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arxiv: 2606.23582 · v1 · pith:23AKBRMTnew · submitted 2026-06-22 · ❄️ cond-mat.supr-con · cond-mat.str-el

Influence of Harris disorder on quantum-critical superconductivity

Serhii Kryhin , Peter Lunts , Aavishkar A. Patel , Subir Sachdev , Pavel A. Nosov This is my paper

Pith reviewed 2026-06-26 06:14 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.str-el
keywords Harris disorderquantum critical superconductivitylocalized bosonic modespairing instabilityUsadel equationsuperconducting puddlesgap inhomogeneitypower-law tails
0
0 comments X

The pith

Harris disorder in quantum-critical metals localizes bosonic modes and produces power-law tails in local pairing scales.

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

The paper investigates the effects of symmetry-preserving disorder on superconductivity near a quantum critical point in the Hertz-Millis framework. Disorder couples to the bosonic order-parameter fluctuations and creates a finite density of localized low-energy bosonic modes. These modes mediate Cooper pairing through a spatially random vertex and an effective random potential for pairs, solved via a real-space linearized Usadel equation. The result is two temperature regimes of instability and a power-law distribution of local pairing scales rather than the stretched-exponential form found in ordinary disordered BCS superconductors. This mechanism supplies a route to mesoscopic gap inhomogeneity and superconducting puddles while fermions remain extended.

Core claim

Localized overdamped bosonic modes generated by Harris disorder produce both a random pairing vertex and a random potential for Cooper pairs. Numerical solutions of the real-space Usadel equation, together with a self-consistent Born approximation and Lifshitz-tail analysis, identify two regimes: at higher temperatures pairing nucleates in compact puddles centered on the strongest localized modes; at lower temperatures an extended pairing eigenstate appears whose transition scale and spatial structure are shaped by mesoscopic returns to favorable regions. The distribution of local pairing scales therefore acquires a power-law tail.

What carries the argument

Localized low-energy overdamped bosonic modes from Harris disorder, which enter the real-space linearized Usadel equation for the two-particle propagator and generate a spatially random pairing vertex plus an effective random potential for Cooper pairs.

If this is right

  • Pairing first appears in compact superconducting puddles nucleated on the strongest localized bosonic modes.
  • An extended pairing eigenstate forms only at lower temperatures and remains spatially inhomogeneous due to mesoscopic correlation effects.
  • The histogram of local pairing scales follows a power-law tail at low energies.
  • Broad gap inhomogeneity and superconducting puddles arise in quantum-critical metals without requiring strong fermionic localization.

Where Pith is reading between the lines

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

  • The same localized-mode mechanism could account for the inhomogeneous gaps seen in STM studies of the cuprates near their quantum critical point.
  • Analogous power-law tails might appear in other disordered quantum-critical systems such as heavy-fermion or iron-based superconductors.
  • A direct comparison of gap histograms between quantum-critical and conventional disordered superconductors could test whether the tail shape distinguishes the two classes.

Load-bearing premise

Fermionic states remain extended and continue to supply Landau damping of the bosonic modes even after the bosonic modes themselves localize.

What would settle it

STM measurements on a quantum-critical metal showing stretched-exponential rather than power-law tails in the histogram of local gap values would falsify the predicted distribution.

Figures

Figures reproduced from arXiv: 2606.23582 by Aavishkar A. Patel, Pavel A. Nosov, Peter Lunts, Serhii Kryhin, Subir Sachdev.

Figure 1
Figure 1. Figure 1: FIG. 1: The properties of the effective bosonic theory of the normal state are displayed. (a) A schematic phase [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Properties of the superconducting state induced by the overdamped localized bosonic modes described in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The first line is a definition for [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Analysis of STM data on Bi [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Overlap of the most localized superconducting puddles Φ [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The figure shows the behavior of [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The boson eigenmode density of states, across the phase diagram. [PITH_FULL_IMAGE:figures/full_fig_p036_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Comparison of the nFL and BCS form of the transition temperature as a function of coupling strength, [PITH_FULL_IMAGE:figures/full_fig_p037_9.png] view at source ↗
read the original abstract

In the Hertz theory, a quantum critical metal is described by the coupling of a Fermi surface to fluctuations of a Landau-damped bosonic field $\phi$, which may represent either an order parameter or a Higgs field for a transition without symmetry breaking. Scattering from $\phi$ produces non-Fermi-liquid behavior in the normal state, while the same fluctuations mediate enhanced Cooper pairing. By the Harris criterion, symmetry-preserving disorder couples most strongly to the coefficient of the $\phi^2$ term, locally tuning the system toward or away from criticality. This random mass (``Harris disorder'') leads to a finite density of localized low-energy $\phi$ modes even when the fermionic states remain extended and continue to provide Landau damping. We study the onset of pairing mediated by these localized overdamped bosonic modes. Starting from a real-space linearized Usadel equation for the two-particle propagator, we show that the localized bosonic wave functions generate both a spatially random pairing vertex and an effective random potential for Cooper pairs. Numerical solution, a self-consistent Born approximation, and Lifshitz-tail analysis reveal two regimes of the pairing instability. At high temperatures, pairing nucleates compact superconducting puddles on the most localized bosonic modes. At lower temperatures, an extended pairing eigenstate appears, but its transition scale and spatial structure remain strongly affected by mesoscopic correlation effects and enhanced probability of returns to favorable regions of the localized bosonic glue. The resulting distribution of local pairing scales has a power-law tail, in contrast to the stretched-exponential tails of disordered BCS superconductors. This mechanism provides a route to broad gap inhomogeneity and superconducting puddles in quantum-critical metals, and offers an interpretation of STM measurements of the cuprates.

