Correlation-driven enhancement of pairing in a nematic Hund's metal

This paper demonstrates that in a nematic Hund's metal, dynamical correlation effects beyond a quasiparticle description are essential for sustaining robust superconductivity by simultaneously enhancing orbital differentiation of the gaps and preventing coherence collapse, leading to a complex, frequency-dependent pairing structure.

The Gist

The Big Picture: A Dance in a Crowded Room

Imagine a ballroom filled with dancers (electrons). In a normal metal, these dancers move freely, like people at a casual mixer. But in superconductors, the dancers pair up and waltz perfectly in sync, allowing them to move without any friction (zero resistance).

In some materials, like iron-based superconductors, things get complicated. The room isn't just a square; it gets stretched into a rectangle. This stretching is called nematicity. It breaks the symmetry, making the "x" direction different from the "y" direction.

The scientists in this paper asked: What happens to the dancing pairs when the room is stretched, and the dancers are also very grumpy and crowded (strongly correlated)?

The Cast of Characters

1. The Dancers (Electrons): They want to pair up to superconduct.
2. The Stretch (Nematicity): The room is distorted. Some dancers are squeezed, others have more space.
3. The "Hund's Rule" (The Grumpy Bouncer): This is the key character. In these materials, electrons have a rule (Hund's coupling) that makes them act like stubborn teenagers who refuse to sit next to each other unless they have to. They prefer to stay in their own "orbitals" (dance floors) and avoid sharing. This creates a state called a "Hund's Metal."

The Problem: When Stretching Breaks the Dance

Usually, when you stretch a room (nematicity), you expect the dancers to just adjust. But in a "Hund's Metal," the grumpy bouncer (Hund's coupling) makes the dancers very sensitive to the stretch.

• The Old Way of Thinking (Quasiparticles): Scientists used to think of electrons as simple, independent dancers. If you used this simple view, they predicted that stretching the room would crush the dance pairs. The "grumpy bouncer" would make the dancers so disorganized that they would stop dancing entirely (superconductivity would die).
• The New Discovery: The authors found that this simple view is wrong. They realized that electrons aren't just simple dancers; they are complex, messy, and constantly interacting. When you look at the full picture (including all the messy interactions), the dance pairs don't just survive the stretch—they actually get stronger.

The Key Findings (The "Aha!" Moments)

1. The "Messy" Dancers Save the Day

The paper shows that you cannot understand these superconductors by only looking at the "clean" dancers (quasiparticles). You have to look at the "messy" ones too.

• Analogy: Imagine trying to predict how a crowd moves by only looking at the people walking in a straight line. You'd miss the people shoving, bumping, and creating a chaotic flow that actually keeps the crowd moving forward. In this metal, the "chaos" (incoherent spectral weight) is actually what helps the pairs form. Without the chaos, the superconductivity collapses.

2. The Bouncer is a Double-Edged Sword

The "Hund's Rule" (the grumpy bouncer) does two opposite things at once:

• It makes the stretch worse: It exaggerates the difference between the "x" dancers and the "y" dancers. The gap between their dance moves gets bigger.
• It saves the dance: If the stretch gets too extreme, the "grumpy bouncer" actually stops the room from collapsing completely. It prevents the dancers from getting so polarized that they stop moving. It acts like a safety valve, keeping the metal conductive and the superconductivity alive even when things get very crowded.

3. The "Frequency Filter" (The Cut-off)

This is the most fascinating part. The scientists realized that the "music" (the energy of the bosons that help electrons pair up) has different frequencies (high pitch vs. low pitch).

• The Analogy: Imagine the pairing mechanism is like a radio. If you tune the radio to only play low notes (low energy), the dancers pair up one way. If you tune it to high notes (high energy), they pair up a completely different way.
• In a normal metal, the music doesn't matter much. But in this "Hund's Metal," the music matters a lot. Depending on which "frequency window" you listen to, the dancers might swap partners or change their dance style. The paper shows that the "order" of the dance gaps (which pair is strongest) can flip depending on the energy range you look at.

The Conclusion: Why This Matters

This paper changes how we understand superconductors in iron-based materials.

1. Don't ignore the mess: To understand these materials, you can't just look at the "clean" electrons. You must include the messy, chaotic interactions.
2. Nematicity isn't a killer: Stretching the material doesn't necessarily kill superconductivity. In fact, with the right amount of "grumpiness" (Hund's coupling), the material becomes more robust against the stretch.
3. Energy matters: The way electrons pair up depends heavily on which energies are involved. Different "tunings" of the interaction can lead to completely different superconducting states.

In short: The scientists discovered that in these complex, crowded, and stretched-out metals, the very things that make the electrons "grumpy" and "messy" are actually the secret ingredients that keep the superconducting dance alive and make it more interesting than we ever thought.

Technical Summary

1. Problem Statement

The paper investigates the interplay between electronic nematicity (spontaneous breaking of rotational symmetry) and superconductivity (SC) in Hund's metals, a class of strongly correlated materials exemplified by iron-based superconductors (FeSC).

While previous studies have explored the nematic-SC interplay using low-energy quasiparticle (QP) descriptions, they often fail to capture the full complexity of Hund's metals. In these systems, the normal state is not a weakly correlated metal but is governed by strong Hund's coupling ($J_H$), which leads to:

• Orbital-selective coherence scales.
• Redistribution of spectral weight over a broad energy range (beyond the Fermi surface).
• Strong frequency dependence of self-energies.

