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arXiv:2502.02645 · Physical review. B./Physical review. B

Monitored interacting Dirac fermions

T. Müller, Michael Buchhold, Sebastian Diehl

Published May 1, 2025• 7 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.111.174201

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We analytically study interacting Dirac fermions, described by the Thirring model, under weak local particle number measurements with monitoring rate \ensuremathγ\ensuremath{\gamma}\ensuremathγ. 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 \ensuremathγc{\ensuremath{\gamma}}_{c}\ensuremathγc​. When \ensuremathγ>\ensuremathγc\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}\ensuremathγ>\ensuremathγc​, the system enters a localized phase characterized by exponentially decaying density-density correlations beyond a finite correlation length; for \ensuremathγ<\ensuremathγc\ensuremath{\gamma}<{\ensuremath{\gamma}}_{c}\ensuremathγ<\ensuremathγc​, 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, \ensuremathγc\ensuremath→0{\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0\ensuremathγc​\ensuremath→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.
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A theory of the measurement-induced entanglement phase transition for free-fermion models in d>1d>1d>1 dimensions is developed. The critical point separates a gapless phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}\mathrm{ln}\ensuremath{\ell}$ scaling of the second cumulant of the particle number and of the entanglement entropy and an area-law phase with ${\ensuremath{\ell}}^{d\ensuremath{-}1}$ scaling, where $\ensuremath{\ell}$ is a size of the subsystem. The problem is mapped onto an SU(R)\mathrm{SU}(R)SU(R) replica nonlinear sigma model in d+1d+1d+1 dimensions, with $R\ensuremath{\rightarrow}1$. Using renormalization-group analysis, we calculate critical indices in one-loop approximation justified for $d=1+\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\ll}1$. Further, we carry out a numerical study of the transition for a d=2d=2d=2 model on a square lattice, determine numerically the critical point, and estimate the critical index of the correlation length, $\ensuremath{\nu}\ensuremath{\approx}1.4$.
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This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKTT_{\text{BKT}}TBKT​ as a function of \emph{attractive} pairing strength by associating it with the onset of quasi-condensation in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established Quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS-BEC crossover theory for fermions, leads to non-monotonic behavior for TBKTT_{\text{BKT}}TBKT​ as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that TBKTT_{\text{BKT}}TBKT​ has an absolute maximum of the order of 0.1EF0.1 E_F0.1EF​. We discuss how our results compare with those derived from the Nelson Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in TBKTT_\text{BKT}TBKT​ followed by an often dramatic decrease before the system enters the BEC regime.
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Abstract This paper addresses the transition from the normal to the superfluid state in strongly correlated two dimensional fermionic superconductors and Fermi gases. We arrive at the Berezinskii–Kosterlitz–Thouless (BKT) temperature T BKT as a function of attractive pairing strength by associating it with the onset of ‘quasi-condensation’ in the normal phase. Our approach builds on a criterion for determining the BKT transition temperature for atomic gases which is based on a well established quantum Monte Carlo analysis of the phase space density. This latter quantity, when derived from BCS–BEC crossover theory for fermions, leads to non-monotonic behavior for T BKT as a function of the attractive interaction or inverse scattering length. In Fermi gases, this implies a robust superconducting dome followed by a long tail from the flat BEC asymptote, rather similar to what is observed experimentally. For lattice systems we find that T BKT has an absolute maximum of the order of 0.1E F. We discuss how our results compare with those derived from the Nelson–Kosterlitz criterion based on the mean field superfluid density and the approach to the transition from below. While there is agreement in the strict mean-field BCS regime at weak coupling, we find that at moderate pairing strength bosonic excitations cause a substantial increase in T BKT followed by an often dramatic decrease before the system enters the BEC regime.