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

This paper demonstrates that including quantum fluctuations in the theoretical description of $s$-wave superconductors shifts the Higgs mode's frequency below the superconducting gap, creating a new undamped pole that results in sharper experimental signatures in Third Harmonic Generation and Raman scattering while explaining potential discrepancies in gap measurements across different techniques.

The Gist

The Big Picture: Finding a Hidden Note in a Symphony

Imagine a superconductor (a material that conducts electricity with zero resistance) as a giant, perfectly synchronized choir. Every singer (an electron) is holding a note perfectly in tune with everyone else. This collective harmony is what makes the superconductor work.

In physics, when this choir gets excited, it can vibrate in two main ways:

1. The Phase (The Timing): If the singers start singing slightly out of sync with each other, that's a "Goldstone mode." In superconductors, this is usually pushed so high in energy that we can't hear it.
2. The Amplitude (The Volume): If the singers all suddenly get louder or softer together, that's the Higgs mode. This is the "volume knob" of the superconductor.

The Problem:
For decades, physicists have been trying to "hear" this Higgs mode. But it's like trying to hear a single violin playing a specific note while a massive drum section (the "quasiparticle continuum") is banging away right next to it. The drum noise drowns out the violin. In standard theory, the violin's note is exactly the same pitch as the loudest part of the drum, making it impossible to distinguish. It's a blurry, messy sound.

The Discovery:
This paper says: "Wait a minute! We've been ignoring a tiny, invisible force called Quantum Fluctuations."

Think of quantum fluctuations as the tiny, jittery movements of the air molecules in the room. They are so small we usually ignore them. But the authors of this paper realized that even these tiny jitters act like a subtle "tuning fork."

When you include these jitters in the math, something magical happens: The violin's note shifts.

Suddenly, the Higgs mode (the violin) moves to a slightly lower pitch, slipping underneath the loud drum noise. It becomes a clear, sharp, distinct note that stands out from the background chaos.

The Analogy: The Crowded Dance Floor

Let's use a dance floor analogy to visualize what the paper found:

• The Superconductor: A crowded dance floor where everyone is dancing in a perfect circle (the Cooper pairs).
• The Higgs Mode: A specific dance move where everyone suddenly jumps up and down in unison.
• The "Gap" (2Δ): The minimum energy required to break the circle and make a dancer leave the floor (a "broken pair").
• The Old Theory (Mean Field): In the old view, the "jumping up and down" move happens at the exact same energy as breaking the circle. So, if you try to watch the jumping, you just see a blur of people breaking apart. You can't tell the jump from the break.
• The New Theory (With Quantum Fluctuations): The authors added "quantum jitter" to the model. They found that this jitter acts like a gentle push. It pushes the "jumping" move to a slightly lower energy level.
  • Now, the jumping happens before anyone breaks the circle.
  • Because it's at a different energy, it creates a sharp, clear signal (a "pole") that doesn't get drowned out by the noise of breaking pairs.

Why Does This Matter?

1. Sharper Signatures (The "Fingerprint")
Because the Higgs mode is now a sharp, distinct note rather than a blurry mess, it leaves a much clearer "fingerprint" in experiments. The paper shows that if you shine light on the superconductor (specifically using a technique called Third Harmonic Generation or Raman Scattering), you will see a sharp spike in the data where the Higgs mode is. Before, this spike was hidden; now, it's visible.

2. Different Gaps for Different Tools
This leads to a confusing but exciting prediction for experimentalists.

• If you use a Scanning Tunneling Microscope (STM) to measure the energy gap (the "breaking point"), you will measure one value.
• If you use Raman Scattering (light) to measure the Higgs mode, you will measure a slightly smaller value.
• Why? Because the Higgs mode has been "pushed down" by the quantum fluctuations, while the STM measures the raw breaking point. It's like measuring the height of a building from the ground (STM) versus measuring the height of a flag on a pole that's been lowered slightly (Higgs mode).

