Superconductivity mediated by nematic fluctuations -- the dispersion of collective modes

This paper derives the pair susceptibility and analyzes the spectral functions of a superconductor mediated by nematic fluctuations, revealing that the presence of gapless Fermi surface arcs leads to a pair susceptibility and collective mode dispersion that are qualitatively distinct from those in conventional BCS superconductors.

The Gist

Imagine a superconductor not as a perfect, uniform block of ice, but as a crowded dance floor where electrons pair up to move without friction. In most famous superconductors, this dance floor is smooth and the rules are the same everywhere. But in the specific material this paper studies (a doped version of Iron Selenide, or FeSe), the dance floor is strangely lumpy.

Here is a simple breakdown of what the authors, Islam and Chubukov, discovered about how these electrons move and vibrate in this "lumpy" environment.

1. The Setting: A Dance Floor with "Hot" and "Cold" Spots

In a normal superconductor, the energy gap (the "glue" holding electron pairs together) is the same strength everywhere on the dance floor.

In this specific material, the "glue" is provided by nematic fluctuations. Think of nematicity like a crowd of people suddenly deciding to all face East instead of North. This creates a special directionality. Because of this, the glue holding the electron pairs together is incredibly strong in some directions (the "hot spots") and incredibly weak in others (the "cold spots").

• The Result: Even though the pairing symmetry is technically "s-wave" (usually meaning a perfect circle), the actual energy gap looks like a four-leaf clover. It is huge at the tips of the leaves (hot spots) and almost vanishes in the valleys between them (cold spots).

2. The Experiment: Shaking the System

The authors wanted to know: "If we shake this superconductor, how does it vibrate?" In physics, these vibrations are called collective modes. They looked at two types of shakes:

• The Transverse Shake (Phase Mode): Imagine the dancers all changing their rhythm slightly together, but not changing their speed. This is like a wave of "phase" moving through the crowd.
• The Longitudinal Shake (Amplitude Mode): Imagine the dancers suddenly getting closer together or further apart, changing the strength of their bond. This is a wave of "amplitude."

3. The Big Discovery: The Vibration is Weird

In a standard, uniform superconductor, these vibrations are predictable.

• Standard Phase Mode: It's like a clear, sharp whistle (a "Goldstone mode"). It has a specific pitch that depends on how fast you shake it.
• Standard Amplitude Mode: It's like a heavy drum beat that only happens above a certain volume (frequency). Below that volume, it's silent.

In this "lumpy" superconductor, the rules change completely:

The Phase Mode (Transverse) Becomes a Muffled Rumble

Instead of one sharp whistle, the authors found that the phase vibration splits into two distinct, damped sounds.

• The Analogy: Imagine shouting in a canyon with two different types of walls. Instead of one clear echo, you hear two overlapping echoes that fade away quickly.
• The Detail: The "pitch" of these sounds depends entirely on the direction you are looking at the material. If you look at the "hot" direction, you hear one tone; if you look at the "cold" direction, you hear another. They merge in the middle, but they never become a sharp, clear note. They are always "damped" (muffled).

The Amplitude Mode (Longitudinal) Becomes a Chaotic Scream

This is where the results get truly unconventional.

• At Zero Momentum (Shaking the whole room at once): In a normal superconductor, the amplitude mode is silent below a certain energy. Here, it is never silent. It is always humming, but the volume changes in a strange way.
  • Near the maximum energy (the "loud" part), the sound doesn't just rise; it hits a "logarithmic singularity." Imagine a speaker that suddenly starts screaming at a specific frequency, but the scream is shaped like a sharp spike rather than a smooth hill.
• At Finite Momentum (Shaking a specific spot): When they looked at vibrations moving through the material, the "loud" part split into two separate spikes.
  • The Analogy: Think of a normal drum that hits one note. This new drum hits two different notes simultaneously, and the pitch of those notes changes depending on which direction you hit the drum.
  • The "Cold" Spots: Because the gap is so small in the "cold" regions, the material allows these vibrations to happen at very low energies, creating sudden "jumps" in the signal that don't exist in normal superconductors.

