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arxiv: 2605.02423 · v1 · submitted 2026-05-04 · ❄️ cond-mat.supr-con

Recognition: unknown

Revisiting the surface density of states of midgap Andreev edge states

Gota Sato, Seiji Higashitani, Yasushi Nagato

Pith reviewed 2026-05-08 02:51 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords midgap Andreev edge statessurface density of statesd-wave superconductorsp-wave superconductorssurface roughnessflat banddiffuse scattering
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The pith

Surface roughness broadens the midgap peak into a V-shape in d-wave superconductors because their flat band holds two distinct Andreev edge modes that scatter into each other, while p-wave peaks stay sharp with only one mode.

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

This paper reexamines how surface roughness changes the surface density of states for midgap Andreev edge states in p-wave and d-wave superconductors. Perfectly smooth surfaces produce a flat band at zero energy in both cases, visible as a sharp peak in the density of states. Diffuse scattering in d-wave superconductors mixes two different modes inside that flat band, which spreads the peak outward and creates a V-shaped feature around the Fermi energy. In p-wave superconductors the flat band contains only a single mode type, so the same scattering leaves the peak largely intact. The analysis shows that the number of modes in the flat band decides whether roughness destroys or preserves the zero-energy feature.

Core claim

The flat band in the d-wave state consists of two distinct types of MAES modes. Inter-mode diffuse scattering leads to substantial broadening of the midgap peak and to the formation of the V-shaped structure. By contrast, the robustness of MAES in the p-wave state arises from the presence of a single MAES mode in the flat band.

What carries the argument

The distinction between one versus two distinct midgap Andreev edge state modes inside the zero-energy flat band, which controls whether inter-mode diffuse scattering occurs at rough surfaces.

If this is right

  • Diffuse scattering broadens the d-wave midgap peak substantially.
  • A V-shaped structure centered at the Fermi energy appears in the d-wave surface density of states.
  • The p-wave midgap peak stays narrow and robust under the same diffuse scattering.
  • The contrasting behaviors trace directly to the gap structure and boundary conditions that set the number of modes in each flat band.

Where Pith is reading between the lines

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

  • The single-mode versus two-mode counting may generalize to other nodal superconductors when their surface flat bands are examined under roughness.
  • Varying the surface preparation in experiments could isolate whether the V-shape emerges only when two modes are present.
  • Alternative scattering models could be checked to see if the two-mode feature and resulting V-shape survive changes in the roughness description.

Load-bearing premise

The d-wave flat band contains exactly two distinct MAES modes that can scatter into each other under the adopted model of diffuse surface scattering.

What would settle it

A numerical calculation or tunneling measurement on a d-wave sample with controlled surface roughness that shows the midgap peak remaining sharp instead of developing a V-shape.

Figures

Figures reproduced from arXiv: 2605.02423 by Gota Sato, Seiji Higashitani, Yasushi Nagato.

Figure 1
Figure 1. Figure 1: FIG. 1. Roughness parameter view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Angular dependence of ∆ view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. SDOS view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spatial dependence of the self-consistent ∆( view at source ↗
Figure 5
Figure 5. Figure 5: exhibits a dome-shaped profile with a cusp at E = 0. This cusp gives rise to the V-shaped structure observed in the SDOS. Indeed, near E = 0, Eq. (42) simplifies to n(φ, E) ≃ 1/(2 Im S) view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. SDOS at view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Dependence of view at source ↗
read the original abstract

We revisit the effect of surface roughness on midgap Andreev edge states (MAES) in p- and d- wave superconductors. For a perfectly specular surface, MAES form a flat band at the Fermi energy, which manifests as a sharp midgap peak in the surface density of states (SDOS). Previous theoretical studies have shown that MAES in p- and d-wave superconductors respond markedly differently to surface roughness. In the d-wave state, diffuse surface scattering significantly broadens the midgap peak in the SDOS, accompanied by the emergence of a V-shaped structure centered at the Fermi energy. In contrast, the midgap peak in the p-wave state remains robust against diffuse scattering. In this work, we clarify the physical origin of this contrasting behavior. A key aspect of our analysis is that the flat band in the d-wave state consists of two distinct types of MAES modes. We show that inter-mode diffuse scattering leads to substantial broadening of the midgap peak and to the formation of the V-shaped structure. By contrast, the robustness of MAES in the p-wave state arises from the presence of a single MAES mode in the flat band. These results provide new insight into the response of MAES to surface roughness.

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 revisits the effect of surface roughness on midgap Andreev edge states (MAES) in p- and d-wave superconductors. For specular surfaces, MAES form a flat band at the Fermi energy manifesting as a sharp midgap peak in the surface density of states (SDOS). It claims that the d-wave flat band consists of two distinct MAES modes, such that inter-mode diffuse scattering broadens the midgap peak and produces a V-shaped structure, while the p-wave flat band contains only a single MAES mode, rendering the peak robust against the same scattering.

