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arxiv: 2604.15120 · v1 · submitted 2026-04-16 · ❄️ cond-mat.supr-con

Quantum fluctuations and the emergence of in-gap Higgs mode in superconductors

Sida Tian , Naoto Tsuji , Dirk Manske This is my paper

Pith reviewed 2026-05-10 09:40 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords Higgs modequantum fluctuationss-wave superconductorsthird harmonic generationRaman scatteringamplitude modessuperconducting gap
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The pith

Quantum fluctuations shift the Higgs mode below the superconducting gap in s-wave superconductors.

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

This paper extends the standard action for the Higgs mode in s-wave superconductors to include quantum fluctuations. It shows that one-loop corrections to the Higgs propagator lower the mode frequency below the gap edge 2Δ. This turns the mode into an undamped pole inside the gap, separate from the quasiparticle continuum. Readers would care because the shift produces much sharper signals in third harmonic generation and Raman scattering experiments. The work also indicates that different probes, such as tunneling versus optical methods, can report different gap values due to these corrections, and it suggests the effect applies to amplitude modes in other weakly fluctuating systems.

Core claim

We extend the well-established action of the Higgs mode in s-wave superconductors to include quantum fluctuations. We find that already one-loop quantum corrections to the Higgs propagator shift its eigenfrequency below the superconducting energy gap 2Δ. Consequently, the Higgs mode appears as an undamped pole below the quasiparticle continuum, leading to drastically sharper experimental signatures. We demonstrate this by calculating two characteristic fingerprints of the Higgs mode, namely in Third Harmonic Generation (THG) and inelastic Raman scattering signals.

What carries the argument

one-loop quantum corrections to the Higgs propagator, which shift its eigenfrequency below 2Δ and produce an undamped in-gap pole

If this is right

  • The Higgs mode appears as an undamped pole below the quasiparticle continuum.
  • Third harmonic generation signals exhibit drastically sharper features.
  • Inelastic Raman scattering reveals distinct in-gap mode signatures.
  • Gaps measured by different techniques such as scanning tunneling microscopy and Raman scattering may differ due to fluctuation corrections.
  • The mechanism sheds light on amplitude modes in other systems with weak quantum fluctuations, including charge density waves, antiferromagnets, and cold atom condensates.

Where Pith is reading between the lines

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

  • This shift could explain apparent discrepancies between gap values obtained from different experimental probes in the same material.
  • Similar in-gap amplitude modes may appear in weakly fluctuating systems when one-loop corrections are included, offering a general way to interpret damped modes in other contexts.
  • Testing the shift in materials where fluctuation strength can be tuned, such as under external fields, would provide a direct check on the one-loop prediction.

Load-bearing premise

The one-loop quantum correction to the Higgs propagator is sufficient to produce the shift and undamped pole, with the initial action for the Higgs mode in s-wave superconductors being an accurate starting point without requiring higher-order terms or material-specific adjustments.

What would settle it

A calculation or measurement showing the Higgs mode remains at or above 2Δ after including one-loop quantum fluctuations would falsify the claim that these corrections create an undamped pole below the gap.

Figures

Figures reproduced from arXiv: 2604.15120 by Dirk Manske, Naoto Tsuji, Sida Tian.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic view for the spectral functions of the Higgs [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical solutions of Eq. (46). (a) and (b)— [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Panels (a) and (b)—The Fermi surface and the den [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

We extend the well-established action of the Higgs mode in $s$-wave superconductors to include quantum fluctuations (QFs). We find that already one-loop quantum corrections to the Higgs propagator shift its eigenfrequency below the superconducting energy gap $2\Delta$. Consequently, the Higgs mode appears as an undamped pole below the quasiparticle continuum, leading to drastically sharper experimental signatures. We demonstrate this by calculating two characteristic fingerprints of the Higgs mode, namely in Third Harmonic Generation (THG) and inelastic Raman scattering signals. More generally, gaps measured in $s$-wave superconductors with different experimental techniques (such as scanning tunneling microscope and Raman scattering) may be different due to fluctuation corrections. Since already arbitrarily weak QFs lead to the shift and to the new pole, our results shed some light on other amplitude modes even for systems with weak QFs, including charge density waves, (anti-) ferromagnets, or cold atom fermionic condensates.

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 extends the standard effective action for the Higgs (amplitude) mode in s-wave superconductors to incorporate quantum fluctuations. Using a one-loop calculation of corrections to the Higgs propagator, the authors report that the mode eigenfrequency is shifted below the gap edge 2Δ, producing an undamped in-gap pole. This is illustrated through explicit computations of the resulting signatures in third-harmonic generation (THG) and inelastic Raman scattering, with the broader claim that fluctuation-induced shifts may cause apparent discrepancies between gap values measured by different probes (e.g., STM vs. Raman) and that the effect appears for arbitrarily weak fluctuations, with implications for other amplitude modes.

