Path-Dependency and Emergent Computing under Vectorial Driving
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.
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BibTeX
@article{meulblok2026pathdependency,
title = {{Path-Dependency and Emergent Computing under Vectorial Driving}},
author = {C. M. Meulblok and A. Singh and M. Labousse and M. van Hecke},
journal = {Physical Review X},
year = {2026},
month = nov,
doi = {10.1103/2pry-2kqq},
url = {https://hal.science/hal-05847967v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Path-Dependency and Emergent Computing under Vectorial Driving
AU - C. M. Meulblok
AU - A. Singh
AU - M. Labousse
AU - M. van Hecke
PY - 2026
DA - 2026/11/02
JO - Physical Review X
DO - 10.1103/2pry-2kqq
UR - https://hal.science/hal-05847967v1
AB - 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.
SN - 2160-3308
ER -