Quantum dynamics of monitored free fermions: Evolution of quantum correlations and scaling at measurement-induced phase transition

This paper investigates the quantum dynamics of monitored free fermions by extending the nonlinear sigma-model field theory to finite evolution times and various initial states, thereby analytically and numerically characterizing the development of quantum correlations and the scaling behavior near the measurement-induced phase transition in two dimensions.

The Gist

Imagine you are watching a complex dance performance by a huge group of dancers (the quantum system). These dancers are "free fermions," which is a fancy way of saying they move around freely but don't bump into each other in complicated ways.

Usually, in quantum physics, we think of two things happening to these dancers:

1. The Music (Unitary Dynamics): They dance to a rhythm, moving smoothly and creating beautiful, complex patterns of entanglement (they hold hands and move as one giant, connected group).
2. The Critics (Measurements): Every now and then, a critic jumps in and shouts, "Stop! What are you doing right now?" This is a measurement. When the critic shouts, the dancers freeze, and their complex, mysterious quantum connection collapses into a simple, definite state.

This paper explores what happens when you have a dance floor where the music plays and critics are shouting randomly.

The Big Question: Chaos vs. Order

The researchers wanted to know: If the critics shout too often, do the dancers stay in small, isolated groups? Or if they shout too rarely, do the dancers stay connected in one giant, chaotic web?

They found that there is a tipping point (a phase transition):

• Too many critics (High measurement rate): The dancers are constantly interrupted. They can't build long connections. They stay in small, local groups. This is the "Area Law" phase (like a neighborhood where everyone only knows their immediate neighbors).
• Too few critics (Low measurement rate): The dancers get to dance for a long time before being interrupted. They build massive, long-distance connections across the whole room. This is the "Volume Law" phase (the whole room is one giant entangled web).

The Journey: How the Dance Evolves

Most previous studies only looked at the dance after it had been going on for a very, very long time (the "steady state"). They asked, "What does the room look like after 100 years?"

This paper asks a different, more dynamic question: "How does the room change as the dance happens?"

They looked at three different starting scenarios:

1. The "Maximally Mixed" Start: Imagine the dancers start in a state of total confusion, with no connections at all (like a chaotic crowd). As the music plays and critics shout, the dancers slowly start to form connections. The paper shows exactly how fast these connections spread out from the chaos.
2. The "Maximally Disentangled" Start: Imagine the dancers start in perfect, tiny pairs (holding hands only with their partner). As the music plays, they start to break those pairs and connect with others. The paper tracks how the "area law" (small groups) slowly grows into something larger.
3. The "Volume Law" Start: Imagine the dancers start already connected in a giant web. As the critics shout, they slowly lose these connections, shrinking back down to small groups.

The Analogy: Think of it like a rumor spreading in a school.

• If you start with no rumors (mixed state), the rumor spreads out over time.
• If you start with a rumor that everyone knows (volume law), the "truth" (or lack of mystery) spreads out, and the rumor dies down.
• The paper calculates the speed limit of this spread and how it changes depending on how often the "principal" (the measurement) interrupts the students.

The "Time Travel" Trick

Here is the clever part of their math. They realized that watching a quantum system evolve over time while being measured is mathematically identical to looking at a disordered system in space.

• Time becomes a new dimension: Imagine the dance floor is 2D (left-right, forward-back). The paper treats the time the dance has been going on as a third dimension (up-down).
• The "Localization" Time: They found a specific time, let's call it $T^*$.
  • If the dance goes on for less than $T^*$, the system remembers its starting state.
  • If it goes on longer than $T^*$, the system "forgets" where it started and settles into its final pattern.
  • This $T^*$ is like the "purification time"—the time it takes for the system to become "pure" (lose its quantum fuzziness) or "sharpen" its charge.

The Discovery: Finding the Tipping Point

The authors used this "time evolution" method to find the exact point where the system switches from "small groups" to "giant web."

