arxiv: 2604.07700 · v1 · submitted 2026-04-09 · ❄️ cond-mat.dis-nn · cond-mat.mes-hall

Recognition: 1 theorem link

· Lean Theorem

Localization--non-ergodic transition in controllable-dimension fractal networks from diffusion-limited aggregation

Oleg I. Utesov , Alexei Andreanov , Tomasz Bednarek , Alexandra Siklitskaya , Sergei V. Koniakhin

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Pith reviewed 2026-05-10 18:10 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mes-hall

keywords fractal networkslocalization-non-ergodic transitiondiffusion-limited aggregationtight-binding modelcritical statesspectral propertiescompact localized states

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The pith

In 3D fractal networks from diffusion-limited aggregation, increasing density triggers a localization-non-ergodic transition with critical states emerging in the spectrum.

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

This paper examines the spectral properties of fractal agglomerates whose effective dimension is tuned by a parameter alpha in the diffusion-limited aggregation algorithm. The authors model electron states using a nearest-neighbor tight-binding Hamiltonian on these structures embedded in two and three dimensions. They find that while all states localize in 2D, in 3D a transition occurs at a critical alpha where a sub-extensive number of critical non-ergodic states appear as the fractals become denser. A reader would care because this links the localization physics of disordered systems to the geometry of self-similar fractals, showing how tunable complexity can control quantum state nature without added randomness.

Core claim

The nearest-neighbor tight-binding model on fractal agglomerates generated by diffusion-limited aggregation and embedded in 3D space exhibits a localization-non-ergodic transition upon increasing the aggregation parameter alpha, which controls the density from sparse to dense: at a critical value of alpha, a sub-extensive number of critical states emerge in the spectrum, while in 2D all eigenstates remain localized. The complex geometry additionally produces a hierarchy of compact localized states and singularities in the density of states similar to those in ordered fractals.

What carries the argument

The diffusion-limited aggregation algorithm parameterized by alpha to produce fractal networks of controllable density and dimension, with the nearest-neighbor tight-binding Hamiltonian defined on the resulting graphs.

If this is right

In two-dimensional embeddings, the spectrum consists entirely of localized states independent of alpha.
In three dimensions, the transition produces critical states that are neither fully localized nor ergodic.
The fractal geometry induces a hierarchy of compact localized states.
Singularities appear in the density of states due to the self-similar structure.

Where Pith is reading between the lines

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

The emergence of critical states suggests that fractal geometry can induce ergodicity breaking similar to many-body localization but in single-particle setting.
Experimental systems like aggregated nanoparticles or designed fractal lattices could be used to observe this transition.
Further analysis of participation ratios or wavefunction multifractality at the critical point could quantify the non-ergodicity.

Load-bearing premise

The nearest-neighbor tight-binding Hamiltonian on the generated fractal graphs faithfully captures the essential physics without long-range hops, on-site disorder, or embedding effects that would alter the observed transition.

What would settle it

Numerical diagonalization of the Hamiltonian for large 3D fractal aggregates at varying alpha values, checking for the sudden appearance of eigenstates with intermediate inverse participation ratios or level statistics indicative of critical behavior at a specific alpha.

Figures

Figures reproduced from arXiv: 2604.07700 by Alexandra Siklitskaya, Alexei Andreanov, Oleg I. Utesov, Sergei V. Koniakhin, Tomasz Bednarek.

**Figure 2.** Figure 2: IPRs of the eigenstates of the tight-binding model on cluster-cluster and particle-cluster fractal agglomerates. Top row: 2D C-C [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: (a) Density of states for the tight-binding model on fractal 3D agglomerates with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** Figure 4: IPR of most delocalized states for 2D agglomerates with [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: IPRs from exact diagonalisation for 3D agglomerates: C-C, [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Time-evolved wavefunction from a random initial site of a 3D agglomerate. For each [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: Fraction of sites on cycles as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 9.** Figure 9: Number of visited sites, defined in the main text, for the [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗

**Figure 8.** Figure 8: 2D P-C agglomerate with N = 512. Spatially separated compact localized states with eigenvalue E = 1 (left) and E = −1 (right). (N - 1)- IPR 0 /ln 1016 =4000 16 348, T=8000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2000 4000 6000 8000 10 000 � 16 348, T [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

**Figure 10.** Figure 10: Localization properties based on IPRs computed from the normalized real-space Green’s function [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗

read the original abstract

Our study connects the physics of disordered integer-dimensional systems and regular self-similar objects by studying spectral properties of fractal agglomerates with tunable dimension. The latter is controlled by parameter $\alpha$ of the algorithm that generates the agglomerates. We consider the nearest-neighbor tight-binding model on the agglomerates embedded in 2D and 3D, and observe that all eigenstates are localized in the 2D case, whereas in the 3D case, there is a localization--non-ergodic transition upon increasing $\alpha$,i.e., going from sparse to dense fractals: a sub-extensive number of critical states emerge in the spectrum at a certain critical value of $\alpha$. The complex geometry of the agglomerates is also responsible for a peculiar hierarchy of compact localized states and singularities in the density of states, which are typical for ordered fractals.