The authors address three critical questions:

1. Does the Hund-driven enhancement of superconductivity persist in the nematic state?
2. How does nematicity reshape the orbital differentiation of superconducting gaps across different Hund regimes?
3. To what extent does the pairing energy cutoff act as a spectral filter in a nematic Hund metal?

2. Methodology

The authors employ a combination of Dynamical Mean-Field Theory (DMFT) and mean-field BCS theory to model a minimal three-orbital system (representing $yz$, $xz$, and $xy$ orbitals) relevant to FeSC.

• Hamiltonian: The model includes a tight-binding kinetic term ($H_0$), a nematic perturbation term ($H_{nem}$) that explicitly breaks $xz/yz$ degeneracy, and a multiorbital Kanamori interaction ($H_{int}$) containing Hubbard repulsion ($U$) and Hund's coupling ($J_H$).
• Normal State Calculation: They solve the DMFT equations using an exact diagonalization solver at zero temperature to obtain the full, frequency-dependent self-energy $\Sigma_\mu(i\omega_n)$ for the nematic normal state.
• Superconductivity Calculation: Superconductivity is treated as a perturbation on top of the correlated normal state. They solve the self-consistent BCS equations for the orbital gaps $\Delta_\mu$.
  • Full Dynamical Approach: The pairing kernel $\Pi_{\mu\nu}$ is constructed using the full frequency-dependent DMFT Green's functions.
  • Quasiparticle (QP) Approximation: For comparison, they also solve the BCS equations using a low-energy linearization of the self-energy, retaining only the quasiparticle weights $Z_\mu$ and static energy shifts.
• Cutoff Analysis: To mimic retarded boson-mediated interactions, they introduce a frequency cutoff $\omega_0$ in the pairing kernel ($|\omega| < \omega_0$). By varying $\omega_0$, they probe which frequency sectors of the correlated spectrum contribute to pairing.

3. Key Contributions and Results

A. Necessity of Dynamical Correlations (Beyond QP)

The study demonstrates that a quasiparticle-only description is insufficient for describing superconductivity in the Hund regime.

• QP Failure: The QP approximation systematically underestimates the superconducting gaps. In the strong coupling regime (large $U$), the QP solution predicts a collapse of superconductivity ($\Delta \to 0$) even when quasiparticle weights $Z_\mu$ are finite.
• Dynamical Robustness: The full DMFT solution remains robust and yields sizable gaps even when $Z_\mu$ is strongly suppressed (approaching zero). This confirms that incoherent spectral weight at low energies, encoded in the frequency-dependent self-energies, is essential for pairing in Hund's metals.

B. Dual Role of Hund's Coupling in the Nematic Phase

Hund's coupling plays a dual, stabilizing role in the nematic superconductor:

1. Enhancement of Orbital Differentiation: $J_H$ amplifies the nematic differentiation of the superconducting gaps. The gaps in the $xz$ and $yz$ orbitals become increasingly distinct as $J_H$ increases.
2. Suppression of Coherence Collapse: In the absence of strong Hund's coupling (low $J_H/U$), nematicity drives a strong charge reorganization that leads to an Orbital-Selective Mott (OSM) state, where coherence is lost in specific orbitals, quenching superconductivity. Strong Hund's coupling suppresses this extreme nematic-driven charge polarization and prevents the OSM transition, thereby stabilizing a metallic state where superconductivity can survive even at strong coupling.

C. Cutoff Dependence and Spectral Filtering

A major finding is the non-trivial, orbital-dependent evolution of gaps with the pairing cutoff $\omega_0$.

• Frequency-Dependent Imbalance: In the Hund regime, the nematic imbalance between $xz$ and $yz$ orbitals is strongly frequency-dependent (changing magnitude and sign across frequencies), unlike the rigid shift seen in low-$J_H$ regimes.
• Gap Hierarchy Inversion: Because the pairing kernel acts as a spectral filter, different cutoffs select different sectors of the nematic imbalance. Consequently, the hierarchy of orbital gaps at small $\omega_0$ can differ from (and even invert) the hierarchy at $\omega_0 \to \infty$.
• Sign Changes: The gap anisotropy $R_\Delta(\omega_0)$ exhibits sign changes as $\omega_0$ increases in the Hund regime, directly linked to the sign change in the real part of the self-energy difference ($\text{Re}\Sigma_{yz} - \text{Re}\Sigma_{xz}$) at intermediate frequencies.

4. Significance and Implications

• Mechanism of Robustness: The paper establishes that superconductivity in nematic Hund's metals is not driven solely by coherent quasiparticles but relies heavily on the redistribution of incoherent spectral weight. This explains why SC can persist in regimes where traditional QP theories predict a collapse.
• Interplay of Nematicity and Correlations: It reveals that Hund's physics acts as a "protective" mechanism against the destructive effects of nematicity (OSM transition) while simultaneously enhancing the orbital selectivity of the pairing state.
• Material Specificity and Tuning: The findings suggest that different pairing mechanisms (characterized by different boson energies/cutoffs) can lead to distinct gap structures and nematic responses in the same underlying material. This provides a theoretical lens to understand why different iron-based superconductors, or the same material under different tuning parameters, exhibit diverse nematic gap hierarchies despite sharing similar normal-state correlations.
• Methodological Insight: The work highlights the critical importance of going beyond static or low-energy approximations when studying superconductivity in strongly correlated, multi-orbital systems.

In summary, the paper provides a comprehensive framework showing that dynamical correlations are the key to understanding the robustness and orbital structure of superconductivity in nematic Hund's metals, challenging the validity of purely quasiparticle-based descriptions in this regime.