• They ran computer simulations (like a video game of the dancers) for a 2D dance floor.
• They measured how long it took ($T^*$) for the system to forget its start, for different sizes of dance floors and different rates of critics shouting.
• The Result: They found that right at the tipping point, the time it takes to forget scales perfectly with the size of the room. This allowed them to calculate the "critical exponent" (a number that describes how sharp the transition is).

Why Does This Matter?

1. It's a New Tool: Instead of waiting forever for a system to settle down (which is hard to simulate), you can just watch how it evolves over a shorter time and use that to predict the final result. It's like predicting the weather by watching the clouds move for an hour rather than waiting for the storm to pass.
2. Universality: They showed that this method works for different types of starting states. Whether you start with chaos or order, the path to the final state follows the same rules.
3. Future Tech: Understanding these transitions is crucial for building quantum computers. If you want to keep a quantum computer working (entangled), you need to know exactly how much "noise" (measurement) it can handle before it collapses.

In a Nutshell

This paper is a guidebook on how quantum information spreads and dies when a system is constantly being watched. By treating time as a spatial dimension, the authors created a powerful new way to map out the "phase diagram" of quantum systems, showing us exactly when a quantum system stays connected and when it falls apart. They proved that watching the process of change is just as powerful as studying the end result.

Technical Summary

1. Problem Statement

The paper investigates the quantum dynamics of many-body free-fermion systems subjected to continuous local density measurements. While previous research has extensively characterized the steady-state properties of such systems (specifically the Measurement-Induced Phase Transition or MIPT between area-law and volume-law entanglement phases), the time-dependent evolution of quantum correlations from an arbitrary initial state to the steady state remains less understood.

The authors address two primary questions:

1. How do quantum correlations evolve from specific initial states (maximally mixed, random area-law, random volume-law) towards the asymptotic steady state as the evolution time $T$ increases?
2. Can the dynamical scaling of the system at large but finite times be used to efficiently determine the critical parameters (transition point $\gamma_c$ and critical exponent $\nu$) of the MIPT, analogous to finite-size scaling in Anderson localization?

2. Methodology

The authors employ a dual approach combining analytical field theory and numerical simulations.

A. Analytical Framework: Non-Linear Sigma Model (NLSM)

• Mapping: The authors extend the existing mapping of monitored free fermions to a Non-Linear Sigma Model (NLSM) field theory. Unlike previous works focusing on the $T \to \infty$ limit, this formulation accounts for finite evolution time $T$.
• Boundary Conditions: A key innovation is the derivation of specific NLSM boundary conditions at the initial time $t = -T$ corresponding to three distinct classes of Gaussian initial states:
  1. Maximally Mixed State: Corresponds to a fully absorbing boundary condition ($\hat{U} = \hat{I}$).
  2. Maximally Disentangled (Area-law) Pure State: Corresponds to a reflecting boundary condition ($\hat{J} = 0$).
  3. Random Volume-law Pure State: Corresponds to an absorbing condition for all non-zero momentum modes but a reflecting condition for the zero mode ($q=0$), due to global charge conservation.
• Diffusive Regime: For intermediate times ($\ell_0/v \ll T \ll T^*$), the NLSM is treated in the Gaussian approximation to derive analytical expressions for the density correlation function $C(q, T)$ and charge variance $C^{(2)}_A(T)$.
• Long-Time Scaling: The system is viewed as a quasi-one-dimensional disordered system in $(d+1)$ space-time dimensions. The "localization length" in the time direction, denoted as $T^*$, is identified as the purification time (or charge-sharpening time).

B. Numerical Simulations

• Model: A 1D and 2D lattice of non-interacting complex fermions with nearest-neighbor hopping ($H$) and random projective measurements of site occupation numbers at rate $\gamma$.
• Observables: The authors simulate the time evolution of the density correlation function $C(q)$ and the variance of charge fluctuations $C^{(2)}_A$ in subsystems.
• Finite-Size Scaling: In 2D, they calculate the purification time $T^*(L, \gamma)$ for various system sizes $L$ and measurement rates $\gamma$. They perform a single-parameter scaling collapse to extract the critical measurement rate $\gamma_c$ and the correlation-length exponent $\nu$.

