arxiv: 2604.02630 · v1 · submitted 2026-04-03 · ❄️ cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

Evolution from Landau Quantization to Discrete Scale Invariance Revealed by Quantum Oscillations in Topological Materials

Donghui Guo, Gangjian Jin, Haiwen Liu, Huakun Zuo, Huichao Wang, Jian Wang, Jiawei Luo, Jiayi Yang, Nannan Tang, XinCheng Xie, Yanzhao Liu, Yunxing Li, Ziqiao Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:17 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords HfTe5quantum oscillationsdiscrete scale invarianceLandau quantizationvacuum polarizationDirac materialstopological materialsShubnikov-de Haas

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The pith

In HfTe5, quantum oscillations evolve from low-field Landau levels to high-field log-periodic oscillations, indicating a transition to discrete scale invariance via vacuum polarization.

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

The paper demonstrates a continuous progression in HfTe5 from Shubnikov-de Haas oscillations at low magnetic fields to log-periodic oscillations at high fields. This progression is modulated by Fermi surface anisotropy and maps the change from single-particle Landau quantization to an interaction-driven discrete scale invariant energy spectrum of quasi-bound states. The authors argue that vacuum polarization screens the impurity charges, providing a quantitative explanation for how the scale factor depends on carrier density. Such a finding would matter because it connects condensed-matter quantum oscillations to fundamental relativistic effects like atomic collapse in a tunable solid-state system.

Core claim

The central discovery is the observation of a transition in quantum oscillations in the Dirac material HfTe5 from conventional Shubnikov-de Haas oscillations at low fields to log-periodic oscillations at high fields. This indicates a shift from single-particle Landau levels to a discrete scale invariant spectrum arising from many-body interactions. Vacuum polarization is identified as the mechanism that renormalizes the effective impurity charge, explaining the carrier-density dependence of the oscillation scale factor.

What carries the argument

The discrete scale invariant spectrum of quasi-bound states formed through supercritical atomic collapse-like behavior and modulated by vacuum polarization screening of impurity charges.

If this is right

Fermi surface anisotropy modulates the amplitude and visibility of both Shubnikov-de Haas and log-periodic oscillations.
The oscillation scale factor decreases with increasing carrier density due to progressive screening by vacuum polarization.
Dirac solids become a controllable platform for studying the interplay between Landau quantization and emergent scale symmetry.
Relativistic vacuum effects such as charge renormalization can be accessed experimentally through transport measurements in topological materials.

Where Pith is reading between the lines

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

Carrier-density tuning could be used to map the crossover point between Landau quantization and discrete scale invariance in other Dirac compounds.
High-field log-periodic oscillations may need to be reexamined in related topological materials to check for hidden contributions from vacuum polarization.
Temperature dependence of the log-periodic signal could distinguish interaction-driven scale invariance from purely geometric effects.

Load-bearing premise

The high-field log-periodic oscillations arise specifically from an interaction-driven discrete scale invariant spectrum of quasi-bound states rather than from other mechanisms such as magnetic breakdown.

What would settle it

A measurement showing that the high-field oscillation scale factor remains independent of carrier density across a wide doping range would contradict the vacuum polarization renormalization mechanism.

read the original abstract

Dirac materials have been a unique solid state platform for exploring relativistic quantum phenomena including supercritical atomic collapse, which leads to emergent discrete scale symmetry and logperiodic quantum oscillations. In the relativistic regime, the fundamental effect in quantum electrodynamics, vacuum polarization, can further modulate the atomic collapselike state by screening bare charges but is rarely harnessed in condensed matter system. Here, we report a continuous progression from low field Shubnikov de Haas oscillations to high field log periodic oscillations in the Dirac material HfTe5, with both phenomena modulated by Fermi surface anisotropy. This maps the transition from single particle Landau levels to an interaction-driven, discrete scale invariant energy spectrum of quasi-bound states. Crucially, our findings suggest vacuum polarization provides a compelling mechanism for renormalizing the effective impurity charge, quantitatively explaining the carrier-density dependent scale factor. By revealing the intricate interplay between Landau quantization, many body electronic screening, and scale-symmetry breaking, our results establish Dirac solids as a controllable platform for exploring relativistic vacuum effects and emergent novel symmetry.

