Interband pairing in two-band superconductors with spin-orbit and Zeeman couplings

This paper demonstrates that a Zeeman magnetic field can stabilize interband pairing in two-band superconductors with spin-orbit coupling, driving a transition from a conventional intraband state to a gapless, interband-dominated mixing state characterized by anomalous thermodynamic properties like a linear specific heat.

The Gist

The Big Picture: A Dance of Electrons

Imagine a superconductor as a giant, perfectly synchronized dance floor where electrons (the dancers) pair up to move without any friction. Usually, these dancers pair up with their immediate neighbors on the same "floor" (band). This is called intraband pairing, and it's the standard, easy way to dance.

However, this paper explores a rare and tricky scenario: Interband pairing. This is like a dancer on the first floor trying to pair up with a dancer on the second floor. Usually, this is hard because the two floors are at different heights (energies), making it energetically "expensive" to bridge the gap. So, physicists often ignore this possibility.

The Twist: The authors show that if you apply a strong magnetic field (a "Zeeman field"), you can force these two floors to line up perfectly, making the "inter-floor" dance not only possible but actually the best way to dance.

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The Setup: The Honeycomb Lattice

The researchers used a model based on a honeycomb lattice (like a beehive or graphene).

• The Structure: Imagine a two-story building where the stairs are slightly twisted (this is Spin-Orbit Coupling). This twist means the "up" and "down" dancers have different rules.
• The Magnetic Field: They turned on a strong magnetic field. Think of this as a strong wind blowing through the building. This wind pushes the "up-spin" dancers to one side and the "down-spin" dancers to the other, effectively splitting the energy levels of the two floors.

The Discovery: The "Mixing" State

Usually, when you apply a magnetic field, it breaks the superconducting dance (this is called the Pauli limit). The dancers get too agitated to pair up, and the superconductivity dies.

But in this specific setup, something magical happens:

1. The Alignment: The magnetic wind pushes the "down-spin" dancers on the top floor down, and the "up-spin" dancers on the bottom floor up, until they are at the exact same height.
2. The Switch: Once they are aligned, the dancers stop pairing with their neighbors on the same floor. Instead, they start pairing across the gap between the floors.
3. The Result: A new state emerges called the "Mixing State." It's a hybrid dance where the electrons are paired in a complex way, mixing traits from both floors.

The Weird Outcome: The "Gapless" Dance

In a normal superconductor, there is a "gap" in energy. It's like a moat around a castle; you need a certain amount of energy to jump over the moat and disturb the dancers. If you don't have that energy, the castle is perfectly safe and quiet.

In this new Mixing State, the moat disappears.

• The Gapless Spectrum: The dancers can be disturbed with zero energy. It's as if the moat has been filled in, and you can walk right into the castle.
• The Consequence: Because there are always dancers ready to move (even at very low temperatures), the material behaves strangely.
  • Normal Superconductor: As it gets colder, the specific heat (how much energy it takes to warm it up) drops to almost zero because the dancers are frozen in place.
  • This Mixing State: As it gets colder, the specific heat stays high and grows linearly. It's like the dancers are still jittery and moving even in the deep freeze.

Why Does This Matter?

This paper is important for two main reasons:

1. It's a New Way to Control Superconductors: It shows that you don't need exotic, complicated materials to get weird superconducting states. You just need a standard multi-band material and a strong magnetic field to "tune" the energy levels into alignment.
2. Experimental Fingerprints: The authors give scientists a way to spot this state in real life. If you see a superconductor that:
   • Survives in very strong magnetic fields (where it should have died).
   • Has a specific heat that goes up linearly with temperature (instead of dropping).
   • Has a finite number of electrons moving at zero energy.
     ...then you have likely found this "Mixing State."

The 3D Extension: The "Charge-Neutral" Analogy

The paper also looks at a 3D version of this model. They clarify that while this might not happen in a solid metal block (because magnetic fields usually mess up the orbits of electrons in 3D), it is perfectly real in ultracold atomic gases.

Think of these gases as "synthetic matter" created in a lab. Scientists can create artificial magnetic fields that push the atoms without the messy orbital effects found in solids. This makes the 3D model a perfect playground to test these ideas with atoms instead of electrons.

Summary in One Sentence

By using a strong magnetic field to align energy levels in a two-band material, the researchers discovered a new type of superconductivity where electrons pair across bands, creating a "gapless" state that stays active and jittery even at absolute zero, defying the rules of traditional superconductors.

Technical Summary

1. Problem Statement

In conventional multiband superconductivity theories, interband pairing (Cooper pairing between electrons in different bands) is typically neglected because it is energetically less favorable than intraband pairing (pairing within the same band). While interband pairing is known to play a role in various unconventional phenomena (e.g., Josephson tunneling, anomalous Hall effects), explicit microscopic realizations where interband pairing becomes the dominant superconducting instability remain underexplored.

The authors address the following questions:

• Can a simple, local attractive interaction stabilize a superconducting state dominated by interband pairing?
• How do Spin-Orbit Coupling (SOC) and an external Zeeman magnetic field influence the competition between intraband and interband pairing?
• What are the unique thermodynamic signatures of such a state, particularly in systems with locally broken inversion symmetry?

