]}$, Gardner
demonstrated for $p$-spin Ising systems that (i) for all $p>2$ the spin glass
transition from paramagnet to spin glass is 1RSB (whereas for p=2 (SK) the
transition is full RSB (FRSB)) and (ii) that for $2<p<\infty$ there is also a
lower temperature (Gardner) transition to FRSB. Both observations have had
further important ramifications, discussed in later chapters.

####  3.2 Spherical models

A solvable extension of the normal ferromagnetic Ising model, known as the
spherical model, was introduced by $\hbox{}{[{\mbox{?}}]}$, replacing the
Ising variables $\sigma=\pm{1}$ on each site $i=1..N$ by a single
(hyperspherical) condition $\sum_{i}^{N}{\sigma_{i}}^{2}=N$, with the
individual $\sigma$ otherwise able to take any real value. A corresponding
spherical extension of the (infinite-range $p=2$) SK Ising spin glass model
was introduced and solved by Kosterlitz et al. $\hbox{}{[{\mbox{?}}]}$,
shortly after SK, using several different mathematical procedures (including
without needing replicas), and shown to exhibit a spin glass phase without
replica symmetry breaking.

In a later study of the equilibrium statistical mechanics (statics) for the
case of a spherical extension of the general-$p$-spin infinite-range (SK-
extended) model, Crisant and Sommers (1992) showed that for $p>2$ the spin
glass state is 1RSB, with onset discontinuous below a critical magnetic field,
continuous above it $\hbox{}{[{\mbox{?}}]}$ . In a further (1993) paper on the
relaxational dynamics with Horner $\hbox{}{[{\mbox{?}}]}$ they demonstrated
several further intriguing features, including that the onset temperatures for
1RSB is higher for dynamics than that given by equilibrium statics. Later work
recognised that the dynamical 1RSB onset corresponds to marginal stability of
the statics.

Also in 1993, Cugliandolo and Kurchan $\hbox{}{[{\mbox{?}}]}$ made a different
study of the dynamics of the $p$-spin spherical model at long but finite
times, demonstrating many unexpected but important new features, including
‘weak’ ergodicity breaking and aging effects, leading to much further activity
and results, which will be expanded upon in later chapters.

Even though its statics do not exhibit RSB, the $p=2$ spherical model was
shown also to have non-trivial dynamics $\hbox{}{[{\mbox{?}}]}$.

Fig. (7) shows examples of several models with various $p$-spin random zero-
mean SK-extended interactions together with 2-spin ferromagnetic interactions,
demonstrating several phases, including glassy ferromagnetism, and
transitions.

Figure 7: Phase diagrams for $p$-spin random zero-mean interactions plus
2-spin ferromagnetic interactions: (i) spherical spins $p=2$ (from
$\hbox{}{[{\mbox{?}}]}$ with permission. ©American Physical Society (1976)),
statics, no RSB; (ii) spherical spins $p=4$, statics (solid lines), dynamics
(fine lines), dot (critical point, discontinuous 1RSB to left, continuous 1RSB
to right.) (from $\hbox{}{[{\mbox{?}}]}$ with permission. ©American Physical
Society (1999) ); (iii) Ising $p=5$ statics, showing also 1RSB to FRSB
transition (from $\hbox{}{[{\mbox{?}}]}$ ©IOP Publishing (2001).

There has also been significant interest in mixed-$p$ spin glasses with
infinite-ranged extended-SK interactions, particularly spherical models
because of their solvability advantages along with realisation that they
(mixed-$p$ systems) can have FRSB as well as 1RSB, along with more complicated
issues, both static and dynamic, of potential relevance to the understanding
of other glasses; see e.g. $\hbox{}{[{\mbox{?}},{\mbox{?}},{\mbox{?}}]}$ and
later chapters in this book.

###  4 Finite-connectivity random graph models

Although the infinite-ranged SK model was introduced for its putative
solvability rather than to mimic specific experimental systems, subsequent
interest (and extension of the model) turned to many other problems where
spatial separation is irrelevant, e.g., communication through the world wide
web and random satisfiability in computer science. Several such problems are
discussed in later chapters, but we here (briefly) discuss a few which
appeared relatively early on in the field.

