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REVIEW 3 major objections 4 minor 57 references

Anderson localization induced by structural disorder

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Pith's one-line read Structural disorder alone can drive the Anderson localization transition in three dimensions.

desk verdict Two clean 3D models show that lattice geometry alone can drive an Anderson transition; the universality-class claim is credible but needs explicit error bars and a corrections-to-scaling check before it is airtight. read the letter →

arxiv 2411.10247 v2 pith:HES2OSRD submitted 2024-11-15 cond-mat.dis-nn cond-mat.stat-mechquant-ph

Sourav Bhattacharjee , Piotr Sierant , Marek Dudyński , Jan Wehr , Jakub Zakrzewski , Maciej Lewenstein This is my paper · ORCID
classification cond-mat.dis-nncond-mat.stat-mechquant-ph
keywords Andersonlocalizationstructuraldisordergapratiofinite-sizescalinguniversalityclassrandomspanningtreelevelstatisticsmobilityedge
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Most discussions of Anderson localization assume the randomness lives in an on-site potential, so a particle scatters off random energies as it hops. This paper argues the disorder can instead be purely structural: the lattice itself is missing sites or connections, yet the same physics emerges. In two different three-dimensional tight-binding models, smoothly increasing the structural disorder drives level statistics from the delocalized random-matrix value to the localized Poisson value, with a sharp transition at a critical strength. Finite-size scaling gives exponents $\nu \approx 1.57$ and $\nu \approx 1.61$, consistent with the standard 3D Anderson transition, so the authors conclude that geometric irregularities alone belong to the same universality class as random-potential disorder.

What carries the argument

The machinery is a pairing of disordered geometric substrates with the spectral gap-ratio probe. The honeycomb substrate is generated by a geometric cluster Monte Carlo algorithm sampling the equilibrium of an Ising-like attraction between occupied sites at fixed density $p$; only the largest cluster of occupied sites is kept. The link substrate is a random spanning tree of the cubic lattice, generated by Wilson's algorithm, with extra nearest-neighbour links added independently with probability $P_{\mathrm{link}}$. On each substrate the observable is the average gap ratio $r$ of consecutive eigenvalues, whose value distinguishes GOE (delocalized) from Poisson (localized) statistics; finite-size scaling of $r$ as $f[(p-p_c)L^{1/\nu}]$ yields the critical point and exponent. The recursive Green's function method supplies an independent localization-length probe for the link model.

What would settle it

Repeat the gap-ratio scaling analysis at system sizes $L\gtrsim 120$ with comparable statistical sampling and test whether all data collapse onto a single curve $r=f[(p-p_c)L^{1/\nu}]$ with no correction term. If the best-fit $p_c$ drifts beyond its error bar, or the same-quality collapse requires a second scaling variable, the claimed universality class is not established.

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Extended reading notes

Core claim

The central claim is that Anderson localization can be induced solely by structural disorder, i.e., by irregularities of the lattice geometry, without any on-site disordered potential. The paper establishes this by analyzing level statistics in two models. In the honeycomb model, sites are removed according to a geometric-cluster Monte Carlo equilibrium at fixed occupation $p$, and the tight-binding Hamiltonian is defined on the largest cluster; the average gap ratio crosses from the Poisson value $r\approx 0.386$ to the GOE value $r\approx 0.531$ at $p_c\approx 0.36$ with $\nu\approx 1.57$. In the link model, a random spanning tree of the cubic lattice is supplemented by extra links with probability $P_{\mathrm{link}}$; the transition occurs at $P_{\mathrm{link},c}\approx 0.024$ with $\nu\approx 1.61$, and is corroborated by recursive Green's function localization-length scaling and by fractal-dimension analysis. Because the extracted exponents agree with the 3D Anderson model, the paper concludes that the transition belongs to the same universality class, despite the long-range correlations introduced by restricting the analysis to the largest cluster.

Load-bearing premise

The load-bearing premise is that the one-parameter finite-size scaling ansatz holds with negligible corrections over the studied system sizes, so the extracted exponents are the true asymptotic ones despite the long-range-correlated structural disorder.

