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Topological Phase Transition under Infinite Randomness

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Pith's one-line read A disordered Su–Schrieffer–Heeger chain changes between topological and trivial phases by tuning the ratio of hopping disorder widths, with the critical point at ratio one.

desk verdict A plausible new axis for tuning disorder-driven topology in 1D, but the SDRG protocol behind one pillar of evidence is nonstandard and needs validation. read the letter →

arxiv 2506.19913 v1 pith:75NOCMFM submitted 2025-06-24 cond-mat.dis-nn cond-mat.stat-mechcond-mat.str-el

Saikat Mondal , Adhip Agarwala This is my paper · ORCID
classification cond-mat.dis-nncond-mat.stat-mechcond-mat.str-el
keywords Su–Schrieffer–Heegerchainstrongdisorderinfiniterandomnessfixedpointtopologicalphasetransitionstrong-disorderrenormalizationgroupGriffithsactivateddynamicalscalingpolarization
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

This paper studies a Su–Schrieffer–Heeger chain whose intra-cell and inter-cell hoppings are independently random, drawn from normal distributions with means $\bar{v}$, $\bar{w}$ and widths $\sigma_v$, $\sigma_w$. It argues that in the strong-disorder regime the topological phase is governed entirely by the ratio $\lambda = \sigma_v/\sigma_w$: for large $\sigma_w$ the chain is topological when $\lambda < 1$ and trivial when $\lambda > 1$, independent of $\bar{w}/\bar{v}$ and of which phase the clean chain would occupy. Both phases turn out to be gapless, with localized zero modes in the bulk, rare regions, and a bulk gap scaling as $\epsilon \sim L^{-z}$ with $z$ diverging at $\lambda=1$. At the critical point $\lambda=1$ the gap obeys activated scaling $\epsilon \sim \exp(-a\sqrt{L})$ and the effective central charge is $c_{\rm eff} = \ln(2)$, identifying the transition as an infinite-randomness fixed point. The claim matters because it shows that competing fluctuation scales, not average parameters, can set the topology of a disordered system.

What carries the argument

The argument is carried by the strong-disorder renormalization-group decimation rules for a one-dimensional tight-binding chain. When the largest remaining bond is an intra-cell hopping $\Omega=v_r$, the two neighboring inter-cell hoppings combine into $\tilde w_{r-1,r+1}= -w_{r-1,r}w_{r,r+1}/\Omega$; when $\Omega=w_{r,r+1}$ is inter-cell, the two adjacent intra-cell hoppings combine into $\tilde v_{r,r+1}= -v_r v_{r+1}/\Omega$. After $L/2$ decimations the remaining hopping lists are doubled, reshuffled, and re-decimated, and the limit of $|w_{\rm eff}/v_{\rm eff}|$ (infinity or zero) labels the phase. The analytical complement is the probability $p_{\rm top} = \int_{-\infty}^{\infty} dw\, p_2(w) \int_{-|w|}^{|w|} dv\, p_1(v)$ that a unit cell satisfies $|w|>|v|$; it equals $1/2$ at $\lambda=1$ for all $\bar w/\bar v$, and the polarization computed from the many-body position operator confirms the same boundary.

What would settle it

Compute the disorder-averaged polarization and bulk-gap distribution for the same Hamiltonian by exact diagonalization at sizes up to $L=400$, without the list-doubling protocol: if the polarization crossing moves away from $\lambda=1$ as $\bar w/\bar v$ is varied, or if the gap distribution at $\lambda=1$ no longer collapses under $\ln(\epsilon)/\sqrt{L}$, the central claim is refuted.

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Extended reading notes

Core claim

The central discovery is that strong randomness replaces the clean SSH transition with an infinite-randomness fixed point whose location is set by the ratio of disorder widths, not by the mean hoppings. For large $\sigma_w$, the SDRG flow drives $|w_{\rm eff}/v_{\rm eff}|$ to infinity when $\lambda<1$ and to zero when $\lambda>1$, for every $\bar w/\bar v$; the disorder-averaged polarization $P$ is correspondingly close to $1$ or $0$. The two phases are gapless: the density of states is nonzero at $E\to 0$ and the bulk gap follows rare-region scaling $\epsilon\sim L^{-z}$, with $z$ diverging as $\lambda\to1$. At $\lambda=1$, the gap shows activated scaling $\epsilon\sim\exp(-a\sqrt{L})$ and the entanglement entropy grows as $(c_{\rm eff}/3)\ln l$ with $c_{\rm eff}\approx\ln(2)$, identifying the critical point as an infinite-randomness fixed point. The same critical behavior is found for transitions driven by $\sigma_w$ at fixed $\lambda$ and for uniform disorder distributions.