In 1985, Viana and Bray introduced a dimension-free but (on-average) finite-
connectivity alternative to the SK spin glass model [38], in which active
spins are effectively located on an Erdös-Renyi graph of average connectivity
$c$ with interactions on linked sites quench-randomly chosen from a
distribution $P_{c}(J)$:

$H=-\sum_{(ij)}J_{(ij)}{{\sigma}}_{i}{{\sigma}}_{j};\hskip
14.22636ptP(J_{ij})=(c/N)P_{c}(J_{ij})+(1-c/N)\delta(J_{ij}).$ (6)

As for EA and SK, a spin glass phase results from sufficient frustration in
$P_{c}(J)$, in purest form with $P_{c}(J)=P_{c}(-J)$, but without need for
$N$-scaling of the mean or variance for $c$ finite.

In 1986 Fu and Anderson $\hbox{}{[{\mbox{?}}]}$ showed its relationship to
graph bipartitioning444In the optimal graph bipartioning problem one seeks a
separation of vertices into two groups of equal size with the minimal number
of graph edges between them.This problem is equivalent to finding the ground
state of of a system of Ising spins on the vertices of the graph interacting
ferromagnetically along the edges but with a frustrating constraint of zero
overall magnetization., including a demonstration of mapping the extensively
connected graph problem to the SK model.

The Erdös-Renyi graph above has variable connectivity on different sites with
only its average given by $c$. However, there is also much interest –
particularly in computer science – in what are known as random regular graphs,
where all vertices have same connectivity $c$, but still quench-disordered.
This problem, with finite $c$, independent of the total number of vertices
$N$, was considered numerically by Banavar et al. $\hbox{}{[{\mbox{?}}]}$ in
connection with optimal graph bi-partitioning, finding evidence for several of
the novel features of the Parisi SK solution. The full analytical solution has
proven harder than for SK, but RSB was found
$\hbox{}{[{\mbox{?}},{\mbox{?}},{\mbox{?}}]}$. Further finite-connectivity
random graph optimization studies, and extensions to classic computer science
problems such as random satisfiability, have led to many new concepts, some of
which will be discussed in later chapters.

###  5 Induced moment spin glasses

Models considered thus far have moments (non-zero spins) even in the absence
of interactions. For some systems, however, interactions are required to
bootstrap-induce moments and exhibit cooperative magnetism. Ferromagnetic
examples have been known for centuries and recognised for many decades, but
here we make the case for spin glasses.

####  5.1 Discrete spins

Stimulated by consideration of local crystal field splitting favouring a
singlet ground state, a discrete/‘hard’-spin example was introduced in 1977 by
Ghatak and Sherrington (GS) $\hbox{}{[{\mbox{?}}]}$, with Hamiltonian

$H=-D\sum_{i}S_{i}^{2}-\sum_{ij}J_{ij}S_{i}S_{j};\ S=0,\pm 1,$ (7)

with the $J_{ij}$ Gaussian-distributed around zero mean. In the absence of the
interaction term, for $D>0$ the local term alone favours $S=\pm 1$, but for
$D<0$ it favours all $S=0$, no moments. Consequently, for $D>0$ the full
system (including the interaction term) behaves qualitatively analogously to
the SK model, while for $D<0$ there is a critical maximum $|D|$ for
interactions to induce bootstrap-induced cooperative order. At the RS level of
description, the phase diagram is as shown in Fig.(8) and demonstrates both
continuous and (Ehrenfest) first order phase transitions from paramagnet to
spin glass, along with re-entrance/inverse-freezing from spin glass to
paramagnet. Including RSB modifies the details of the first order transition
line but retains its qualitative character $\hbox{}{[{\mbox{?}}]}$.

Figure 8: Phase diagram of Ghatak-Sherrington model; figure from
$\hbox{}{[{\mbox{?}}]}$ ©IOP Publishing (1985) : The solid curve indicates a
continuous transition between paramagnet and spin glass, the dashed curve a
thermodynamically first order transition. The spin glass phase is full RSB.