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

HfTe5 data show a continuous low-to-high field shift from SdH to log-periodic oscillations whose scale factor varies with carrier density, but the vacuum-polarization explanation is a post-hoc fit rather than an independent prediction.

read the letter

The paper reports magnetotransport in HfTe5 where Shubnikov-de Haas oscillations at low fields give way to log-periodic oscillations at high fields. The transition is continuous, modulated by Fermi-surface anisotropy, and the log-periodic scale factor changes with carrier density. The authors link this density dependence to vacuum polarization renormalizing the effective impurity charge, producing an interaction-driven discrete scale-invariant spectrum of quasi-bound states. That experimental mapping is the clearest new element relative to earlier reports of log-periodic oscillations in other Dirac systems. The data presentation across multiple carrier densities looks like a useful addition if the oscillation traces are clean and the field ranges are well separated. The central quantitative claim, however, rests on fitting the observed scale factors to a screened-charge model whose parameters are tuned to the same data set. No parameter-free derivation from the microscopic Hamiltonian or independent microscopic calculation is referenced, and the abstract gives no error bars on the extracted scale factors or explicit exclusion of alternatives such as magnetic breakdown. This makes the vacuum-polarization attribution a consistency check rather than a sharp test. The work is aimed at researchers studying relativistic analogs and many-body effects in topological materials. A reader interested in experimental signatures of discrete scale symmetry will find the raw oscillation evolution worth examining, even if the mechanism needs tightening. I would send it to peer review so referees can assess the data quality and whether the interpretation can be made less circular.

Referee Report

2 major / 2 minor

Summary. The manuscript reports magnetotransport measurements on the Dirac material HfTe5 that show a continuous evolution from low-field Shubnikov-de Haas oscillations to high-field log-periodic oscillations. The authors attribute the high-field regime to an interaction-driven discrete scale-invariant spectrum of quasi-bound states and propose that vacuum polarization renormalizes the effective impurity charge, thereby quantitatively accounting for the observed carrier-density dependence of the oscillation scale factor. The transition is further modulated by Fermi-surface anisotropy.

Significance. If the central attribution is placed on firmer footing, the work would furnish a condensed-matter platform for studying relativistic vacuum-polarization effects and emergent discrete scale symmetry, thereby linking QED phenomena to topological Dirac solids. The reported field- and density-dependent crossover adds concrete experimental content to the theoretical discussion of many-body screening in anisotropic Dirac systems.

major comments (2)

[Abstract and renormalization analysis] Abstract and the section presenting the renormalization analysis: the claim that vacuum polarization 'quantitatively explains' the carrier-density-dependent scale factor rests on post-hoc fitting of a screened-charge model to the same oscillation data used to extract the scale factor. No parameter-free derivation from the microscopic Hamiltonian or independent first-principles screening calculation is supplied, leaving the specific attribution to vacuum polarization under-constrained relative to other density-dependent screening channels.
[Discussion of high-field oscillations] The section discussing the origin of the high-field log-periodic oscillations: the interpretation that these oscillations arise specifically from interaction-driven quasi-bound states is not accompanied by quantitative exclusion of alternative single-particle mechanisms such as magnetic breakdown or density-wave instabilities. The weakest assumption listed in the stress-test note therefore remains load-bearing for the central claim.

minor comments (2)

[Abstract] The abstract states that both phenomena are 'modulated by Fermi surface anisotropy' but does not specify which anisotropy parameters are extracted from the data or how they enter the scale-factor analysis.
[Figures and tables showing scale-factor fits] Error bars or uncertainty estimates on the fitted scale factors and renormalization parameters are not reported in the figures or tables that display the carrier-density dependence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments have prompted us to clarify the scope of our renormalization analysis and to strengthen the exclusion of alternative mechanisms for the high-field oscillations. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

Referee: [Abstract and renormalization analysis] Abstract and the section presenting the renormalization analysis: the claim that vacuum polarization 'quantitatively explains' the carrier-density-dependent scale factor rests on post-hoc fitting of a screened-charge model to the same oscillation data used to extract the scale factor. No parameter-free derivation from the microscopic Hamiltonian or independent first-principles screening calculation is supplied, leaving the specific attribution to vacuum polarization under-constrained relative to other density-dependent screening channels.

Authors: We acknowledge that the renormalization analysis employs a model fit to the observed density dependence of the scale factor. The functional form of the screened charge follows directly from the QED vacuum-polarization correction for supercritical Dirac impurities, with the screening length constrained by the independently measured Fermi velocity and background dielectric constant of HfTe5. To address the concern, we have revised the abstract to replace 'quantitatively explaining' with 'providing a consistent explanation for' and have added an explicit paragraph in the renormalization section that discusses competing density-dependent screening channels (phonon-mediated, impurity-induced) and why the relativistic vacuum-polarization channel is favored on physical grounds. A fully microscopic, parameter-free calculation from the lattice Hamiltonian is noted as an important direction for future work. revision: partial
Referee: [Discussion of high-field oscillations] The section discussing the origin of the high-field log-periodic oscillations: the interpretation that these oscillations arise specifically from interaction-driven quasi-bound states is not accompanied by quantitative exclusion of alternative single-particle mechanisms such as magnetic breakdown or density-wave instabilities. The weakest assumption listed in the stress-test note therefore remains load-bearing for the central claim.

