Entanglement phases and phase transitions in monitored free fermion system due to localizations

TL;DR

We address how continuous measurement competes with localization to shape entanglement phases in a one-dimensional free-fermion chain. We simulate quantum trajectories under a stochastic Schrödinger equation, extract the steady-state entanglement entropy and an effective central charge, and perform finite-size/BKT scaling to locate the phase boundary. The results reveal a measurement–localization driven entanglement transition from a log-law to an area-law regime, with SP and QPP producing qualitatively similar phase diagrams but different boundary bending, consistent with a shared BKT universality class. The work yields a unified phase diagram across localization mechanisms and provides experimental criteria for cold-atom, trapped-ion, and quantum-dot platforms.

Abstract

In recent years, the presence of local potentials has significantly enriched and diversified the entanglement patterns in monitored free fermion systems. In our approach, we employ the stochastic Schrödinger equation to simulate a one-dimensional spinless fermion system under continuous measurement and local potentials. By averaging the steady-state entanglement entropy over many quantum trajectories, we investigate its dependence on measurement and localization parameters. We used a phenomenological model to interpret the numerical results, and the results show that the introduction of local potentials does not destroy the universality class of the entanglement phase transition, and that the phase boundary is jointly characterized by the measurement process and the localization mechanism. This work offers a new perspective on the characterization of the entanglement phase boundary arising from the combined effects of measurement and localization, and provides criteria for detecting this novel phase transition in cold atom systems, trapped ions, and quantum dot arrays.

Figures (10)

• Figure 1: (a): Monitored half-filled fermions in SP (brown) and QPP (purple). (b): The phase boundaries in the phase diagram are formed by different localization mechanisms (Wannier-Stark localization and Anderson localization). We present the general solution form, with specific parameters given below. The grey region corresponds to the case of a very weak localization potential, where we tend to revert to the results obtained from the Keldysh field theory Igo_prx; the black dashed line corresponds to the phase boundary within this region.
• Figure 2: Phase diagrams and phase boundaries extracted from the effective central charge. (a) Phase diagram under joint tuning of the measurement strength and SP strength; (b) Phase diagram under joint tuning of the measurement strength and the QPP strength. Colors indicate the effective central charge $\bar{c}_{\mathrm{eff}}$, extracted from the finite-size scaling of half-chain entanglement entropy: green corresponds to large $\bar{c}_{\mathrm{eff}}$ values (log-law-like regime), while brown corresponds to $\bar{c}_{\mathrm{eff}} \rightarrow 0$ (area-law regime). Dashed curves denote the semi-analytical expressions for phase boundaries derived by Eq. \ref{['fs']}. Black solid dots mark four phase transition points obtained through finite-size scaling analysis and data collapse. The grey region corresponds to the case of a very weak localization potential, where we tend to revert to the results obtained from the Keldysh field theory Igo_prx; therefore, it is not shown in the diagram. In the subsequent finite-size scaling analysis, four representative points on each phase diagram (marked in red, green, blue, and purple, respectively) are selected to characterize the entanglement scaling behavior at the phase boundary of that point.
• Figure 3: Entanglement entropy versus system size for representative points. (a) Results corresponding to the four selected points from Fig. 1(a): $\Delta=0.1, \gamma=0.1$, $\Delta=0.1, \gamma=0.8$, $\Delta=0.8, \gamma=0.1$, and $\Delta=0.8, \gamma=0.8$; (b) Results corresponding to the four selected points from Fig. 1(b): $V=0.3, \gamma=0.1$, $V=0.3, \gamma=0.6$, $V=2.5, \gamma=0.1$, and $V=2.5, \gamma=0.6$. The plots show the half-chain entanglement entropy $\bar{S}_{L/2}$ as a function of system size $L$, revealing the scaling behavior under different potential and measurement strengths. Log-log axes are used throughout for visual clarity.
• Figure 4: Effective central charge $\bar{c}_{\mathrm{eff}}$ versus various control parameters with entanglement scaling insets. (a) Fixed $\Delta = 0.1$, varying measurement strength ; (b) fixed $V = 0.3$, varying measurement strength ; (c) fixed $\gamma = 0.1$, varying SP strength ; (d) fixed $\gamma = 0.1$, varying QPP strength. Insets display entanglement entropy versus $\sin(\pi \ell / L)$ on a logarithmic $x$-axis to emphasize scaling behavior. Shades of blue from light to dark correspond to system sizes $L = 12, 24, 48, 64, 80, 96, 112$, the variation in system size is also reflected in the extent of the x-axis in the insets.
• Figure 5: Mutual information under varied control parameters, with scaling insets. (a) Fixed $\Delta = 0.1$, varying measurement strength ; (b) fixed potential strength $V = 0.3$, varying measurement strength; (c) fixed measurement strength $\gamma = 0.1$, varying SP strength; (d) fixed $\gamma = 0.1$, varying QPP strength. Main panels show mutual information as a function of each parameter. Insets present correlation functions versus $\sin(\pi \ell / L)$, log-scaled on the $x$-axis to highlight scaling transitions. Shades of blue from light to dark correspond to system sizes $L = 12, 24, 48, 64, 80, 96, 112$, the variation in system size is also reflected in the extent of the x-axis in the insets.