####  5.2 Itinerant spin glasses

A simple ‘soft’-spin induced-moment spin glass model was initially proposed in
1973 by Sherrington and Mihill $\hbox{}{[{\mbox{?}},{\mbox{?}}]}$, to try to
explain the magnetic behaviour of the transition-metal spin-glass alloy RhCo;
see Fig(10) $\hbox{}{[{\mbox{?}}]}$555Pure bulk Rh is non-magnetic, pure bulk
Co is an itinerant ferromagnet, but isolated Co ions in Rh do not carry a
long-lived moment and at very low concentrations of Co RhCo is Pauli
paramagnetic. However, at intermediate concentrations a spin glass was
observed, along with magnetic clustering features.. Starting from a disordered
Hubbard model alloy of ions with different Coulomb $U$, transforming to
auxiliary local magnetization fields and with some simplification, including
static approximation, leads to an effective Hamiltonian of the form

$H=\sum_{i}\\{{\kappa}_{i}|{\bf{m}}_{i}|^{2}+\lambda_{i}|{\bf{m}}_{i}|^{4}\\}-\sum_{(ij);i\neq
j}J({\bf{R}}_{ij}){\bf{m}}_{i}.{\bf{m}}_{j},$ (8)

where the $i,j$ label sites, the $\bf{m_{i}}$ are unconstrained local
magnetization variables, the local coefficients $\kappa_{i}$ and $\lambda_{i}$
depend on the type of atom at site $i$, and the $J({\bf{R}}_{ij})$ are inter-
site interaction energies666In terms of the Hubbard model parameters,
$\kappa_{i}=(1-U_{i}\chi_{ii})$ and $J({\bf{R}}_{ij})=(U_{i}U_{j}\chi_{ij})$,
where the $U$ are the appropriate (site-occupation dependent) Hubbard
potentials and $\chi_{ij}$ is the two-site conduction-band susceptibility..

$\kappa<0$ would favour local moments (ground state with $m\neq 0$ in the
absence of interactions), the analogue of $D>0$ in the GS model;
cf$\hbox{}{[{\mbox{?}}]}$. But for both Rh and Co the $\kappa$ are positive,
so that the local harmonic terms alone favour no moment, $|{\bf{m}}|=0$.
Inclusion of the interaction term, however, offers the potential for
bootstrapped collective order if the resultant binding energy can overcome the
local cost.

Figure 9: RhCo; Paramagnet, spin Glass, Ferromagnet. Figure from
$\hbox{}{[{\mbox{?}}]}$ with permission. ©IOP Publishing (1975)

Figure 10: Schematic prediction for RhCo-type alloy, from
$\hbox{}{[{\mbox{?}}]}$ with permission ©European Physical Journals.

For pure Co this results in ferromagnetism, but the same is not true of Rh777
$(1-U_{Co}\sum_{j}{\chi_{ij}})<0$, $(1-U_{Rh}\sum_{j}{\chi_{ij}})>0$. which
consequently is not magnetic. Hence, already without need for computation, in
analogies with local moment spin glasses (such as Au Fe) and itinerant
ferromagnets (such as Ni), one can immediately anticipate a phase diagram with
three low temperature phases as a function of increasing Co concentration of
Pauli paramagnet, itinerant spin glass, and itinerant ferromagnet; see
Fig(10)888Pre-EA, Sherrington and Mihill $\hbox{}{[{\mbox{?}},{\mbox{?}}]}$
were motivated by a desire to include statistically occurring local moment-
like clustering. Post-EA, Hertz $\hbox{}{[{\mbox{?}}]}$ examined a similar
starting model, concentrating on prediction of an extended itinerant spin
glass phase and proposed naming it Stoner glass in recognition of the
pioneering work of Stoner $\hbox{}{[{\mbox{?}}]}$ on itinerant (conduction
electron) magnetism..

####  5.3 Relaxor ferroelectrics as ‘soft’ pseudo-spin glasses

A similar reasoning provides a likely explanation for a long-standing puzzle
in some displacive ferroelectric alloys and also opens up a potentially
interesting new direction for random-field magnets.

