Path-Dependency and Emergent Computing under Vectorial Driving

Physical Review X, vol. 16

Abstract

The sequential response of frustrated materials—ranging from crumpled sheets and amorphous media to metamaterials—reveals their memory effects and emergent computational potential. Despite their spatial extension, most studies rely on a single global stimulus, such as compression, effectively reducing the problem to scalar driving. Here, we introduce vectorial driving of frustrated materials by applying multiple spatially localized stimuli to explore path-dependent, sequential responses. We uncover a wealth of phenomena absent in scalar driving, including non-Abelian responses, mixed-mode behavior, and chiral loop transients. We show that fold singularities connect three states—ancestor, descendant, and sibling. This recurring pattern serves as the elementary building block of all sequential paths. We then introduce three levels of description of sequential, path-dependent responses. At the most fundamental level, path-dependent transition graphs (pt-graphs) and strain maps capture the response under arbitrary vectorial driving and connect pathways to the underlying singularities. They provide a complete description analogous to transition graphs (t-graphs) for scalar driving. However, as pt-graphs and strain maps become unwieldy for high-dimensional driving, we introduce b-graphs—graphs whose nodes and transitions encode the system response to binarized vectorial driving. These present a less complete but much simpler second-level description by restricting attention to binary input and their induced transitions. The resulting graphs form subgraphs of the pt-graph and naturally connect to sequential computing and Boolean circuits. We show that a single sample can encode multiple b-graphs that are selectable through the driving strength and can be reprogrammed via additional inputs. Finally, we introduce graph-based motifs that enable a systematic analysis of b-graphs and provide a practical tool for characterizing the statistical response of very high-dimensional systems, where a full pt- or b-graph description is neither feasible nor particularly insightful. As statistical measures of pathway complexity, these motifs can be obtained in systems of any size or complexity. Our work paves the way for strategies to explore, harness, and understand complex materials and memory while advancing embodied intelligence and computing.

Authors 4

1. Colin M. Meulblok Aachen

   Leiden University · Institute for Atomic and Molecular Physics

   Affiliation as printed

   AMOLF

   Universiteit Leiden

2. A. Singh Aachen

   Leiden University · Institute for Atomic and Molecular Physics

   Affiliation as printed

   AMOLF

   Universiteit Leiden

3. Matthieu Labousse Aachen

   Leiden University · Centre National de la Recherche Scientifique · Université Paris Sciences et Lettres · Institute for Atomic and Molecular Physics · Gulliver · ESPCI Paris

   Affiliation as printed

   AMOLF

   CNRS

   ESPCI Paris

   Gulliver

   Universiteit Leiden

   Université PSL

4. Martin van Hecke Aachen

   Leiden University · Institute for Atomic and Molecular Physics

   Affiliation as printed

   AMOLF

   Universiteit Leiden