4. The "Series vs. Parallel" Analogy

The authors use a clever electrical analogy to explain why this happens.

• Normal Superconductor (Parallel Circuit): Imagine many resistors connected in parallel. If one path is blocked, the current just flows through the others. The system averages everything out, leading to smooth, uniform behavior.
• This Superconductor (Series Circuit): Here, the different parts of the Fermi surface (the dance floor) are connected in series. If one part of the chain is weak (the cold spots), it drags down the whole system. The behavior of the "weak" parts dominates the whole, creating these sharp, jagged, and highly directional vibrations.

Summary

The paper claims that in a superconductor driven by nematic fluctuations, the collective vibrations of the electron pairs are highly anisotropic (direction-dependent) and unconventional.

• Instead of sharp, clear notes, you get muffled, split tones.
• Instead of a quiet zone below a certain energy, you get a constant, strange hum that spikes dramatically at specific frequencies.
• These features are a direct fingerprint of the "lumpy" gap caused by the nematic order, distinguishing it clearly from standard superconductors.

The authors suggest that scientists could detect these unique "sounds" using spectroscopic tools like Raman scattering or THz conductivity, essentially "listening" to the material to confirm this exotic state of matter.

Technical Summary

Technical Summary: Superconductivity Mediated by Nematic Fluctuations – The Dispersion of Collective Modes

Problem Statement
The interplay between superconductivity and electronic nematicity is a central theme in correlated quantum materials, particularly in iron-based superconductors like FeSe and its doped counterparts (FeSe$_{1-x}$S$_x$, FeSe$_{1-x}$Te$_x$). In doped FeSe at concentrations $x \ge x_c$ (where the nematic transition temperature $T_p$ vanishes), experimental evidence suggests a qualitative change in the superconducting state compared to the pure or weakly doped regimes. Specifically, the pairing mechanism is believed to shift from spin-fluctuation mediation to nematic-fluctuation mediation. A key feature of nematic-fluctuation-mediated pairing (NFMS) is the generation of a highly anisotropic superconducting gap, even within an $s$-wave symmetry channel. This gap vanishes on four arcs of the Fermi surface ("cold spots") while remaining large at "hot spots." While previous studies have analyzed the thermodynamic and spectroscopic consequences of this gap structure, the structure of collective excitations (pairing fluctuations) in such a system remained unexplored. This paper addresses the dispersion and spectral properties of collective modes in a 2D NFMS, contrasting them with conventional BCS superconductors.

Methodology
The authors employ Gorkov's diagrammatic technique to compute the dynamic pair-pair susceptibility $\chi(q, \Omega)$ at finite momentum $q$ and frequency $\Omega$ deep within the superconducting phase ($T=0$).

1. Model: The study utilizes an effective one-band model for 2D electrons with a momentum-dependent short-range interaction mediated by nematic fluctuations. The pairing interaction is approximated as $V(\theta_k, \theta_p) = -V_0 \cos^2(2\theta_k)\delta(\theta_k - \theta_p)$, reflecting the orbital nature of the nematic order in FeSe.
2. Gap Structure: Solving the non-linear gap equation yields a highly anisotropic gap function $\Delta(\theta_k) = \Delta_0 \exp(-\tan^2(2\theta_k)/g)$, where $\Delta_0$ is the maximum gap at hot spots ($\theta_k = n\pi/2$) and the gap is exponentially small at cold spots ($\theta_k = (2n+1)\pi/4$).
3. Susceptibility Calculation: The authors calculate the two-point pair-pair correlation function using the ladder approximation. The susceptibility is decomposed into transverse (phase) $\chi_T$ and longitudinal (amplitude) $\chi_L$ components. The calculation involves summing ladder diagrams for the two-fermion vertices and evaluating the resulting integrals over frequency and energy, followed by analytic continuation to real frequencies.
4. Analytical and Numerical Analysis: The authors perform analytical expansions for small $q$ and $\Omega$, particularly focusing on the behavior near the cold spots (where the gap is small) and the hot spots. These analytical results are supplemented by numerical evaluations of the remaining integrals to map out the full frequency and momentum dependence.