Examples of pure displacive ferroelectrics are found in ionic crystalline
compounds based on the generic formula $\rm{ABO_{3}}$ with charges
${\rm{A^{2+},B^{4+}}}$, and ${\rm{O^{2-}}}$, and at high temperature have
cubic perovskite structure; see Fig (11). Inter-ionic interactions can lead to
spontaneous structural distortions to lower symmetry at lower temperature if
the consequential interaction energy lowering is great enough to overcome
local displacement costs. With several ionic types within the unit cell, with
different harmonic displacement coefficients, the displacements of the
different ion types relative to their locations in the higher-temperature
phase are themselves different, leading to spontaneous intra-cell electric
dipole moments and hence ferro-electricity999Displacive ferroelectricity is
thus an analogue of itinerant ferromagnetism.. $\rm{BaTiO_{3}}$ (BT) and
${\rm{PbTiO_{3}}}$ (PT) are two classic examples and have been studied
theoretically quantitatively using non-disordered and three-dimensional
analogues of eqn.(8), with the local magnetizations replaced by displacements;
e.g by $\hbox{}{[{\mbox{?}},{\mbox{?}}]}$101010First-principles quantum-
mechanical calculations are used to determine the parameters in effective
classical Hamiltonians whose subsequent thermodynamical consequences are then
explored through computer statistical mechanical simulations.; see again Fig
(11).

Figure 11: $\rm{ABO_{3}}$ perovskite.
Circles show high T cubic cell: open circles = A, central black dot = B, dark
shaded circles = O.
Unit cell distorts to tetragonal ($c>a=b$) beneath $T_{c}$. Arrows show ionic
displacements in the ferroelectric phase relative to the tetragonal cell.
${\rm{PbTiO_{3}}}$, from $\hbox{}{[{\mbox{?}}]}$ with permission ©American
Physical Society (1996).

The present interest, however, concerns substitutional alloys.

Already in the 1950s, experimental behaviour $\hbox{}{[{\mbox{?}}]}$
retrospectively reminiscent of corresponding behaviour seen in the 1970s and
80s in experimental spin glasses $\hbox{}{[{\mbox{?}},{\mbox{?}}]}$ was
observed in the alloy $\rm{Pb(Mg_{1/3}Nb_{2/3})O_{3}}$ (PMN), with the Mg and
Nb ions randomly distributed on the B-sites of the $\rm{ABO_{3}}$ lattice, ;
high, sharpish, strongly frequency-dependent peaks in AC susceptibility, see
Figs. (12(i) and (ii), but with no change in overall lattice structure and no
ferroelectricity, again reminiscent of early spin glass experimentation that
demonstrated spin freezing without periodicity. Other characteristic spin
glass features, such as differences between FC and ZFC static susceptibilities
beneath the static/low frequency susceptibility peak temperature, were
observed later; see e.g. Fig. (13). Initially referred to simply as a
ferroelectric with diffuse phase transitions, later the name relaxor was
attributed to the new phase. Given their experimental similarity, theoretical
comparison with spin glasses, with induced electric dipoles the analogues of
local magnetizations, seems natural
$\hbox{}{[{\mbox{?}},{\mbox{?}},{\mbox{?}},{\mbox{?}}]}$.

Figure 12:  AC susceptibilities: (i) heterovalent relaxor
${\rm{PbMg_{1/3}Nb_{2/3}O_{3}}}$ (PMN), from $\hbox{}{[{\mbox{?}}]}$ with
permission ©Physical Society of Japan (1970); (ii) spin glass CuMn, from
$\hbox{}{[{\mbox{?}}]}$, with permission ©American Physical Society (1981);
(iii) homovalent relaxor ${\rm{BaZr_{0.35}Ti_{0.65}O_{3}}}$ (BZT), from
$\hbox{}{[{\mbox{?}}]}$, with permission ©IOP Publishing (2004); (iv)
ferroelectric ${\rm{BaZr_{0.20}Ti_{0.80}O_{3}}}$ from $\hbox{}{[{\mbox{?}}]}$,
with permission ©IOP Publishing (2004)  Figure 13: Field cooled (FC) vs. zero-
field cooled (ZFC) static susceptibilities; (i) PMN from
$\hbox{}{[{\mbox{?}}]}$, with permission ©American Physical Society (1998);
(ii)${\rm{BZT}}_{50/50}$ simulation, from $\hbox{}{[{\mbox{?}}]}$, with
permission ©American Physical Society;(2012) (iii) CuMn spin glass, from
$\hbox{}{[{\mbox{?}}]}$, with permission ©American Physical Society (1979).

