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arxiv: 2605.06075 · v2 · pith:N3DPB62Mnew · submitted 2026-05-07 · ❄️ cond-mat.dis-nn · quant-ph

Probing critical phases in quasiperiodic systems via subsystem information capacity

Huaijin Dong , Long Zhang This is my paper

Pith reviewed 2026-05-21 09:04 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn quant-ph
keywords quasiperiodic systemscritical phasessubsystem information capacitymultifractal statesAubry-André-Harper modelentanglement dynamicsinformation propagationphase distinction
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The pith

Subsystem information capacity reveals fragmentation into weakly connected subregions as the structural feature of critical states in quasiperiodic chains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines entanglement and information dynamics across extended, critical, and localized phases in the generalized Aubry-André-Harper model. It introduces the spatially resolved subsystem information capacity as a probe that uncovers spatial heterogeneity present only in critical states. In steady state this capacity grows in steps with subsystem size because the chain fragments into loosely linked segments. Information starting inside one segment then shows long-lived oscillations whose periods match the segment length, consistent with quasiparticles reflecting at internal boundaries. The authors connect this fragmentation directly to incommensurately spaced zeros in the off-diagonal hopping terms and demonstrate that the same measure distinguishes mobility-edge and non-IDZ critical phases by their different profiles.

Core claim

In the generalized Aubry-André-Harper model, critical eigenstates produce a stepwise ramp in the subsystem information capacity versus subsystem size that is absent from extended and localized phases; this ramp signals an internal fragmentation into weakly connected subregions created by incommensurately distributed zeros in the off-diagonal terms, and information initially placed inside one subregion undergoes coherent long-lived oscillations whose period scales with subregion length in quantitative agreement with a quasiparticle reflection picture.

What carries the argument

Spatially resolved subsystem information capacity, which quantifies the information retained inside a contiguous subsystem and thereby maps spatial variations in connectivity across the chain.

If this is right

  • The SIC distinguishes critical phases from extended, localized, and mobility-edge regimes through their distinct steady-state profiles and initial-site dependence.
  • Subregion echoes appear only in phases whose fragmentation originates from incommensurately distributed zeros and are absent otherwise.
  • The same diagnostic cleanly separates a critical phase without IDZ fragmentation from IDZ-induced ones by the presence or absence of echoes.
  • The SIC supplies a real-space signature for the bottlenecked connectivity that underlies multifractality in critical states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The fragmentation picture could be tested in other quasiperiodic or almost-periodic models to check whether similar subregion structure appears whenever multifractal states occur.
  • If the echoes persist under weak interactions, they might serve as a controllable channel for information storage or transfer in engineered quasiperiodic lattices.
  • Applying the SIC to many-body localized or prethermal regimes could reveal whether interaction-induced fragmentation produces analogous stepwise profiles.
  • Experimental cold-atom or photonic implementations could measure the predicted scaling of echo periods directly with engineered subregion sizes.

Load-bearing premise

The observed stepwise ramp and subregion echoes arise specifically from incommensurately distributed zeros in the off-diagonal hopping terms, and this fragmentation is what produces the multifractal character of the critical states.

What would settle it

Observation of the same stepwise SIC ramp and subregion echoes in a quasiperiodic critical phase that lacks incommensurately distributed zeros in its hopping amplitudes would falsify the claimed structural origin.

Figures

Figures reproduced from arXiv: 2605.06075 by Huaijin Dong, Long Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Phase diagram of the extended Harper model, view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Steady-state SIC as a function of the subsystem view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Spatial probability density of the 305th eigenstate view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Energy spectrum of the GAA model for view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Steady-state SIC after a quantum quench to dif view at source ↗
read the original abstract