Editorial extensions

If this is right

  • Irregular connectivity alone is enough to localize single particles in three dimensions; random potential energy is not required.
  • The two models, honeycomb site removal and tree-plus-links, become new members of the 3D Anderson universality class, with critical exponents $\nu\approx 1.57$ and $\nu\approx 1.61$.
  • A mobility edge exists between localized and delocalized states in the energy spectrum, so structural disorder produces energy-dependent localization.
  • The link model shows that a very small density of added loops, $P_{\mathrm{link}}\approx 2.4\%$, is enough to delocalize a random spanning tree, quantifying how connectivity controls transport.
  • Materials whose structural disorder comes from irregular platelet or graphene networks could exhibit Anderson localization even if they are chemically clean.
  • The same gap-ratio and localization-length machinery can be applied to other irregular 3D lattices to test whether the mechanism is generic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the universality-class claim survives, it implies that the long-range correlations induced by the largest-cluster restriction do not change the critical behavior, which is not what generic correlated-disorder arguments would predict; a renormalization-group check of this point would be valuable.
  • The honeycomb construction is essentially a correlated quantum-percolation model, so comparing its critical $p_c$ with uncorrelated quantum percolation thresholds could isolate the role of geometric correlations.
  • A testable extension would be to add weak interactions to these structurally disordered lattices and look for signatures of many-body localization without on-site disorder.
  • Applying the same probes to 3D hyperbolic or curved lattices could show whether the 'geometric disorder alone' mechanism persists when the background geometry itself carries curvature.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper claims that Anderson localization in three-dimensional tight-binding systems can be induced solely by structural disorder, i.e., by irregularities of the lattice geometry, without any on-site disordered potential. Two models are analyzed: a honeycomb lattice in which sites are removed according to a classical statistical model at fixed temperature T=1 and only the largest connected cluster is kept, and a link model built from a random spanning tree of the cubic lattice with additional links added with probability P_link. Using exact diagonalization of systems up to about 10^6 sites, the authors compute the averaged gap ratio and observe a localization-delocalization transition as the structural disorder strength is tuned (p_c ≈ 0.36 and P_link,c ≈ 0.024). Finite-size scaling of the gap ratio gives critical exponents ν ≈ 1.57 and ν ≈ 1.61, which are quoted as consistent with the standard three-dimensional Anderson universality class. The supplementary material adds fractal-dimension analysis for the link model (ν ≈ 1.53) and a recursive Green's function study of the localization length.

Significance. If the universality-class claim holds, the paper identifies a genuinely new family of structurally disordered lattice models in which destructive interference alone produces a metal-insulator transition in three dimensions, without onsite disorder. The numerical evidence is genuinely multi-pronged: gap-ratio scaling in two independent models, a crossing point that is stable with system size, a separate fractal-dimension analysis, and a qualitative recursive Green's function check. The analysis is not circular: the critical parameters and exponents are extracted from standard finite-size scaling and then compared with literature values, not fitted to them. The main weakness is that the universality-class identification rests on a one-parameter scaling collapse over a limited range of system sizes, with no demonstrated control of corrections to scaling and no error bars quoted in the main text. This makes the central claim plausible but not yet fully established; the requested revisions are therefore focused on error reporting and scaling-robustness checks.