However, it is instructive to consider first the more recently discovered
alloy ${\rm{Ba{Zr_{x}Ti_{1-x}}O_{3}}}$ (BZT)$\hbox{}{[{\mbox{?}}]}$, in which
the Ti and Zr B-ions are homovalent (both 4+) and so without the extra,
potentially complicating, charge- (and hence effective field-) disorder of
PMN. The AC susceptibility for BZT with $x=0.35$ is shown in Fig. (12(iii). It
looks very similar to those shown for PMN and CuMn, suggesting similar
physics. (Again, the average crystal structure remains cubic perovskite and
without global electric order.) The preparation-dependence of static
susceptibilities also look similar, with a separation between field cooled and
zero field cooled susceptibilities onsetting at the transition and growing as
the temperature is reduced; see Fig (13).

Noting that pure $\rm{BaZrO}_{3}$ is experimentally found not to exhibit
ferroelectricity, in contrast to $\rm{BaTiO}_{3}$, one can already deduce that
the local harmonic displacement coefficient of ${\rm{Zr^{4+}}}$ must be
significantly greater than that of ${\rm{Ti^{4+}}}$, too large for the
interaction terms to overcome it 111111Note that, since ${\rm{Zr^{4+}}}$ and
${\rm{Ti^{4+}}}$ are homovalent, the strengths of the ionic Coulombic
interactions in BT and BZ are essentially the same. 121212In fact the
coefficients can be calculated from numerical first-principles quantum
mechanics. Another crude indicator comes from comparison of ionic radii..

The relaxor phase of BZT, as discussed above, is straightforwardly explainable
by analogy with the induced-moment discussion above for a spin glass phase in
RhCo, with Zr the analogue of Rh, Ti of Co, and the relaxor the analogue of a
spin glass, with in-cell ionic displacements analogues of local
magnetizations; see Figs. (14) and (14) Simulations of ${\rm{BZT}}_{50/50}$
using a model as discussed above have been performed by
$\hbox{}{[{\mbox{?}}]}$ (FC/ZFC; Fig (13)(ii)) and $\hbox{}{[{\mbox{?}}]}$ (AC
susceptibility), in good accord with experiments.

Figure 14:  Left: Heuristic phase diagram for BZT, by analogy with spin glass.
PNRs refer to locally correlated statistical clusters (polar nano-regions)
above the extended cooperative transition $\hbox{}{[{\mbox{?}}]}$. F-R refers
to a possible mixed ferroelectric-relaxor.
Right: Experimental phase diagram for BZT, from $\hbox{}{[{\mbox{?}}]}$, with
permission ©American Ceramics Society (2008)

It is also interesting to compare the frequency-dependences of the location of
the peaks in experimental AC susceptibility curves of
${\rm{BaZr_{x}Ti_{1-x}O_{3}}}$ for concentrations of the Zr ions above and
below the critical concentration separating ferroelectric and relaxor. The
ferroelectric transition peak temperature of the AC susceptibility for
$x=0.20$ (ferroelectric; Fig (12)(iv)) is essentially frequency-independent,
as for a normal pure ferroelectric transition, while that for $x=0.035$
(relaxor; Fig(12)(iii)) is highly frequency-dependent, as indeed are the peaks
for spin-glass onset in low concentration CuMn (Fig(12(ii)) and heterovalent
relaxor PMN (Fig(12(i)). It seems likely that the greater dispersion of the
relaxors and spin glass, compared with their ferro counterparts, are
consequences of the rugged and chaotically-evolving free energy landscape that
underlies RSB.

Let us now return to the original, canonical and much studied relaxor PMN. It
differs from BZT in that its site-occupation disorder is heterovalent, with
the two types of B ions carrying different charges, Mg being 2+ and Nb 5+,
rather than the 4+ of the normal pure ($\rm{ABO_{3}}$) template. Being also
randomly distributed, they effectively lead to extra quenched random fields in
an extension of eqn.(8), in three dimensions.