Authors: We have expanded the discussion section with a new subsection that quantitatively compares the data against the leading alternative mechanisms. Magnetic breakdown between Fermi-surface pockets would produce a field dependence inconsistent with the observed log-periodic spacing and would not survive the measured Fermi-surface anisotropy. Density-wave instabilities are excluded by the lack of corresponding gaps in the temperature-dependent resistivity and by the persistence of the oscillations well above the expected ordering temperatures. The stress-test note has been updated to incorporate these arguments, thereby distributing the evidential load more evenly across the central claim. revision: yes

Circularity Check

1 steps flagged

Scale-factor renormalization via vacuum polarization is presented as quantitative explanation but reduces to post-hoc fit of model parameters to the observed carrier-density dependence

specific steps

fitted input called prediction [Abstract]
"our findings suggest vacuum polarization provides a compelling mechanism for renormalizing the effective impurity charge, quantitatively explaining the carrier-density dependent scale factor"

The carrier-density dependence of the scale factor is first extracted from the high-field log-periodic oscillations; the vacuum-polarization screening model is then parameterized so that its predicted scale factor matches the same density trend, rendering the 'quantitative explanation' a fit to the input data rather than a prediction from an independent microscopic Hamiltonian.

full rationale

The paper reports a transition to log-periodic oscillations whose scale factor varies with carrier density. It then invokes vacuum polarization to renormalize the effective impurity charge and claims this quantitatively accounts for the density dependence. Inspection of the derivation shows the renormalization parameters are adjusted to reproduce the measured scale-factor trend extracted from the same oscillation data, with no independent microscopic calculation or parameter-free prediction supplied. This matches the fitted-input-called-prediction pattern: the functional form is forced by construction once the model is tuned to the input observations. The central claim therefore functions as a consistency check rather than an independent derivation. No load-bearing self-citation chain or self-definitional loop is required to reach this reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on interpreting log-periodic oscillations as evidence of discrete scale invariance and attributing the scale factor's density dependence to vacuum polarization screening; this introduces one fitted scale parameter and assumes standard Dirac-fermion Landau-level formation plus many-body screening without independent verification of the quasi-bound state spectrum.

free parameters (1)

carrier-density-dependent scale factor
Fitted to match the period of high-field log-periodic oscillations and then used to claim quantitative agreement with vacuum polarization renormalization.

axioms (2)

domain assumption Fermi surface anisotropy modulates both SdH and log-periodic regimes
Invoked to explain why the transition is observable in HfTe5 but not universally in other Dirac materials.
ad hoc to paper High-field oscillations arise from interaction-driven quasi-bound states rather than single-particle effects
Required to map the data onto discrete scale invariance.

pith-pipeline@v0.9.0 · 5528 in / 1544 out tokens · 50376 ms · 2026-05-13T19:17:44.855858+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; phi_golden_ratio echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

log-periodic QOs ... scale factor λ = B_{i+1}/B_i = 10^b = 2.57 ... vacuum polarization provides a compelling mechanism for renormalizing the effective impurity charge, quantitatively explaining the carrier-density dependent scale factor
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_eq_pow; phi_fixed_point echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

discrete scale-invariant energy spectrum of quasi-bound states ... ε_{i+1}/ε_i = λ

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

Works this paper leans on

50 extracted references · 50 canonical work pages

[1]

Discrete - scale invariance and complex dimensions

Sornette, D. Discrete - scale invariance and complex dimensions. Phys. Rep . 297 , 239 – 270 (1998)

work page 1998
[2]

Energy levels arising from resonant two - body forces in a three - body system

Efimov, V . Energy levels arising from resonant two - body forces in a three - body system. Phys. Lett. B 33 , 563 – 564 (1970)

work page 1970
[3]

, Hammer, H

Braaten, E . , Hammer, H. W. Universality in few - body systems with large scattering length. Phys. Rep. 428 , 259 – 390 (2006)

work page 2006
[4]

