No measurement induced phase transition in the entanglement dynamics of monitored non-interacting one-dimensional fermions in a disordered or quasiperiodic potential

This paper demonstrates that monitored non-interacting one-dimensional fermions in disordered or quasiperiodic potentials do not exhibit a measurement-induced phase transition, as previously claimed results were finite-size artifacts and both large-scale numerical simulations and analytical nonlinear sigma model calculations confirm the system remains in an area-law phase for all monitoring strengths.

The Gist

The Big Picture: A Game of "Whac-A-Mole" with Quantum Particles

Imagine a long line of people (quantum particles) standing in a hallway. They are holding hands, creating a complex web of connections called entanglement. In a perfect, empty hallway, these connections spread out easily, covering the whole line.

Now, imagine a game where a referee (the "monitor") constantly checks on these people to see who is standing where. Every time the referee looks, the "magic" of their connections gets disrupted. In the world of quantum physics, this is called measurement.

Scientists have been asking a big question: If we keep watching these particles, will their connections eventually snap completely, or will they stay connected?

• Volume Law: If they stay connected, the amount of "connection" grows with the size of the line (like a volume).
• Area Law: If the connections snap, the "connection" only exists at the edges or breaks into small chunks (like an area).

A Measurement-Induced Phase Transition (MIPT) is the tipping point where changing how often you watch the particles switches the system from "Volume" to "Area."

The Controversy: A Case of "Too Small a Sample"

Recently, other scientists studied this game using a hallway with disorder (random obstacles) or quasiperiodic patterns (a repeating but non-repeating rhythm). They claimed they found a tipping point: if they watched the particles hard enough, the connections would break (a phase transition).

This paper argues they were wrong.

The authors say the previous scientists made a classic mistake: they didn't look at a big enough hallway.

• The Analogy: Imagine trying to predict the weather by looking at a single puddle. If the puddle is small, you might think it's raining everywhere. But if you look at the whole continent, you see that the rain is actually just a local storm, and the rest is sunny.
• The Problem: The previous studies used a system size of about 500 particles. However, when the "watching" is very gentle, the "connection" (correlation length) can stretch out to thousands of particles. Because the previous systems were too small, they only saw the "storm" (the volume law) and missed the fact that the "sun" (the area law) was actually winning in the long run.

What This Paper Did: The Super-Sized Simulation

To settle the debate, the authors built a much, much bigger simulation.

1. Supercomputers: They used powerful Graphics Processing Units (GPUs)—the same chips used for high-end video games—to simulate systems with up to 18,000 particles. This is more than 30 times larger than the previous studies.
2. Two Scenarios: They tested two types of "hallways":
   • Random Disorder: Like a hallway with random furniture scattered everywhere.
   • Quasiperiodic: Like a hallway with a specific, repeating pattern that never quite repeats itself (like the rhythm of a Fibonacci sequence).
3. The Result: No matter how strong the disorder was, or how hard they watched the particles, the system always ended up in the "Area Law" phase. The connections never broke in a way that created a new phase.

The Conclusion: There is no tipping point (no phase transition) in these specific systems. The previous claim of a transition was just an illusion caused by the system being too small to show the true behavior.

The "Why": A Change in the Rules

The paper also explains why this happens using a mathematical tool called a Non-Linear Sigma Model (NLSM). Think of this as a rulebook for how the particles move.

• In a clean hallway (no obstacles): The rulebook has a specific symmetry (called BDI).
• In a messy hallway (with disorder): The obstacles break one of the rules, changing the symmetry to a different type (AIII).

This change in the rulebook actually makes the "connection" (correlation length) stronger and longer when there is a little bit of disorder. It's counter-intuitive: usually, messiness breaks things. Here, the specific type of messiness actually helps the connections stretch further before they finally snap.

Because these connections stretch so far, you need a massive system (like the 18,000 particles they used) to finally see that they do eventually snap back to the "Area Law."

Summary in One Sentence

By simulating a quantum system with a massive number of particles (up to 18,000), this paper proves that disorder does not create a new phase transition in monitored fermions; instead, the system always settles into a state where connections are limited (Area Law), and previous claims of a transition were simply due to looking at systems that were too small to see the full picture.

Technical Summary

Technical Summary: No Measurement Induced Phase Transition in Monitored Non-Interacting 1D Fermions with Disorder

Problem Statement
The paper investigates the entanglement dynamics of one-dimensional (1D) non-interacting fermions with $U(1)$ symmetry (Dirac fermions) subject to continuous monitoring in the presence of either a disordered or a quasiperiodic potential. While recent literature suggested that such systems undergo a Measurement-Induced Phase Transition (MIPT) from a volume-law to an area-law entanglement phase at a finite critical monitoring strength, this work challenges those findings. The central question is whether the introduction of disorder or quasiperiodicity, which alters the clean system's dynamics, induces a MIPT or if the system remains in an area-law phase for any finite monitoring strength, consistent with the clean limit.

