Measurement-Induced Phase Transitions

• Measurement-induced phase transitions (MIPTs) are characterized by a transition from logarithmic to area-law entanglement scaling driven by continuous measurements in quasiperiodic free fermion systems.
• The interplay between measurement backaction and the Aubry–André–Harper model shifts the localization threshold to Vc/J ∼ 2.3, indicating a distinct BKT-like universal critical behavior.
• Experimental platforms using ultracold atoms and engineered quasiperiodic potentials validate these findings through observed changes in entanglement and correlation scaling.

Measurement-induced phase transitions (MIPTs) describe entanglement transitions driven by continuous or repeated measurements in non-equilibrium quantum many-body systems. In the context of free fermions in a quasiperiodic potential, these transitions fundamentally reorganize the scaling properties of entanglement and correlation under competition between localization and measurement backaction, leading to distinct phases separated by sharp critical behaviors.

1. Phase Diagram and Entanglement Entropy Scaling

The prototypical model is the Aubry–André–Harper (AAH) Hamiltonian: $$H = -J \sum_{j} (c_j^\dagger c_{j+1} + \text{h.c.}) + V \sum_j \cos(2\pi j/\tau + \theta)\, n_j,$$ where $J$ is the hopping, $V$ the quasiperiodic potential strength, $n_j = c_j^\dagger c_j$, and $\tau = (\sqrt{5}+1)/2$ is the irrational period parameter.

The steady-state entanglement entropy $S_A = -\text{tr}_A(\rho_A \log \rho_A)$ for a subsystem $A$ is analyzed in the presence of continuous measurement of the local occupation number (implemented via quantum trajectories). The main findings are:

• For $V < V_c$ (with $V_c/J \sim 2.3$), and weak measurement strength, the entropy shows robust logarithmic scaling:

$$S(L) = \frac{c_{\mathrm{eff}}}{3} \log\left[\frac{L}{\pi} \sin\left(\frac{\pi \ell}{L}\right)\right] + s_0$$

where $c_{\mathrm{eff}}$ is an effective central charge and $s_0$ a non-universal constant. This logarithmic law appears for a wide range $0 < V < V_c$.

• For $V > V_c$, any nonzero measurement strength immediately stabilizes area-law entanglement ($c_{\mathrm{eff}} = 0$).
• The critical point $V_c/J \sim 2.3$ (in the monitored case) is shifted upward compared to the unitary (Hamiltonian-only) AAH model transition at $V/J = 2$, emphasizing that measurement changes the universality boundary.

The logarithmic-law phase is thus a critical phase, induced and protected by the interplay of measurement and the underlying quasiperiodic Hamiltonian.

2. Measurement-Induced Phase Transitions (MIPTs) and Criticality

In the unitary limit ($\gamma/J = 0$), the AAH model supports a volume-law to area-law entanglement transition at $V/J = 2$, corresponding to a delocalization-localization crossover. With continuous measurement, volume-law scaling ($S \sim L$) is absent even in the extended regime; measurements promote a new regime with $S \sim \log L$ for $V < V_c$, leading, upon increasing measurement rate, to a transition into an area-law ($S \sim \text{const}$) phase.

This transition can be characterized by a critical measurement rate $\gamma_c$ (for fixed $V$) via a Berezinskii–Kosterlitz–Thouless (BKT)–like finite-size scaling: $$S(\gamma) - S(\gamma_c) = F\left[(\gamma - \gamma_c)(\log L)^2\right]$$ reflecting the exponential divergence of the correlation length near the transition.

When $V > V_c$, measurement localizes the system immediately. Hence, the area-law phase is stabilized by any nonzero $\gamma$ in this regime (even for infinitesimal monitoring strength).

3. Connected Correlation Functions

To further substantiate the phases, the connected density–density correlation function,

$$C(r) = \langle n_j n_{j+r} \rangle - \langle n_j \rangle \langle n_{j+r} \rangle,$$

is evaluated.

• In the logarithmic-law phase ($V < V_c$, small $\gamma$), $C(r)$ decays algebraically:

$$C(r) \propto \left[ \frac{L}{\pi} \sin\left(\frac{\pi r}{L}\right) \right]^{-2},$$

which yields $C(r) \sim r^{-2}$ at large distances, characteristic of scale-invariant (critical) states.

• In the area-law regime (strong measurement or $V > V_c$), $C(r)$ decays rapidly, consistent with an exponential law, marking the loss of long-range quantum correlations as expected for localized, weakly entangled states.

This dramatic change in the spatial decay of correlations is a direct diagnostic of the MIPT.

4. Distinction from Unitary AAH Model and Implications

In the absence of measurement, the AAH model's self-dual point at $V/J=2$ separates extended states (volume-law entanglement, $V/J < 2$) from localized states (area-law, $V/J > 2$). Continuous local monitoring fundamentally alters this landscape:

• The volume law is generically absent: even for $V/J < 2$, any nonzero measurement pushes the system out of the volume-law phase into the logarithmic-law regime.
• The critical point for the onset of localization (in the sense of area-law entanglement) is shifted to higher $V$ by measurement, $V_c/J \sim 2.3$, and is governed by distinct universality (BKT-like scaling).
• Measurement induces a new universal scaling regime not present in the underlying closed Hamiltonian evolution.

This highlights the nontrivial interplay between measurement-induced decoherence and quasiperiodicity-driven localization.

5. Experimental Considerations and Realization

The predicted phase structure is accessible in ultracold atom experiments. Quasiperiodic potentials can be engineered via optical lattices, allowing precise control of the potential strength $V$. Continuous monitoring of the onsite number can be performed using off-resonant probe light, thus directly realizing the theoretical model in experiment.

Tuning $V$ and $\gamma$ in these platforms should enable observation of:

• A transition from logarithmic-law to area-law entanglement scaling.
• A corresponding change from algebraically to exponentially decaying correlation functions.
• The shift of the localization threshold from the unitary AAH value.

Such observations would establish the universal structure of MIPTs in quasiperiodic systems and verify the theoretical framework.

6. Mathematical and Computational Methods

Numerical results are obtained using quantum trajectory methods simulating continuous measurement dynamics in free fermion models. The evolution of the quantum state under monitoring is governed by stochastic evolution equations for the state vector $|\Psi(\xi)\rangle$, reflecting measurement backaction via random variables $\xi_{j,t}$ linked to local occupation measurements.

The steady-state entanglement entropy is extracted using large ensembles of such trajectories, enabling reconstruction of $S_A$ and the effective central charge $c_\mathrm{eff}$ via fitting to CFT-inspired formulas. Finite-size scaling analyses and scaling collapse extract the critical point $V_c$ and support the underlying BKT nature.

The key theoretical distinction is that the monitored AAH model displays MIPTs where logarithmic entanglement scaling is stabilized by measurement—in stark contrast to both the volume-law of metallic states and the area-law of localization—enabling a unique form of long-range quantum criticality in open, monitored systems.

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Overall, measurement-induced phase transitions in free fermions with a quasiperiodic potential manifest as a novel logarithmic-law phase, a measurement-induced shift of localization thresholds, and nontrivial scaling properties of both entanglement and correlation, with clear prospects for experimental observation in ultracold atom platforms (Matsubara et al., 31 Mar 2025).