Pairing-induced Momentum-space Magnetism and Its Implication In Optical Anomalous Hall Effect In Chiral Superconductors

This paper generalizes Onsager's relation for a single-orbital spinful Hamiltonian to identify two distinct mechanisms of pairing-induced momentum-space magnetism—arising from Cooper pair angular momentum and spin-orbit coupling, respectively—that drive the optical anomalous Hall effect in chiral superconductors, thereby highlighting the essential role of spin degrees of freedom.

The Gist

Imagine you are trying to understand a mysterious dance happening inside a special kind of material called a chiral superconductor.

In the world of physics, these materials are like a ballroom where electrons (the dancers) pair up to move in perfect unison without any friction. This is "superconductivity." But in a chiral superconductor, these pairs don't just dance; they spin in a specific direction, breaking the symmetry of time (like a clock that only runs forward, never backward).

Scientists have been trying to see this dance using a special camera called the Magneto-Optical Kerr Effect (MOKE). This camera shines light on the material and watches how the light twists. If the light twists, it's a sign that the material is chiral.

However, there's a big problem. For a long time, physicists thought you needed a complex, multi-layered dance floor (multiple "orbitals" or paths for electrons) to make the light twist. They believed that if the dance floor was simple (just one path), the light would go straight through, no matter how much the electrons spun.

This paper says: "Wait a minute! We missed a crucial partner in the dance: Spin."

Here is the breakdown of their discovery, using simple analogies:

1. The Missing Ingredient: Spin

Think of an electron not just as a ball, but as a ball with a tiny, spinning top attached to it. This is its spin.
Previous theories treated the spin as a minor accessory, like a hat on a dancer. This paper argues that the spin is actually a lead dancer. Even in a simple, single-path system, if you pay attention to how the spin interacts with the electron's movement, you can get the light to twist.

2. The "Ghost Magnet" (Momentum-Space Magnetism)

In normal magnets (like a fridge magnet), the magnetism comes from the electrons lining up their spins in the same direction, creating a real magnetic field.

In these superconductors, there is no real magnetic field. Instead, the authors discovered a "Ghost Magnet" that exists only in the momentum of the electrons.

• The Analogy: Imagine a crowd of people running in a stadium. In a normal magnet, everyone is facing North. In this "Ghost Magnet," the people aren't facing North, but their running paths create a swirling pattern that acts as if there is a magnetic field. It's a magnetic effect that lives in the "map of where they are going," not in the physical space they occupy.

3. Two Ways to Create the Ghost Magnet

The paper identifies two different ways this "Ghost Magnet" is created, depending on how the electron pairs dance:

• Type A: The Non-Unitary Dance (The Spin-Heavy Pair)
  Imagine two dancers holding hands and spinning so fast that they generate their own angular momentum. This creates a strong, unified "Ghost Magnet" that points in one direction (Ferromagnetism). This was known before, but the paper confirms it works even in simple systems.

• Type B: The Unitary Dance (The Hidden Partner)
  This is the paper's big new discovery. Imagine two dancers who are perfectly synchronized (no net spin), but they are dancing on a floor that is slightly tilted or twisted (due to Spin-Orbit Coupling).

  • The Analogy: Think of a spinning top on a wobbly table. Even if the top spins perfectly, the wobble of the table makes it lean.
  • In this case, the "wobble" is the interaction between the electron's spin and its path. Even though the pair is "balanced," the tilt of the floor creates a "Ghost Magnet." The authors realized this mechanism was largely overlooked because people thought it required complex multi-layered floors, but it works on simple ones too.

4. The Surprise: The Sideways Twist

The most exciting part is what happens when you have this "Ghost Magnet."

Usually, if you have a magnet pointing Up, the light twists in a specific way. But the authors found that with the "Unitary Dance" (Type B), you can create a "Ghost Magnet" that points sideways (in the plane of the material).

• The Result: This creates a strange optical effect where the light twists in a way that doesn't fit the old rules. It's like a magnet pointing East, but the light reacts as if the magnet is pointing North. This is called the In-Plane Optical Anomalous Hall Effect.

Why Does This Matter?

• Solving a Mystery: It explains why materials like Strontium Ruthenate (Sr2RuO4) show these twisting light signals, even if they don't have the complex structures scientists thought were necessary.
• Quantum Computing: Chiral superconductors are the holy grail for building quantum computers (specifically for storing "Majorana fermions," which are robust quantum bits). Understanding exactly how they work helps us build better quantum computers.
• New Physics: It unifies the physics of magnets and superconductors, showing that "spin" is the universal key to unlocking these strange optical effects.

In a nutshell:
The authors found that you don't need a complicated dance floor to make light twist in superconductors. You just need to realize that the electrons' internal "spin" creates a magnetic ghost in their movement patterns. This ghost can point in weird directions, causing light to twist in new and unexpected ways, opening the door to better quantum technologies.

Technical Summary

1. Problem Statement

The intrinsic mechanism behind the magneto-optical Kerr effect (MOKE) observed in chiral superconductors (e.g., Sr$_2$RuO$_4$, heavy fermion systems) remains a subject of debate.