Load-bearing premise

The central assumption is that the SDRG list-doubling procedure — decimating $L/2$ bonds, then doubling and reshuffling the remaining hoppings and repeating — converges to the true infinite-randomness fixed point, so that the fate of $|w_{\rm eff}/v_{\rm eff}|$ reflects the actual phase and not an artifact of the protocol.

Editorial extensions

If this is right

  • In the strong-disorder regime both the topological and trivial phases are gapless, so topological order is carried by edge-localized zero modes sitting on a featureless bulk rather than by a finite gap.
  • The phase boundary at $\lambda=1$ is independent of $\bar w/\bar v$; a chain whose clean limit is trivial becomes topological for $\lambda<1$, and one whose clean limit is topological becomes trivial for $\lambda>1$.
  • The critical point exhibits activated dynamical scaling $\epsilon\sim\exp(-a\sqrt{L})$ and effective central charge $c_{\rm eff}=\ln(2)$, the signatures of an infinite-randomness fixed point.
  • Transitions can also be driven by $\sigma_w$ alone when the mean hoppings favor the opposite phase, and those critical points show the same infinite-randomness signatures.
  • The results are qualitatively unchanged when the normal disorder distributions are replaced by uniform distributions of the same width.

Reading between the lines

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

  • If the central claim is right, the clean-limit topology is irrelevant in the strong-disorder regime: a chain whose clean limit is deep in the trivial phase should become topological as soon as $\sigma_v<\sigma_w$. This is a sharp prediction one could test with exact diagonalization without relying on the SDRG doubling step.
  • The same ratio-of-fluctuations mechanism may organize other strongly disordered topological systems, such as disordered Chern insulators, where competing hopping dispersions would play the role of $\sigma_v$ and $\sigma_w$; this is a direction the paper names but does not develop.
  • A practical observable for cold-atom or mechanical SSH realizations is the disorder-averaged polarization as a function of $\lambda$ at fixed mean hoppings: a sharp crossing from $P\approx1$ to $P\approx0$ at $\lambda=1$ would confirm the phase diagram, and measuring the sample-to-sample variance of $P$ could expose the rare-region regime.
  • The power-law divergence of the dynamical exponent of the rare-region scaling, $z \sim |\lambda-1|^{-1}$ on one side, suggests a universal exponent that a future analytic strong-disorder RG treatment could be checked against.
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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 / 5 minor

Summary. This manuscript studies a one-dimensional Su-Schrieffer-Heeger (SSH) chain with intra-cell and inter-cell hoppings drawn from normal distributions with means \bar v, \bar w and standard deviations \sigma_v, \sigma_w. For large \sigma_w, the authors report that the phase is controlled only by \lambda = \sigma_v/\sigma_w: topological for \lambda<1 and trivial for \lambda>1, independent of the mean ratio \bar w/\bar v. They support this with three methods: (i) a numerical strong-disorder RG (SDRG) procedure that yields |w_eff/v_eff| tending to infinity or zero depending on \lambda; (ii) exact-diagonalization polarization P, which jumps from approximately 1 to approximately 0 at \lambda=1; and (iii) an analytic single-cell probability p_top. In both phases the bulk gap is reported to be gapless with Griffiths scaling \epsilon~L^{-z}; at \lambda=1 the authors find activated scaling \epsilon~exp(-a\sqrt L) and entanglement entropy S_l \approx (\ln 2)/3 \, \ln l, which they interpret as an infinite-randomness fixed point with central charge \ln 2.

