Entanglement behavior and localization properties in monitored fermion systems

This paper investigates the asymptotic bipartite entanglement and Hilbert space localization in monitored fermion systems, proposing a characterization of entanglement phases through fitting parameters that reveal distinct volume-law and transition behaviors in nonintegrable models while demonstrating that anomalous delocalization does not necessarily correlate with entanglement properties.

The Gist

The Big Picture: A Quantum Tug-of-War

Imagine a group of tiny, invisible dancers (fermions) on a stage. They are constantly moving, spinning, and holding hands with each other. In the quantum world, when they hold hands, they become entangled. This means their movements are perfectly synchronized, no matter how far apart they are.

Usually, if you let these dancers move freely for a long time, they get so tangled up that the whole group becomes a giant, messy knot. The amount of "tangling" (entanglement) grows as the stage gets bigger. This is called a volume law.

However, in this paper, the scientists introduce a "watcher" (the environment). Every now and then, the watcher peeks at the dancers to see where they are. In the quantum world, looking at something changes it. When the watcher checks a dancer, it forces them to stop dancing with their partners and stand still. This "peeking" tries to untangle the group.

The paper asks a simple question: What happens when the dancers try to get tangled up, but a watcher keeps trying to untangle them? Does the group stay messy, or does it become orderly? And how does the size of the stage (the number of dancers) change the answer?

The Main Discovery: A New Way to Measure the Knot

The researchers studied many different types of dance floors (models), some where the dancers follow strict, predictable rules (integrable) and some where they move chaotically (non-integrable).

They found that the amount of entanglement doesn't just jump from "messy" to "orderly." Instead, it follows a very specific curve that looks like a smooth slide. They proposed a mathematical formula (Equation 1 in the paper) that acts like a universal ruler for this situation.

Think of this formula as a smart thermostat for entanglement:

• On a small stage: The dancers can easily hold hands with everyone. The entanglement grows in a straight line (linear) as you add more dancers.
• On a huge stage: The watcher's peeks become too frequent for the dancers to hold hands across the whole room. The entanglement growth slows down and follows a curved path (power-law).

The researchers found that this single "thermostat" formula fits almost every scenario they tested, whether the dancers were following strict rules or moving chaotically.

The Different Dance Floors

The paper tested several specific scenarios:

1. The Strict Dancers (Integrable Models):

   • The Tight-Binding Chain: Imagine dancers in a line passing a ball. If the watcher checks them often, they eventually stop passing the ball across the whole line. The entanglement stays small (Area Law).
   • The Kitaev Chain: This is a special dance where partners can swap places. The researchers found that depending on how strong the "watcher" is, the dancers could be in a state where they are partially tangled (Sub-volume law), which is a middle ground between being fully messy and fully orderly.

2. The Chaotic Dancers (Non-Integrable Models):

   • The SYK Model: This is a group of dancers who are all connected to everyone else in a random, chaotic way. Even with the watcher peeking, these dancers are so naturally chaotic that they stay fully tangled (Volume Law) no matter how big the stage gets.
   • The Staggered t-V Model: This is a mix of order and chaos. Here, the researchers saw a hint of a "crossover." If the watcher is weak, the dancers get tangled; if the watcher is strong, they stay orderly.

The "Ghost" Connection: Localization vs. Entanglement

The paper also looked at something called localization. Imagine a crowd of people in a room.

• Localized: Everyone is stuck in one corner, unable to move.
• Delocalized: Everyone is running around the whole room.

Usually, scientists think that if people are stuck in a corner (localized), they can't get tangled up. But the researchers found something surprising: The dancers could be running around the whole room (delocalized) but still not be tangled up.

They found a strange "anomalous delocalization" where the dancers are spread out but behave in a complex, fractal-like way. Crucially, this "spreading out" had no direct relationship to how tangled they were. You can have a spread-out crowd that is either very tangled or very orderly. This suggests that "being stuck" and "being tangled" are two different things in this quantum world.

The Ladder Experiment

Finally, they tested a more complex setup: a ladder made of two parallel chains of dancers. One chain is the "System" and the other is an "Ancilla" (a helper chain). They watched how the two halves of the System chain got tangled.

Even in this complex geometry, their "thermostat" formula worked perfectly. It could predict whether the dancers would be tangled or not, proving that their method is a robust tool for understanding these quantum systems.

Summary

In short, the paper shows that when you watch quantum particles, you create a tug-of-war between chaos (entanglement) and order (measurement). The researchers found a universal mathematical shape that describes exactly how this tug-of-war plays out, regardless of whether the particles are following strict rules or acting chaotically. They also discovered that how spread out the particles are is a separate issue from how tangled they are, challenging some previous ideas about how these systems work.

Technical Summary

Technical Summary: Entanglement behavior and localization properties in monitored fermion systems

Problem and Motivation
The paper investigates the stationary bipartite entanglement entropy (EE) and localization properties in monitored fermionic systems evolving along quantum trajectories. While unitary dynamics in short-range systems typically lead to volume-law entanglement, and projective measurements alone induce area-law behavior, the interplay between Hamiltonian dynamics and continuous monitoring leads to complex dynamical phases and entanglement transitions. Previous studies have identified transitions between volume-law, area-law, and intermediate (subvolume or logarithmic) phases in various models, including quantum circuits and integrable/non-integrable Hamiltonian systems. However, analytical models allowing for a priori determination of scaling regimes are scarce, and numerical studies are often limited by system size. The authors aim to characterize these behaviors across a wide range of system sizes and model types (integrable and non-integrable) using a unified fitting approach, while also examining the relationship between entanglement and Hilbert space localization.

