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260 Order and disorder in spin systems

31 32 33 34 35 36 + −−− +
+ −−−− + J 30,36 = J 36,30 = +1
25 26 27 28 29 30 +++ − +
−− + −−−
19 20 21 22 23 24 − ++++
+ − +++ −
13 14 15 16 17 18 −−−− +
+++ − + −
789 10 11 12 + −− ++
−− + − + − J = J = −1
123456 − + − ++ 6,12 12,6
sites couplings J kl
Fig. 5.26 Neighbor scheme and coupling strengths of a two-dimensional
±1 spin glass sample without periodic boundary conditions.

Table 5.9 Logarithm of the partition with the ferromagnetic Ising model resides in the existence ofa large
function and mean energy per particle number ofground states.In ourexample, there are 672 ground states;
of the two-dimensional spin glass shown
in Fig. 5.26 (frommodified Alg. 5.3 some ofthemshownin Fig.5.27.The thermodynamics ofthis model
(enumerate-ising)) followsfrom N (E), using Alg.5.4 (thermo-ising)(see Table 5.9). For
a quantitativestudyofspinglasses, we wouldhaveto average the energy,
T log Z E /N the free energy,etc., over many realizations ofthe {J kl }.However, this
is beyond the scope ofthis book.
1. 46.395 −0.932
2. 31.600 −0.665
3. 28.093 −0.495
4. 26.763 −0.389
5. 26.126 −0.319

Fig. 5.27 Several of the 672 ground states (with E = −38) of the two-
dimensional spin glass shown in Fig. 5.26.

Our aim is to check how the computational algorithms carryover from
the Ising model to the Ising spin glass.Wecan easily modify the local
Monte Carlo algorithm (see Alg.5.10 (markov-spin-glass)),and re-
produce the data in Table 5.9. For larger systems, the localMonte Carlo
algorithm becomes very slow. This is due, roughly,to the existence of
a large number ofground states, which lie at the bottoms of valleys
in a very complicated energylandscape.At low temperature, Alg.5.10
(markov-spin-glass) becomes trapped in these valleys, so that the lo-
cal algorithm takes a long time to explore a representativepart ofthe
configurationspace.Inmore than two dimensions, this time is solarge
that the algorithm, in the language of Subsection 1.4.1,is practically
nonergodic for large system sizes.
The cluster algorithm of Subsection 5.2.3 can be generalized to the
case ofspin glasses (see Alg.5.11 (cluster-spin-glass)) by changing
asingle line in Alg.5.9 (cluster-ising)(instead ofbuilding a cluster
with spins ofsamesign, weconsider neighboring spins σ j and σ k that
satisfy σ j J jk σ k > 0). Algorithm 5.11 (cluster-spin-glass) allowsto