Monitored interacting Dirac fermions

Abstract

We analytically study interacting Dirac fermions, described by the Thirring model, under weak local particle number measurements with monitoring rate ${\hat{I}}^3$. This system maps to a bosonic replica field theory, analyzed via the renormalization group. For a nonzero attractive interaction, a phase transition occurs at a critical measurement strength ${\hat{I}}_c^3$. When ${\hat{I}}^3>{\hat{I}}_c^3$, the system enters a localized phase characterized by exponentially decaying density-density correlations beyond a finite correlation length; for ${\hat{I}}^3<{\hat{I}}_c^3$, the correlations decay algebraically. The transition is of BKT-type, reflected by a characteristic scaling of the correlation length. In the non-interacting limit, ${\hat{I}}_c^3\to 0$ shifts to zero, reducing the algebraic phase to a single point in parameter space. This identifies weak measurements in the free case as an implicit double fine-tuning to the critical endpoint of the BKT phase transition. Along the non-interacting line, we compute the entanglement entropy from density-density correlation functions and find no entanglement transition at nonzero measurement strength in the thermodynamic limit.