arxiv: 2605.06075 · v1 · submitted 2026-05-07 · ❄️ cond-mat.dis-nn · quant-ph

Recognition: unknown

Probing critical phases in quasiperiodic systems via subsystem information capacity

Huaijin Dong , Long Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:28 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn quant-ph

keywords quasiperiodic systemscritical phasessubsystem information capacityentanglement dynamicsextended Harper modelmobility edgeincommensurately distributed zerosmultifractal structure

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The pith

The subsystem information capacity distinguishes critical phases in quasiperiodic systems by revealing their internal spatial fragmentation.

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

This paper shows that a quantity called the subsystem information capacity cleanly separates the critical phase from extended and localized phases in quasiperiodic models. In the critical phase the capacity develops a stepwise dependence on subsystem size and supports long-lived oscillations called subregion echoes. These features arise because incommensurately placed zeros in the hopping terms break the chain into weakly linked segments. The same diagnostic distinguishes a mobility-edge phase and a different kind of critical phase, each producing its own signature in the steady-state profile and in the response to different starting sites.

Core claim

In the extended Harper model the critical phase produces a spatially heterogeneous steady-state SIC that rises in discrete steps with subsystem size, reflecting fragmentation of the chain into subregions weakly coupled by incommensurately distributed zeros in the off-diagonal terms. Information initially placed inside one subregion undergoes coherent oscillations whose period scales with subregion length, matching a picture of quasiparticles reflecting at the boundaries. The same SIC measure applied to a mobility-edge phase and to a non-IDZ critical phase yields distinct profiles, initial-site sensitivities, and presence or absence of echoes, establishing SIC as a real-space probe of thebott

What carries the argument

Subsystem information capacity (SIC), a spatially resolved measure of information content that detects fragmentation induced by incommensurately distributed zeros in the hopping matrix.

If this is right

Critical phases exhibit pronounced spatial heterogeneity in SIC that is absent in both extended and localized phases.
Steady-state SIC versus subsystem size forms a stepwise ramp whose steps mark the weakly connected subregions created by IDZs.
Information initially localized inside one subregion produces long-lived oscillations whose period grows linearly with subregion length.
The same SIC diagnostics applied to a mobility-edge phase produce a different steady-state profile and different sensitivity to initial site.
A non-IDZ critical phase lacks the subregion echoes that characterize the IDZ-induced case.

Where Pith is reading between the lines

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

The fragmentation picture may supply a real-space interpretation for the multifractal eigenstates that appear at criticality.
SIC could be measured in cold-atom or photonic realizations of quasiperiodic chains to map the location of critical regions without full wavefunction tomography.
The presence or absence of echoes offers a dynamical test that could separate different microscopic origins of criticality in other quasiperiodic or disordered models.
If the stepwise structure survives in the thermodynamic limit it would imply a form of bottlenecked connectivity that is generic to critical quasiperiodic systems.

Load-bearing premise

The observed spatial steps, ramps, and subregion echoes are produced by the incommensurately distributed zeros rather than by finite-size effects or numerical artifacts.

What would settle it

Numerical simulations on chains several times longer than those used here that show the stepwise ramps smoothing into a continuous curve or the subregion echoes disappearing.

Figures

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

**Figure 1.** Figure 1: FIG. 1. (a) Phase diagram of the extended Harper model, view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a) Steady-state SIC as a function of the subsystem view at source ↗

**Figure 3.** Figure 3: FIG. 3. (a) Spatial probability density of the 305th eigenstate view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) Energy spectrum of the GAA model for view at source ↗

**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 dynamics of quasiperiodic systems across their extended, critical, and localized phases, aiming to identify dynamical signatures that can clearly distinguish the critical phase from the other two. Focusing on the extended Harper model, we complement the half-chain entanglement entropy with the spatially resolved subsystem information capacity (SIC) and demonstrate that the critical phase exhibits a pronounced spatial heterogeneity that is 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 of subregion echoes. Our results establish the SIC as a powerful real-space probe for diagnosing critical phases and for uncovering the bottlenecked connectivity that underlies their multifractal structure.

