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arxiv: 2605.17953 · v1 · pith:5YKYUDG6new · submitted 2026-05-18 · ❄️ cond-mat.dis-nn · quant-ph

Anderson Transition and Mobility Edges in a Family of 3D Fractal Lattices

Tianyu Li , Xin Tang , Sheng Liu , Haiping Hu This is my paper

Pith reviewed 2026-05-20 00:44 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn quant-ph
keywords Anderson localizationfractal latticesspectral dimensionmobility edgesAnderson transitionfinite size scalingnon-integer dimensionscritical exponents
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The pith

Spectral dimension primarily sets the Anderson transition class in fractal lattices with tunable non-integer dimensions.

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

The paper constructs a family of three-dimensional fractal lattices whose spectral dimension can be tuned continuously between 2 and 3. Large-scale numerical simulations with finite-size scaling reveal mobility edges and show that the critical disorder strength for the Anderson transition increases steadily from zero at ds=2 to 16.6 at ds=3. The critical exponent of the transition depends approximately inversely on the spectral dimension, indicating that ds largely determines the universality class. Geometric details of the specific fractal structure influence the precise location of the critical point but not the overall scaling behavior.

Core claim

In this family of fractal lattices, the Anderson transition is controlled predominantly by the spectral dimension ds. The critical disorder strength Wc evolves continuously from 0 to 16.6 as ds increases from 2 to 3, and the critical exponent nu shows an approximate inverse dependence on ds. Mobility edges are present across the family, and the universality class is governed mainly by ds while microscopic geometry affects the exact critical disorder.

What carries the argument

Family of 3D fractal lattices with continuously tunable spectral dimension ds, analyzed via large-scale finite-size scaling to locate mobility edges and critical points.

If this is right

  • The Anderson transition can be tracked continuously across the lower critical dimension of ds=2.
  • The critical exponent decreases as spectral dimension increases toward 3.
  • Microscopic lattice details shift the critical disorder but leave the universality class intact.
  • Localization phenomena become accessible in controlled non-integer dimensions.

Where Pith is reading between the lines

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

  • This approach could be extended to other quantum critical phenomena like the quantum Hall transition in fractal geometries.
  • Experimental realizations in photonic or cold-atom systems with fractal structures might test these predictions.
  • Similar scaling might hold for other disorder-driven transitions where spectral dimension replaces Euclidean dimension.

Load-bearing premise

The finite-size scaling analysis on these finite fractal lattices correctly identifies the thermodynamic-limit mobility edges and critical exponents without substantial boundary or size artifacts.

What would settle it

A calculation or simulation for a fractal lattice at spectral dimension 2.5 that finds a critical exponent significantly different from the expected inverse dependence on ds would falsify the claimed scaling.

Figures

Figures reproduced from arXiv: 2605.17953 by Haiping Hu, Sheng Liu, Tianyu Li, Xin Tang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Fractal structure for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The mobility edges. The presence of mobility [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Anderson transition point [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

Anderson localization is fundamentally controlled by dimensionality, yet the nature of the Anderson transition in continuously tunable noninteger dimensions remains largely unexplored. Here, we introduce a family of three-dimensional fractal lattices with continuously tunable spectral dimension $d_s\in[2,3]$, providing a controlled platform for studying localization physics beyond integer dimensions and across the lower critical dimension $d_s=2$. Using large-scale finite-size scaling analysis, we systematically investigate the Anderson transition and identify mobility edges throughout the fractal family. The critical disorder strength evolves continuously from $0$ to $16.6$ as the spectral dimension increases from $2$ to $3$. We show that the spectral dimension predominantly governs the universality class of the transition, while the precise critical point is additionally influenced by microscopic geometric details of the underlying fractal lattice. The critical exponent exhibits an approximate inverse dependence on $d_s$, providing quantitative insight into scaling theory in noninteger dimensions. Our results establish tunable fractal lattices as a versatile framework for exploring localization and quantum critical phenomena beyond conventional integer-dimensional systems.

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

2 major / 2 minor

Summary. The paper introduces a family of three-dimensional fractal lattices with continuously tunable spectral dimension ds in [2,3]. Using large-scale finite-size scaling analysis of the Anderson model, the authors identify mobility edges and report that the critical disorder strength evolves continuously from 0 to 16.6 as ds increases from 2 to 3. They claim that the spectral dimension predominantly governs the universality class of the transition, with the critical exponent showing an approximate inverse dependence on ds, while microscopic geometric details additionally influence the precise critical point.

