Measurement-Induced Dark State Phase Transitions in Long-Ranged Fermion Systems

We identify an unconventional algebraic scaling phase in the quantum dynamics of long-range hopping, free fermions, which are exposed to continuous local measurements. The algebraic phase occurs for hopping decay exponents 1 < p less than or similar to 3/2, and features an algebraic entanglement entropy growth, and a slow algebraic decay of the density-density correlation function, both with a fractional exponent. It is separated 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 predicts that the transitions to the long-range phase correspond to an unconventional, modified sine-Gordon theory. Exact numerical simulations of the monitored wave functions are in excellent agreement with an analytical replica field theory approach, which confirms the view of the measurement-induced phase transition as a quantum phase transition in the dark state of an effective, non-Hermitian Hamiltonian.