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5.3 Generalized Ising models 259

5.3 Generalized Ising models

Interactions in nature vary in strength with distance but usually do not
change sign.This applies to the fourfundamental forces in nature, and
also to many effective (induced) forces.The exchange interaction, due
to the overlap of d-electron orbitals ondifferent lattice sites, which is
responsibleforferromagnetism, is ofthis type.Itstrives to align spins.
In addition, it falls off veryquickly with distance.This explains why,in
the Ising model,the effectivemodel forferromagnetism, only nearest-
neighborinteractions are retained and longer-range interactions are ne-
glected.
Some other interactions are more complicated. One example will be
the depletioninteraction ofcolloids considered in Chapter 6. Weshall
see that at some distances, particles are attracted to each other, and at
other nearby distances they are repelled fromeach other.The dominant
interactionbetween ferromagnetic impurities in many materialsis also
ofthis type.Itcouples spins over intermediate distances, butsometimes
favors them to be aligned, sometimes to be of opposite sign.Materials
for which this interactionisdominant are called spin glasses.
The theoryofspin glasses, and more generally ofdisordered systems, is
an activefield of research whose basic model is the Ising spin glass, where
the interaction parameter J is replaced by aterm J kl which is different
forany two neighbors k and l.More precisely, each piece ofmaterial
(each experimental“sample”) is modeled by aset ofinteractions {J kl },
which are random, because their values depend on the precise distances
between spins, which differ fromsampleto sample. Each experimental
samplehas its ownset ofrandom parameters which do not change during
the life ofthe sample (the {J kl } are “quenched” random variables). Most
commonly,the interaction J kl between neighboring sites is taken as ran-
domly positive and negative, ±1. One ofthe long-lasting controversies
in the field ofspinglasses concerns the nature ofthe low-temperature
spin-glass phase in three spatial dimensions.

5.3.1 The two-dimensional spin glass

Inthissubsection, wepay a lightning visit to the two-dimensional spin
glass.Among the many interesting problems posed by this model, were- Table 5.8 Number of configurations
strict ourselves to running through the batteryofcomputational meth- with energy E of the two-dimensional
ods, enumeration (listing and counting), localMonte Carlo sampling, spin glass shown in Fig. 5.26 (from
modified Alg. 5.3 (enumerate-ising))
and cluster sampling.Forconcreteness, weconsider a single Ising spin
glass sample ona 6×6lattice withoutperiodic boundary conditions (see
E N (E)= N (−E)
Fig.5.26) with an energy
0 6 969 787 392

E = −J kl σ k σ l , −2 6 754 672 256
. .
k,l . .
. .
−34 59 456
where the parameters J kl are defined in Fig.5.26.
−36 6 912
Algorithm 5.3 (enumerate-ising) is easily modified to generate the −38 672
densityof states N (E)ofthis system (see Table 5.8). The main difference