Key Contributions and Results

• Transverse (Phase) Susceptibility ($\chi_T$):

  • Unlike conventional $s$-wave BCS superconductors, which host a sharp, gapless Anderson-Bogolubov (AB) Goldstone mode with dispersion $\Omega \propto |q|$, the NFMS does not support a sharp phase mode when Coulomb interactions are neglected.
  • Instead, the imaginary part of the transverse susceptibility, $\text{Im}\chi_T(q, \Omega)$, exhibits two broad maxima at frequencies $\Omega_1 = v_F|q||\cos\theta_q|$ and $\Omega_2 = v_F|q||\sin\theta_q|$, where $\theta_q$ is the direction of momentum $q$.
  • These maxima correspond to anisotropic, sound-like damped collective modes. The peaks merge at $\theta_q = \pi/4$, but the mode remains damped. The authors note that including Coulomb interactions would likely modify the dispersion to $\Omega \propto \sqrt{q}$, but the modes would remain damped rather than sharp.

• Longitudinal (Amplitude) Susceptibility ($\chi_L$):

  • At $q=0$: The spectral function $\text{Im}\chi_L(0, \Omega)$ is non-zero for all frequencies. It vanishes at $\Omega=0$ but rises rapidly as $1/\sqrt{\log(\Delta_0/\Omega)}$ at exponentially small frequencies, driven by the exponentially small gap in the cold regions. Near the maximum gap energy $\Omega = 2\Delta_0$, $\text{Im}\chi_L$ displays a two-sided logarithmic singularity, $\text{Im}\chi_L \propto \log|\Omega - 2\Delta_0|$, while the real part $\text{Re}\chi_L$ exhibits a finite jump.
  • At Finite $q$: The structure becomes more complex. The logarithmic singularity at $2\Delta_0$ splits into two distinct logarithmic singularities at:
    $$\Omega_{\text{peak},1}(q) = 2\Delta_0 + \frac{v_F^2|q|^2}{4\Delta_0}\cos^2\theta_q$$
    $$\Omega_{\text{peak},2}(q) = 2\Delta_0 + \frac{v_F^2|q|^2}{4\Delta_0}\sin^2\theta_q$$
    These correspond to two dispersing longitudinal modes.
  • Threshold Behavior: At low frequencies, $\text{Im}\chi_L(q, \Omega)$ vanishes below an angle-dependent threshold frequency proportional to $q$. Depending on the direction $\theta_q$, the susceptibility exhibits one or two discontinuities (jumps) in the imaginary part and corresponding logarithmic divergences or jumps in the real part.

• Comparison with Conventional BCS Superconductors:

  • The authors highlight a fundamental structural difference in the pair susceptibility. In a conventional BCS superconductor with a constant gap, the inverse susceptibility is an integral over the Fermi surface (analogous to resistors in parallel), leading to a single sharp Higgs mode at $2\Delta_0$ and a gapless phase mode.
  • In the NFMS, the susceptibility is a direct integral of local susceptibilities (analogous to resistors in series). This "series" summation preserves the anisotropy of the gap function, preventing the formation of sharp collective modes and instead yielding the damped, multi-peak structures described above.

Significance and Claims
The paper claims that the analytic structure of the pair susceptibility in both transverse and longitudinal channels in an NFMS is qualitatively distinct from that of both gapped and nodal conventional superconductors. The highly unconventional dispersion of the collective modes arises directly from the unique interplay between the nematic-fluctuation-mediated pairing interaction and the resulting anisotropic gap structure (specifically the presence of exponentially small gaps in cold regions).

The authors assert that these distinct spectral features—specifically the damped transverse modes and the split, logarithmic singularities in the longitudinal response—should be detectable in spectroscopic probes such as Raman scattering or THz conductivity. The work aims to provide a theoretical framework for understanding the dynamics of superconducting states driven by nematic quantum criticality, distinguishing them from standard BCS scenarios. The paper concludes modestly, suggesting these findings open a new avenue for studying the interplay between electronic criticality and superconducting dynamics mediated by strongly anisotropic interactions.