3. Key Contributions

1. Extension of NLSM to Finite Time: The paper generalizes the NLSM formalism to finite evolution times by rigorously deriving boundary conditions for different initial states. This bridges the gap between initial state properties and the asymptotic steady state.
2. Dynamical Evolution of Correlations: The authors analytically and numerically demonstrate how quantum correlations "propagate" from the initial state. They show that correlations develop gradually, transitioning from initial-state-dependent behaviors (e.g., volume-law or area-law) to the universal steady-state scaling (area $\times$ log-law) as $T$ increases.
3. Dynamical Determination of MIPT Criticality: The paper establishes that the purification time $T^*$ (the time scale for the system to lose memory of the initial state) serves as a robust proxy for the localization length in the time direction. This allows for the determination of MIPT critical parameters using dynamical observables, offering an alternative to steady-state covariance analysis.
4. Universality and Consistency: The results confirm that the dynamical approach yields critical exponents consistent with those obtained from steady-state studies, validating the universality of the MIPT in free-fermion systems.

4. Key Results

A. Short-to-Medium Time Dynamics (Diffusive Regime)

• Correlation Function: The density correlation function $C(q, T)$ evolves from its initial value towards the steady-state form $C(q) \propto q$.
  • For a maximally mixed state, $C(q, T) \propto q \coth(vqT)$. At short times, charge fluctuations decay as $1/T$ (volume law), eventually crossing over to the steady-state area-law behavior.
  • For a maximally disentangled state, $C(q, T) \propto q \tanh(vqT)$. Fluctuations grow logarithmically with time before saturating to the steady state.
• Real-Space Correlations: The power-law correlations characteristic of the steady state ($\sim 1/x^2$ in 1D) emerge ballistically, propagating to distances $x \sim vT$.

B. Long-Time Dynamics and Scaling (2D System)

• Purification Time Scaling: The purification time $T^*$ scales with system size $L$ differently depending on the phase:
  • Delocalized Phase ($\gamma < \gamma_c$): $T^* \propto L^d$ (specifically $L^2$ in 2D).
  • Critical Point ($\gamma = \gamma_c$): $T^* \propto L$.
  • Localized Phase ($\gamma > \gamma_c$): $T^*$ becomes independent of $L$ ($T^* \sim \xi_{loc}/v$).
• Critical Parameters: By fitting the numerical data for $T^*(L, \gamma)$ to the scaling form $vT^*/L = \Psi(L^{1/\nu}(\gamma - \gamma_c))$, the authors determined:
  • Critical measurement rate: $\gamma_c = 2.847 \pm 0.024$
  • Correlation-length exponent: $\nu = 1.397 \pm 0.053$
• Validation: These values are in excellent agreement with previous steady-state studies (e.g., Ref. [23] and [41]), confirming the consistency of the dynamical approach.

5. Significance and Outlook

• Methodological Advancement: The work provides a powerful dynamical framework for studying MIPTs. By utilizing the purification time, researchers can extract critical exponents without waiting for the system to reach a steady state, which is computationally expensive for large systems.
• Physical Insight: It clarifies the mechanism of information scrambling and purification in monitored systems, showing how the "memory" of the initial state is exponentially lost over the time scale $T^*$.
• Future Directions: The authors suggest extending this dynamical analysis to:
  • Other universality classes (different symmetries, topologies).
  • Systems with weak interactions (where charge and entanglement transitions may split).
  • Non-uniform measurement protocols and non-Gaussian initial states.
  • The study of mesoscopic fluctuations and multifractality at the transition point.

In conclusion, this paper successfully unifies the field-theoretic description of monitored free fermions with dynamical scaling analysis, providing a comprehensive picture of how quantum correlations evolve and how critical phenomena can be probed via time-dependent observables.