3. Where to Look
The paper suggests that this effect is strongest in 2D materials (like a single layer of atoms) rather than thick 3D blocks. They specifically point to Monolayer FeSe on SrTiO3 (a very thin iron-based superconductor) as the perfect place to test this. In this material, the "shift" is big enough to be seen with current technology.

The "So What?" Conclusion

For a long time, physicists thought the Higgs mode in superconductors was just a theoretical ghost—hard to find and impossible to pin down.

This paper says: It's not a ghost; it's just hiding.

By accounting for the tiny, unavoidable "jitters" of the quantum world, the Higgs mode reveals itself as a stable, long-lived particle sitting just below the energy gap. This changes how we interpret experiments. It suggests that if we see a sharp peak in our data that doesn't match the standard "breaking energy," we aren't seeing a mistake—we are seeing the Higgs mode, finally clear and distinct, thanks to the subtle push of quantum fluctuations.

In short: The universe is jittery. That jitter pushes the Higgs mode out of the noise, giving us a clear window to study the fundamental "heartbeat" of superconductors.

Technical Summary

1. Problem Statement

In conventional $s$-wave superconductors, the Higgs mode (amplitude mode of the order parameter) is theoretically predicted to exist exactly at the pair-breaking energy edge, $\omega_H = 2\Delta$ (where $\Delta$ is the superconducting gap). Within the standard Random Phase Approximation (RPA) or mean-field theory:

• The Higgs mode does not possess a well-defined pole; instead, its spectral function exhibits a square-root divergence ($1/\sqrt{4\Delta^2 - \omega^2}$) at the edge of the quasiparticle (QP) continuum.
• This lack of a sharp pole, combined with the fact that the Higgs mode couples only quadratically to light (yielding weak signals often overshadowed by QP excitations), makes experimental identification extremely difficult.
• While "in-gap" Higgs modes (where $\omega_H < 2\Delta$) have been observed in specific strong-coupling regimes or systems with magnetic impurities, it was unclear if such a shift occurs in the weak-coupling limit (standard BCS regime) due to intrinsic quantum fluctuations.

The Core Question: Do quantum fluctuations (QFs) perturbatively shift the Higgs mode eigenfrequency below the superconducting gap in standard $s$-wave superconductors, thereby creating a sharp, undamped pole?

2. Methodology

The authors employ a field-theoretical approach extending beyond the mean-field approximation:

• Model: They utilize the attractive Hubbard model at zero temperature.
• Formalism:
  • They perform a Hubbard-Stratonovich (HS) transformation to introduce a complex scalar field $\Delta$ representing the order parameter.
  • They decompose $\Delta$ into a homogeneous mean-field value $\Delta_0$ and amplitude fluctuations $h$ (the Higgs mode), ignoring phase fluctuations (which are gapped by the Anderson-Higgs mechanism in charged systems).
  • Luttinger-Ward (LW) Functional: Instead of standard perturbation theory, they construct a self-consistent theory using the LW functional $\Phi[G, H]$, defined as the sum of all closed, two-particle-irreducible skeleton diagrams.
• Expansion and Approximation:
  • The theory is expanded in terms of a small parameter $\alpha/\Delta_0$, where $\alpha$ is related to the coherence length $\xi_0$ and density of states $N_F$. This parameter quantifies the strength of quantum fluctuations.
  • They calculate the self-energy $\Sigma$ of the Higgs propagator up to one-loop order in the collective mode momentum.
  • The Dyson equation for the dressed Higgs propagator is solved: $H^{-1}(i\omega) = H_0^{-1}(i\omega) - \Sigma(i\omega)$.
• Numerical Implementation:
  • The self-energy integrals are evaluated numerically for both 2D and 3D systems.
  • Analytic continuation is performed from imaginary to real frequencies using a specific integration contour to handle branch cuts correctly.
• Response Functions: They compute the Third Harmonic Generation (THG) and Inelastic Raman scattering ($A_{1g}$ symmetry) response functions using the dressed Higgs propagator to predict experimental signatures.