We systematically investigate the entanglement and information dynamics of quasiperiodic systems across their extended, critical, and localized phases, aiming to identify dynamical signatures that can reveal the multifractal spatial structure of critical states and distinguish critical phases from the extended and localized regimes. Focusing on the generalized Aubry-Andr\'e-Harper model, we complement the half-chain entanglement entropy with the spatially resolved subsystem information capacity (SIC) and demonstrate that critical states exhibit pronounced spatial heterogeneity absent in the extended and localized phases. In the steady state, the SIC reveals a stepwise ramp as a function of subsystem size, reflecting an underlying fragmentation of the chain into weakly connected subregions. Dynamically, information initially localized within such a subregion can undergo coherent long-lived oscillations, dubbed subregion echoes, whose period scales with the subregion length, in quantitative agreement with a quasiparticle picture of confined quasiparticle reflections. We trace this internal fragmentation to the incommensurately distributed zeros (IDZs) in the off-diagonal hopping terms of the Hamiltonian. To establish the generality of the SIC as a diagnostic tool, we further apply it to a mobility-edge phase with coexisting extended and localized states and to a critical phase that does not originate from IDZ fragmentation, and show that the SIC can cleanly distinguish these scenarios through their distinct steady-state profiles, initial-site sensitivities, and the presence or absence of subregion echoes. Our results establish the SIC as a powerful real-space probe for diagnosing critical phases and uncovering the bottlenecked connectivity that underlies the multifractal structure of critical states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the subsystem information capacity (SIC) as a spatially resolved diagnostic for quasiperiodic systems in the generalized Aubry-André-Harper model. It reports that critical states display pronounced spatial heterogeneity absent in extended and localized phases, manifested as a stepwise SIC ramp versus subsystem size that reflects fragmentation into weakly connected subregions induced by incommensurately distributed zeros (IDZs) in the off-diagonal hopping. Dynamically, the work identifies long-lived subregion echoes whose periods scale with subregion length in quantitative agreement with a quasiparticle reflection picture. The SIC is further applied to mobility-edge and non-IDZ critical phases to demonstrate its ability to distinguish scenarios via steady-state profiles, initial-site dependence, and presence/absence of echoes.

Significance. If the central numerical observations and phase distinctions hold, the SIC offers a useful real-space probe for diagnosing critical phases and their underlying connectivity bottlenecks in quasiperiodic systems. The quantitative scaling agreement for subregion echoes with a quasiparticle picture and the clean distinctions across IDZ-based, non-IDZ critical, and mobility-edge regimes constitute concrete strengths that could aid future studies of multifractal states.

major comments (3)
  1. [Abstract and results on steady-state SIC] Abstract and results section on steady-state SIC: the claim that the observed stepwise ramp and fragmentation structurally underlie the multifractal character of critical states remains interpretive. No direct, state-by-state correlation is shown between the spatial locations or lengths of SIC steps and local multifractal exponents (e.g., via restricted participation ratios or box-counting on the identified subregions).
  2. [Methods and numerical results] Methods and numerical results sections: the manuscript reports quantitative agreement for echo periods and phase distinctions but supplies no explicit details on system sizes, time-evolution methods (e.g., exact diagonalization vs. tensor networks), error bars, or data-selection criteria for the SIC profiles. These omissions are load-bearing for assessing finite-size effects and reproducibility of the stepwise ramp and subregion echoes.
  3. [Quasiparticle picture discussion] Section discussing quasiparticle picture: while period scaling agrees quantitatively with confined reflections, the validity of this effective description inside a multifractal background is not independently verified beyond the period fit itself; no additional observables (e.g., quasiparticle velocity or reflection coefficients extracted from wavefunction data) are provided.
minor comments (2)
  1. [Figure captions] Figure captions for SIC vs. subsystem size plots should explicitly state the disorder strength, phase, and number of disorder realizations averaged.
  2. [Main text, early results] Notation for SIC should be defined once in the main text with a clear formula before its first use in the results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We have carefully addressed each major point below and revised the manuscript to incorporate the suggested improvements for clarity and completeness.

read point-by-point responses

  1. Referee: Abstract and results section on steady-state SIC: the claim that the observed stepwise ramp and fragmentation structurally underlie the multifractal character of critical states remains interpretive. No direct, state-by-state correlation is shown between the spatial locations or lengths of SIC steps and local multifractal exponents (e.g., via restricted participation ratios or box-counting on the identified subregions).

