Interacting Hysterons with Asymptotically Small or Large Spans

TL;DR

Interacting hysterons model bistable units with hysteresis to capture memory and complex pathways in multistable materials. The authors analyze two physically motivated limits—the large-span limit and the zero-span spin limit—establishing concrete mappings between hysteron and spin descriptions and deriving scaffold-inequality structures that govern transition graphs. In the large-span limit, type-II inequalities dependent on the mean span control state stability and can truncate mixed avalanches, while the scaffold remains governed by type-I inequalities; in the zero-span limit, nontrivial t-graphs require avalanches, and any set of $n$ hysterons can be mimicked by $2n$ spins via two mappings (symmetric and asymmetric pairs), with the asymmetric construction avoiding race conditions. These results connect hysteron- and spin-based models, clarifying when avalanches are essential and offering practical routes to design metamaterials with memory and controlled avalanche dynamics.

Abstract

Models of interacting hysteretic elements, called hysterons, capture the sequential response and complex memory effects in a wide range of complex systems and can guide the design of intelligent metamaterials. However, even simple models with few hysterons feature a bewildering number and variety of behaviors. Here we study the hysteron model in two physically relevant limits, where {the} response {of a hysteron system} is easier to understand. First, when the hysteron span - the gap between its two hysteretic transitions - dominates all other scales, the range of pathways encoded in transition graphs (t-graphs) becomes limited because many avalanches {are} absent. Second, when the hysteron span becomes vanishingly small, hysterons behave as interacting binary spins, {which require avalanches in order to} exhibit nontrivial pathways. Finally we show that hysterons can be mimicked by pairs of strongly interacting spins, {such} that collections of $n$ interacting hysterons can be mapped to $2n$ interacting spins, albeit {via} highly specific interactions. {Altogether,} our work provides a deeper understanding of the role of the hysteron parameters on their collective behavior, and points to connections and differences between spin- and hysteron-based models of complex matter.

Figures (4)

• Figure 1: The spin limit of the hysteron model in the presence and absence of interactions. a) Schematic representation of spins within the parameter space of interacting hysterons: the space of interacting hysterons encompasses that of interacting spins, and the behaviour of independent spins is trivial. b) Non-interacting spins can only exhibit trivial t-graphs, where up and down transitions for each spin are paired. c) Scaffold for independent spins.
• Figure 2: Illustration of the impossibility of nontrivial scaffolds in spin systems without avalanches. a) A single reversible spin flip for a t-graph without avalanches, occurring at $U_1$. b) Posited part of a t-graph for a spin system without avalanches, consisting of two transitions and three states. Since there are two up transitions from state $S'$ this t-graph is inconsistent, as indicated by the exclamation mark; thus, as stated in the main text, it is not possible to realize nontrivial t-graphs in spin systems without avalanches.
• Figure 3: Two constructions to replicate hysteron behavior using interacting spins. (a) Trivial scaffold for two spins. (b) Schematic of a symmetrically coupled pair of spins that realizes the trivial scaffold. (c) T-graph for symmetrically coupled spins featuring hysteretic avalanche transitions between $(00)$ and $(11)$ with the same intermediate state $(01)$. (d) Nontrivial scaffold for two spins. (e) Schematic of an asymmetrically coupled pair of spins that realizes the nontrivial scaffold. (f) T-graph for asymmetrically coupled spins featuring hysteretic avalanche transitions between $(00)$ and $(11)$ with different intermediate states $(01)$ and $(10)$.
• Figure 4: Replicating a system of interacting hysterons using spin pairs. a-b) T-graphs for two-hysteron systems as discussed in the main text, for $c_{21}>-3$ (a) and $c_{21}<-3$ (b) c) T-graph for $n=4$ spins which mimic the $n=2$ t-graph in panel (b).