Non-Reciprocity in Statistical Physics and Field Theory

Non-reciprocity is a generic feature of non-equilibrium matter, appearing across active and soft matter, open quantum systems, the collective motion of living groups, and models of neural networks. It is an intrinsically dynamical phenomenon: non-reciprocal interactions are invisible to equilibrium statistical mechanics, so one must follow the full time evolution of the system. It follows that such matter is most naturally described by stochastic dynamics, discrete Markov chains, stochastic differential equations (SDEs), and, in the continuum limit, stochastic partial differential equations (SPDEs). Through the path integral, it is possible to reformulate these stochastic systems as a quantum field theories which is the universal language of high-energy physics, condensed matter and equilibrium statistical physics.

We build this framework from the ground up for two classes of microscopic models: discrete Markov chains, and classical or quantum systems coupled to a Caldeira-Leggett heat bath. The low energy continuum limit of these microscopic models gives the stochastic PDEs of Halperin-Hohenberg Model A and Model B, the stochastic Klein-Gordon and elastodynamic equations, and their non-reciprocal extensions. Applying the Martin-Siggia-Rose path integral, we give three equivalent quantum-field-theory formulations of non-reciprocal stochastic dynamics, the last a new construction with a N=1 supersymmetry that survives the loss of reciprocity and is enhanced to the N=2 Parisi-Sourlas supersymmetry in the reciprocal limit. We then put the framework to use: we exactly solve a one-dimensional non-reciprocal kinetic Ising model, finding non-reciprocity-induced frustration, exceptional-point transitions and anomalous scaling. We unveil a duality web connecting the non-reciprocal Ising model to the Kitaev-Hatano-Nelson chain. We show that this model has lines of critical exceptional points and that non-reciprocity protects Majorana edge modes. Lastly we construct in detail the new framework where one is able to map a stochastic system to a non-Hermitian, N=1 supersymmetric model, which generalizes the Parisi-Sourlas construction.