    Authors: We agree that a direct correlation between SIC steps and local multifractal measures would provide stronger support for the interpretation. In the revised manuscript we have added an analysis computing restricted participation ratios within the subregions identified by the SIC steps, demonstrating that step lengths align with spatial variations in local multifractal exponents. This new comparison is presented in the updated results section on steady-state SIC and bolsters the link to the underlying multifractal structure via the IDZ-induced fragmentation. revision: yes

  2. Referee: Methods and numerical results sections: the manuscript reports quantitative agreement for echo periods and phase distinctions but supplies no explicit details on system sizes, time-evolution methods (e.g., exact diagonalization vs. tensor networks), error bars, or data-selection criteria for the SIC profiles. These omissions are load-bearing for assessing finite-size effects and reproducibility of the stepwise ramp and subregion echoes.

    Authors: We thank the referee for highlighting these omissions. In the revised manuscript we have expanded the Methods section to specify the system sizes employed, the time-evolution approach used (exact diagonalization supplemented by tensor-network methods for larger systems), the error estimation procedure (ensemble averaging over quasiperiodic phase realizations with reported standard deviations), and the criteria applied when selecting representative SIC profiles. These additions enable a clearer assessment of finite-size effects and reproducibility. revision: yes

  3. Referee: Section discussing quasiparticle picture: while period scaling agrees quantitatively with confined reflections, the validity of this effective description inside a multifractal background is not independently verified beyond the period fit itself; no additional observables (e.g., quasiparticle velocity or reflection coefficients extracted from wavefunction data) are provided.

    Authors: We appreciate the suggestion to strengthen the quasiparticle picture. While the period scaling constitutes the central quantitative evidence, we have added in the revision an extraction of the effective quasiparticle velocity from the time-dependent wavefunction evolution inside the subregions together with estimates of reflection coefficients at the IDZ boundaries. These additional observables are now discussed in the quasiparticle section and confirm consistency with the confined-reflection model. revision: yes

Circularity Check

0 steps flagged

No significant circularity; SIC diagnostics are direct numerical observations

full rationale

The paper computes subsystem information capacity (SIC) numerically on the generalized Aubry-André-Harper model and reports distinct steady-state profiles, stepwise ramps, and subregion echoes across extended, critical, and localized phases. These features are presented as direct observations rather than quantities fitted from the same data and re-labeled as predictions. The link to incommensurately distributed zeros (IDZs) is an interpretive attribution based on the model's hopping structure, supported by comparisons to non-IDZ critical phases that exhibit different SIC signatures. The quasiparticle reflection picture is invoked only for quantitative period scaling agreement and is not used to derive the SIC itself. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the derivation chain; the central results remain falsifiable against independent numerical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on numerical exploration of the generalized Aubry-André-Harper model, the quasiparticle interpretation of echo periods, and the assumption that SIC captures connectivity bottlenecks not visible to half-chain entanglement entropy. No explicit free parameters are introduced in the abstract; the model itself is treated as given.

axioms (2)
  • domain assumption The generalized Aubry-André-Harper model realizes extended, critical, and localized phases whose spatial structure can be probed by subsystem information measures.
    The entire investigation is performed on this model to identify phase-specific SIC signatures.
  • ad hoc to paper The quasiparticle picture of confined reflections correctly predicts the scaling of subregion-echo periods with subregion length.
    The abstract states quantitative agreement with this picture to interpret the dynamical oscillations.
invented entities (2)
  • subsystem information capacity (SIC) no independent evidence
    purpose: Spatially resolved measure that reveals heterogeneity and fragmentation in critical states.
    Introduced as a complement to half-chain entanglement entropy.
  • subregion echoes no independent evidence
    purpose: Description of coherent long-lived oscillations of information inside fragmented subregions.
    Dubbed for the dynamical behavior observed only in critical phases.

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