Topological Transitions with Continuously Monitored Free Fermions

We study a free fermion model where two sets of non-commuting continuous measurements induce a transition between area-law entanglement scaling phases of distinct topological order. We find that, in the presence of unitary dynamics, the two topological phases are separated by a region with sub-volume scaling of the entanglement entropy and that the transition universality class of the measurement-only model differs from that in interacting models with stroboscopic dynamics and projective measurements. We further show that the phase diagram is qualitatively captured by an analytically tractable non-Hermitian Hamiltonian model obtained via post-selection. By the introduction of a partial-post-selection continuous mapping, we show that the topological distinct phases of the fully stochastic measurement-induced dynamics can be continuously deformed to phases of the post-selected model with different topological numbers.