Absence of measurement- and unraveling-induced entanglement transitions in continuously monitored one-dimensional free fermions

Using replica Keldysh field theory and numerical simulations, this paper demonstrates that continuously monitored one-dimensional free fermions do not exhibit genuine measurement- or unraveling-induced entanglement phase transitions, as their steady-state entanglement ultimately obeys an area law beyond exponentially large length scales despite displaying critical-like behavior at intermediate scales.

The Gist

The Big Picture: Watching a Quantum System Without Breaking It

Imagine you have a long line of people (these are free fermions, or quantum particles) holding hands and passing a secret message down the line. This is their natural "dance" or movement.

Now, imagine a group of spies (the measurements) standing next to every person, constantly checking what they are doing. In the quantum world, checking someone's state usually disturbs them. If the spies check too often, the secret message gets scrambled, and the people stop passing it along. This is called "disentangling."

For a while, scientists were confused. Some simulations suggested that if the spies checked at just the right speed, the system would enter a special "critical" state where the secret message could travel infinitely far, and the people would be deeply connected in a complex way. Others thought the spies would always win, eventually breaking all connections.

This paper settles the debate. The authors say: There is no special "critical" phase. No matter how you tune the spies, if you wait long enough, the connections will eventually break down. The "special state" people thought they saw was just an illusion caused by looking at the system for too short a time.

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The Key Concept: The "Unraveling" Dial

The paper introduces a clever tool called the unraveling phase ($\phi$). Think of this as a dial on the spies' equipment.

• Setting the dial to 0 (The Strict Spy): The spies are very precise. They look at the particles and say, "I see you are here." This is a standard measurement. It tends to break the quantum connections (entanglement) between particles.
• Setting the dial to 90 degrees (The Chaotic Spy): The spies aren't looking to measure; they are just adding random noise. Imagine them randomly pushing the people in the line. This "noise" actually creates connections and entanglement, making the system very messy and highly connected.
• Turning the dial in between: You can smoothly slide between these two extremes.

The Discovery: The authors tested every setting on this dial. They found that for almost every setting (from 0 up to, but not including, 90 degrees), the system eventually settles into a state where connections are short and weak (an Area Law). The "critical" state where connections stretch forever only appears to exist for a while, but it is a temporary trick.

The "Long Wait" Analogy

Why did previous studies think there was a special phase?

Imagine you are watching a marathon runner.

• The Illusion: For the first 10 miles, the runner is sprinting incredibly fast. If you only watch for 10 miles, you might conclude, "This runner is a superhuman who will never get tired!"
• The Reality: The runner is actually slowing down. If you watch them for 100 miles, you see them eventually stop or walk.

In this paper, the "superhuman sprint" is the logarithmic growth of entanglement (the critical phase). The authors proved that this sprint only lasts for a specific distance. Beyond that distance, the runner (the quantum system) inevitably slows down to a walk (the area law).

The distance the runner can sprint before slowing down depends on how fast the spies are checking.

• If the spies check very slowly, the runner can sprint for a huge distance (mathematically, an exponentially large distance).
• If the spies check quickly, the runner slows down almost immediately.

Because the "sprint" distance can be so huge (like the distance to the moon), computer simulations (which are like short video clips) often only see the sprint and miss the slowdown. This paper used advanced math to predict the slowdown and confirmed it with simulations that looked at the right scales.

The "Noise" Exception

There is one special setting on the dial: 90 degrees.
At this exact setting, the "spies" are just adding pure random noise (like static on a radio). In this specific case, the system does stay in a highly connected, "volume-law" state forever. The noise keeps the connections alive. However, this is a very specific, fragile point. The moment you turn the dial even slightly away from 90 degrees, the system eventually collapses back into the short-connection state.

Summary of Findings

1. No Phase Transition: Changing how often you measure or how you "unravel" the measurement (the dial) does not create a permanent new phase of matter.
2. The "Critical" Phase is Temporary: The complex, long-range connections people thought they saw are just a temporary crossover. They look like a new phase, but they eventually fade away.
3. The Scale of the Illusion: The distance over which this "fake" critical phase lasts is exponentially large. It's so big that it's very hard to see the end of it in computer simulations, which is why the confusion existed for so long.
4. The Math: The authors used a sophisticated mathematical framework (called Replica Keldysh field theory) to describe the system as a "Nonlinear Sigma Model." This model predicted that the connections would eventually break, and their computer simulations confirmed this prediction.

In short: The quantum system is like a rubber band. You can stretch it (measure it) or shake it (add noise), and it might look like it's holding together forever for a while. But if you wait long enough, the rubber band always snaps back to a short, relaxed state. There is no magic setting that keeps it stretched forever, except for one very specific, noisy exception.