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 manuscript studies Harris disorder (random mass term for the bosonic field) in a quantum-critical metal described by Hertz theory. It claims that this disorder produces a finite density of localized low-energy bosonic modes even while fermionic states remain extended and supply Landau damping. Using a real-space linearized Usadel equation for the two-particle propagator, together with self-consistent Born approximation and Lifshitz-tail analysis, the authors identify two pairing regimes: high-T nucleation of compact superconducting puddles on the most localized bosonic modes, and a lower-T extended pairing eigenstate whose scale and structure are influenced by mesoscopic return probabilities. The resulting distribution of local pairing scales exhibits a power-law tail, in contrast to the stretched-exponential tails found in disordered BCS superconductors. This is proposed as a mechanism for broad gap inhomogeneity and puddles in quantum-critical metals, with possible relevance to cuprate STM data.

Significance. If the central results hold, the work supplies a concrete, disorder-driven route to power-law distributed pairing scales and mesoscopic inhomogeneity that is specific to the quantum-critical setting and does not require the fermions themselves to localize. The contrast with BCS tails is a clear, falsifiable distinction. The use of the standard Usadel framework and Lifshitz-tail methods is a methodological strength that keeps the calculation within established tools of the field.

major comments (2)
  1. [Abstract / model section] Abstract and model construction: the statement that 'fermionic states remain extended and continue to provide Landau damping' is introduced as an input rather than derived or verified within the disordered model. Because the entire subsequent analysis (localized bosonic modes, random pairing vertex, and power-law tail) rests on this premise, its consistency with Harris disorder should be demonstrated, for example by an explicit check that the fermionic self-energy remains non-localizing at the relevant energies.
  2. [Numerical / Lifshitz-tail analysis] Section on numerical solution and Lifshitz-tail analysis: the claim of a power-law tail in the distribution of local pairing scales is load-bearing for the central contrast with BCS superconductors. The manuscript should specify the fitting range, the extracted exponent, and how the tail is distinguished from possible stretched-exponential or other forms; without these quantitative details the distinction remains qualitative.
minor comments (2)
  1. [Introduction / model] Notation for the random mass term and the bosonic propagator should be introduced with an explicit equation early in the text to aid readability.
  2. [Figures] Figure captions for the spatial maps of pairing eigenstates should state the temperature or energy scale at which each map is evaluated.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the positive assessment of the work's significance and for the constructive comments. We respond to each major point below.

read point-by-point responses

  1. Referee: [Abstract / model section] Abstract and model construction: the statement that 'fermionic states remain extended and continue to provide Landau damping' is introduced as an input rather than derived or verified within the disordered model. Because the entire subsequent analysis (localized bosonic modes, random pairing vertex, and power-law tail) rests on this premise, its consistency with Harris disorder should be demonstrated, for example by an explicit check that the fermionic self-energy remains non-localizing at the relevant energies.

    Authors: The assumption follows from the Harris criterion, under which disorder couples primarily to the bosonic mass term while fermionic states at the relevant energies remain extended due to weaker direct coupling. An explicit calculation of the fermionic self-energy in the disordered Hertz model to confirm this would require a separate, computationally intensive analysis outside the scope of the present work, which focuses on bosonic-mediated pairing. We will add a clarifying paragraph in the model section justifying the assumption via scaling arguments and literature references. revision: partial

  2. Referee: [Numerical / Lifshitz-tail analysis] Section on numerical solution and Lifshitz-tail analysis: the claim of a power-law tail in the distribution of local pairing scales is load-bearing for the central contrast with BCS superconductors. The manuscript should specify the fitting range, the extracted exponent, and how the tail is distinguished from possible stretched-exponential or other forms; without these quantitative details the distinction remains qualitative.

    Authors: We agree that quantitative details are needed to make the distinction rigorous. In the revised manuscript we will report the fitting range used for the tail, the extracted power-law exponent, and include a direct comparison (e.g., via log-log plots and goodness-of-fit metrics) showing that the distribution is inconsistent with stretched-exponential forms over the relevant range. revision: yes

standing simulated objections not resolved
  • Explicit verification of fermionic non-localization via self-energy calculation under Harris disorder.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from an explicit model premise (fermions remain extended and supply Landau damping) via standard tools: real-space linearized Usadel equation, self-consistent Born approximation, and Lifshitz-tail analysis. The power-law tail in local pairing scales is obtained as an output of these calculations on the localized bosonic modes; it is not presupposed by definition or by a fitted parameter renamed as a prediction. No self-citation chain is load-bearing for the central result, and the contrast with stretched-exponential BCS tails follows directly once the stated premise is granted. The paper is self-contained against external benchmarks with no reduction of claims to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger extracted from abstract description only; full model details unavailable.

axioms (2)
  • domain assumption Landau damping of bosonic fluctuations by fermions in Hertz theory
    Invoked to describe non-Fermi-liquid behavior and damping in quantum critical metals.
  • domain assumption Harris criterion for disorder coupling to phi^2 coefficient
    Used to argue that disorder creates random mass for bosonic field.

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discussion (0)

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Reference graph

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    Numerical solution Here we provide the technical details of the numerical solution to Eq. (4.7), along with some additional data plots that did not fit into the main text. The dynamical Usadel equation in Eq. (4.7) is an eigenvalue equation, and can be solved efficiently using iterative methods. We write it again below, to make this more transparent, mmax...

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