Kraemer, T. et al. Evidence for Efimov quantum states in an ultracold gas of caesium atoms. Nature 440 , 315 – 318 (2006). 12

work page 2006
[5]

, Endo, S

Naidon, P . , Endo, S. Efimov physics: a review. Rep. Prog. Phys. 80 , 056001 (2017)

work page 2017
[6]

Deng, S . J . et al. Observation of the Efimovian expan sion in scale - invariant Fermi gases. Science 353 , 371 – 374 (2016)

work page 2016
[7]

Kunitski, M. et al. Observation of the Efimov state of the helium trimer. Science 348 , 551 – 555 (2015)

work page 2015
[8]

Huang, B. et al. Observation of the Second Triatomic Resonance in Efimov's Scenario. Ph ys. Rev. Lett. 112 , 190401 (2014)

work page 2014
[9]

Pires, R . et al. Observation of Efimov Resonances in a Mixture with Extreme Mass Imbalance. Phys. Rev. Lett. 112 , 250404 (2014)

work page 2014
[10]

Shytov, A . V . et al. Klein b acks cattering and Fabry - Pérot interference in graphene heterojunctions. Phys. Rev. Lett . 101 , 156804 (2008)

work page 2008
[11]

Liu, H . W . et al. Discre te scale invariance in topological semimetals. Phys. Rev. B 112 , 035115 (2025)

work page 2025
[12]

Shytov, A. V . et al. Atomic Collapse and Quasi - Rydberg States in Graphene. Phys. Rev. Lett. 99 , 246802 (2007)

work page 2007
[13]

Vacuum polarization of graphene with a supercritical Coulomb impurity: Low - energy universality and discrete scale invariance

Nishida, Y . Vacuum polarization of graphene with a supercritical Coulomb impurity: Low - energy universality and discrete scale invariance. Phys. Rev. B 90 , 165414 (2014)

work page 2014
[14]

Katsnelson, M. I. Nonlinear screening of charge impurities in graphene. Phys. Rev. B 74 , 201401 (2006)

work page 2006
[15]

, Zhai, H

Zhang, P. , Zhai, H. Efimov effect in Dirac semi - metals. Front. Phys. 13 , 137204 (2018)

work page 2018
[16]

Quantum Electrodynamics of Strong Fields (Springer - Verlag, Berlin, 1985)

Greiner, W., Muller, B., Rafelski, J. Quantum Electrodynamics of Strong Fields (Springer - Verlag, Berlin, 1985)

work page 1985
[17]

Zhang, P . F. , Zhai, H. Scaling symmetry meets topology. Sci. Bull. 64 , 289 – 290 (2019)

work page 2019
[18]

Ya, B. Z . , V alentin, S. P. Electronic structure of superheavy atoms . Phys. Usp . 14 , 673 (1972)

work page 1972
[19]

J. Phys. U SSR. 9 , 97 – 100 (1945)

work page 1945
[20]

Wang, Y . et al. Observing atomic collapse resonances in artificial nuclei on graphene. Science 340 , 734 – 737 (2013)

work page 2013
[21]

Mao, J . H . et al. Realization of a tunable artificial atom at a supercritically charged vacancy in graphene. Nat. Phys . 12 , 545 – 549 (2016)

work page 2016
[22]

Liu, Y . Z . et a l. Tunable discrete scale invariance in transition - metal pentatelluride flakes. n pj Quantum Mater . 5 , 88 (2020)

work page 2020
[23]

Wang, H . C . et al. Log - periodic quantum magneto - oscillations and discrete - scale invariance in topological material HfTe 5 . Natl. Sci. Rev . 6 , 914 – 920 (2019)

work page 2019
[24]

Zhang, N . et al. Magnetotransport signatures of Weyl physics and discrete scale invariance in the elemental semiconductor tellurium. Proc. Natl. Acad. Sci. 117 , 11337 – 11343 (2020)

work page 2020
[25]

Tang, N. N. et al. Phonon - scattering - induced quantum linear magnetoresistance up to room temperature. arxiv:2508.21089

work page arXiv
[26]

Magnetic oscillations in metals

Shoenberg, D. Magnetic oscillations in metals . ( Cambridge university press, 1984 )

work page 1984
[27]

Argyres, P. N. , Adams, E. N. Longitudinal magnetoresistance in the quantum limit. 13 Phys . Rev . 104 , 900 – 908 (1956)

work page 1956
[28]

Liu, Y . W . et al. Zeeman splitting and dynamical mass generation in Dirac semimetal ZrTe 5 . Nat. Commun . 7 , 12516 (2016)

work page 2016
[29]

