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

Abstract

In classical finite-range spin systems, especially those with disorder such as spin glasses, a lowtemperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly as the logarithm of this number, assuming the question is meaningful in a finite system. As non-trivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)] Höller and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all T ≥ 0. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are 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 or continuous spins (such as m-vector models), but only for T > 0. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.

Metastates and Replica Symmetry Breaking

“…( 1), a simple argument, similar to counting boundary conditions, then produces a bound on the κ-expectation of the complexity K Γ (Λ W ) of the Gibbs states by a constant times the surface area W d−1 [48]. More generally, a bound of the same form can be obtained for short-range models of classical spins [49], though the argument was valid only for T > 0.…”

“…The weights and the pure states allow us to consider the joint distribution w Γ (α)Γ α (σ ) of pure states α and spin configurations σ (for given Γ). If we restrict σ to the spin configuration σ Λ in the hypercube Λ = Λ W as before, then the mutual information I(σ Λ ; α) ≥ 0 between σ Λ and α can be defined in terms of the joint distribution of those random variables, and we now call this Λ W -dependent quantity the complexity K Γ (Λ W ) of the Gibbs state Γ [48,49]:…”

“…As W → ∞, K Γ (Λ W ) increases monotonically, and tends to the same value as for Palmer's definition of complexity. We can similarly define the complexity K ρ J (Λ W ) of the MAS ρ J in place of Γ, and also the complexity K κ J (Λ W ) of the metastate κ J itself: K κ J (Λ W ) is defined as the mutual information between the Gibbs (not pure) state Γ and σ Λ , using the joint distribution κ J (Γ)Γ(σ Λ ) [49]. These three quantities are related (for given J and Λ W ) by…”

“…It was later argued that in a short-range spin glass, this phase, which appeared to possess extensive complexity, would be destroyed by entropic effects [60,61], so that there would be no transition into it on lowering the temperature (there was still supposed to be another transition at lower temperature, which was termed a "random first-order transition" [60,61]). However, it does not seem clear that such effects must completely destroy such a frozen phase; instead they could leave a similar phase, still in class 3), now having a subextensive complexity that obeys the bounds discussed above [48,49].…”