Methodology
The authors study systems of spinless fermions on a lattice described by Hamiltonians containing quadratic (integrable) and quartic (non-integrable) terms. The dynamics are governed by a Lindblad master equation, simulated via the quantum state diffusion (QSD) protocol, which describes the stochastic evolution of pure states (quantum trajectories) under continuous weak measurements.

1. Models Studied:

   • Integrable Models: (i) Tight-binding chain with onsite dephasing; (ii) Kitaev chain with onsite dephasing; (iii) Kitaev chain with long-range dissipators (power-law decaying jump operators).
   • Non-integrable Models: (iv) Staggered $t$-$V$ model; (v) Sachdev-Ye-Kitaev (SYK) model.
   • Ladder Model: A non-interacting fermionic ladder with stroboscopic projective measurements, analyzed using Fermionic Logarithmic Negativity (FLN) due to the mixed state of the subsystem.

2. Numerical Techniques:

   • For integrable models (Gaussian states), the authors utilize the property that the unraveled state remains a Slater determinant or a Gaussian state, allowing simulations up to system sizes $L \sim 10^3$.
   • For non-integrable models, exact diagonalization (Krylov algorithm) is used, limiting simulations to $L \lesssim 20$.
   • The asymptotic bipartite EE is calculated by averaging over quantum trajectories and performing a time average in the steady state.

3. Fitting Strategy:

   • Integrable Models: The steady-state EE ($S_\ell$) versus system size ($L$) is fitted using the function:
     $$f(L) = \frac{A L}{1 + C L^b}$$
     This function interpolates between linear growth ($L \ll C^{-1/b}$) and power-law behavior ($L \gg C^{-1/b}$). The exponent $b$ characterizes the phase: $b=0$ (volume law), $0 < b < 1$ (subvolume law), and $b \ge 1$ (area law).
   • Non-integrable Models: Due to small accessible $L$, the authors first fit the EE versus measurement strength $\gamma$ using a generalized Lorentzian function. They then analyze the scaling of the fitting parameters with $L$ to recover the $L$-dependence form of Eq. (1).
   • Localization: The Inverse Participation Ratio (IPR) in the Hilbert space is computed to study localization/delocalization properties.

Key Results

1. Integrable Models:

   • Tight-binding chain: The fit confirms an asymptotic area-law behavior ($b=1$). The characteristic length scale $L_0 = C^{-1/b}$ scales as a power law with the measurement strength $\gamma$ ($L_0 \sim \gamma^{-0.88}$), consistent with theoretical predictions of a finite-size crossover to an area law.
   • Kitaev chain (onsite dephasing): The fit reveals a regime where $b < 1$ for small chemical potential $h$, suggesting subvolume scaling, and $b \ge 1$ for large $h$ (area law). The transition point is difficult to pinpoint precisely due to finite-size effects but is consistent with previous literature.
   • Kitaev chain (long-range dissipators): The model exhibits three regimes: volume law ($b \approx 0$) for small power-law exponent $\alpha$, area law ($b > 1$) for large $\alpha$, and an intermediate subvolume regime ($0 < b < 1$) where $b$ forms a plateau ($\approx 0.8$). The length scale $L_0$ diverges as $\alpha \to 1^+$, consistent with a transition from volume-law to power-law behavior.

2. Non-integrable Models:

   • SYK Model: The analysis of fitting parameters indicates that the EE increases linearly with system size ($L$) for all measurement strengths, reflecting the robust chaotic nature of the SYK model where eigenstates exhibit volume-law entanglement.
   • Staggered $t$-$V$ Model: The results suggest an entanglement crossover. For strong measurements ($\gamma \gtrsim 1$), the system tends toward an area law, while for weaker measurements, there are traces of an entanglement increase, though finite-size effects prevent a definitive conclusion on the asymptotic phase.

3. Localization Properties:

   • The authors find that the logarithm of the IPR scales linearly with the logarithm of the Hilbert space dimension for both integrable and non-integrable models.
   • The slope of this scaling indicates "anomalous delocalization" (multifractal behavior), distinct from perfect localization or perfect delocalization.
   • Crucially, this localization behavior is found to be independent of the entanglement scaling (volume vs. area law), suggesting that localization properties in these monitored systems are not directly related to entanglement transitions.

4. Fermionic Logarithmic Negativity (Ladder Model):

   • The proposed fitting function (Eq. 1) successfully describes the system-size dependence of the FLN in a two-leg ladder model with stroboscopic measurements.
   • The fit parameters correctly identify different dynamical regimes (area law vs. logarithmic growth), demonstrating the applicability of the fitting ansatz to mixed-state entanglement measures.

Significance and Claims
The paper claims that the fitting function $f(L) = \frac{A L}{1 + C L^b}$ provides a remarkably good description of steady-state entanglement across a variety of monitored fermionic models, spanning integrable and non-integrable cases, and different entanglement measures (EE and FLN).

• Unified Description: The function effectively interpolates between linear and power-law behaviors, capturing area-law, subvolume-law, and volume-law regimes, as well as finite-size crossovers.
• Numerical Reliability: The authors emphasize that their findings are based on purely numerical analysis. They explicitly state that Eq. (1) should not be used to classify entanglement phases without further analytical support, as the logarithmic growth often observed in literature might be a finite-size effect mimicked by a power law with a small exponent ($b \approx 0.8$).
• Independence of Localization: A key finding is the lack of correlation between the entanglement scaling laws and the localization properties (IPR) in the Hilbert space, challenging the assumption that entanglement transitions are driven by localization-delocalization transitions in these specific monitored systems.
• Future Utility: The authors suggest that while the current results do not predict new phases, the robustness of the fitting formula across different models and system sizes may motivate future theoretical investigations into entanglement dynamics, particularly in understanding short-size behaviors that are more accessible to current experimental technologies.