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Path-Dependency and Emergent Computing under Vectorial Driving

C. M. Meulblok1,2, A. Singh1,2, M. Labousse1,2,3, and M. van Hecke1,2

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Phys. Rev. X 16, 031023 – Published 30 July, 2026

DOI: https://doi.org/10.1103/2pry-2kqq

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 in materia computing.

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Popular Summary

Physicists characterize materials by how they respond to external driving, such as stretching or compression. While simple materials—like rubber—behave predictably, complex materials—like sand piles or crumpled sheets—exhibit behavior that remains poorly understood. We study complex materials under sequential driving at different locations and uncover behaviors absent in simple global compression. In particular, we observe non-Abelian behavior: Compressing at location A then B produces a different outcome than compressing B then A. We develop theoretical and experimental tools to understand and describe these path-dependent responses. Path-dependency is a key feature of complex materials, and our work lays the groundwork for a new approach to studying them. This path dependency is also central to computing, where order matters, such as in typing a password. Although demonstrated with mechanical compression, our approach applies to other systems, opening new avenues for exploring memory, emergent behavior, and smart materials.

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Phys. Rev. X 16, 031023– Published 30 July, 2026
  • Received 12 March 2025
  • Accepted 8 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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