Critical dynamics of non-reciprocally coupled conserved systems

Non-reciprocal systems have been shown to sustain time-dependent patterns, most prominently travelling waves. The transition into these time-dependent states generally breaks time-translational invariance, representing a clear deviation from equilibrium dynamics. Though common implementations of non-reciprocity lead to such phenomenology, these spatio-temporal patterns are absent in other models. In the same vein, the ensuing scaling behaviour also depends on the precise way non-reciprocity is implemented. To better understand the effects of different non-reciprocal interactions, we study the critical conserved dynamics of non-reciprocally coupled spin systems. Specifically, we consider the dynamics of two $n$-component order parameter fields $\boldsymbolϕ_i$ with $i \in\{1,2\}$. Unlike the common implementations of non-reciprocal interactions, we introduce the non-reciprocity solely through the non-linear interaction between the distinct species. Using the field-theoretic renormalisation group (RG) procedure, we perform a one-loop analysis and show that at one-loop level, the critical behaviour depends on the microscopic value of certain quantities. Using the flow functions, we elucidate the behaviour of the fixed points for different bare microscopic values. We also show that for $n \geq 4$, there is a fixed point where the ensuing critical dynamics asymptotically obey detailed-balance, implying the emergent dynamics are agnostic to the microscopic non-reciprocity on large scales. Finally, we show that the conserved dynamics reduces the number of independent scaling exponents, mimicking the effect of a standard fluctuation-dissipation relation.