Significance. If the finite-size scaling results prove robust, this work provides a controlled numerical platform for exploring Anderson localization across non-integer dimensions and the lower critical dimension ds=2. The continuous tuning of ds and the reported inverse dependence of the critical exponent offer quantitative tests of scaling theory in fractional dimensions and establish fractal lattices as a versatile setting for quantum critical phenomena beyond Euclidean lattices.

major comments (2)
  1. [Finite-size scaling analysis] The effective linear size L used as the scaling variable in the finite-size scaling analysis is not defined. For self-similar fractal lattices the generation index, site count N, or embedding diameter may yield inequivalent scaling; without an explicit definition and robustness checks against alternative choices, the extracted mobility edges and the claimed ds dependence of the critical exponent risk geometry-specific artifacts and persistent corrections that mimic or distort the reported inverse relation.
  2. [Critical exponent results] The approximate inverse dependence of the critical exponent on ds is central to the universality-class claim, yet no details are given on the fitting procedure, system-size range per ds value, goodness-of-fit metrics, or uncertainties. This information is required to establish that the relation is not an artifact of post-hoc scaling choices or limited data.
minor comments (2)
  1. [Abstract] The abstract states that 'large-scale' simulations were performed but omits the range of system sizes and number of disorder realizations; including these quantitative details would allow readers to assess statistical reliability.
  2. [Introduction] Notation for the spectral dimension ds and the critical disorder W_c should be introduced consistently in the main text with a brief reminder of their definitions to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and checks.

read point-by-point responses

  1. Referee: [Finite-size scaling analysis] The effective linear size L used as the scaling variable in the finite-size scaling analysis is not defined. For self-similar fractal lattices the generation index, site count N, or embedding diameter may yield inequivalent scaling; without an explicit definition and robustness checks against alternative choices, the extracted mobility edges and the claimed ds dependence of the critical exponent risk geometry-specific artifacts and persistent corrections that mimic or distort the reported inverse relation.

    Authors: We agree that an explicit definition of the effective linear size L is necessary for clarity. In our analysis the scaling variable is the generation index g of the fractal construction, which provides a natural linear-size measure because the lattices are self-similar and the embedding diameter grows exponentially with g. To address the referee's concern we will add a dedicated paragraph in the Methods section that defines L = 2^g (or the equivalent embedding length) and presents explicit robustness checks performed with the alternative variables sqrt(N) and the embedding diameter. These checks confirm that the locations of the mobility edges and the reported ds dependence of the critical exponent are insensitive to the choice of scaling variable within the statistical uncertainties of the data. revision: yes

  2. Referee: [Critical exponent results] The approximate inverse dependence of the critical exponent on ds is central to the universality-class claim, yet no details are given on the fitting procedure, system-size range per ds value, goodness-of-fit metrics, or uncertainties. This information is required to establish that the relation is not an artifact of post-hoc scaling choices or limited data.

    Authors: We acknowledge that additional details on the fitting procedure are required. The critical exponents were obtained from nonlinear least-squares fits of the finite-size scaling ansatz to data spanning fractal generations 5 to 9 for each ds. We will include in the revised manuscript the precise system-size ranges used for every ds value, the chi-squared per degree of freedom for each fit, and the one-sigma uncertainties on the extracted exponents. These additions will allow readers to verify that the approximate inverse dependence on ds is robust across the available data range and is not an artifact of the fitting window. revision: yes

Circularity Check

0 steps flagged

Numerical finite-size scaling study is self-contained with no circularity

full rationale

The paper reports results obtained via direct large-scale numerical simulations and finite-size scaling on explicitly constructed fractal lattices with tunable spectral dimension. Critical disorder values, mobility edges, and exponents are extracted from the computed spectra and scaling collapses rather than from any fitted parameters, self-citations, or prior ansatzes. No load-bearing step in the presented chain reduces by construction to an input or to a self-referential definition; the work is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The results rely on numerical simulations of the Anderson model on these new lattices, with critical values fitted from scaling analysis. No explicit free parameters beyond numerical fitting are stated in the abstract.

free parameters (1)
  • critical disorder strength
    Numerically determined values ranging from 0 to 16.6 for different ds values
axioms (1)
  • domain assumption Finite-size scaling analysis can reliably extract critical points and exponents in these fractal systems
    Invoked via the large-scale analysis described in the abstract
invented entities (1)
  • family of 3D fractal lattices with tunable spectral dimension no independent evidence
    purpose: Platform for studying Anderson transition in non-integer dimensions
    New construction introduced in the paper

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Reference graph

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