Quantum phase transition and resurgence: Lessons from three-dimensional N=4 supersymmetric quantum electrodynamics

We study a resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular, we focus on three-dimensional N = 4 supersymmetric quantum electrodynamics (SQED) with multiple hypermultiplets, where a second-order quantum phase transition has recently been proposed in the large-flavor limit. We provide interpretations of the phase transition from the viewpoints of Lefschetz thimbles and resurgence. For this purpose, we study the Lefschetz thimble structure and properties of the large-flavor expansion for the partition function obtained by the supersymmetric localization. We show that the second-order phase transition is understood as a phenomenon where a Stokes and an anti-Stokes phenomenon occur simultaneously. The order of the phase transition is determined by how saddles collide at the critical point. In addition, the phase transition accompanies an infinite number of Stokes phenomena due to the supersymmetry. These features are appropriately mapped to the Borel plane structures as the resurgence theory expects. Given the lessons from SQED, we provide a more general discussion on the relationship between the resurgence and phase transitions. In particular, we show how the information on the phase transition is decoded from the Borel resummation technique.