2. Methodology

The authors employ a minimal theoretical framework using tight-binding models on specific lattice geometries:

• 2D Model: A honeycomb lattice (monolayer) with two sublattices (A and B). This system naturally lacks local inversion symmetry, inducing intrinsic SOC.
• 3D Model: A non-symmorphic two-layer hexagonal lattice (similar to SrPtAs), where individual layers lack inversion symmetry but the crystal as a whole preserves it. This serves as a model for charge-neutral fermionic superfluids in optical lattices.
• Hamiltonian:
  • Normal State: Includes kinetic energy, inter-sublattice hybridization ($\epsilon_{ck}$), intrinsic SOC, and a Zeeman term ($h$) representing an external magnetic field. Orbital effects (vector potential) are neglected, assuming the Zeeman effect dominates (valid for ultrathin films or synthetic systems).
  • Interaction: A minimal on-site attractive interaction ($g$) is introduced. While local in real space, its projection onto the band basis generates multiple pairing channels due to the multiband structure and SOC.
• Analysis:
  • The problem is solved using the mean-field approximation in the band basis.
  • The superconducting order parameter is decomposed into two symmetry-distinct components: $\Delta_0$ (conventional intraband $s$-wave) and $\Delta_1$ (a mixing channel involving interband pairing).
  • The linearized gap equations are solved to determine the transition temperature ($T_c$) and phase stability.
  • Thermodynamic properties (Density of States and Specific Heat) are calculated numerically for various Fermi surface configurations.

3. Key Contributions

1. Zeeman-Driven Stabilization: The paper demonstrates that a Zeeman magnetic field can stabilize a Mixing state where interband pairing dominates, even with a simple on-site attractive interaction. This occurs when the Zeeman splitting brings spin-split branches from different bands into near degeneracy at the Fermi level.
2. Dual Role of the Magnetic Field: The Zeeman field acts dually:
   • It suppresses the conventional intraband $s$-wave state (via the Pauli limiting effect).
   • It promotes interband pairing by reducing the energy mismatch between the spin-down branch of one band and the spin-up branch of another.
3. Role of SOC and Hybridization: The authors clarify that while SOC (from broken inversion symmetry) and inter-sublattice hybridization are necessary to generate interband pairing components, they do not inherently enhance the stability of the Mixing state. The primary driver for the phase transition is the Zeeman-induced band degeneracy.
4. Gapless Excitations: The Mixing state is shown to possess an intrinsically gapless Bogoliubov quasiparticle spectrum, distinct from the fully gapped conventional $s$-wave state.

4. Key Results

Phase Diagram and Transition

• Low Field: The system favors the conventional intraband $s$-wave state ($\Delta_0$).
• High Field: As the magnetic field approaches and exceeds the Pauli limit of the $s$-wave state, the system undergoes a transition to the Mixing state ($\Delta_1$).
• Condition for Stability: The Mixing state becomes energetically favorable only when the Zeeman field creates a near-degeneracy between the spin-down branch of one band and the spin-up branch of the other. This effect is significantly amplified if this degeneracy occurs near a Van Hove singularity, where the Density of States (DOS) is high.

Quasiparticle Spectrum and Density of States (DOS)

• Gapless Nature: Unlike conventional superconductors, the Mixing state exhibits a finite zero-energy DOS.
• Temperature Dependence: Remarkably, the zero-energy DOS increases as the temperature is lowered. This is contrary to fully gapped superconductors, where the DOS is suppressed at low temperatures.
• Origin: The gapless nature arises from the crossing of Zeeman-split bands and the specific branch structure of the Bogoliubov spectrum, not from topological protection (as the state preserves time-reversal symmetry explicitly broken only by the external field).

Thermodynamic Signatures

• Specific Heat: The most distinct experimental signature is the specific heat behavior at low temperatures.
  • Conventional $s$-wave: Exponential suppression ($C_s \sim e^{-\Delta/T}$).
  • Mixing State: Linear dependence ($C_s \propto T$).
• This $T$-linear specific heat is a direct consequence of the finite residual DOS at zero energy.

5. Significance and Implications

• Experimental Fingerprint: The paper provides clear, experimentally accessible signatures (specifically the $T$-linear specific heat and gapless DOS) to distinguish the Mixing state from conventional superconductors in monolayer honeycomb systems (e.g., graphene on WS$_2$, stanene) or ultracold atomic gases.
• Mechanism for Gapless Superconductivity: It offers a clean mechanism for realizing gapless superconductivity driven by band structure engineering and magnetic fields, distinct from impurity-induced gapless states (Fulde-Maki) or time-reversal symmetry breaking states.
• Synthetic Platforms: The 3D model suggests that these phenomena can be explored in ultracold atomic systems (charge-neutral fermionic superfluids) where effective Zeeman fields can be tuned without orbital coupling complications.
• Theoretical Insight: The work challenges the assumption that interband pairing is always a minor correction, showing it can become the dominant instability under specific magnetic and structural conditions.

In summary, Shingu and Goryo establish that Zeeman-driven near-degeneracy in multiband systems with broken inversion symmetry can stabilize a novel Mixing superconducting state characterized by interband dominance, gapless excitations, and anomalous linear-in-temperature specific heat.