major comments (3)
  1. [Fig. 2 and Supplement 'Estimation of the critical exponents'] The universality-class claim rests on the values ν ≈ 1.57 (honeycomb) and ν ≈ 1.61 (link), yet the main text quotes these without error bars and the collapse is a one-parameter scaling form r = f[(p − p_c)L^{1/ν}] with no corrections-to-scaling term. The accessible L range (30–80 for both models) is limited, and the largest-cluster restriction produces long-range correlated disorder, so leading irrelevant scaling fields could plausibly be significant. Please report the error intervals (the Supplement's W < 1.3W* contours are not quoted in the main text), and test stability by adding a correction-to-scaling term or by dropping the smallest L from the fit; if the exponents drift, the claim of belonging to the standard 3D Anderson universality class should be softened.
  2. [The link model (main text) and Fig. 4 caption] The text states that for P_link < P_link,c the ratio λ_M/L_M vanishes with increasing L_M, implying localized eigenstates, while for P_link > P_link,c the localization length increases with L_M, implying delocalization. The caption of Fig. 4 states the opposite for P_link > P_link,c: 'the localization length remains finite and approaches ξ as L_M → ∞ which is characteristic of a localized phase'. Please reconcile this inconsistency; as written, the recursive Green's function confirmation supports both readings and needs to be unambiguous.
  3. [Supplement, 'Analysis of fractal dimension', Eq. (8a)] Eq. (8a) reads D_q = (S_q(L+ΔL) − S_q(L)) / (3 log(L/(L+ΔL))). Since the denominator is negative for ΔL > 0 while a delocalized state has S_q increasing with L, this formula gives negative D_q for delocalized states, contrary to the stated D_q = 1. If the implemented denominator is instead 3 log((L+ΔL)/L), please correct the equation; if not, the D_2 values in Fig. 10 and the derived ν ≈ 1.53 need to be re-examined.
minor comments (4)
  1. [Honeycomb model (main text)] In the honeycomb-model section, the sentence 'we plot a phase diagram in the p vs T plane in Fig. 11' should refer to Fig. 3, which is the phase diagram in the main text; Fig. 11 in the Supplement appears to be a duplicate of the same phase diagram.
  2. [Abstract] The phrase 'two models with distinct types of lattice regularities' should read 'lattice irregularities', matching the terminology used throughout the rest of the paper.
  3. [The honeycomb model, gap-ratio paragraph] The sentence 'we find ⟨r⟩ → r_PS and ⟨r⟩ → r_GOE, respectively' should specify which side of p_c corresponds to which limit, since the two limits are associated with different phases.
  4. [Supplement, 'Recursive Green's function method'] The supplement cites 'these lecture notes' without a full reference; please provide the complete citation for the numerical code example.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: critical exponents are fitted from gap-ratio scaling and compared with external 3D Anderson benchmarks; self-citations are only to supplementary numerical details.

full rationale

The paper's derivation chain is: construct a single-particle tight-binding Hamiltonian on an irregular lattice; compute the average gap ratio r; assume the standard one-parameter finite-size scaling form r ~ f[(p - p_c)L^{1/ν}] (main text, Fig. 2; Supplement Eq. 4); fit p_c and ν by minimizing the cost function W; and compare the fitted ν ≈ 1.57 (honeycomb) and ν ≈ 1.61 (link) with the critical exponent of the 3D Anderson model from Refs. [39,40]. The scaling form is a generic hypothesis for continuous transitions, not an input that fixes the Anderson universality class; the fit does not constrain ν to the Anderson value, so the comparison is an external benchmark rather than a self-consistency check. The RGF analysis of the link model uses a standard localization-length scaling collapse and independently shows two branches around P_link,c ≈ 0.024; it does not import a result from the authors' prior work. Self-citations appear only as pointers to the authors' own Supplementary Material (cluster MC, error estimates, RGF equations) and to a quantum-algorithm reference for participation entropies; none is load-bearing for the universality-class conclusion. No equation defines its target in terms of itself, no fitted parameter is renamed as a prediction, and no uniqueness or ansatz is imported from self-citation. The statistical caveats (single-parameter collapse without explicit corrections to scaling, main-text exponents quoted without error bars) affect the strength of the universality-class inference but are not circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard level-statistics diagnostic, the single-parameter scaling hypothesis, the largest-cluster modeling choice, and the external benchmark value of the 3D Anderson exponent. These are reasonable but are assumptions the reader must accept; the paper does not derive them.

free parameters (5)
  • p_c (honeycomb model critical occupation) = ~0.36 at E=2.0
    Critical fraction of occupied sites at the localization transition, extracted from the finite-size scaling collapse of the gap ratio. This is a fitted parameter central to the claim.
  • ν (honeycomb model critical exponent) = ~1.57
    Correlation length exponent extracted from the same scaling collapse; comparison with the 3D Anderson exponent is the basis of the universality class claim.
  • Plink,c (link model critical link probability) = ~0.024
    Critical probability of adding links to the random spanning tree, extracted from gap ratio and fractal dimension scaling.
  • ν (link model critical exponent) = 1.61 (gap ratio), 1.53 (fractal dimension)
    Correlation length exponent for the link model, from two independent analyses; the main text quotes 1.61.
  • T (classical model temperature) = 1.0
    Hand-chosen temperature above the classical order-disorder transition (Tc ≈ 0.85); the localization transition is present for a range of T, so this is not a tuned critical value.
assumptions (4)
  • domain assumption The gap ratio across neighboring energy levels distinguishes localized (Poisson, r ≈ 0.386) from delocalized (GOE, r ≈ 0.531) phases.
    This diagnostic is standard in the Anderson localization literature and is used to locate the transition in both models. It is not proven in the paper but is widely accepted.
  • domain assumption The single-parameter scaling form r = f[(p - p_c)L^{1/ν}] holds with negligible corrections to scaling.
    Used to extract p_c and ν; the validity of this assumption over the studied system sizes is not independently demonstrated.
  • domain assumption Restricting the analysis to the largest connected cluster C of occupied sites yields a valid model of the structurally disordered lattice and does not alter the universality class.
    The Hamiltonian is defined on C; the cluster boundary is part of the structural disorder. This is the paper's modeling choice for the honeycomb model.
  • standard math The standard 3D Anderson transition has the critical exponent ν ≈ 1.57 (from Slevin and Ohtsuki 2018).
    Used as the external benchmark for the universality class claim; the paper relies on this known result.