3. Key Contributions

• Theoretical Breakthrough: The paper demonstrates that arbitrarily weak quantum fluctuations are sufficient to shift the Higgs mode eigenfrequency below the pair-breaking gap ($\omega_H < 2\Delta_0$) in $s$-wave superconductors, even in the weak-coupling BCS limit.
• Mechanism Identification: The shift arises because the one-loop self-energy $\Sigma$ is proportional to the density-density correlator. This interaction pushes the pole of the Higgs propagator away from the quasiparticle continuum edge, converting the square-root divergence into a Lorentzian pole.
• Dimensional Dependence: The magnitude of the shift is derived as $\omega_H \approx 2\Delta_0 - C \cdot \Delta_0 \delta^{d-1}$, where $\delta = \Delta_0/E_F$ and $d$ is the spatial dimension. This implies the effect is significantly more prominent in 2D systems than in 3D.
• Experimental Predictions: The authors provide concrete fingerprints for detecting this mode:
  • A sharp peak in Raman and THG spectra slightly below $2\Delta_0$.
  • A distinct phase jump in the THG signal: while the mean-field divergence yields a $\pi/2$ phase jump, the new pole structure yields a $\pi$ phase jump between $\omega_H$ and $2\Delta_0$.

4. Key Results

• Spectral Shift: Numerical solutions confirm that for $\delta \neq 0$ (finite QFs), the real part of the inverse propagator crosses zero at $\omega_H < 2\Delta_0$. The imaginary part of the self-energy vanishes in this region, indicating the mode is undamped (long-lived).
• Magnitude of Shift:
  • In 2D: $\omega_H \approx 2\Delta_0 (1 - 0.39\delta)$.
  • In 3D: $\omega_H \approx 2\Delta_0 (1 - 0.43\delta^2)$.
• Response Functions:
  • Raman Scattering: The inclusion of QFs leads to a sharp, dominant peak in the $A_{1g}$ Raman susceptibility just below the gap, distinct from the broad pair-breaking background.
  • THG: The THG signal shows a significantly enhanced resonance at $\omega_H$. Crucially, the phase of the THG signal jumps by $\pi$ as the frequency crosses $\omega_H$, a signature distinct from the $\pi/2$ jump expected for a continuum edge.
• Material Candidates:
  • The authors analyze various materials (NbN, MgB$_2$, FeSe, Magic-Angle Twisted Bilayer Graphene).
  • They identify monolayer FeSe on SrTiO$_3$ as a prime candidate, predicting a shift of $\approx 0.54$ meV (or $0.43$ cm$^{-1}$ in Raman units), which is experimentally resolvable.
  • They note that standard "strong-coupling" materials (like NbN) may not show this effect if their coherence length is large, emphasizing that the parameter $\alpha/\Delta_0$ (not just $2\Delta/k_B T_c$) determines the observability.

5. Significance

• Resolution of Experimental Discrepancies: The paper offers a theoretical explanation for why gaps measured by different techniques (e.g., STM measuring single-particle gaps vs. Raman measuring collective mode gaps) might differ in superconductors. The Raman gap would correspond to $\omega_H$, while STM measures $2\Delta_0$.
• Universality: The findings suggest that the "in-gap" Higgs mode is not restricted to exotic strong-coupling systems but is a generic feature of superconductors with finite quantum fluctuations.
• Broader Impact: The mechanism described (fluctuation-induced pole shifting) is applicable to other symmetry-breaking states, including charge density waves (CDW), (anti-)ferromagnets, and cold atom condensates.
• Experimental Guidance: By predicting specific phase jumps in THG and sharp Raman peaks, the work provides clear, testable criteria for experimentalists to confirm the existence of the Higgs mode in standard superconductors, potentially resolving long-standing difficulties in its detection.

In summary, this work fundamentally revises the understanding of the Higgs mode in superconductors, establishing that quantum fluctuations naturally generate a sharp, observable, in-gap mode, transforming the Higgs mode from a theoretical curiosity into a robust experimental probe of the superconducting ground state.