Open quantum spin chains with non-reciprocity: a theoretical approach based on the time-dependent generalized Gibbs ensemble

We study an open quantum spin chain with non-reciprocal dissipation using a theoretical approach known as time-dependent generalized Gibbs ensemble. In the regime of weak dissipation the system is fully characterized by its rapidity distribution and we derive a closed set of coupled differential equations governing their time evolution. We check the accuracy of this theory by benchmarking the results against numerical simulations. Using this framework we are able to compute both the magnetization density and current dynamics, identifying some relations between the two. The problem of the anomalous power-law exponents identified in a previous work is discussed. Our work constitutes a theoretical approach that is able to describe the physics of non-reciprocal open quantum spin chains beyond analyses based on non-interacting fermions.