Significance. The claimed phenomenon—a topological transition driven purely by the relative fluctuation scales of the two hoppings, with the quantum critical point at an infinite-randomness fixed point—is novel and would significantly extend the SDRG/IRFP paradigm to fermionic topological systems. The manuscript's strongest evidence is the direct polarization calculation, which is a standard, parameter-free numerical probe and shows a clear P=1 to P=0 transition at \lambda=1 for several \bar w/\bar v values. The activated-scaling collapse and the logarithmic entanglement growth at \lambda=1 are also convincing qualitative signatures. The p_top calculation is analytic but relies on the clean local criterion; the SDRG protocol contains a nonstandard step that requires justification. Overall, if the identified issues are addressed, the result would be of broad interest to the disordered-topological-phases community.

major comments (3)
  1. [SDRG methods, steps 4 and 5] The list-doubling and random reshuffling step is not a legitimate renormalization-group transformation. After L/2 decimations, the remaining coupling lists are doubled and independently shuffled, which destroys the spatial adjacency pattern that distinguishes intra-cell from inter-cell bonds in the SSH model; the renormalized chain is thus replaced by independent resampling from the empirical marginal distributions of v and w. No argument or numerical test is provided that the fixed point of this map coincides with the true infinite-randomness fixed point of the model. Since Fig. 2(c,d) and the |w_eff/v_eff| limit are used to identify the phases, this is a load-bearing issue; please validate the protocol (e.g., by comparing against exact RG decimation without doubling or against an alternative scheme) or appropriately weaken the SDRG claims.
  2. [Determination of QCP from probability distributions, Eq. (7)] The calculation of p_top builds the clean limit criterion |w|>|v| into the definition of a topological unit cell. Consequently, p_top=1/2 at \lambda=1 in the large-\sigma_w limit is a consequence of the assumed local criterion together with equal variances, not an independent derivation of the transition. This is a consistency check rather than a confirmation. Please either derive p_top from a criterion that emerges from the disordered system or explicitly present it as a heuristic consistency check.
  3. [Fig. 5(b) and SM Sec. III] The claim c_eff \approx \ln(2) is made without any uncertainty estimate, and the dynamical exponent z=2.1 in Fig. 4(b)/SM Fig. S2(a) is also quoted without error bars. Because the "irrational central charge" and the divergence of z at \lambda\to1 are central claims, please provide fit errors (e.g., bootstrap over realizations) and show the sensitivity of the extracted slopes to the chosen fit range.
minor comments (5)
  1. [Fig. 4(a)] Specify explicitly which \lambda values are shown and confirm that both curves are for the same \bar w/\bar v; the text says "both \lambda<1 and \lambda>1" but the caption does not list the curves.
  2. [General text] The phrase "for large \sigma_w" is used throughout but never quantified; please state the range of \sigma_w for which the \lambda=1 boundary and the p_top=1/2 condition hold.
  3. [General claims] The universality claim "regardless of the value of \bar w/\bar v" is demonstrated for \bar w/\bar v=0.5, 1.0, and 2.0; please state this explicitly and, if possible, add one more value or a scan.
  4. [Eq. (5)] The constant a is introduced as "a>0" but not defined; define it as a nonuniversal constant.
  5. [SM Eq. (S4)] Clarify the definition of \tilde p (density versus cumulative probability) and the range of \epsilon used for the linear fits from which z is extracted.

Circularity Check

1 steps flagged · score 4.0 of 10

The central λ=1 phase boundary is supported by independent Hamiltonian numerics, but the analytical p_top 'confirmation' is self-definitional: it builds the clean |w|>|v| criterion into the definition of p_top, so p_top=1/2 at λ=1 follows from the symmetry of equal-width normal distributions rather than from the disordered Hamiltonian.

  1. self definitional [Determination of QCP from probability distributions, Eq. (7) and Fig. 5(c)]
    "Here, a unit cell in the disordered chain can be topological if the associated w and v satisfy |w|>|v| and thus, p_top = ... (7). Performing numerical integration, we find that for large σ_w, p_top > 1/2 in topological phase (λ<1) and p_top < 1/2 in trivial phase (λ>1). Interestingly, p_top ∼ 1/2 at λ=1 for all w/v, thus confirming topological phase transition at λ=1 under strong disorder."

    Eq. (7) defines p_top as the probability that a single unit cell satisfies the clean SSH criterion |w|>|v|, evaluated with the bare normal distributions. At λ=1 (σ_v=σ_w) and large σ_w compared with the means, the joint distribution of (v,w) is symmetric under v↔w, so P(|w|>|v|)=1/2 as a mathematical identity. The calculation uses no information from the Hamiltonian (1), no spectrum, no inter-cell correlations, and no renormalization. Thus the statement that p_top=1/2 'confirms' a topological phase transition at λ=1 is not a prediction from the model: the QCP location is built into the chosen local topological criterion and the symmetry of the input distributions. The analytical p_top result therefore reduces, by construction, to its own definitional input.

