Nonreciprocal many-body physics

TL;DR

This review surveys nonreciprocal many‑body physics by organizing it around five core definitions: nonvariational dynamics, violations of Newton's third law, broken detailed balance, nonreciprocal responses, and reciprocity in linear operators. It combines mathematical decomposition tools (Conley, Helmholtz–Hodge, transverse/quasipotential), stochastic frameworks, and spectral theories (PT symmetry, exceptional points) to link microscopic nonreciprocity to macroscopic phenomena such as nonequilibrium phase transitions, time crystals, and nonnormal amplification. It details how nonreciprocity arises across diverse platforms from hydrodynamics and electromagnetism to neural networks and quantum open systems, and discusses universal consequences like dynamical phase transitions, anomalous noise amplification, and scale‑dependent behavior via renormalization group concepts. The article emphasizes that while a universal classification remains elusive, structured approaches that combine dynamical systems theory, large deviation principles, and nonequilibrium field theory pave the way for understanding and engineering complex, irreversible collective states. Overall, nonreciprocity provides a unifying lens for studying driven, dissipative many‑body systems with broad implications for physics, engineering, and beyond, including active matter, quantum transport, and neuromorphic computation.

Abstract

Reciprocity is a fundamental symmetry present in many natural phenomena and engineered systems. Distinct situations where this symmetry is broken are typically grouped under the umbrella term "nonreciprocity", colloquially defined by: the action of A on B $\neq$ the action of B on A. In this review, we elucidate what nonreciprocity is by providing an introduction to its most salient classes: nonvariational dynamics, violations of Newton's third law, broken detailed balance, nonreciprocal responses and nonreciprocity of arbitrary linear operators. Next, we point out where to find these manifestations of non-reciprocity, from ensembles of particles with field mediated interactions to synthetic neural networks and open quantum systems. Given this breadth of contexts and the lack of an all-encompassing definition, it makes it all the more intriguing that some general conclusions can be gathered, when distinct definitions of nonreciprocity overlap. We explore what these universal consequences are with a special emphasis on collective phenomena that arise in nonreciprocal many-body systems. The topics covered include nonreciprocal phase transitions and non-normal amplification of noise and perturbations. We conclude with some open questions.

Figures (22)

• Figure 1: Classes of nonreciprocity. (a) Different classes of nonreciprocal behavior can be distinguished when the action of A on B has a strength that can be "positive" or "negative". (See Fig. \ref{['nonreciprocal_scattering_elements']} for a less binary notion of phase nonreciprocity.) We can then distinguish (i) reciprocal behavior where sign and amplitude are the same, (ii) weakly asymmetric behavior where amplitudes are different but signs identical, (iii) unidirectional nonreciprocity when A has an action on B and B has no action on A, and (iv) antagonistic nonreciprocity where the signs are different. (b) In addition, nonreciprocity may be completely unstructured within degrees of freedom, structured with two or more species, or structured in space, in several different ways. (c) Finally, nonreciprocity can occur within a species of agents, between different fields or modes, or between different species. Details are given in Sec. \ref{['overview_classification']} of the main text.
• Figure 2: Volume-preserving versus gradient-descent dynamics. Sketch Eq. \ref{['example_mix']} When $V = 0$, the phase space is made of concentric periodic orbits centered around a fixed point called a center. The motion occurs on a given periodic orbit selected by the initial value of $H$, and any perturbation of the system will make it go from one periodic orbit to the other (panel a). When $H = 0$, the variational dynamics drives the system towards the circle of stable fixed points ($r \equiv \lVert x \rVert = \sqrt{-a/b}$) at the bottom of the wine-bottle potential, where there is no motion (panel c).
• Figure 3: Scattering. We send incoming waves (in red) onto the scattering region, and receive outgoing waves (in blue).
• Figure 4: Nonreciprocal scattering elements. Standard symbols and canonical scattering matrices for a few basic nonreciprocal scattering elements. In the symbols (loosely following ANSI-Y32/IEC standards), each leg (numbered $1$, $2$, $3$, ...) describes a port with an incoming and an outgoing mode. For instance, dipoles correspond to the schematic of Fig. \ref{['fig_scattering_simple']}. The isolator describes "amplitude nonreciprocity". The scattering matrix of an isolator (or diode) is a Jordan block of size two (i.e. it is at an exceptional point). The gyrator is a particular case of the nonreciprocal phase shifter with differential phase shift $\phi = \pi$, and both describe "phase nonreciprocity". Finally, a circulator (here with three ports, but it can have any number $n \geq 3$ of ports) describes a "chiral nonreciprocity" capturing an imbalance between the clockwise and counterclockwise flow of signal or energy in the system.
• Figure 5: Exceptional point marking the boundary between PT-exact and PT-inexact regions. The eigenvalue spectrum corresponds to Eq. \ref{['L_EP_PT']} with $\delta = \kappa - \epsilon$ and fixed $\kappa > 0$.