Methodology
The authors employ a dual approach combining large-scale numerical simulations and analytical field theory:

1. Numerical Simulations:

   • Models: The study considers a 1D free fermionic chain with on-site potentials $V_i$. Two potential types are analyzed: (i) Random disorder (box distribution) and (ii) Quasiperiodic potential (Aubry-André model with golden mean frequency).
   • Protocols: Two monitoring schemes are implemented: Projective Measurements (PM) for the quasiperiodic case and Quantum State Diffusion (QSD) for the disordered case. Both monitor the local occupation number $n_i$.
   • Observables: The entanglement entropy (EE) is computed via the correlation matrix $D_{ij}(t)$. To avoid direct EE calculation bottlenecks, the authors utilize the cumulant expansion, focusing on the second cumulant proportional to the density-density correlation function $C(r, t)$. The correlation length $l_{cor}$ is extracted from the exponential decay of $C(r)$.
   • Scale and Hardware: A critical aspect of the methodology is the system size. Previous studies were limited to $L \sim 500$. This work utilizes Graphics Processing Units (GPUs) to simulate systems up to $L \le 18,000$. This scale is necessary because the correlation length in the weak monitoring limit ($\gamma \to 0$) grows exponentially, and previous studies likely observed a "critical" region simply because $L < l_{cor}$.
   • Analysis: Finite-size scaling is performed by fitting the extracted $l_{cor}$ to the ansatz $l_{cor} \sim \exp(a/|\gamma - \gamma_c|)$ to determine the critical monitoring strength $\gamma_c$.

2. Analytical Approach:

   • Field Theory Mapping: For the disordered case, the entanglement dynamics are mapped onto a Non-Linear Sigma Model (NLSM) using the Keldysh path-integral formalism and replica trick.
   • Symmetry Analysis: The authors identify that the clean system belongs to the BDI universality class, while the addition of diagonal disorder breaks chiral symmetry, shifting the system to the AIII class.
   • Renormalization Group (RG): A one-loop RG analysis is performed on the NLSM, which includes an additional mass-like term arising from the interplay of disorder and monitoring. This allows for the derivation of the correlation length as a function of disorder strength $W$ and monitoring strength $\gamma$.

Key Results

• Absence of MIPT: Both numerical and analytical results confirm that the system remains in the area-law phase for any finite monitoring strength $\gamma$ and disorder strength $W$. The critical monitoring strength is found to be consistent with zero ($\gamma_c \approx 0$).
• Finite Size Effect Resolution: The previously reported MIPT in Refs. [59, 60] is identified as a finite-size artifact. In those studies, the system size ($L \sim 500$) was smaller than the correlation length ($l_{cor}$) in the weak monitoring regime, leading to an apparent power-law (volume-law-like) scaling. By reaching $L \le 18,000$, the authors observe the true exponential decay characteristic of the area law.
• Symmetry and Correlation Length:
  • Clean Limit (BDI): $l_{cor} \sim \frac{1}{\gamma} \exp\left(\frac{\sqrt{2\pi}}{2\gamma}\right)$.
  • Disordered Limit (AIII): The symmetry change to class AIII modifies the exponent, yielding $l_{cor} \sim \frac{1}{\gamma} \exp\left(\frac{\sqrt{2\pi}}{\gamma}\right)$. This results in a significantly larger correlation length for the same $\gamma$ compared to the clean case.
  • Disorder Dependence: Contrary to intuition that disorder might suppress entanglement, the analytical solution reveals that in the weak disorder limit, the correlation length increases with disorder strength. This is attributed to the mass term in the NLSM and the symmetry change.
• Quasiperiodic Potential: Similar to the disordered case, the quasiperiodic potential induces a transition to the AIII symmetry class. Numerical results for the correlation length confirm the AIII scaling and show no evidence of a phase transition.

Significance and Claims
The paper claims to definitively resolve the controversy regarding MIPT in monitored 1D non-interacting fermions with disorder. The authors assert that:

1. No MIPT Exists: The entanglement dynamics of these systems do not exhibit a phase transition; they are always in the area-law phase regardless of the monitoring or disorder strength.
2. Previous Claims Were Artifacts: The observation of a MIPT in prior works was due to insufficient system sizes relative to the exponentially large correlation length in the weak monitoring regime.
3. Analytical Confirmation: The analytical mapping to the NLSM with the AIII symmetry class provides a theoretical underpinning for the numerical findings, explaining the enhanced correlation length and the absence of a transition.
4. Methodological Necessity: The work highlights the critical importance of large-scale simulations (enabled by GPUs) and careful finite-size scaling in studying measurement-induced phenomena, where correlation lengths can diverge rapidly.

The authors conclude that the presence of disorder or quasiperiodicity does not alter the fundamental scaling of the entanglement entropy from the clean case, preserving the area-law behavior for all finite parameters.