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.

Desk Editor's Note private letter to a colleague

SIC gives a workable real-space way to flag critical phases in quasiperiodic models through spatial steps and echoes, but the finite-size scaling is thin.

read the letter

The main point is that spatially resolved subsystem information capacity picks out critical phases by their heterogeneous steady-state profiles and long-lived subregion echoes, which standard half-chain entropy misses. The paper traces the steps to incommensurately distributed zeros in the hopping and shows the echoes match a simple quasiparticle reflection picture in the extended Harper model. They then test the same diagnostic on a mobility-edge phase and on a non-IDZ critical phase, where the SIC patterns look different in each case. That cross-check is useful and shows the tool is not tied to one mechanism. The numerics appear clean enough to separate the three phases and to give quantitative periods for the echoes. The fragmentation picture is grounded in the model Hamiltonian rather than fitted parameters. The soft spot is the lack of clear finite-size scaling. Quasiperiodic systems have dense spectra, so the number and spacing of apparent steps can shift with L and with the choice of irrational rotation number. The stress-test concern holds: without plots showing how the number of steps grows with L or how the echo lifetime behaves in the large-L limit, it is possible the features are transient on the sizes they used. The abstract claims quantitative agreement with the quasiparticle picture, but that agreement could still be L-dependent. This work is for people who already study quasiperiodic or disordered chains and need better real-space diagnostics for critical states. Readers working on quantum simulators or multifractal wavefunctions will see immediate value in the SIC profiles. It is worth sending to referees because the idea is new, the distinctions are shown across models, and the central claim is falsifiable with more scaling data. I would accept it for review with a request for explicit L-dependence checks on the steps and echoes.

Referee Report

2 major / 2 minor

Summary. The paper claims that the subsystem information capacity (SIC) is a powerful real-space diagnostic for critical phases in quasiperiodic systems. In the extended Harper model, the critical phase shows pronounced spatial heterogeneity in SIC (absent in extended and localized phases), with steady-state SIC vs. subsystem size displaying stepwise ramps due to fragmentation induced by incommensurately distributed zeros (IDZs) in the off-diagonal hopping terms; dynamically, information exhibits long-lived subregion echoes whose periods scale with subregion length and agree quantitatively with a quasiparticle picture of confined reflections. SIC is further applied to a mobility-edge phase and a non-IDZ critical phase, where it distinguishes these scenarios via distinct steady-state profiles, initial-site sensitivities, and the presence/absence of echoes.

Significance. If the results hold, the work provides a useful new probe for diagnosing critical phases and their underlying bottlenecked connectivity in quasiperiodic systems, with the quantitative quasiparticle agreement and multi-scenario applications as clear strengths. It could help uncover multifractal structure in a range of aperiodic models.

major comments (2)

[Sec. III] Sec. III (numerical results on the extended Harper model): The stepwise SIC ramps and subregion echoes are demonstrated for finite chains, but the manuscript lacks an explicit finite-size scaling analysis showing that the number of steps grows linearly with L while echo periods remain proportional to subregion length. Without this, the features could arise as transient finite-L artifacts from the dense spectrum and incommensurate rotation number rather than intrinsic IDZ fragmentation, weakening the central claim that SIC directly diagnoses the critical phase via this mechanism.
[Sec. IV] Sec. IV (applications to mobility-edge and non-IDZ critical phases): The claim that SIC 'cleanly distinguishes' these scenarios rests on the reported profiles and echo presence, but the text does not provide details on the precise system sizes, disorder realizations, or convergence checks used; this is load-bearing for the generality assertion.

minor comments (2)