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

Tunable DLA fractals with alpha give a 3D localization-non-ergodic transition and sub-extensive critical states, but the scaling evidence for those states looks thin.

read the letter

The main point is that they generate DLA clusters where alpha tunes the effective dimension, embed the graphs in 2D and 3D, and diagonalize the nearest-neighbor tight-binding Hamiltonian. In 2D everything localizes. In 3D they see a transition at some critical alpha where a sub-extensive set of critical states appears as the clusters get denser. The compact localized states and DOS singularities are noted as expected for fractal geometry. The controllable-dimension trick is the clearest new element; most fractal localization work fixes the dimension, so varying it continuously and watching the spectral change is a useful addition. The numerics are direct on the generated structures and the 2D-3D contrast is clean. The soft spot is the identification of the critical states. The claim rests on a sub-extensive number emerging at the transition, but without explicit finite-size scaling that shows their count stays o(N) in the large-N limit, it is difficult to be sure this is not a finite-size effect tied to the irregular embedding and branching. Standard IPR diagnostics are probably used, yet the stress-test concern about thermodynamic-limit behavior holds weight here. Nearest-neighbor hopping is the natural starting point, though longer-range terms could shift the transition in real systems. This is for people who study Anderson localization on complex or fractal graphs and want a tunable model between regular fractals and disordered lattices. The thinking is straightforward and the setup is reproducible, so the work deserves referee time even if the scaling section needs tightening.

Referee Report

1 major / 1 minor

Summary. The manuscript examines spectral properties of the nearest-neighbor tight-binding Hamiltonian on fractal agglomerates generated via a diffusion-limited aggregation algorithm whose parameter α tunes the effective dimension. In 2D embeddings all eigenstates are localized; in 3D embeddings, increasing α (from sparse to dense fractals) produces a localization–non-ergodic transition at a critical α, with a sub-extensive number of critical states appearing in the spectrum, together with a hierarchy of compact localized states and singularities in the density of states.

Significance. If the reported transition survives the thermodynamic limit, the work would usefully connect the physics of disordered integer-dimensional systems with controllable self-similar geometries, showing how fractal dimension can drive the emergence of non-ergodic critical states. The numerical construction is parameter-free and directly observable on generated graphs, which is a methodological strength.

major comments (1)

[Abstract and main results] Abstract and main results: the central claim that a sub-extensive number of critical states emerge in the 3D spectrum at a critical α rests on the identification of these states (presumably via IPR or similar diagnostics) but provides no finite-size scaling to demonstrate that their number is o(N) as N→∞. Without such scaling the observation remains vulnerable to finite-size artifacts arising from the complex fractal geometry and embedding.

minor comments (1)

[Abstract] The abstract states that the transition occurs 'at a certain critical value of α' but does not indicate the diagnostic or fitting procedure used to locate this value.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point regarding the thermodynamic limit. We address the major comment below.

read point-by-point responses

Referee: Abstract and main results: the central claim that a sub-extensive number of critical states emerge in the 3D spectrum at a critical α rests on the identification of these states (presumably via IPR or similar diagnostics) but provides no finite-size scaling to demonstrate that their number is o(N) as N→∞. Without such scaling the observation remains vulnerable to finite-size artifacts arising from the complex fractal geometry and embedding.

Authors: We agree that a direct finite-size scaling analysis is required to rigorously establish that the number of critical states is sub-extensive (o(N)) in the thermodynamic limit. In the present manuscript the critical states are identified via the inverse participation ratio (IPR) together with level-spacing statistics and wave-function diagnostics on finite agglomerates of varying size. While these diagnostics are consistent across the sizes we have accessed, they do not yet include an explicit extrapolation of the critical-state count normalized by N. In the revised version we will add a dedicated finite-size scaling section that plots the fraction of critical states versus system size N for several values of α near the transition, demonstrating that this fraction vanishes as N→∞ at the critical α while remaining finite in the non-ergodic phase. This addition will directly address the concern about possible finite-size artifacts arising from the fractal geometry. revision: yes

Circularity Check

0 steps flagged

No circularity: claims are direct numerical observations on generated graphs

full rationale

The paper generates fractal agglomerates via a tunable algorithm controlled by α, constructs the nearest-neighbor tight-binding Hamiltonian on the resulting graphs embedded in 2D or 3D, and computes their eigenstates and spectra. The reported localization in 2D and the localization-non-ergodic transition (with sub-extensive critical states) in 3D at a critical α are presented as direct computational observations, not as outputs of any analytical derivation. Standard diagnostics such as inverse participation ratio are applied to classify states, with no reduction of the central claim to a fitted parameter, self-referential definition, or load-bearing self-citation. The work is self-contained against external benchmarks and contains no ansatz smuggling or renaming of known results as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The tight-binding model and DLA generation are treated as standard tools.

pith-pipeline@v0.9.0 · 5473 in / 1104 out tokens · 24959 ms · 2026-05-10T18:10:55.216168+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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Relation between the paper passage and the cited Recognition theorem.

We consider the nearest-neighbor tight-binding model on the agglomerates embedded in 2D and 3D, and observe that all eigenstates are localized in the 2D case, whereas in the 3D case, there is a localization–non-ergodic transition upon increasing α

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Localisation–non-ergodic transition in controllable-dimension fractal networks from diffusion-limited aggregation

D. Thongjaomayum, S. Flach, and A. Andreanov, Multifrac- tality of correlated two-particle bound states in quasiperiodic chains, Phys. Rev. B101, 174201 (2020). 8 End Matter Degenerate states with E = ±1— Typical E = ±1 eigen- states are shown in Fig. 8. Note that they come in pairs, i.e., if there is a state with E =1 made of the same-sign dimers, there ...

2020