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Cite this review

Pith. "Pith review of Anderson localization induced by structural disorder." pith.science (2026). https://pith.science/paper/HES2OSRD

@misc{pith2026241110247,
  author       = {Pith},
  title        = {Pith review of: Anderson localization induced by structural disorder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HES2OSRD}},
  note         = {Machine review of arXiv:2411.10247}
}
read the original abstract

We examine the onset of Anderson localization in three-dimensional systems with structural disorder in the form of lattice irregularities and in the absence of any on-site disordered potential. Analyzing two models with distinct types of lattice regularities, we show that the Anderson localization transition occurs when the strength of the structural disorder is smoothly increased. Performing finite-size scaling analysis of the results, we show that the transition belongs to the same universality class as regular Anderson localization induced by onsite disorder. Our work identifies a new class of structurally disordered lattice models in which destructive interference of matter waves may inhibit transport and lead to a transition between metallic and localized phases.

Figures

Figures reproduced from arXiv: 2411.10247 by the authors.

Figure 1
Figure 1. Destructive interference arising from structural [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (a) The averaged gap ratio ⟨r⟩ captures the AL transition on C for different system sizes L in the honeycomb model as a function of p, where p is the fraction of sites remaining in the lattice. The inset shows a critical scaling collapse with exponent ν = 1.57. (b) A similar AL transition is also observed for the link model in which case Plink represents the probability of added links in the random spanning tree. 0… view at source ↗
Figure 3
Figure 3. Phase diagram in the p vs T plane showing localized and delocalized phases of the honeycomb model. lar to the transition observed in [33]. For completeness, we plot a phase diagram in the p vs T plane in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Finite size scaling analysis of the AL transition [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Connected correlation function averaged over disorder realizations at (a) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: (a) Ratio of the largest cluster size NC to the lattice size NL as as function of p at T = 1.0. The blue dashed line marks pc (for E = 0.5) at which the Anderson localization transition is observed, and the red dash-dotted line marks pcs below which the correlations be…
Figure 7
Figure 7. Figure 7: Density of states g(E) corresponding to the single-particle Hamiltonian Hsp of the honeycomb model discussed in the main text for L = 30, T = 1.0 and (a) p = 0.275 and (b) p = 0.475. The spectrum is symmetric about E = 0 and has a singularity at E = 0. The maxima of th…
Figure 8
Figure 8. Figure 8: Gap ratio averaged over 100 nearest energy eigenstates around (a) [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: Cost function defined in Eq. 4 for estimating the critical exponent and the critical disorder strength reported in the main text in the case of the (a) honeycomb model and (b) link model [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: The fractal dimension D2 calculated using Eq. 8 for the link model. An AL transition is identified at Plink,c ≈ 0.024 with ν ≈ 1.53, thus corroborating the result obtained from analysing the gap ratio in the main text. The D2 values have been averaged over the same di…
Figure 11
Figure 11. Figure 11: Phase diagram in the p vs T plane showing localized and delocalized phases of the honeycomb model. enable to assess whether the eigenstate |ϕn⟩ is localized or delocalized, and, hence, played a vital role in the studies on Anderson transitions [1, 2, 47]. In the follo…

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    Repeat 4 until S is empty. The above algorithm guarantees convergence to the equilibrium distribution. As the occupations are always swapped in the above algorithm, the constraintp = const is always satisfied. In our case, we have randomly chosen reflection along cartesianx, y...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.