Process-tensor approach to full counting statistics of charge transport in quantum many-body circuits

Abstract: We introduce a numerical tensor-network method to compute the statistics of the charge transferred across an interface partitioning an interacting one-dimensional many-body lattice system with $U(1)$ symmetry. Our approach is based on a matrix-product state representation of the process tensor (also known as influence functional or influence matrix) describing the effect of the bulk system on the degrees of freedom at the interface, allowing us to evaluate a multi-time correlation function that yields the moment-generating function of charge transfer. We develop a scheme to truncate non-Markovian correlations which preserves the proper normalization of the process tensor and ensures the correct physical properties of the generating function. We benchmark our approach by simulating magnetization transport within the Heisenberg spin-$1/2$ XXZ brickwork circuit model at infinite temperature. Our results recover the correct transport exponent describing ballistic, superdiffusive, and diffusive transport in different regimes of the model. We also demonstrate anomalous transport encoded by a self-similar scaling form of the moment-generating function outside of the ballistic regime. In particular, we confirm the breakdown of Kardar-Parisi-Zhang universality in higher-order transport cumulants at the isotropic point. Our work paves the way for process-tensor descriptions of non-Markovian open quantum systems to address current fluctuations in strongly interacting systems far from equilibrium.