Collective Modes in Weyl Superconductors and the Axial Anomaly

This paper develops a covariant Lagrangian framework for time-reversal symmetry-broken Weyl superconductors to demonstrate that FFLO pairing spontaneously breaks axial $U(1)_A$ symmetry, generating a pseudo-scalar Nambu-Goldstone mode that couples to gauge fields via the axial anomaly and predicts additional vector and axial-vector collective excitations analogous to mesonic modes in QCD.

The Gist

Imagine a bustling city where everyone is dancing in perfect pairs. In a normal superconductor, this is like a ballroom where partners are always facing each other, moving in perfect sync but with zero net momentum. They are "BCS pairs."

But in this paper, the author, Mehran Abyaneh, explores a much stranger, more exotic dance floor: a Weyl Superconductor. Here, the dancers (electrons) have a special property called "chirality" (think of it as being either a "left-handed" or "right-handed" dancer). In this city, the left-handed and right-handed dancers live in different neighborhoods, separated by a distance in their "dance map."

Here is the story of what happens when these dancers try to pair up, told through simple analogies:

1. The Two Ways to Dance (BCS vs. FFLO)

The paper discusses two ways these electrons can pair up:

• The Standard Dance (BCS): A left-handed dancer pairs with a right-handed dancer. They cancel each other out, and the whole city becomes a smooth, quiet superconductor. This is the "boring" way we know well.
• The Exotic Dance (FFLO): This is the star of the show. Here, two dancers of the same handedness (two lefties or two righties) decide to pair up. But because they are so similar, they can't stand still; they have to move together with a specific, non-zero speed. This is called FFLO pairing.

2. The Broken Rule and the New Ghost

In the standard dance, there is a rule called "Chiral Symmetry." It's like a law saying, "Left-handed dancers and right-handed dancers are perfectly balanced and independent."

When the FFLO dance happens, this rule gets broken. The author shows that this breaking creates a new, invisible "ghost" in the system.

• The Analogy: Imagine a spinning top. If it spins perfectly, it's stable. If you break the symmetry, it starts to wobble. That wobble is a new kind of vibration.
• In physics terms, this wobble is a Nambu-Goldstone mode. Usually, we expect one kind of wobble (related to the electric charge). But because of the FFLO dance, a second, special kind of wobble appears. The author calls this a pseudo-scalar mode. Think of it as a "secret handshake" vibration that only exists because the dancers are moving in this exotic, finite-momentum way.

3. The "Pion" Connection (The QCD Analogy)

This is where the paper gets really clever. The author compares this superconductor to the world of Particle Physics (QCD), specifically the world of protons and neutrons.

• In the subatomic world, there are particles called pions. They are the "wobbles" that appear when the symmetry between matter and antimatter is broken.
• The author says: "Hey, the secret handshake vibration in our Weyl superconductor is exactly like a pion in the particle world!"
• Just like pions have a tiny mass because the symmetry isn't perfectly broken (there's a tiny bit of "explicit breaking" from the environment), this new vibration in the superconductor also gets a tiny, tiny mass.

4. The Magic Trick: Turning into Light

Here is the most exciting part. In the world of particles, a neutral pion can spontaneously turn into two flashes of light (photons). This happens because of a weird quantum glitch called the Axial Anomaly.

The author predicts that this new "pion-like" vibration in the Weyl superconductor can do the same thing!

• The Magic Trick: This vibration can decay and turn into two photons (light).
• The Catch: Inside the bulk (the middle) of the superconductor, this is very hard to see because the superconductor acts like a shield (the Meissner effect) that blocks electromagnetic waves. It's like trying to see a firefly inside a thick steel box.
• The Loophole: However, at the surface or edges of the material, the shield is weaker. The author suggests that if you shine the right kind of light or look at the surface, you might see this "pion" turn into light.

5. Why Does This Matter?

Why should we care about a dancing electron turning into light?

1. Proof of the Exotic Dance: Finding this "pion" turning into light would be the "smoking gun" proof that the material is actually doing the exotic FFLO pairing. Scientists have been trying to prove FFLO pairing exists for decades; this gives them a new way to look for it.
2. A Bridge Between Worlds: It connects the physics of superconductors (condensed matter) with the physics of the early universe (particle physics). It shows that the same mathematical rules govern both a block of metal on a table and the fundamental forces of the cosmos.
3. New Particles: The paper predicts not just this one "pion," but a whole family of new vibrations (vector and axial-vector modes), like finding a whole new zoo of particles just by looking at a superconductor.

Summary in One Sentence

This paper proposes that in a special type of superconductor where electrons pair up while moving, a new "wobble" appears that acts exactly like a subatomic pion, capable of magically turning into light at the material's surface, offering a new way to detect these exotic quantum states.

Technical Summary

1. Problem Statement

The paper addresses the theoretical characterization of collective excitations in three-dimensional Weyl superconductors (3DWS), specifically focusing on systems where time-reversal symmetry is broken. While the spontaneous breaking of $U(1)$ gauge symmetry in conventional superconductors is well-understood (leading to the Anderson-Higgs mechanism and the Meissner effect), the fate of the axial $U(1)_A$ chiral symmetry in Weyl superconductors remains largely unexplored.

The central problem is to determine:

1. How different pairing mechanisms (BCS inter-node vs. FFLO intra-node) affect the symmetry structure of the superconducting state.
2. Whether the spontaneous breaking of axial symmetry in these systems generates new collective modes analogous to mesons in Quantum Chromodynamics (QCD).
3. Whether these modes can couple to electromagnetic fields via the axial anomaly, potentially leading to observable decay channels (e.g., into photons) similar to neutral pion decay ($\pi^0 \to 2\gamma$).