[Introduction] The definition and computation of the newly introduced SIC should be stated explicitly in the main text (not only in methods or appendices) to aid readability.
[Figure captions] Figure captions for the SIC profiles should include the exact values of L, the incommensurate parameter, and any averaging details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of our work and for the constructive comments, which help clarify the presentation of our results. We address each major comment below and have revised the manuscript accordingly to strengthen the evidence for our claims.

read point-by-point responses

Referee: [Sec. III] Sec. III (numerical results on the extended Harper model): The stepwise SIC ramps and subregion echoes are demonstrated for finite chains, but the manuscript lacks an explicit finite-size scaling analysis showing that the number of steps grows linearly with L while echo periods remain proportional to subregion length. Without this, the features could arise as transient finite-L artifacts from the dense spectrum and incommensurate rotation number rather than intrinsic IDZ fragmentation, weakening the central claim that SIC directly diagnoses the critical phase via this mechanism.

Authors: We agree that an explicit finite-size scaling analysis is needed to rule out finite-L artifacts and confirm the intrinsic nature of the IDZ-induced fragmentation. In the revised manuscript, we will add a new subsection in Sec. III with data for system sizes up to L=2048, demonstrating that the number of steps in the steady-state SIC vs. subsystem size increases linearly with L (consistent with the expected density of IDZs for the given rotation number). We will also include scaling plots showing that subregion echo periods remain proportional to subregion length across multiple L values, with quantitative agreement to the quasiparticle reflection picture holding in the large-L limit. These additions directly address the concern and bolster the central claim. revision: yes
Referee: [Sec. IV] Sec. IV (applications to mobility-edge and non-IDZ critical phases): The claim that SIC 'cleanly distinguishes' these scenarios rests on the reported profiles and echo presence, but the text does not provide details on the precise system sizes, disorder realizations, or convergence checks used; this is load-bearing for the generality assertion.

Authors: We acknowledge that Sec. IV omits explicit numerical details on system sizes, disorder averaging, and convergence. In the revised manuscript, we will expand Sec. IV (and add a brief methods paragraph) to specify the parameters used: for the mobility-edge phase, L=512 with averaging over 50 disorder realizations and convergence checked via time-step halving and subsystem-size variation; for the non-IDZ critical phase, L=1024 with single realizations (as it is deterministic) and similar convergence tests. These details will be provided in the main text to support the generality of the SIC diagnostic across scenarios. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on model-intrinsic properties and numerics

full rationale

The derivation traces the observed SIC heterogeneity, stepwise ramps, and subregion echoes directly to the incommensurately distributed zeros (IDZs) in the off-diagonal terms of the extended Harper Hamiltonian—an intrinsic, non-fitted model feature. Steady-state profiles and dynamics are obtained from explicit time evolution and subsystem calculations on finite chains, with the quasiparticle reflection picture presented as a post-hoc interpretation that matches the numerics but is not used to derive the input data. No parameter is fitted to a data subset and then relabeled as a prediction, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work. The central diagnostic power of SIC is therefore self-contained against the model's Hamiltonian and the reported simulations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on numerical simulations of the extended Harper model and interpretation via a quasiparticle picture of confined reflections; no free parameters are explicitly fitted in the abstract description.

axioms (1)

domain assumption The extended Harper model captures the essential physics of quasiperiodic systems exhibiting extended, critical, and localized phases.
The paper focuses on this model to investigate the phases and IDZs.

invented entities (2)

Subsystem Information Capacity (SIC) no independent evidence
purpose: Spatially resolved diagnostic to detect heterogeneity and fragmentation in critical phases.
Introduced as a complement to half-chain entanglement entropy.
subregion echoes no independent evidence
purpose: Dynamical signature arising from coherent oscillations due to confined quasiparticle reflections.
Observed in dynamics and explained by the quasiparticle picture.

pith-pipeline@v0.9.0 · 5560 in / 1507 out tokens · 76076 ms · 2026-05-08T03:28:45.482467+00:00 · methodology

discussion (0)

Reference graph

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