• The Constraint: Due to Galilean invariance, a clean single-band chiral superconductor cannot support a non-zero Kerr signal. Previous theories relied on multi-orbital degrees of freedom (inter-orbital pairing) to break this invariance, but this approach faces skepticism regarding the smallness of required interband pairing and often treats spin as merely an auxiliary renormalization factor.
• The Gap: There is a lack of a complete understanding of how the spin degree of freedom (specifically spin-orbit coupling) drives the optical anomalous Hall effect (OAHE) in single-orbital chiral superconductors. The relationship between time-reversal symmetry (TRS) breaking in the Cooper channel and the resulting Hall conductivity needs a rigorous theoretical framework.

2. Methodology

The authors employ a combination of symmetry analysis and effective Hamiltonian techniques:

• Generalized Onsager Relations: They derive generalized Onsager reciprocal relations for the optical anomalous Hall conductivity ($\sigma_{xy}^H$) in single-orbital, spinful systems. By utilizing time-reversal ($T$) and gauge symmetries, they establish constraints on how $\sigma_{xy}^H$ depends on the pairing potential $\Delta(k)$.
• Down-folding Method: To isolate the role of pairing, they use a down-folding procedure to project out the hole sector of the Bogoliubov-de Gennes (BdG) Hamiltonian. This yields an effective electron Hamiltonian ($\hat{H}_{eff}^e$) that incorporates the effects of pairing as a perturbation.
• Symmetry Analysis: They analyze the structure of the effective Hamiltonian to identify terms that break time-reversal symmetry and act as effective magnetic fields in momentum space.

3. Key Contributions

The paper introduces the concept of "Momentum-space Magnetism" induced by superconducting pairing, unifying the mechanisms of OAHE in magnetic materials and chiral superconductors.

• Identification of Two Mechanisms: The authors identify two distinct sources of momentum-space magnetism ($\mathbf{m}$) in the effective Hamiltonian:
  1. Non-unitary Pairing ($m_{NU}$): Arises from the angular momentum of the Cooper pair (where $\mathbf{d} \neq \mathbf{d}^*$). This is analogous to standard ferromagnetism.
  2. Unitary Pairing ($m_{U}$): A previously overlooked mechanism. It arises from the joint effect of spin-orbit coupling (SOC) and unitary pairing (where the pairing is a mix of even and odd parity, e.g., $s$-wave + $p$-wave). This does not require a net Cooper pair angular momentum.
• Generalized Onsager Constraints: They prove that the optical Hall conductivity is an odd function of the effective magnetic order parameters derived from the pairing ($\mathbf{d} \times \mathbf{d}^*$ and $\text{Im}(d_0 d^*)$). This establishes a direct link between the TRS-breaking pairing and the Hall response.
• In-plane OAHE in Superconductors: They demonstrate that unitary pairing can generate a momentum-space magnetism that leads to an in-plane optical anomalous Hall effect, where the Hall deflection is perpendicular to the magnetization direction, analogous to recent findings in normal magnetic metals.

4. Key Results

The authors validate their theoretical framework using concrete model systems:

• Ferromagnetism from Non-unitary Pairing:

  • In systems like Sr$_2$RuO$_4$ (with $D_{4h}$ symmetry), non-unitary $p$-wave pairing generates a momentum-space magnetism $\mathbf{m}_{NU} \propto (0, 0, \Delta_1 \Delta_2 k^2)$.
  • The resulting total spin magnetization scales quadratically with both the pairing strength and SOC strength.
  • The optical Hall conductivity follows the expected linear dependence on this magnetization.

• Ferromagnetism from Unitary Pairing:

  • In non-centrosymmetric systems like $\alpha$-CaPtAs (with $C_{4v}$ symmetry), a mixed $s$-wave and $p$-wave pairing (unitary) combined with Rashba SOC generates $\mathbf{m}_{U} \propto (0, 0, -\lambda \Delta_s \Delta_p k^2)$.
  • Crucially, the magnetization here is an odd function of the pairing strength and SOC, distinct from the non-unitary case.
  • This mechanism produces a non-zero Kerr signal without requiring multi-orbital physics.

• Non-collinear Antiferromagnetism:

  • In $C_{3v}$ symmetric systems (e.g., Bi/Ni heterostructures), the mismatch between the momentum dependence of SOC and the pairing vector leads to a complex, non-collinear antiferromagnetic spin texture in momentum space.
  • In-plane OAHE: The resulting optical Hall conductivity ($\sigma_{xy}$) shows a linear dependence on the in-plane spin polarization ($S_y$) rather than the out-of-plane component ($S_z$). This represents a "superconductor version" of the in-plane anomalous Hall effect, breaking the conventional empirical law where Hall signals are typically tied to out-of-plane magnetization.

5. Significance

• Resolving the Single-Orbital Paradox: The work demonstrates that a single-orbital chiral superconductor can exhibit a Kerr signal, provided spin-orbit coupling and specific pairing symmetries are present. This removes the strict necessity for multi-orbital interband pairing.
• Re-evaluating Spin's Role: It elevates the spin degree of freedom from an auxiliary role to a central driver of the optical anomalous Hall effect in superconductors.
• New Experimental Signatures: The prediction of an in-plane optical anomalous Hall effect (where the Hall signal depends on in-plane magnetization) offers a new, distinct experimental signature to distinguish between non-unitary and unitary pairing mechanisms in candidate chiral superconductors.
• Theoretical Unification: By framing the effect as "momentum-space magnetism," the paper unifies the understanding of Hall effects in magnetic metals and chiral superconductors, showing they share a common origin in spin-splitting of the electronic Hamiltonian.