Quantum entanglement phase transitions have provided new insights into quantum many-body dynamics. Both disorders and measurements are found to induce similar entanglement transitions. Here, we provide a theoretical framework that unifies these two seemingly disparate concepts and discloses their internal connections. Specifically, we analytically analyze a
d-dimensional free-fermion gas subject to continuous projective measurements. By mapping the Lindblad master equation to the functional Keldysh field theory, we develop an effective theory termed as the time-local Keldysh nonlinear sigma model, which enables us to analytically describe the physics of the monitored system. Our effective theory resembles that used to describe the disordered fermionic systems. As an application of the effective theory, we study the transport property and obtain a Drude-form conductivity where the elastic scattering time is replaced by the inverse measurement strength. According to these similarities, two different concepts, measurements and disorders, are unified in the same theoretical framework. A numerical verification of our theory and predictions is also provided.
In this thesis, we consider two different topics: the probes of out-of-equilibrium topological matter and many-body-localization (MBL) in higher dimensions. The first part focuses on the two-dimensional (2D) topological Floquet p+ip superfluid induced by an instantaneous quench of the interaction strength. We investigate radio-frequency (rf) spectroscopy, metal-to-superconductor tunneling, and angle-resolved photoemission spectroscopy (ARPES) as probes of this out-of-equilibrium quantum system in both the cold-atomic and solid-state realizations. We notice a single avoided crossing in the Floquet band structure and a population inversion in the occupation of these bands. In addition, the rf spectrum is well captured by a quasiequilibrium approximation and shows a robust gap. We also compute the rf signal when only the lower Floquet band is occupied as well as the tunneling signal. In both cases, the spectra show wildly different behaviors compared with the rf signal from the post-quench state, and the gap disappears for strong quenches. We attribute this to the fact that the distribution function influences the rf but not the tunneling signal. Finally, we look into the ARPES spectrum which exhibits a clear series of Floquet copies. The second part is devoted to interacting fermions in the presence of quenched disorder. We derive a finite-temperature Keldysh response theory for such a system in the delocalized phase, in the form of the Finkel'stein nonlinear sigma model (FNLσM). Applicable to any symmetry class with at least a U(1) symmetry, our formulation automatically incorporates the correct infrared cutoffs for quantum corrections to transport, and provides a framework to study the real and virtual scattering processes. In particular, we are able to rederive from the FNLσM the Altshuler-Aronov-Khmelnitsky equations for dephasing. We then propose a strategy to approach the 2D ergodic-to-MBL transition from the ergodic side. Such a transition is explored as a dephasing catastrophe in an isolated system with short-range interactions. In this case, the self-generated heat bath responsible for dephasing exhibits diffusive (non-Markovian) density fluctuations. We recast the dephasing of quantum interference corrections as a self-interacting polymer loop, and study its critical behavior using renormalization group approach. In dimensions d=4-ε, we identify a nontrivial fixed point corresponding to a finite temperature where the dephasing time diverges. This fixed point can be associated with a toy version of 2D ergodic-to-MBL transition, provided that it survives to ε=2.