Noting that ${\rm{Mg^{2+}}}$ has a similar ionic radius to ${\rm{Zr^{4+}}}$
and ${\rm{Nb^{5+}}}$ a similar ionic radius to ${\rm{Ti^{4+}}}$, one can
expect some B-ordering tendencies in PMN similar to those in BZT
$\hbox{}{[{\mbox{?}}]}$. However, since ${\rm{Pb{Zr_{x}Ti_{1-x}}O_{3}}}$ (PZT)
has not been observed to exhibit relaxor behaviour, the random fields appear
to be also necessary for the favouring of spin glass behaviour in PMN131313An
experimentally-based case for the important role of random fields in PMN was
originally made by $\hbox{}{[{\mbox{?}}]}$. 141414We might also note that
${\rm{Pb^{2+}}}$ is significantly more easily displaceable than
${\rm{Ba^{2+}}}$; in ferroelectric BT the relative displacements of Ba:Ti is
1:4 while in PT the relative displacements of Pb:Ti is 7:3, so that the
undiluted A-site (Pb) ions play the dominant role in PMN, in contrast to the
diluted B-site Ti-ions in BZT. This is in accord with the smaller ionic radius
of ${\rm{Pb^{2+}}}$ compared with ${\rm{Ba^{2+}}}$. This further raises the
issue of possible (quasi-)spin-glass behaviour in even-simpler systems with a
combination of frustrated but non-random interactions and quenched local
disorder only through random fields. Given that truly 3-dimensionally random
local magnetic fields are impossible to produce experimentally, heterovalent
relaxor alloys may provide fruitful alternative experimental laboratories for
the study of their consequences.

A third feature observed experimentally in relaxors, and considered as one of
their key characteristics, is of clusters of locally ordered regions above the
(susceptibility-peak) transition temperature, generally referred to as ‘polar
nano-regions’ (PNRs). Within the picture outlined above they are simply
explainable for homovalent systems (such as BZT) as quench-statistically-
occurring localized states in an analogue of Anderson localization
$\hbox{}{[{\mbox{?}}]}$ in which a ‘mobility edge’ determines the separation
of cooperative extended order and insufficiently correlated PNRs, while a
density-of-states ’band edge’ emulates the limit of PNR observability
$\hbox{}{[{\mbox{?}},{\mbox{?}}]}$.

A minimal SK-analogue induced-moment model is

$H=\sum_{i}(rm_{i}^{2}+um_{i}^{4})-\sum_{(ij)}J_{ij}m_{i}m_{j},$ (9)

$\hbox{}{[{\mbox{?}}]}$, with the {$J_{ij}$} drawn independently randomly from
the infinite-ranged SK distribution, offering the possibility of exact
solution151515It has also recently been proposed and studied in connection
with low temperature vibrational excitations of structural glasses
$\hbox{}{[{\mbox{?}}]}$.. However, since all the eigenstates of the SK-
distribution for {$J_{(ij)}$} are extended, no PNRs should result.

In summary, in this sub-section we have drawn attention to displacive relaxor
ferroelectrics that display features that are reminiscent of spin
glasses161616For a discussion of pseudo-spin glass-like behaviour in material
systems with pre-formed moments, see
$\hbox{}{[{\mbox{?}},{\mbox{?}},{\mbox{?}}]}$.. Although these features were
observed already six decades ago and have led to considerable practical
applications, their understanding nevertheless remains controversial. We have
argued for an explanation as induced pseudo-spin glasses. Furthermore, by
comparison of two different relaxor examples, homovalent and heterovalent, we
speculate on an important role for quenched random fields in systems with
spatially frustrated interactions, and on heterovalent relaxors as potentially
valuable experimental ‘laboratories’ that do not exist for magnetic systems.

###  6 Short-range and lattice models

The discussions above have been principally in terms of range-free systems.
Real experimental systems are normally lattice-based in physical dimensions,
normally $d=3$, sometimes with relatively short-ranged, but still frustrated,
interactions (e.g $\rm{Eu_{x}Sr_{1-x}S}$), sometimes long-ranged but decaying
with separation (e.g AuFe, CuMn). There has been much theoretical
investigation concerning whether such systems, particularly short-ranged, can
have RSB, whether there is a lower critical dimension for an AT transition in
a field and, if so, what. These issues will not however be further discussed
here, but left to later chapters, as will many applications and extensions of
concepts derived from infinite-ranged modelling.

###  7 Conclusion

We have introduced several extensions of the Ising SK spin glass model, to
different kinds of variables/spins/pseudo-spins and several types of
interactions, mostly still randomly drawn from range-free intensive
distributions, indicating several of the new features that have been exposed
by theorists’ ‘blue sky’ curiosity, mainly from the first few decades of spin
glasses. We have also noted two examples of conceptual and technical
transfers, to graph partitioning and relaxor ferroelectrics.

Later chapters will extend the developments and expose much further
theoretical enlightenment and fruitful conceptual and practical progress in
application to problems far beyond understanding the metallic alloys that
stimulated the theory of spin glasses half a century ago.

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## Index