Probing critical phases in quasiperiodic systems via subsystem information capacity

This article establishes subsystem information capacity (SIC) as a powerful real-space diagnostic tool that distinguishes critical phases in quasiperiodic systems from extended and localized phases by revealing unique spatial heterogeneity, stepwise stationary profiles, and coherent echo effects in subsystems arising from incommensurably distributed zeros in the Hamiltonian.

The Gist

Imagine you are trying to understand how a crowd moves through a city. In some cities, everyone flows freely like water in a river (this is the extended phase). In others, people are stuck in their houses and cannot move at all (this is the localized phase). But there is a third, mysterious state: a city where people can move, but the streets are so strangely designed that they get trapped in loops, wander aimlessly, or bounce back and forth within certain neighborhoods. This is the critical phase, and scientists have long struggled to distinguish it from the other two using conventional tools.

This article introduces a new, sharper tool called Subsystem Information Capacity (SIC) to solve this puzzle. Here is how the authors explain it using simple concepts:

1. The Old Tool versus the New Tool

The researchers first tried the standard method, which involves viewing the city from a helicopter and measuring the "total noise level" (entanglement entropy) of the entire crowd.

• The Problem: From a great height, the "critical" city looks very similar to the "freely flowing" city. Both eventually fill up with noise. The helicopter view overlooks the chaotic, complicated details on the ground.
• The New Tool (SIC): Instead of looking from above, SIC is like placing a tiny, invisible microphone at a single street corner, listening to how a whispered secret spreads there. It tracks precisely how this secret distributes across neighborhoods of different sizes.

2. The "Staircase Rise" (the Stationary State)

When the researchers whispered a secret in the extended phase, it spread smoothly and evenly, like ink falling into a glass of water. The graph showing how much of the city heard the secret looked like a smooth, straight line.

When they whispered it in the localized phase, the secret barely moved. It stayed exactly where it was. The graph looked like a flat line that suddenly jumped up at the very end.

But in the critical phase, they saw something unique: a staircase.

• The secret did not spread smoothly. It traveled a bit, hit a wall, traveled a bit more, hit another wall, and so on.
• The graph looked like a ramp with distinct steps.
• The Analogy: Imagine a hallway whose floor consists of alternating smooth tiles and sticky patches. You can walk, but every few steps you get stuck for a moment before moving again. The "steps" in the graph reveal that the city is actually broken down into weakly connected neighborhoods.

3. The "Subregion Echoes" (the Dynamics)

The most exciting discovery happened when they observed how the secret moved over time.

• In the critical phase, the secret did not simply fade away. It got trapped in one of these "neighborhoods" (created by the sticky patches) and began to bounce back and forth like a ball in a box.
• The researchers call these "subregion echoes".
• The Analogy: Imagine shouting into a long, narrow tunnel divided into separate chambers. Your voice hits the wall of the chamber, bounces back, hits the other wall, and bounces again. You hear your own voice repeating in a rhythmic pattern.
• The article found that the time these "echoes" took to repeat perfectly matched the size of the chamber. If the chamber was small, the echo returned quickly; if it was large, it took longer. This proved that the "critical" state is indeed an accumulation of tiny, isolated spaces where information gets trapped and bounces around.

4. Why Does This Happen? (the "Invisible Walls")

The authors attributed these "sticky patches" and "walls" to a specific mathematical feature in the system's design: incommensurably distributed zeros (IDZs).

• The Metaphor: Imagine the connections between the houses (the streets) as invisible "traffic lights." In the critical phase, these traffic lights stand at very specific, irregular intervals on "red." These red lights act as bottlenecks, cutting the long city street into short, isolated segments.
• Since the red lights are placed in a strange, non-repeating pattern, all segments have different sizes, creating this unique "staircase" and "echo" pattern.

5. Does This Work Everywhere?

To ensure their new tool (SIC) was not just a coincidence for this specific city, they tested it on two other types of "cities":

1. The Mixed City (SPME): A place where some streets are wide highways and others are dead ends. Here, the secret spread in a straight line but with a sudden jump at the beginning. No stairs, no echoes.
2. The "Smooth" Critical City: A place that is critical but does not possess these invisible red-light walls. Here, the secret spread in a wavy, bumpy line, but it had no distinct "steps" or rhythmic "echoes."

The Conclusion

The article concludes that Subsystem Information Capacity (SIC) is a powerful magnifying glass. It can uncover the hidden "spaces" and "bottlenecks" within a critical phase that other tools miss.

• Smooth line? You are in a freely flowing city.
• Flat line with a jump? You are in a stuck city.
• Staircase with bouncing echoes? You are in the mysterious critical city where the streets are secretly cut into isolated neighborhoods.

This method allows scientists not only to identify these phases but also to understand why they behave this way by mapping the invisible walls that trap information.