2. Methodology

The author employs a field-theoretic approach combining the Bogoliubov–de Gennes (BdG) formalism with the Nambu–Jona-Lasinio (NJL) model framework.

• Covariant Formulation: The BdG Hamiltonian for a 3DWS is rewritten in a covariant 8-dimensional representation. This involves defining 8-component Nambu spinors and constructing $8 \times 8$ Dirac matrices ($\Gamma_\mu, \Gamma_5, \Sigma_i$) that unify particle-hole and chirality spaces.
• Symmetry Analysis: The study distinguishes between two pairing channels:
  • BCS Pairing: Inter-node pairing (opposite chirality, zero momentum).
  • FFLO Pairing: Intra-node pairing (same chirality, finite momentum).
    The author demonstrates that while BCS pairing only breaks the gauge $U(1)$ symmetry, FFLO pairing spontaneously breaks both the gauge $U(1)$ and the axial $U(1)_A$ symmetries.
• NJL Framework: An effective Lagrangian with four-fermion interactions is constructed to analyze the spectrum of collective modes. By applying Fierz transformations and analyzing the T-matrix (Bethe-Salpeter equation) in various interaction channels (Scalar, Pseudo-scalar, Vector, Axial-vector), the authors identify the bound states (collective modes).
• Anomaly Calculation: The paper incorporates explicit chiral symmetry breaking (via a small bare mass term $M_0$) to calculate the mass of the pseudo-scalar mode using a superconducting analogue of the Gell-Mann–Oakes–Renner (GOR) relation. It then estimates the decay width of this mode into two photons via the axial anomaly, drawing parallels to the triangle anomaly in QCD.

3. Key Contributions

1. Unified Covariant BdG Formulation: The paper provides a novel 8-dimensional covariant formulation of the BdG Hamiltonian that treats both BCS and FFLO pairing channels on equal footing, explicitly encoding the finite-momentum character of the FFLO state through an axial shift ($q\Gamma_5$).
2. Identification of Axial Symmetry Breaking: It establishes that FFLO pairing is the specific mechanism that spontaneously breaks the axial $U(1)_A$ symmetry in Weyl superconductors, a feature absent in conventional BCS inter-node pairing.
3. Discovery of a Pseudo-Scalar Nambu-Goldstone (NG) Mode: The breaking of $U(1)_A$ gives rise to a pseudo-scalar NG mode (analogous to the pion in QCD). This mode corresponds to relative phase fluctuations between chiral pairing components.
4. Systematic Classification of Modes: Using the 8$\times$8 NJL framework, the paper predicts a full spectrum of collective excitations:
   • Pseudo-scalar mode (Phase mode, NG boson).
   • Scalar mode (Amplitude/Higgs mode).
   • Vector and Axial-vector modes (Analogous to $\rho$ and $a_1$ mesons).
5. Anomaly-Induced Decay Mechanism: The paper demonstrates that under explicit chiral symmetry breaking, the pseudo-scalar mode acquires a small mass and couples to gauge fields via the axial anomaly, allowing for a radiative decay channel into two photons.

4. Key Results

• Symmetry Breaking: In the FFLO state, the pairing term $\Delta_F \Sigma_1$ breaks both $U(1)$ gauge and $U(1)_A$ axial symmetries. In contrast, BCS pairing ($\Delta_B \Gamma_0$) breaks only the gauge symmetry.
• Mass Generation:
  • In the chiral limit ($M_0 = 0$), the pseudo-scalar mode is massless (Nambu-Goldstone).
  • With explicit symmetry breaking ($M_0 \neq 0$), the mode acquires a mass $M_{ps}$ governed by the relation:
    $$M_{ps}^2 F_{ps}^2 = M_0 \langle \bar{\Psi}\Psi \rangle$$
    (Analogous to the GOR relation).
  • Estimate: For typical parameters ($\Delta_F \approx 1$ meV), the mass is estimated at $M_{ps} \approx 3.3 \times 10^{-2}$ meV.
• Decay Rate: The mode decays into two photons via the axial anomaly. The estimated decay width is:
  $$\Gamma_{ps}^D \approx 9.74 \times 10^{-14} \text{ eV}$$
  This corresponds to a lifetime of $\tau \approx 6.8 \times 10^{-3}$ s.
• Bulk vs. Surface: The decay is strongly suppressed in the bulk due to the Meissner effect (Anderson-Higgs mechanism screening the photon). However, the paper suggests this decay could be observable via surface electromagnetic fields or at interfaces where screening is incomplete.

5. Significance

• QCD Analogy in Condensed Matter: The work successfully establishes a rigorous correspondence between low-energy QCD phenomena (chiral symmetry breaking, pion physics, axial anomalies) and topological superconductivity. It provides a "tabletop" realization of mesonic modes and anomaly-induced decays.
• Experimental Signature for FFLO: The existence of the pseudo-scalar mode and its specific anomaly-induced coupling to photons serves as a unique theoretical signature for FFLO pairing in Weyl superconductors. Since FFLO states are experimentally elusive in these materials, this decay channel offers a potential detection method (e.g., via nonlinear optical responses or surface photon emission).
• Unified Framework: The 8-dimensional covariant approach offers a powerful algebraic tool for classifying collective excitations in complex superconducting systems, bridging the gap between relativistic field theory and non-relativistic condensed matter physics.

In conclusion, the paper predicts that Weyl superconductors in the FFLO phase host a rich spectrum of collective modes, including a massive pseudo-scalar boson that decays into photons via the axial anomaly, providing a novel experimental probe for topological superconductivity and chiral symmetry breaking.