Quantum Critical Dynamics Induced by Topological Zero Modes

Ilia Komissarov; Raquel Queiroz; Tobias Holder

arxiv: 2502.12233 · v2 · submitted 2025-02-17 · ❄️ cond-mat.mes-hall · cond-mat.dis-nn

Quantum Critical Dynamics Induced by Topological Zero Modes

Ilia Komissarov , Tobias Holder , Raquel Queiroz This is my paper

Pith reviewed 2026-05-23 02:42 UTC · model grok-4.3

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

keywords topological zero modesac conductivitySSH modelchiral disorderdelocalization transitionstretched-exponential decayquantum critical dynamicshybridization

0 comments

The pith

Hybridized pairs of topological zero modes produce logarithmic scaling of ac conductivity at the critical point in disordered SSH chains.

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

The paper studies ac transport in the Su-Schrieffer-Heeger chain with chiral disorder close to the topological delocalization transition. It establishes that hybridized pairs of topological domain wall zero modes lead to ac conductivity scaling as the logarithm of frequency exactly at criticality. This scaling originates from the stretched-exponential decay of the zero-mode wavefunctions in space. A reader would care because it provides a direct link between topological features and measurable electrical response in a quantum critical regime.

Core claim

At the topological delocalization transition in the SSH chain with chiral disorder, the ac conductivity scales as σ(ω) ∼ log ω because hybridized pairs of topological domain wall zero modes form, and this form follows directly from the stretched-exponential spatial decay ψ(x) ∼ e^{-s √x} of the zero-mode wavefunctions. The derivation combines real-space renormalization group analysis with qualitative hybridization arguments. Away from criticality the scaling changes to σ(ω) ∼ ω^{2 δ} log² ω.

What carries the argument

Hybridized pairs of topological domain wall zero modes enabled by the stretched-exponential decay of their wavefunctions at the critical point.

If this is right

The ac conductivity follows σ(ω) ∼ log ω precisely at the critical point.
Away from criticality the conductivity acquires an additional power-law factor and becomes σ(ω) ∼ ω^{2 δ} log² ω.
The stretched-exponential wavefunction decay determines the logarithmic form of the conductivity scaling.
Real-space renormalization group analysis combined with hybridization arguments suffices to obtain the scaling without further details on disorder.

Where Pith is reading between the lines

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

This scaling might appear in other one-dimensional systems with topological domain walls and disorder.
Measuring frequency-dependent conductivity in mesoscopic SSH samples could confirm the logarithmic dependence.
The mechanism suggests that zero-mode hybridization could affect other dynamical quantities such as noise spectra.
Similar anomalous scalings may occur in related models with different types of disorder.

Load-bearing premise

The real-space renormalization group analysis and qualitative hybridization arguments are enough to derive the conductivity scaling from the wavefunction decay without requiring quantitative details on disorder strength or higher-order interactions.

What would settle it

A direct measurement showing that ac conductivity at the critical point does not scale logarithmically with frequency, or that zero-mode wavefunctions do not exhibit stretched-exponential decay, would disprove the central claim.

Figures

Figures reproduced from arXiv: 2502.12233 by Ilia Komissarov, Raquel Queiroz, Tobias Holder.

**Figure 2.** Figure 2: a)Dynamical conductivity in the SSH chain with chiral box disorder at half filling for different values of the parameter δ, which encodes the distance to the topological phase transition. The numerical data is fitted to σ(ω) = c1ω 2δfit log2 (c2/ω), and δfit = 0.78, 1.01, 1.23 indicate a good agreement with the actual values of δ, confirming our expression for ac conductivity in this regime (2). The crosso… view at source ↗

**Figure 3.** Figure 3: Dipole transition amplitude between states separated [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

read the original abstract

We investigate the low-frequency ac transport in the Su-Schrieffer-Heeger (SSH) chain with chiral disorder near the topological delocalization transition. Our key finding is that the formation of hybridized pairs of topological domain wall zero modes leads to the anomalous logarithmic scaling of the ac conductivity $\sigma(\omega) \sim \log \omega$ at criticality, and $\sigma(\omega) \sim \omega^{2 \delta} \log ^2 \omega$ away from it. Using the combination of real-space renormalization group analysis and qualitative hybridization arguments, we demonstrate that the form of the scaling of ac conductivity at criticality stems directly from the stretched-exponential ($\psi(x) \sim e^{-s \sqrt{x}}~\,$) spatial decay of zero-mode wavefunctions at the critical point.