Technical Summary

Technical Summary: Investigation of Critical Phases in Quasiperiodic Systems via Subsystem Information Capacity

Problem Statement
Quasiperiodic systems occupy a unique regime between perfectly periodic and fully random systems, supporting not only extended and localized states but also mobility edges and multifractal critical phases. While theoretical frameworks for characterizing these critical states—often defined by multifractal wavefunctions and local scale invariance—have advanced, distinguishing critical phases from extended and localized regimes in realistic settings remains a significant challenge. Conventional diagnostic tools, such as the fractal dimension derived from inverse participation ratios (IPR), provide static, global characterizations of the spectrum. These metrics offer only limited insight into the dynamical transport properties that fundamentally distinguish critical phases. Furthermore, while the growth pattern of entanglement entropy (EE) for extended (volume-law) and localized (area-law) phases is well understood, a systematic dynamical characterization of critical phases remains incomplete. In particular, it is unclear how the multifractal nature of critical wavefunctions manifests in real-time entanglement dynamics or how such signatures can be utilized to distinguish critical phases from others.

Methodology
The authors systematically investigate the entanglement dynamics of the extended Harper model, which exhibits pure extended, critical, and localized phases without mobility edges. The study employs two primary probes:

1. Half-chain Entanglement Entropy (EE): Serves as a baseline to track global entanglement growth following a quantum quench from a bipartite product state.
2. Subsystem Information Capacity (SIC): A spatially resolved diagnostic tool introduced in recent literature. SIC tracks how quantum information, initially encoded at a single lattice site (via a Bell pair with a reference qubit), diffuses through the system as a function of subsystem size.

The authors analyze stationary SIC profiles and their dynamical evolution across the three phases. They further extend this analysis to two additional scenarios to test the generality of SIC:

• A Single-Particle Mobility Edge (SPME) phase (using the generalized Aubry-André model), where extended and localized states coexist.
• A Non-IDZ Critical Phase (using a spin-orbit coupled atomic chain), where criticality arises without the specific mechanism of incommensurably distributed zeros (IDZs) in the hopping terms.

Key Results

• Limitations of Half-chain EE: While half-chain EE can distinguish localized from delocalized behavior, it fails to resolve the rich internal structure of the critical phase. In the critical phase, EE grows slower than in the extended phase but eventually saturates at a comparable value, offering only a coarse-grained view.
• Stationary SIC Signatures:
  • Extended Phase: SIC shows a linear increase with subsystem size, reflecting ballistic transport.
  • Localized Phase: SIC exhibits a step-like shape, indicating strong information trapping.
  • Critical Phase (IDZ-induced): SIC displays a characteristic stepwise increase. This profile reflects an underlying fragmentation of the chain into weakly connected subregions. Crucially, the SIC profile in the critical phase is highly sensitive to the initial position of the encoded information (spatial offset), unlike in the extended and localized phases, where the profile remains robust against such shifts.
• Subregion Echoes: The dynamical evolution of SIC in the critical phase reveals coherent, long-lived oscillations termed "subregion echoes." These oscillations correspond to quasiparticles confined within subregions bounded by incommensurably distributed zeros (IDZs) in the off-diagonal hopping terms. The oscillation period scales linearly with subregion length and quantitatively matches a quasiparticle picture of confined reflections.
• Distinction of Scenarios:
  • SPME Phase: The stationary SIC shows an abrupt initial rise followed by a linear increase, distinct from the stepwise increase of the IDZ-critical phase. It exhibits no subregion echoes, and sensitivity to the initial lattice site affects only the magnitude of the early rise.
  • Non-IDZ Critical Phase: SIC displays a wavelike increase that grows somewhat faster than in the extended phase but lacks the characteristic stepwise structure and subregion echoes of the IDZ-critical phase. Although it exhibits spatial heterogeneity (sensitivity to the initial lattice site), it shows no regular oscillatory dynamics associated with IDZ fragmentation.

Main Contributions

1. Identification of Dynamical Signatures: The work establishes that SIC enables clear dynamical distinction between critical, extended, and localized phases, specifically identifying the "stepwise increase" and "subregion echoes" as unique fingerprints of IDZ-induced critical phases.
2. Elucidation of Mechanism: The study attributes the internal fragmentation of critical states to IDZs in the hopping terms, providing a physical mechanism for the observed restricted connectivity and multifractal structure.
3. Generality of the Diagnostic Tool: By applying SIC to SPME and Non-IDZ critical phases, the authors demonstrate that the tool can cleanly distinguish various localization scenarios based on stationary profiles, sensitivities to the initial lattice site, and the presence or absence of subregion echoes.

Significance
The work posits that Subsystem Information Capacity (SIC) serves as a powerful real-space probe for diagnosing critical phases. In contrast to global static measures, SIC reveals the "restricted connectivity" underlying the multifractal structure of critical states. The ability to distinguish critical phases from mobility edge scenarios and Non-IDZ critical phases through unique spatial and dynamical signatures offers a refined framework for identifying criticality. The authors note that SIC is experimentally accessible with current platforms (cold atoms, photonic lattices, superconducting circuits), requiring only a single ancilla qubit and forward time evolution, making it a practical tool for the experimental verification of complex quantum phases.