⚛️ quantum physics

Entanglement growth and information capacity in a quasiperiodic system with a single-particle mobility edge

This study demonstrates that a one-dimensional quasiperiodic system with a single-particle mobility edge exhibits a smooth dynamical crossover in entanglement entropy and subsystem information capacity, providing distinct non-interacting benchmarks for understanding mixed-phase quantum dynamics.

Original authors: Yuqi Qing, Yu-Qin Chen, Shi-Xin Zhang

Published 2026-02-20

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Yuqi Qing, Yu-Qin Chen, Shi-Xin Zhang

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a long, narrow hallway filled with people (these are our "particles" or electrons). In a normal hallway, people can walk freely from one end to the other. But in this paper, we are looking at a special kind of hallway where the floor has a weird, repeating pattern of bumps and dips (a "quasiperiodic potential").

The scientists wanted to see how these people move and interact when the floor gets really bumpy. They compared two types of hallways:

The "Standard" Hallway: Here, the bumps are uniform. If the bumps get too high, everyone suddenly stops moving and gets stuck in place. It's an all-or-nothing switch: either everyone runs free, or everyone is frozen.
The "Special" Hallway (The GAA Model): This is the star of the show. Here, the floor is shaped in a way that creates a Mobility Edge. Think of this as a "speed limit line" drawn on the floor.
- People with high energy (fast runners) can jump over the bumps and keep running freely.
- People with low energy (slow walkers) get stuck in the deep dips and can't move.
- Crucially: Both groups exist at the same time in the same hallway.

The Experiment: The "Quantum Quench"

To test this, the scientists started with everyone standing still in a specific pattern (like a checkerboard). Then, they hit "Go" (a quantum quench) and watched what happened over time. They used two special tools to measure the chaos:

Tool 1: Entanglement Entropy (The "Social Network" Meter)

Imagine "entanglement" as how much people in the hallway are talking to each other and sharing secrets.

In the Standard Hallway: When the floor gets bumpy, the conversation stops abruptly. The "Social Network" collapses instantly from a massive web of connections to just neighbors whispering.
In the Special Hallway: Because fast runners are still moving, they keep carrying secrets across the room. The "Social Network" doesn't collapse; it just gets smaller.
- The Analogy: Imagine a party where half the guests are stuck in a room, but the other half are still dancing in the main hall. The party isn't dead (it's not a total freeze), but it's quieter than a full party. The more people get stuck, the quieter the party gets, but it never goes completely silent as long as some dancers remain.

Tool 2: Subsystem Information Capacity (The "Message Delivery" Test)

This tool asks: "If I whisper a secret to one person in the middle of the hallway, how much of that secret can be recovered by looking at a group of people nearby?"

In the Standard Hallway:
- Free Phase: The secret travels instantly to everyone (a straight line of delivery).
- Frozen Phase: The secret gets trapped right where it started. No one else hears it (a flat line).
In the Special Hallway: The result is a hybrid.
- The secret gets trapped immediately around the starting person (because the slow walkers are stuck).
- But, it slowly leaks out to the rest of the hallway because the fast runners are still carrying it.
- The Visual: Instead of a straight line or a flat line, the graph looks like a staircase. It jumps up a bit (trapped info) and then slowly climbs (leaking info). This "staircase" is the unique fingerprint of the Mobility Edge.

The Big Discovery

The main takeaway is that nature doesn't always switch on and off like a lightbulb. Sometimes, it's a dimmer switch.

In the "Special Hallway," the transition from "free movement" to "stuck" isn't a sudden crash. It's a smooth slide. As the floor gets bumpier, the number of "fast runners" slowly decreases, and the number of "stuck walkers" slowly increases. The system's behavior changes gradually, blending the two states.

Why Does This Matter?

This is like finding a new rule for traffic. If you understand how cars behave when some are stuck in traffic jams while others are speeding on the highway, you can better understand complex systems like:

Quantum Computers: How to protect information (keep it from getting lost) even when parts of the system are noisy.
New Materials: Designing materials that conduct electricity in some parts but act as insulators in others, all within the same piece of matter.

In short: The paper shows that when you have a mix of "free" and "stuck" particles, the system doesn't break; it adapts. It creates a unique, middle-ground state where information is partially trapped and partially flowing, offering a new way to control and understand the quantum world.

1. Problem Statement

The paper addresses the fundamental challenge of understanding quantum thermalization and its breakdown in isolated systems, specifically focusing on the dynamical signatures of Single-Particle Mobility Edges (SPME).

Context: While the standard Aubry-André (AA) model exhibits a sharp localization transition (all states localized or all extended), the Generalized Aubry-André (GAA) model features a more complex spectral structure where localized and extended eigenstates coexist, separated by an energy threshold (the mobility edge).
Gap: Previous theoretical work on the GAA model has largely focused on static spectral properties (eigenstates). It remains an open question how this mixed spectral structure translates into non-equilibrium dynamics starting from a product state. Specifically, does the system obey a volume law (due to extended states), an area law (due to localized states), or does a new intermediate behavior emerge?
Goal: To characterize the dynamical response of the non-interacting GAA model using quantum information probes to establish unambiguous fingerprints of the SPME.