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

The paper claims log ω ac conductivity at the SSH delocalization transition from hybridized zero modes with stretched-exp decay, but the step from wavefunction overlap to that exact scaling looks qualitative rather than fully derived.

read the letter

The punchline is that hybridized topological zero-mode pairs produce σ(ω) ∼ log ω at criticality in the disordered SSH chain, with the form tied to the stretched-exponential wavefunction decay obtained from real-space RG. Off criticality they get a modified power-law times log squared. This scaling is presented as the main new result and is not just a restatement of standard critical transport forms. The RG part that produces the exp(-s √x) decay is the concrete technical step they contribute. The hybridization argument then connects that decay to the conductivity via pair formation and level spacing. That link is the part that stands out as potentially useful for AC measurements near the transition. The paper does a reasonable job motivating why the zero modes matter for low-frequency transport and why the decay is not the usual exponential. The soft spot is exactly the one in the stress-test note. The abstract asserts the log form “stems directly” from the decay, but without an explicit sum or integral over pair separations, current matrix elements, and the resulting density of states, the mapping stays at the level of a plausible argument rather than a controlled derivation. If the full text only invokes qualitative hybridization without showing that integral or equivalent, the central claim rests on an unverified step. That is a moderate rather than fatal issue, but it is the load-bearing one. The work is aimed at specialists in mesoscopic topological systems and disordered critical points. Someone already thinking about AC response in similar models could extract the proposed scaling and test it numerically or against data. It is coherent enough on its own terms to merit referee time, even if the hybridization step needs tightening. Recommendation: send to peer review.

Referee Report

2 major / 0 minor

Summary. The manuscript studies low-frequency AC transport in the Su-Schrieffer-Heeger chain with chiral disorder near the topological delocalization transition. It claims that hybridized pairs of topological domain-wall zero modes produce the anomalous scaling σ(ω) ∼ log ω exactly at criticality (and σ(ω) ∼ ω^{2δ} log² ω away from it), with this form arising directly from the stretched-exponential spatial decay ψ(x) ∼ e^{-s √x} of the zero-mode wave functions, as obtained from a combination of real-space renormalization-group analysis and qualitative hybridization arguments.

Significance. If the explicit mapping from the stretched-exponential overlaps to the conductivity scaling can be supplied, the result would furnish a parameter-free derivation of logarithmic AC conductivity at a topological critical point, linking zero-mode hybridization to transport without additional fitting parameters or disorder-strength tuning.

major comments (2)

[Abstract] Abstract: the central claim that the log ω scaling 'stems directly' from the stretched-exponential decay rests on the unshown step that qualitative hybridization of pairs with overlap ∼ exp(−s √x) produces σ(ω) ∼ log ω. No explicit integral or sum (e.g., ∫ dx P(x) |⟨j⟩|² δ(ω − Δ(x)) with Δ(x) = exp(−s √x)) is indicated, leaving the pair density of states, current-matrix-element scaling, and cutoff on multi-pair interactions unspecified; this step is load-bearing for the asserted conductivity form.
[Abstract] Abstract: the real-space RG analysis is stated to renormalize the disorder distribution, yet the abstract provides no demonstration that this automatically supplies the required pair statistics or matrix elements needed to convert the wave-function decay into the conductivity scaling; the sufficiency of the RG-plus-qualitative-hybridization combination therefore remains to be verified quantitatively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for identifying points where the abstract could more explicitly connect the stretched-exponential decay to the conductivity scaling. We address each major comment below and will revise the abstract accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the log ω scaling 'stems directly' from the stretched-exponential decay rests on the unshown step that qualitative hybridization of pairs with overlap ∼ exp(−s √x) produces σ(ω) ∼ log ω. No explicit integral or sum (e.g., ∫ dx P(x) |⟨j⟩|² δ(ω − Δ(x)) with Δ(x) = exp(−s √x)) is indicated, leaving the pair density of states, current-matrix-element scaling, and cutoff on multi-pair interactions unspecified; this step is load-bearing for the asserted conductivity form.

Authors: We agree the abstract is too concise on this mapping. The main text derives the log ω form at criticality by combining the RG-obtained stretched-exponential overlaps with the resulting pair density of states and current matrix elements; the qualitative hybridization argument is made quantitative via the change of variables from separation x to energy gap Δ(x) ∼ exp(−s √x), which directly produces the logarithmic scaling without additional parameters. We will revise the abstract to include a one-sentence outline of this integral transformation and the resulting pair statistics. revision: yes
Referee: [Abstract] Abstract: the real-space RG analysis is stated to renormalize the disorder distribution, yet the abstract provides no demonstration that this automatically supplies the required pair statistics or matrix elements needed to convert the wave-function decay into the conductivity scaling; the sufficiency of the RG-plus-qualitative-hybridization combination therefore remains to be verified quantitatively.

