Keldysh Nonlinear Sigma Model for a Free-Fermion Gas under Continuous Measurements

Abstract

We analytically analyze the quantum dynamics of a $d$-dimension free-fermion gas subject to continuous projective measurements. By mapping the Lindblad master equation to the functional Keldysh field theory, we observe that the Keldysh Lindblad partition function resembles that in the Keldysh treatment of the disordered fermionic systems. Based on this observation, we develop an effective theory termed as the Keldysh nonlinear sigma model to describe the low-energy physics. Two types of diffuson correlators, similar to those in the disordered fermionic systems, are derived. In addition, up to the one-loop level of the effective theory, we obtain a Drude-form conductivity where the elastic scattering time is replaced by the inverse measurement strength. According to these similarities, these two different concepts, i.e., projective measurements and disorders, are unified in the same framework.Comment: 10 pages, updated the reference list in V