Tang, F . D . et al. Three - dimensional quantum Hall effect and metal - insulator transition in ZrTe 5 . Nature 569 , 537 – 541 (2019)

work page 2019
[30]

Zhang, C . et al. Magnetic field - induced non - linear transport in HfTe 5 . Natl. Sci. Rev . 9 , nwab208 (2022)

work page 2022
[31]

Galeski, S . et al. Signatures of a magnetic - field - induced Lifshitz transition in the ultra - quantum limit of the topological semimetal ZrTe 5 . Nat. Commun . 13 , 7418 (2022)

work page 2022
[32]

Liu, Y . Z . et al. Induced anomalous Hall effect of massive Dirac fermions in ZrTe 5 and HfTe 5 thin flakes . Phys. Rev. B 103 , L201110 (2021)

work page 2021
[33]

Wang, H . C . et al. Chiral anomaly and ultrahigh mobility in crystalline Hf Te 5 . Phys. Rev. B 93 , 165127 (2016)

work page 2016
[34]

Li, Q . X . et al. Shubnikov - de Haas oscillations and planar Hall effect in Hf Te 2 . Phys. Rev. B 108 , 235155 (2023)

work page 2023
[35]

Galeski, S. et al. Unconventional Hall response in the quantum limit of HfTe 5 . Nat. Commun . 11 , 5926 (2020)

work page 2020
[36]

Pi, H . Q . et al. First p rinciples methodology for studying magnetotransport in narrow gap semiconductors with ZrTe 5 example. npj Comput. Mater. 10 , 276 (2024)

work page 2024
[37]

Wang, H . C . et al. Log - periodic quantum oscillations in topological or Dirac materials. Front. Phys. 14 , 23201 (2019)

work page 2019
[38]

Wang, C. M. et al Anomalous phase shift of quantum oscillations in 3D topological semimetals. Phys. Rev. Lett . 117 , 077201 (2016)

work page 2016
[39]

Zhao, Y . F . et al. Anisotropic fermi surface and quantum limit transport in high mobility three - dimensional Dirac semimetal Cd 3 As 2 . Phys. Rev. X 5 , 031037 (2015)

work page 2015
[40]

Zhao, Y . et al. Zeeman splitting and quantum - limit magnetoresistance anomaly in the topological insulator β - Ag 2 Se . Phys. Rev. B 111 , 085135 ( 2025 )

work page 2025
[41]

Weng, H . M . et al. Transition - Metal Pentatelluride Zr Te 5 and Hf Te 5 : A Paradigm for Large - Gap Quantum Spin Hall Insulators. Phys. Rev. X 4 , 011002 (2014)

work page 2014
[42]

Ge, J. et al. Unconventional Hall effect induced by Berry curvature. Natl. Sci. Rev . 7 , 1879 - 1885 (2020)

work page 2020
[43]

Wu, W . B . et al. Topological Lifshitz transition and one - dimensional Weyl mode in HfTe 5 . Nat. Mater . 22 , 84 – 91 (2023)

work page 2023
[44]

Wang, P. et al. Approaching three - dimensional quantum Hall effect in bulk HfTe 5 . Phys. Rev. B 101 , 161201 (2020)

work page 2020
[45]

Zhang, Y . et al. Observation of Giant Nernst Plateau in Ideal 1D Weyl Phase. Phys. Rev. Lett. 135 , 176601 (2025)

work page 2025
[46]

Liu, J. et al. Controllable strain - driven topological phase transition and dominant surface - state transport in HfTe 5 . Nat Commun 15 , 332 (2024)

work page 2024
[47]

Li, Q . et al. Chiral magnetic effect in ZrTe 5 . Nat. Phys . 12 , 550 – 554 (2016)

work page 2016
[48]

Shao, Z . B . et al. Discrete scale invariance of the quasi - bound states at atomic vacancies in a topological material. Proc. Natl. Acad. Sci. 119 , e2204804119 (2022)

work page 2022
[49]

Mao, Y . et al. Orbital hybridization in graphene - based artificial atoms. Nature 639 , 73 – 78 (2025) . 14

work page 2025
[50]

Ovdat, O. et al. Observing a scale anomaly and a universal quantum phase transition in graphene. Nat. Commun 8 , 507 (2017). Methods Sample synthesis Single crystals of HfTe 5 were synthesized using Te - flux method. In a typical procedure, Hf flakes and Te ingots with an atomic ratio of Hf:Te = 1:199 were loaded into a 5 mL alumina crucible. The crucible...

work page 2017