Schwinger-Keldysh field theory for operator R\'{e}nyi entropy and entanglement growth in non-interacting systems with sub-ballistic transports

TL;DR

This paper develops a Schwinger-Keldysh field theory framework to analyze operator Rényi entropy and entanglement growth in non-interacting disordered systems, revealing their connection to different transport regimes.

Contribution

It introduces a unified SK field theory formalism for operator and entanglement entropies, linking them to transport phenomena in non-interacting quantum systems with disorder.

Findings

Subsystem operator Rényi entropy captures transport behavior.

Method applies to quasiperiodic and random disorder models.

Growth of entropies distinguishes ballistic, diffusive, and localized phases.

Abstract

The notion of operator growth in quantum systems furnishes a bridge between transport and the generation of entanglement between different parts of the system under quantum dynamics. We define a measure of operator growth in terms of subsystem operator R\'{e}nyi entropy, which provides a state-independent measure of operator growth, unlike entanglement entropies, and the usual measures of operator growth like out-of-time-order correlators. We show that the subsystem operator R\'{e}nyi entropy encodes both spatial and temporal information, and thus can directly connect to transport for a local operator related to a conserved quantity. We construct a unified Schwinger-Keldysh (SK) field theory formalism for the time evolution of operator R\'{e}nyi entropy and entanglement entropies of initial pure states. We use the SK field theory to obtain the operator R\'{e}nyi and state entanglement entropies in terms of infinite-temperature and vacuum Keldysh Green's functions, respectively, for non-interacting systems. We apply the method to explore the connection between operator and entanglement growth, and transport in non-interacting systems with quasiperiodic and random disorder, like the one- and two-dimensional Aubry-Andr\'{e} models and the two-dimensional Anderson model. In particular, we show that the growth of subsystem operator R\'{e}nyi entropy and state von Neumann and R\'{e}nyi entanglement entropies can capture both ballistic and sub-ballistic transport behaviors, like diffusive and anomalous diffusive transport, as well as localization in these systems.