Monitored interacting Dirac fermions

We analytically study interacting Dirac fermions, described by the Thirring model, under weak local particle number measurements with monitoring rate $\ensuremath{\gamma}$. 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 ${\ensuremath{\gamma}}_{c}$. When $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, the system enters a localized phase characterized by exponentially decaying density-density correlations beyond a finite correlation length; for $\ensuremath{\gamma}<{\ensuremath{\gamma}}_{c}$, the correlations decay algebraically. The transition is of Berezinskii-Kosterlitz-Thouless (BKT) type, reflected by a characteristic scaling of the correlation length. In the noninteracting limit, ${\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0$ 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 end point of the BKT phase transition. Along the noninteracting line, we compute the entanglement entropy from density-density correlation functions and find no entanglement transition at nonzero measurement strength in the thermodynamic limit.