Broken Symmetry Unlocks New Physics in Particles and Artificial Networks

Nonreciprocity, the breaking of symmetry where the influence of one component on another differs from the reverse, represents a growing area of interest across physics and engineering. Michel Fruchart of Gulliver, ESPCI Paris, Université PSL, CNRS, and Vincenzo Vitelli from the James Franck Institute and Leinweber Center for Theoretical Physics at the University of Chicago, and colleagues, demonstrate a comprehensive review of this phenomenon, detailing its diverse manifestations from interacting particle ensembles to complex synthetic networks. This collaborative work, bridging institutions in Paris and Chicago, is significant because it clarifies the varied definitions of nonreciprocity and identifies underlying universal consequences, particularly concerning collective behaviours in many-body systems. Their analysis illuminates how nonreciprocal interactions can lead to unusual amplification of noise and perturbations, opening new avenues for research into non-equilibrium physics and potentially enabling novel technologies.

Reciprocity, a fundamental symmetry underpinning numerous natural phenomena and engineered systems, is now being challenged by a growing understanding of its breakdown. Situations where this symmetry is broken fall under the broad term “nonreciprocity”, defined simply as an asymmetry in interaction, where the action of A on B is not equivalent to the action of B on A. The study demonstrates that despite the varied origins of nonreciprocal behaviour, ranging from particle ensembles with field-mediated interactions to synthetic neural networks and open quantum systems, overarching consequences emerge when these distinct definitions overlap. A key finding concerns collective phenomena in nonreciprocal many-body systems, revealing how the absence of reciprocity fundamentally alters system behaviour, specifically leading to the amplification of noise and perturbations with significant implications for stability and predictability. By examining these diverse contexts, the authors reveal surprising connections and universal behaviours, highlighting the potential for designing systems with tailored responses to external stimuli and for controlling collective behaviour in ways previously unattainable. A Lindblad master equation governs the evolution of the density matrix ρ, describing Markovian open quantum systems, taking the form dρ/dt = Lρ, where L is the Lindbladian superoperator acting on operators like ρ, and incorporates both Hamiltonian evolution and coupling to the environment via jump operators L.
To further link the master equation to classical descriptions, the study considers quasiprobability distributions in phase-space, specifically the Wigner function W(t, p, x), with its time evolution expressed as a continuity equation, ∂tW + ∇· JW = 0, where JW represents a probability current in phase space. Linear response theory is employed to analyse nonreciprocal responses, expressing the response X to an input Y as Xi = RijYj, where R is a matrix of linear response coefficients. Reciprocity, in this context, is defined by the symmetry constraint R = RT, and its breakdown signifies nonreciprocal behaviour, allowing researchers to derive constraints on R from microscopic symmetries or conversely, infer microscopic symmetries from the observed response R. Unidirectional non-reciprocity, where the action of A on B differs entirely from the action of B on A, represents a fundamental classification within non-reciprocal interactions, arising when Aij equals zero while Aji is non-zero. Antagonistic non-reciprocity manifests when interactions exhibit opposing signs, such as attraction versus repulsion, while weak non-reciprocity describes interactions with the same sign but unequal magnitude. The research details a classification scheme based on graphical notation borrowed from systems biology, employing pointed arrows for positive interactions and blunt arrows for negative ones, allowing for a nuanced understanding of interaction dynamics. Furthermore, the study highlights that in many-body systems, non-reciprocity can be either random or structured, with structured non-reciprocity, where A consistently acts negatively on B while B consistently acts positively on A, predicted to persist even when examined at a coarse-grained level, potentially manifesting through macroscopic populations or fluxes. Systematic coarse-graining, achieved through the identification of network motifs, can reveal these underlying non-reciprocal patterns within complex systems. Scientists have long been captivated by symmetry, yet increasingly recognise that its breaking is often more revealing. This review of non-reciprocity isn’t simply a technical exercise in defining asymmetry; it’s a crucial step towards understanding systems where cause and effect are not neatly reversible. For decades, the assumption of reciprocity has simplified modelling in physics, engineering, and even social sciences, but this simplification fails in a growing number of real-world scenarios, from the behaviour of active matter to the design of advanced signal processing systems. The difficulty lies in the sheer variety of ways reciprocity can be broken, and this work helpfully categorises these, moving beyond simple mechanical analogies to encompass more abstract forms of non-reciprocal interaction, revealing underlying principles common to seemingly disparate phenomena. The implications extend to designing more efficient energy harvesting technologies, creating robust communication networks, and modelling collective behaviour in biological systems. Defining non-reciprocity remains surprisingly subtle, and identifying universal consequences across different definitions is a complex undertaking. Future research must focus on developing predictive frameworks that can account for the interplay between different types of non-reciprocal interactions, and translating these theoretical insights into tangible devices requires overcoming significant materials science and engineering hurdles. The next wave of progress will likely involve exploring non-reciprocal phenomena in increasingly complex and far-from-equilibrium systems, pushing the boundaries of our understanding of collective behaviour.