Authors: The RG flow in the manuscript renormalizes the disorder to yield the precise stretched-exponential wave-function decay ψ(x) ∼ e^{-s √x} at criticality; the pair statistics then follow directly from the spatial distribution of domain walls under this decay, with matrix elements set by the overlap. This connection is shown quantitatively in the body. We will add a clarifying clause to the abstract stating that the RG-renormalized disorder distribution supplies the pair density and matrix-element scaling. revision: yes

Circularity Check

0 steps flagged

No circularity: scaling derived from independent RG wavefunction decay via qualitative hybridization

full rationale

The paper's central claim rests on real-space RG analysis producing the stretched-exponential zero-mode decay ψ(x)∼e^{-s√x} at criticality, followed by a separate qualitative hybridization argument linking pair overlaps to σ(ω)∼logω. No equation reduces the conductivity scaling to a fitted parameter or to the decay form by algebraic identity; the RG step is presented as an independent computation of the wavefunction envelope, and the hybridization step is explicitly labeled qualitative rather than a quantitative integral over matrix elements. No self-citations appear in the provided abstract or derivation outline, and the result is not obtained by renaming a known empirical pattern or by smuggling an ansatz. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that zero-mode wavefunctions exhibit stretched-exponential decay at criticality and that RG plus hybridization arguments capture the transport without further parameters.

axioms (2)

domain assumption Zero-mode wavefunctions decay as ψ(x) ∼ e^{-s √x} at the critical point
Invoked in the abstract as the direct source of the log scaling.
domain assumption Real-space RG combined with qualitative hybridization arguments suffices to obtain the conductivity scaling
Stated as the method used to demonstrate the result.

pith-pipeline@v0.9.0 · 5665 in / 1292 out tokens · 49697 ms · 2026-05-23T02:42:33.274565+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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the form of the scaling of ac conductivity at criticality stems directly from the stretched-exponential (ψ(x)∼e^{-s√x}) spatial decay of zero-mode wavefunctions at the critical point
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

real-space renormalization group analysis and qualitative hybridization arguments

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

39 extracted references · 39 canonical work pages · 2 internal anchors

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[1] [1]

Evers and A

F. Evers and A. D. Mirlin, Anderson transitions, Reviews of Modern Physics 80, 1355 (2008)

work page 2008

[2] [2]

Huckestein, Scaling theory of the integer quantum hall effect, Rev

B. Huckestein, Scaling theory of the integer quantum hall effect, Rev. Mod. Phys. 67, 357 (1995)

work page 1995

[3] [3]

KRAMER, T

B. KRAMER, T. OHTSUKI, and S. KETTEMANN, Random network models and quantum phase transitions in two dimensions, Physics Reports 417, 211–342 (2005)

work page 2005

[4] [4]

A. M. M. Pruisken, On localization in the theory of the quantized hall effect: A two-dimensional realization of the θ-vacuum, Nuclear Physics B 235, 277 (1984)

work page 1984

[5] [5]

Das Sarma, Localization, metal-insulator transitions, and quantum hall effect (2007) pp

S. Das Sarma, Localization, metal-insulator transitions, and quantum hall effect (2007) pp. 1–36

work page 2007

[6] [6]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42, 1698 (1979)

work page 1979

[7] [7]

Theodorou and M

G. Theodorou and M. H. Cohen, Extended states in a one- demensional system with off-diagonal disorder, Physical Review B 13, 4597 (1976)

work page 1976

[8] [8]

Altland, D

A. Altland, D. Bagrets, and A. Kamenev, Topology versus Anderson localization: Nonperturbative solutions in one dimension, Physical Review B 91, 085429 (2015)

work page 2015

[9] [9]

S. N. Evangelou, Spectral singularities and localisation in one- and quasi-one-dimensional systems with off-diagonal disorder, Journal of Physics C: Solid State Physics 19, 4291 (1986)

work page 1986

[10] [10]

M. Inui, S. A. Trugman, and E. Abrahams, Unusual properties of midband states in systems with off-diagonal disorder, Phys. Rev. B 49, 3190 (1994)

work page 1994

[11] [11]

C. M. Soukoulis and E. N. Economou, Off-diagonal disorder in one-dimensional systems, Physical Review B 24, 5698 (1981)

work page 1981

[12] [12]

Zhang and A

H. Zhang and A. Kamenev, Anatomy of topological An- derson transitions, Physical Review B 108, 224201 (2023), arXiv:2301.04565 [cond-mat]

work page arXiv 2023

[13] [13]

Balents and M

L. Balents and M. P. A. Fisher, Delocalization transition via supersymmetry in one dimension, Physical Review B 56, 12970 (1997)

work page 1997

[14] [14]

Gogolin, Electron localization in quasi-one-dimensional organic conductors, Physics Reports 166, 269 (1988)

A. Gogolin, Electron localization in quasi-one-dimensional organic conductors, Physics Reports 166, 269 (1988)

work page 1988

[15] [15]

E. J. Meier, A. F. An, A. Dauphin, M. Maffei, P. Massig- nan, T. L. Hughes, and B. Gadway, Observation of the topological anderson insulator in disordered atomic wires, Science 362, 929 (2018), epub 2018 Oct 11, 30309909

work page 2018

[16] [16]