Significance. If the central result holds under self-consistent treatment, the work would provide a concrete mechanism by which quantum fluctuations qualitatively alter the spectrum of amplitude modes even at weak coupling, potentially explaining experimental discrepancies in gap measurements and offering sharper, undamped signatures in nonlinear and spectroscopic probes. The explicit THG and Raman calculations supply falsifiable predictions that could be tested in conventional superconductors.

major comments (2)
  1. [one-loop Higgs propagator calculation (following the extended action)] The one-loop shift of the Higgs pole below the bare 2Δ (computed from the fluctuation-free starting action) is presented as sufficient to produce an undamped mode, but the same one-loop diagrams that correct the Higgs propagator also renormalize the gap equation that determines Δ. Without a self-consistent solution in which both the propagator pole and the continuum threshold 2Δ are evaluated at the same order, it remains unclear whether the pole stays below the renormalized threshold. This consistency check is load-bearing for the claim that arbitrarily weak QFs produce the in-gap mode.
  2. [model and action extension] The initial action is taken as the standard fluctuation-free form for the Higgs mode; the one-loop corrections are then added perturbatively. If the gap renormalization at one-loop order is comparable in magnitude to the propagator shift, the perturbative starting point may require justification or resummation, particularly since the abstract asserts the result holds for arbitrarily weak QFs.
minor comments (2)
  1. [Introduction] Notation for the Higgs field and fluctuation corrections should be defined more explicitly at first use to aid readability for readers outside the immediate subfield.
  2. [THG and Raman calculations] The THG and Raman response functions would benefit from an additional panel or inset showing the bare (no-QF) versus corrected spectra for direct visual comparison.

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 two major concerns point by point below, clarifying the perturbative framework and the regime of validity of our results.

read point-by-point responses

  1. Referee: The one-loop shift of the Higgs pole below the bare 2Δ (computed from the fluctuation-free starting action) is presented as sufficient to produce an undamped mode, but the same one-loop diagrams that correct the Higgs propagator also renormalize the gap equation that determines Δ. Without a self-consistent solution in which both the propagator pole and the continuum threshold 2Δ are evaluated at the same order, it remains unclear whether the pole stays below the renormalized threshold. This consistency check is load-bearing for the claim that arbitrarily weak QFs produce the in-gap mode.

    Authors: We agree that a fully self-consistent treatment at one-loop order, with both the Higgs self-energy and the gap equation solved simultaneously, would strengthen the analysis. Our present calculation employs the standard mean-field starting action with fixed Δ and computes the leading one-loop correction to the Higgs propagator. The gap renormalization arises from related but distinct diagrams (primarily tadpole contributions). In the weak-fluctuation limit the leading shift of the pole is linear in the fluctuation strength, while the gap correction is also linear; however, the coefficient of the pole shift is larger in magnitude because of the momentum structure of the Higgs bubble diagram. We will add an explicit estimate of the one-loop gap shift in the revised manuscript and demonstrate that the pole remains below the renormalized continuum edge for the parameter range considered. This addresses the consistency concern without requiring a full resummation. revision: yes

  2. Referee: The initial action is taken as the standard fluctuation-free form for the Higgs mode; the one-loop corrections are then added perturbatively. If the gap renormalization at one-loop order is comparable in magnitude to the propagator shift, the perturbative starting point may require justification or resummation, particularly since the abstract asserts the result holds for arbitrarily weak QFs.

    Authors: The perturbative expansion around the mean-field action is controlled for weak quantum fluctuations, as higher-order diagrams are suppressed by additional powers of the fluctuation strength. The abstract statement that the effect appears for arbitrarily weak QFs refers to the fact that the downward shift of the pole is a non-vanishing first-order correction that moves the mode below the bare gap edge even as the fluctuation parameter approaches zero. We will revise the manuscript to include a short paragraph justifying the perturbative starting point, clarifying that the gap renormalization, while present, does not cancel the pole shift at leading order, and stating the range of validity more precisely. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation extends established action with explicit one-loop corrections

full rationale

The paper begins from the well-established fluctuation-free action for the Higgs mode in s-wave superconductors, then adds one-loop quantum corrections specifically to the Higgs propagator. The resulting eigenfrequency shift below the bare 2Δ and the emergence of an undamped pole are computed outcomes of those diagrams, not redefinitions of the input action or gap parameter. THG and Raman signatures are derived directly from the corrected propagator. No step equates a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames a known result; the central result remains an independent perturbative calculation whose validity can be checked against external benchmarks or higher-order terms.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, the paper relies on the well-established action for the Higgs mode and adds one-loop quantum fluctuations. No explicit free parameters, axioms, or invented entities are described.

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Forward citations

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