full rationale

The paper's main phase diagram rests on three pillars: polarization (Fig. 3), the SDRG ratio |weff/veff| (Fig. 2), and the analytical p_top (Fig. 5(c)). The first two are direct numerical computations on the disordered SSH Hamiltonian and are independent of the p_top construction; the energy-gap scaling and entanglement-entropy results (Figs. 4-5) also provide independent evidence for the λ=1 IRFP. The p_top calculation, however, is circular in the sense defined above: it defines a 'topological unit cell' by the clean |w|>|v| criterion and then finds p_top=1/2 when the two normal distributions have equal width, which is a symmetry property of the input distributions, not a consequence of the Hamiltonian. The SDRG doubling-and-shuffling protocol (main text steps 4-5) is nonstandard and not shown to reproduce the true infinite-randomness fixed point, but that is a validity/correctness concern rather than a circularity: the protocol does not claim to derive the phase boundary from an independent theorem, and its ratio diagnostic is consistent with, rather than the source of, the numerical phase boundary. Overall, one supporting analytical result is self-defined while the central claim retains substantial independent numerical content, so the circularity is partial: score 4.

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

The central claim rests on imported SDRG decimation rules, IRFP scaling forms, the local p_top criterion, and the nonstandard list-doubling step. There are no fitted free parameters in the model, but the analytical determination of the phase boundary assumes the clean |w|>|v| criterion at the single-cell level. No new entities are invented.

assumptions (5)
  • domain assumption SDRG decimation rules for the disordered tight-binding chain (Eqs. (2) and (3)) are valid for the SSH model and preserve the topological content.
    The paper imports the decimation rules from Javan Mard et al. (2014) and uses |weff/veff| as the phase indicator; no derivation from the SSH Hamiltonian is given.
  • domain assumption The local criterion p_top=P(|w|>|v|) for a unit cell being topological determines the global phase in the strong-disorder limit.
    Eq. (7) defines a topological unit cell by the clean condition |w|>|v|; the analytical phase boundary follows from this assumed criterion rather than from a derivation.
  • domain assumption The activated scaling form epsilon~exp(-a sqrt L) and the Griffiths form epsilon~L^{-z} apply at the IRFP and in the gapless phases respectively.
    Imported from Fisher (1994) and Young-Rieger (1996); the paper uses these functional forms to fit the gap distributions.
  • ad hoc to paper The list-doubling and reshuffling step in the SDRG procedure is equivalent to continuing the exact RG flow.
    Step 4 of the SDRG protocol doubles the remaining hopping lists and shuffles them; this is not a standard SDRG operation and no justification is provided.
  • standard math Entanglement entropy of the free-fermion ground state obeys S_l=(c_eff/3) ln(l)+c_0 at criticality.
    Vidal/Calabrese-Cardy scaling used in Eq. (6) to extract c_eff.

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Pith. "Pith review of Topological Phase Transition under Infinite Randomness." pith.science (2026). https://pith.science/paper/75NOCMFM

@misc{pith2026250619913,
  author       = {Pith},
  title        = {Pith review of: Topological Phase Transition under Infinite Randomness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/75NOCMFM}},
  note         = {Machine review of arXiv:2506.19913}
}
read the original abstract

In clean and weakly disordered systems, topological and trivial phases having a finite bulk energy gap can transit to each other via a quantum critical point. In presence of strong disorder, both the nature of the phases and the associated criticality can fundamentally change. Here we investigate topological properties of a strongly disordered fermionic chain where the bond couplings are drawn from normal probability distributions which are defined by characteristic standard deviations. Using numerical strong disorder renormalization group methods along with analytical techniques, we show that the competition between fluctuation scales renders both the trivial and topological phases gapless with Griffiths like rare regions. Moreover, the transition between these phases is solely governed by the fluctuation scales, rather than the means, rendering the critical behavior to be determined by an infinite randomness fixed point with an irrational central charge. Our work points to a host of novel topological phases and atypical topological phase transitions which can be realized in systems under strong disorder.

Figures

Figures reproduced from arXiv: 2506.19913 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]
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Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

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    From the slope of linear fit using Eq. (S4), we obtain z= 2.1. (b)zexponent as a function ofλ, where diverging behavior ofzatλ→1 is indicated. (c) ln(z) as a function of ln(1−λ) forλ<1 where slope of linear fit is−1.0461±0.0021. (d) ln(z) as a function of ln(λ−1) forλ >1 where...

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Reviewed August 15, 2026 · model on record in the stance chip above.