D. S. Fisher, Random transverse field ising spin chains, Phys. Rev. Lett. 69, 534 (1992)

work page 1992

[17] [17]

D. S. Fisher, Critical behavior of random transverse-field ising spin chains, Phys. Rev. B 51, 6411 (1995)

work page 1995

[18] [18]

Bouchaud, A

J. Bouchaud, A. Comtet, A. Georges, and P. Le Dous- sal, Classical diffusion of a particle in a one-dimensional random force field, Annals of Physics 201, 285 (1990)

work page 1990

[19] [19]

Sinai Diffusion at Quasi-1D Topological Phase Transitions

D. Bagrets, A. Altland, and A. Kamenev, Sinai Diffu- sion at Quasi-1D Topological Phase Transitions, Physical Review Letters 117, 196801 (2016), arXiv:1605.01657 [cond-mat]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[20] [20]

N. F. Mott and E. A. Davis, Electronic processes in non-crystalline materials , 2nd ed., International series of monographs on physics (Clarendon Press, Oxford, 2012)

work page 2012

[21] [21]

Abrikosov and I

A. Abrikosov and I. Ryzhkin, Conductivity of quasi-one- dimensional metal systems, Advances in Physics 27, 147 (1978)

work page 1978

[22] [22]

V. L. Berezinski˘ ı, Kinetics of a quantum particle in a one- dimensional random potential, in World Scientific Series in 20th Century Physics , Vol. 11 (WORLD SCIENTIFIC,

work page

[23] [23]

T. P. Eggarter and R. Riedinger, Singular behavior of tight-binding chains with off-diagonal disorder, Phys. Rev. B 18, 569 (1978)

work page 1978

[24] [24]

Resta, Drude weight and superconducting weight, Jour- nal of Physics: Condensed Matter 30, 414001 (2018)

R. Resta, Drude weight and superconducting weight, Jour- nal of Physics: Condensed Matter 30, 414001 (2018)

work page 2018

[25] [25]

See supplemental material, where we explain the main ingredients of the real-space renormalization group in one dimension, and give instructions on how to apply the scaling analysis to other models

work page

[26] [26]

Del Maestro, B

A. Del Maestro, B. Rosenow, J. A. Hoyos, and T. Vojta, Dynamical conductivity at the dirty superconductor-metal quantum phase transition, Physical Review Letters 105, 10.1103/physrevlett.105.145702 (2010)

work page doi:10.1103/physrevlett.105.145702 2010

[27] [27]

Gopalakrishnan, M

S. Gopalakrishnan, M. M¨ uller, V. Khemani, M. Knap, E. Demler, and D. A. Huse, Low-frequency conductivity in many-body localized systems, Physical Review B 92, 10.1103/physrevb.92.104202 (2015)

work page doi:10.1103/physrevb.92.104202 2015

[28] [28]

Motrunich, K

O. Motrunich, K. Damle, and D. A. Huse, Dynamics and transport in random quantum systems governed by strong-randomness fixed points, Physical Review B 63, 10.1103/physrevb.63.134424 (2001)

work page doi:10.1103/physrevb.63.134424 2001

[29] [29]

J. K. Asb´ oth, L. Oroszl´ any, and A. P´ alyi,A Short Course on Topological Insulators (Springer International Publish- ing, 2016)

work page 2016

[30] [30]

Phases and phase transitions in disordered quantum systems

T. Vojta, Phases and phase transitions in disordered quantum systems (2013) pp. 188–247, arXiv:1301.7746 [cond-mat]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[31] [31]

B. M. McCoy and T. T. Wu, Theory of a two-dimensional ising model with random impurities. i. thermodynamics, Phys. Rev. 176, 631 (1968)

work page 1968

[32] [32]

R. B. Griffiths, Nonanalytic behavior above the critical point in a random ising ferromagnet, Phys. Rev. Lett. 23, 17 (1969)

work page 1969

[33] [33]

Since the matrix element of the position operator for asym- metric in energy transitions is regular in ω, we estimate from (6) that σasym(ω)≲ω ∫ ω 0 dE ν(E)ν(E+ ̵hω)∼ω4δ

work page

[34] [34]

Queiroz, R

R. Queiroz, R. Ilan, Z. Song, B. A. Bernevig, and A. Stern, Ring states in topological materials, arXiv preprint arXiv:2406.03529 (2024)

work page arXiv 2024

[35] [35]

Javan Mard, J

H. Javan Mard, J. A. Hoyos, E. Miranda, and V. Do- brosavljevi´ c, Strong-disorder renormalization-group study of the one-dimensional tight-binding model, Physical Re- view B 90, 10.1103/physrevb.90.125141 (2014)

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