LAUSR

We identify an unconventional algebraic scaling phase in the quantum dynamics of free fermions with long range hopping, which are exposed to continuous local density measurements. The unconventional phase is characterized by an algebraic entanglement entropy growth, and by a slow algebraic decay of the density-density correlation function, both with a fractional exponent. It occurs for hopping decay exponents 1 < p . 3/2 independently of the measurement rate. The algebraic phase gives rise to two critical lines, separating it from a critical phase with logarithmic entanglement growth at small, and an area law phase with constant entanglement entropy at large monitoring rates. A perturbative renormalization group analysis suggests that the transitions to the long-range phase are also unconventional, corresponding to a modified sine-Gordon theory. Comparing exact numerical simulations of the monitored wave functions with analytical predictions from a replica field theory approach yields an excellent quantitative agreement. This confirms the view of a measurement-induced phase transition as a quantum phase transition in the dark state of an effective, non-Hermitian Hamiltonian.