Technical Summary

Problem Statement
The paper addresses the ongoing debate regarding the existence of measurement-induced entanglement transitions in one-dimensional (1D) free fermionic systems with conserved particle numbers. Previous numerical studies on similar models suggested the presence of a Kosterlitz-Thouless (KT) transition from an area-law phase to a critical phase characterized by logarithmic entanglement growth and algebraic correlations. However, other theoretical works utilizing nonlinear sigma models (NLSM) predicted that such critical phases are unstable in the thermodynamic limit, with entanglement ultimately obeying an area law beyond an exponentially large length scale. A specific point of contention involves the "unraveling" of the underlying Lindblad master equation: while some studies indicated that tuning the unraveling scheme (specifically an unraveling phase $\varphi$) could induce a phase transition, the universality of this effect in free fermion systems remained unclear. The central question is whether the apparent critical behavior observed in simulations reflects a genuine phase of nonequilibrium quantum matter or merely a finite-size crossover phenomenon.

Methodology
The authors investigate a 1D lattice of free fermions subject to continuous monitoring of lattice-site occupations. The system is described by a stochastic Schrödinger equation that incorporates an unraveling phase $\varphi$, allowing for interpolation between conventional quantum state diffusion ($\varphi=0$) and a unitary unraveling corresponding to randomly fluctuating onsite potentials ($\varphi=\pi/2$).

The study employs a dual approach:

1. Analytical Framework: The authors develop a replica Keldysh field theory to describe the dynamics. They derive a nonlinear sigma model (NLSM) describing the long-wavelength physics. A key technical contribution is the construction of the Keldysh functional integral from the linear stochastic master equation for the unnormalized density matrix, which allows for a transparent symmetry classification. They analyze the model within a Gaussian approximation and then incorporate renormalization group (RG) flow corrections derived from the NLSM to account for strong fluctuations of Goldstone modes.
2. Numerical Simulations: The authors perform direct numerical simulations of the stochastic Schrödinger equation for system sizes up to $L=1000$. They calculate the connected density correlation function and the von Neumann entanglement entropy for various measurement rates $\gamma$ and unraveling phases $\varphi$. They specifically analyze the scaling of crossover scales and the behavior of the effective central charge to test analytical predictions.

Key Contributions and Results

• Symmetry Classification: The analytical derivation reveals that the model belongs to the chiral orthogonal symmetry class BDI for $\varphi=0$ and the chiral unitary class AIII for $\varphi \neq 0$. This distinction affects the numerical prefactor of the weak-localization correction but does not alter the qualitative absence of a phase transition.
• Absence of Entanglement Transitions: The NLSM analysis demonstrates that for any $0 \le \varphi < \pi/2$, the system does not exhibit a genuine measurement- or unraveling-induced entanglement transition. Instead, the entanglement entropy obeys an area law beyond a crossover scale $l_{\varphi, \ast}$, which is exponentially large in the inverse measurement rate: $\ln(l_{\varphi, \ast}) \sim J/[\gamma \cos(\varphi)]$.
• Finite-Size Crossover: The apparent critical behavior (logarithmic entanglement growth and algebraic correlations) observed in previous numerical studies is identified as a finite-size crossover phenomenon. These signatures persist only below a crossover scale $l_c \sim \gamma^{-2}$, which is algebraically large and thus accessible in numerical simulations, unlike the exponentially large $l_{\varphi, \ast}$.
• Weak-Localization Correction: The authors derive and numerically verify the "weak-localization correction" to the density correlation function. The slope of this correction as a function of $\ln(q l_0)$ is universal and distinguishes between symmetry classes AIII ($\beta=1$) and BDI ($\beta=1/2$), providing strong evidence for the NLSM description and the absence of a transition.
• Scaling of Critical Signatures: The study confirms that the scale $l_c$ (where deviations from critical behavior begin) and the scale $l_m$ (where the effective central charge peaks) scale algebraically with $\gamma$ (specifically $l_c \sim \gamma^{-2}$ and $l_m \sim \gamma^{-3/2}$) for small $\varphi$. For larger $\varphi$ (specifically $\varphi = 5\pi/12$), non-universal features at short scales modify these scalings to $l_c \sim \gamma^{-2/3}$ and $l_m \sim \gamma^{-1}$, masking the universal behavior but not changing the ultimate area-law outcome.
• Unitary Unraveling: At the singular point $\varphi = \pi/2$, the system corresponds to classical noise, and the entanglement entropy exhibits volume-law scaling, consistent with random Gaussian states. This behavior is distinct from the $\varphi < \pi/2$ regime.

Significance
The paper concludes that for continuously monitored 1D free fermions with conserved particle number, neither varying the measurement rate $\gamma$ nor tuning the unraveling phase $\varphi$ can drive a genuine entanglement transition. The apparent KT transition to a critical phase reported in previous literature is reinterpreted as a finite-size crossover effect. The findings suggest that the stability of the critical phase in these models is an artifact of limited system sizes and that the true thermodynamic behavior is an area law. The work establishes that the choice of unraveling (including the specific interpolation between measurement schemes) does not determine the presence of a phase transition in this class of models, attributing the absence of a transition to a common underlying symmetry structure. The results provide a unified understanding of measurement-induced dynamics in free fermionic systems, reconciling conflicting numerical observations with field-theoretic predictions.