Magnetic field induced transition from nodeless to nodal superconductivity in $β$-PdBi$_{2}$

This paper presents a microscopic theory explaining the magnetic field-induced transition from nodeless s-wave to nodal p-wave superconductivity in $\beta$-PdBi$_2$ as a result of Zeeman splitting altering the pairing landscape and creating gapless excitations.

The Gist

Imagine a superconductor as a busy dance floor where electrons pair up to move in perfect unison, gliding without any friction. In the material β-PdBi2, these electrons are dancing to two different tunes, or "channels," depending on the environment.

Here is the story of how a magnetic field changes the music, forcing the dancers to switch styles, and eventually causing some of them to trip and stop moving smoothly.

The Two Dance Styles

In this material, there are two ways the electrons can pair up:

1. The "S-Wave" Dance (The Smooth Glide):

   • How it works: At low or no magnetic fields, the electrons pair up in a simple, round, "s-wave" style. Think of this like a perfectly synchronized ballroom dance where everyone holds hands in a circle.
   • The Result: The dance floor is completely smooth. There are no "holes" or gaps in the rhythm. In physics terms, this is a fully gapped state, meaning it takes a certain amount of energy to break the dance. It's very stable.

2. The "P-Wave" Dance (The Twisting Spin):

   • How it works: There is a second, more complex style called "p-wave." Imagine this as a dance where partners spin and twist, changing direction based on where they are on the floor.
   • The Result: This style is a bit more fragile at first but has a secret superpower: it is much better at handling magnetic fields.

The Magnetic Field: The DJ Changing the Music

The researchers discovered what happens when you turn up the magnetic field (the "DJ" changing the vibe):

• At Zero Field: The "S-Wave" dance is the winner. It's the most comfortable style, so the electrons stick to it. The energy required to stop them is high, and the dance floor is solid.
• At Low Magnetic Fields: The magnetic field starts to pull the electrons apart. It's like the DJ suddenly playing a song that makes the S-Wave dancers stumble. Because the magnetic field splits the energy levels of the electrons (like separating two groups of dancers who were previously standing on the same spot), the S-Wave dance becomes very difficult to maintain.
• The Switch: As the magnetic field gets stronger, the S-Wave dance becomes too hard to keep up. The electrons suddenly switch to the P-Wave style. This style is more robust against the magnetic pull.

The "Nodal" Twist: When the Floor Gets Bumpy

Here is the most surprising part of the story. When the electrons switch to the P-Wave dance under a strong magnetic field, the dance floor doesn't stay smooth forever.

• The Split: The magnetic field splits the "Fermi surfaces" (the boundaries of the dance floor) into two separate rings.
• The Gap Appears: In the P-Wave state, the electrons can dance perfectly on these rings, but in the space between the rings, the dance breaks down.
• The "Nodal" Points: Imagine the dance floor has four specific spots between the rings where the music stops completely. At these spots, the electrons can move without any energy cost. In physics, we call these nodes.
  • At first, as the field increases, these "stop spots" appear.
  • If the field gets even stronger, each of these four spots splits into two, creating eight spots where the dance floor is "bumpy" or "gapless."

Why This Matters (According to the Paper)

The paper explains that recent experiments on β-PdBi2 showed exactly this behavior:

1. Low Field: The data looked like a smooth, solid U-shape (indicating a fully gapped, S-wave state).
2. High Field: The data changed to a V-shape with a spike in the middle (indicating the presence of those "stop spots" or nodes).

The authors' theory is that this isn't magic; it's a natural competition. The magnetic field kills the easy S-wave dance, forcing the electrons into the complex P-wave dance. But the P-wave dance has a flaw: under strong magnetic pressure, it develops "holes" (nodes) in the energy spectrum between the split electron rings.

Summary Analogy

Think of the electrons as cars on a highway:

• Low Magnetic Field: All cars are driving in a single, wide, smooth lane (S-wave). No traffic jams, no potholes.
• Increasing Magnetic Field: The road starts to split into two lanes. The cars trying to stay in the single lane (S-wave) crash or stop.
• High Magnetic Field: The cars switch to a new driving pattern (P-wave) that handles the split lanes well. However, in the middle of the road, between the two new lanes, the pavement disappears. These are the nodes. The cars can drive freely there, but the smooth, protected surface is gone.

The paper concludes that this specific mechanism—where a magnetic field splits the electron bands and forces a switch to a P-wave state that naturally develops gaps in the middle—is the reason we see this transition in β-PdBi2.