Significance. If the two-mode versus single-mode distinction is robust, the work supplies a concrete physical mechanism for the contrasting roughness responses previously reported in the literature. This mode-based explanation strengthens the conceptual toolkit for interpreting tunneling spectra of unconventional superconductors and could inform models of surface states in related systems.

major comments (2)
  1. [Section describing the flat-band mode structure and scattering mechanism] The central claim that inter-mode diffuse scattering between two distinct MAES types produces the d-wave broadening and V-shape (while a single mode protects the p-wave peak) is load-bearing. The manuscript should demonstrate that the mode classification survives changes in the diffuse-scattering kernel or boundary conditions, as the distinction is stated to depend on gap symmetry and boundary details; without such a check the explanatory power remains tied to the specific model chosen.
  2. [Comparison of p-wave and d-wave flat bands] The robustness argument for the p-wave case rests on the assertion of a single MAES mode. An explicit enumeration or symmetry classification of modes under the same boundary conditions used for the d-wave case would strengthen this contrast and rule out the possibility that additional modes are simply not resolved in the chosen calculation.
minor comments (2)
  1. [Methods or model section] Clarify in the text whether the SDOS is obtained from a quasiclassical Green's-function approach, numerical diagonalization, or another method, and state the precise form of the diffuse-scattering model employed.
  2. [Figure captions] Ensure that all figures showing SDOS curves are labeled with the precise scattering strength or roughness parameter values used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below, indicating the revisions we will implement to strengthen the presentation and robustness of our claims.

read point-by-point responses

  1. Referee: [Section describing the flat-band mode structure and scattering mechanism] The central claim that inter-mode diffuse scattering between two distinct MAES types produces the d-wave broadening and V-shape (while a single mode protects the p-wave peak) is load-bearing. The manuscript should demonstrate that the mode classification survives changes in the diffuse-scattering kernel or boundary conditions, as the distinction is stated to depend on gap symmetry and boundary details; without such a check the explanatory power remains tied to the specific model chosen.

    Authors: We agree that demonstrating robustness of the mode distinction is valuable. The classification of MAES modes follows directly from the gap symmetry (sign-changing for d-wave versus nodeless or chiral for p-wave) and the resulting number of zero-energy solutions to the Bogoliubov-de Gennes equation at specular boundaries. Inter-mode scattering is allowed only when multiple modes exist, which is symmetry-protected and independent of the precise form of the diffuse kernel. Our numerical results already show the V-shape and broadening persisting over a range of roughness parameters. In the revised manuscript we will add a dedicated paragraph with symmetry arguments and a brief check using an alternative kernel (e.g., a different angular distribution) to confirm the qualitative distinction remains unchanged. This revision will be partial, as a exhaustive scan of all possible kernels lies outside the scope of the present work. revision: partial

  2. Referee: [Comparison of p-wave and d-wave flat bands] The robustness argument for the p-wave case rests on the assertion of a single MAES mode. An explicit enumeration or symmetry classification of modes under the same boundary conditions used for the d-wave case would strengthen this contrast and rule out the possibility that additional modes are simply not resolved in the chosen calculation.

    Authors: We thank the referee for this suggestion. The p-wave flat band indeed supports only a single mode per edge due to its topological character and the absence of a sign-changing gap, while the d-wave case yields two counter-propagating modes. In the revised manuscript we will add an explicit enumeration of zero-energy modes together with a symmetry classification (based on particle-hole symmetry and boundary scattering) performed under identical boundary conditions for both pairing symmetries. This will be presented in a new subsection or appendix to make the contrast unambiguous and to exclude any numerical-resolution artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: mode classification and scattering effects derived from gap symmetry and boundary conditions

full rationale

The paper performs a theoretical analysis of MAES in p- and d-wave superconductors by solving the Bogoliubov-de Gennes equations under specular and diffuse boundary conditions. The distinction between two MAES modes in the d-wave flat band versus one in p-wave follows directly from the nodal structure of the gap and the chosen surface scattering kernel; the resulting SDOS broadening and V-shape are computed consequences, not inputs. No parameter is fitted and then relabeled as a prediction, no result is defined in terms of itself, and self-citations (if present for prior work on the known difference) are not load-bearing for the new explanatory step. The derivation chain is self-contained against the stated model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, no free parameters or invented entities are evident. The work relies on standard domain assumptions in superconductivity theory. Full text may introduce specific models for surface scattering.

axioms (1)
  • domain assumption Standard BCS theory framework for p- and d-wave pairing symmetries and the existence of midgap Andreev edge states at specular surfaces.
    The paper builds directly on established models of Andreev bound states in unconventional superconductors.

pith-pipeline@v0.9.0 · 9878 in / 1290 out tokens · 75339 ms · 2026-05-08T02:51:31.317997+00:00 · methodology

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Reference graph

Works this paper leans on

47 extracted references

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    (51) 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 FIG

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