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Quantum entanglement phase transitions have provided new insights to quantummany-body dynamics. Both disorders and measurements are found to induce similarentanglement transitions. Here, we provide a theoretical framework that unifiesthese two seemingly disparate concepts and discloses their internalconnections. Specifically, we analytically analyze a $d$-dimension free-fermiongas subject to continuous projective measurements. By mapping the Lindbladmaster equation to the functional Keldysh field theory, we develop an effectivetheory termed as the time-local Keldysh nonlinear sigma model, which enables usto analytically describe the physics of the monitored system. Our effectivetheory resembles to that used to describe the disordered fermionic systems. Asan application of the effective theory, we study the transport property andobtain a Drude-form conductivity where the elastic scattering time is replacedby the inverse measurement strength. According to these similarities, twodifferent concepts, measurements and disorders, are unified in the sametheoretical framework. A numerical verification of our theory and predictionsis also provided.

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