Technical Summary

Technical Summary: Magnetic Field Induced Transition from Nodeless to Nodal Superconductivity in $\beta$-PdBi$_2$

Problem Statement
Recent tunneling spectroscopy measurements on the layered compound $\beta$-PdBi$_2$ have revealed a magnetic field-induced phase transition from a fully gapped superconducting state to a nodal state. At low in-plane magnetic fields, the material exhibits a fully gapped spectrum consistent with $s$-wave pairing. Above a critical field, the spectrum develops a V-shape and zero-bias conductance, indicative of nodal quasiparticles. This behavior contradicts the expectations of a single-channel BCS framework, where a magnetic field typically suppresses superconductivity rather than altering the gap symmetry to induce nodes. The central problem addressed is to provide a microscopic theory explaining this field-induced transition from a nodeless to a nodal state in a multi-band system with strong spin-orbit coupling (SOC).

Methodology
The authors develop a microscopic theory for $\beta$-PdBi$_2$ using an effective low-energy continuum model derived from a tight-binding description of Bi $p$-orbitals. The model Hamiltonian includes kinetic energy, interlayer coupling ($\varepsilon$), Rashba spin-orbit coupling ($\alpha$), and a Zeeman term ($h$) directed along the $x$-axis.

The theoretical approach involves the following steps:

1. Diagonalization: The non-interacting Hamiltonian is diagonalized first at zero field (yielding two doubly degenerate bands) and then at finite magnetic field (yielding four non-degenerate, non-circular bands).
2. Interaction Analysis: A repulsive four-fermion density-density interaction is introduced. The authors analyze the pairing channels by projecting this interaction into the BCS channel within the diagonalized basis.
3. Renormalization: The authors incorporate Kohn-Luttinger renormalization (particle-hole channel corrections) to determine the effective attractive and repulsive components of the interaction. They also consider electron-phonon interactions as a mechanism to enhance $s$-wave pairing at zero field.
4. Gap Equations: The authors solve linearized and non-linear gap equations for both $s$-wave (interband) and $p$-wave (intraband) channels.
5. Free Energy and Spectral Analysis: The free energies of the competing states are computed to determine the ground state stability. The Bogoliubov-de Gennes (BdG) excitation spectrum is analyzed to identify the emergence of nodal points. Finally, the differential tunneling conductance ($dI/dV$) is calculated to compare directly with experimental data.

Key Contributions and Results

• Competition Between Pairing Channels: The study identifies two distinct attractive pairing channels:

  • $s$-wave (Interband): Involves fermions from different bands. At zero field, the bands are degenerate, and electron-phonon interactions (assumed to dominate over Kohn-Luttinger repulsion) stabilize this channel, resulting in a fully gapped state.
  • $p$-wave (Intraband): Involves fermions from the same band. The attraction in this channel emerges solely from Kohn-Luttinger renormalization, where the dressed attractive component exceeds the repulsive one.

• Field-Induced Symmetry Transition:

  • At zero field, the $s$-wave channel is energetically favorable.
  • As an in-plane magnetic field is applied, the Zeeman splitting lifts the degeneracy of the bands involved in $s$-wave pairing. This strongly suppresses the $s$-wave channel (similar to the Clogston-Chandrasekhar limit in conventional superconductors).
  • The $p$-wave channel, involving intraband pairing, remains robust against this splitting. Consequently, the system undergoes a transition where the $p$-wave channel becomes the ground state at higher fields.

• Emergence of Nodal Points:

  • In the high-field $p$-wave phase, the excitation spectrum evolves. As the field increases, the minimum excitation energy reaches zero at four specific points in momentum space, located between the split Fermi surfaces of the normal state.
  • Upon further increasing the field, these four nodal points split into eight, creating a nodal Bogoliubov quasiparticle spectrum. The authors emphasize that these are point-like nodes, not extended Bogoliubov Fermi surfaces.

• Comparison with Experiment:

  • The calculated free energy difference confirms that the $s$-wave state is stable at low fields, while the $p$-wave state becomes favorable at higher fields.
  • The calculated differential tunneling conductance reproduces the experimental observations: a U-shaped spectrum at low fields (fully gapped) transitioning to a V-shaped spectrum with increasing zero-bias conductance at higher fields.
  • The model accounts for the transition occurring at a critical field ($h_c \approx 0.2$ T) by phenomenologically introducing a magnetic-field-dependent reduction in the effective $s$-wave interaction strength, likely due to orbital pair-breaking effects not explicitly included in the mean-field Zeeman treatment.

Significance
The paper argues that the observed field-induced nodal superconductivity in $\beta$-PdBi$_2$ is a natural consequence of the competition between $s$-wave and $p$-wave pairing channels with different sensitivities to Zeeman-induced band splitting. The transition to a nodal state does not require additional symmetry breaking; rather, it arises intrinsically from the multi-band nature of the material and the specific topology of the $p$-wave gap function in the presence of a magnetic field. The authors conclude that Zeeman-induced Fermi surface splitting serves as a general mechanism for the development of nodal excitations in multi-band superconductors, offering a framework to interpret field-tuned phase diagrams in related materials.