2. Methodology

The authors investigate the unitary time evolution of non-interacting spinless fermions on a 1D lattice with open boundary conditions, governed by the GAA Hamiltonian:
$H = -t \sum_{i=1}^{L-1} (c^\dagger_i c_{i+1} + \text{h.c.}) + \sum_{i=1}^L \mu_i c^\dagger_i c_i$
where the on-site potential $\mu_i$ is quasiperiodic and deformed by a parameter $a$ (with $a=0$ recovering the standard AA model).

Key Probes Used:

Entanglement Entropy (EE):
- Setup: Quantum quench from a Néel state ( $|1010\dots\rangle$ ) at half-filling.
- Metric: Half-chain EE, $S(t) = -\text{Tr}[\rho_{L/2}(t) \ln \rho_{L/2}(t)]$ .
- Analysis: Examined both the early-time growth velocity ( $v_S$ ) and the long-time saturation value ( $S_{\text{sat}}$ ). Finite-size scaling was performed to extract the scaling exponent $\alpha$ ( $S_{\text{sat}} \propto L^\alpha$ ) to distinguish between area-law ( $\alpha \approx 0$ ) and volume-law ( $\alpha \approx 1$ ) phases.
Subsystem Information Capacity (SIC):
- Setup: A reference qubit $R$ is maximally entangled with a single site $E$ (input), while the rest of the chain is in a product state.
- Metric: Mutual information $I(A:R)$ between the reference $R$ and a subsystem $A$ of size $|A|$ centered on $E$ after time evolution.
- Significance: SIC provides a spatially resolved picture of information propagation. A linear ramp indicates ballistic transport (extended), while a step function indicates information trapping (localized).

Simulation Tools:

Numerical simulations were performed using the TensorCircuit-NG quantum software framework.
System sizes up to $L=200$ were used, with calculations performed in the half-filling sector.

3. Key Contributions and Results

A. Dynamics of Entanglement Entropy (EE)

Early-Time Growth: The initial growth velocity $v_S$ decreases monotonically with increasing potential strength $\lambda$ for both standard AA and GAA models. Crucially, $v_S$ shows no plateaus corresponding to spectral gaps, indicating that early-time dynamics are governed by local kinetic constraints rather than global spectral topology.
Saturation Behavior (The Crossover):
- Standard AA ( $a=0$ ): Exhibits a sharp transition at $\lambda=t$ from volume-law to area-law scaling.
- GAA ( $a>0$ ): Exhibits a smooth crossover. In the SPME phase, the system maintains a persistent volume-law scaling ( $\alpha \approx 1$ ), but the saturation entropy value decreases continuously as $\lambda$ increases.
Mechanism: The saturation entropy $S_{\text{sat}}$ is strongly correlated with the fraction of extended states ( $n_e$ ). As $\lambda$ increases, the mobility edge shifts, reducing the number of available extended modes. The system's capacity to generate entanglement is directly controlled by this fraction, proving that the volume law persists as long as any extended states exist.

B. Subsystem Information Capacity (SIC) Profiles

Standard AA: Shows a sharp transition from a perfect linear ramp (extended) to a step function (localized).
GAA (SPME Phase): The SIC profile exhibits a hybrid behavior that interpolates between the two extremes:
- Initial Jump: A steep rise for small subsystem sizes $|A|$ , caused by information trapping by localized modes near the initial site.
- Slow Ramp: A subsequent gradual increase for larger $|A|$ , driven by ballistic spreading through the remaining extended modes.
Quantitative Correlation: The magnitude of the initial SIC jump ( $SIC_{\text{jump}}$ ) is directly proportional to the fraction of localized states ( $n_l$ ). This confirms that the "mixed" SIC profile is a direct spatial visualization of the coexisting localized and extended modes.

C. Phase Diagram Characterization

The study maps out three distinct dynamical regimes based on potential strength $\lambda$ and deformation $a$ :

Completely Extended: Volume law, linear SIC ramp.
Mobility-Edge Phase (Intermediate): Persistent volume law (with tunable entropy), hybrid SIC profile (jump + ramp).
Completely Localized: Area law, step-function SIC profile.

4. Significance and Implications

Definitive Non-Interacting Benchmark: This work provides the first clear, unambiguous dynamical fingerprints of a mobility edge in a non-interacting system. It establishes that the SPME does not merely "smooth" the transition but creates a distinct phase with tunable entanglement and hybrid information transport.
Insight into Interacting Systems: The findings serve as a crucial baseline for understanding Many-Body Localization (MBL) in interacting systems that may host mobility edges. The debate on whether MBL can coexist with mobility edges in interacting systems can now be compared against this non-interacting reference.
Quantum Information Control: The results demonstrate that an SPME allows for precise control over information flow, ranging from ballistic transport to perfect trapping. This has potential relevance for quantum technologies, such as designing systems for information storage or controlled transport.
Methodological Advancement: The paper successfully demonstrates the utility of Subsystem Information Capacity (SIC) as a diagnostic tool. Unlike EE, which gives a global scalar, SIC reveals the spatial distribution of information, directly visualizing the coexistence of different dynamical phases within a single system.

In conclusion, the paper establishes that the coexistence of localized and extended states in the GAA model fundamentally reshapes quantum dynamics, replacing abrupt phase transitions with smooth crossovers characterized by persistent volume-law entanglement and spatially hybrid information propagation.