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In classical finite-range spin systems, especially those with disorder suchas spin glasses, a low-temperature Gibbs state may be a mixture of a number ofpure or ordered states; the complexity of the Gibbs state has been defined inthe past roughly as the logarithm of this number, assuming the question ismeaningful in a finite system. As non-trivial mixtures of pure states do notoccur in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)]H\"oller and the author introduced a definition of the complexity of aninfinite-size Gibbs state as the mutual information between the pure state andthe spin configuration in a finite region, and applied this also within ametastate construction. (A metastate is a probability distribution on Gibbsstates.) They found an upper bound on the complexity for models of Ising spinsin which each spin interacts with only a finite number of others, in terms ofthe surface area of the region, for all $T\geq 0$. In the present paper, thecomplexity of a metastate is defined likewise in terms of the mutualinformation between the Gibbs state and the spin configuration. Upper boundsare found for each of these complexities for general finite-range (i.e. short-or long-range, in a sense we define) mixed $p$-spin interactions of discrete orcontinuous spins (such as $m$-vector models), but only for $T>0$. Forshort-range models, the bound reduces to the surface area. For long-rangeinteractions, the definition of a Gibbs state has to be modified, and for thesemodels we also prove that the states obtained within the metastateconstructions are Gibbs states under the modified definition. All results arevalid for a large class of disorder distributions.

Complexity as information in spin-glass Gibbs states and metastates: upper bounds at nonzero temperature and long-range models