Semiclassical theory for the orbital magnetic moment of superconducting quasiparticles

This paper employs a semiclassical approach to derive and verify the orbital magnetic moment of Bogoliubov quasiparticles, demonstrating that a nontrivial superconducting pairing gap alone cannot generate this moment and applying the resulting formula to analyze its effects on energy spectra, local density of states, and the orbital Nernst effect in chiral $d$-wave superconductors.

The Gist

The Big Picture: The "Ghost" Dancers in a Superconductor

Imagine a superconductor as a giant, perfectly synchronized dance floor. In a normal metal, electrons are like individual dancers bumping into each other. But in a superconductor, they pair up (Cooper pairs) and move as a single, fluid unit, gliding without friction.

However, if you poke this perfect dance floor (by adding energy or a magnetic field), you create "quasiparticles." Think of these quasiparticles as ghost dancers. They aren't just electrons; they are a quantum superposition—a mix of an electron and a "hole" (the absence of an electron). They behave like waves that can spin and orbit.

The authors of this paper wanted to answer a specific question: When these ghost dancers spin around, do they create their own tiny magnetic field?

In regular materials, we know that spinning electrons create magnetism (like a tiny bar magnet). The authors discovered that in superconductors, this "magnetic spin" is much more complicated and behaves very differently than we expected.

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Key Concept 1: The "Self-Rotating" Wave

The Analogy: The Spinning Top vs. The Swirling Vortex

In regular physics, we think of an electron's magnetic moment like a spinning top. If it spins, it has a magnetic field.

In this paper, the authors used a "semiclassical" approach. Imagine the quasiparticle not as a single point, but as a swirling vortex of water (a wavepacket).

• As this vortex moves across the dance floor, it also spins on its own axis.
• This self-rotation creates a tiny magnetic moment.
• The authors calculated exactly how strong this spin is based on the shape of the dance floor (the energy bands) and the pairing rules of the dancers.

Key Concept 2: The "Charge Confusion" Problem

The Analogy: The Two-Faced Coin

Here is where it gets tricky. A normal electron has a fixed negative charge. A "hole" has a positive charge.

• A quasiparticle is a mix of both. It's like a coin that is half-heads and half-tails, and it keeps flipping back and forth.
• Because its charge is constantly shifting between positive and negative, its "center of charge" (where the magnetism comes from) is in a different spot than its "center of probability" (where the dancer actually is).

The Discovery:
The authors found that because of this "charge confusion," the magnetic moment of a superconducting quasiparticle is not just a simple copy of the magnetic moment of a regular electron.

• Regular Electron: Spin = Magnetic Moment.
• Superconducting Quasiparticle: Spin $\neq$ Magnetic Moment. The math is messier because the "charge" part of the dancer is playing hide-and-seek.

Key Concept 3: The "Chiral" Surprise

The Analogy: The Twisted Scarf

Scientists often study "chiral" superconductors, where the electron pairs are twisted like a spiral or a scarf.

• Old Expectation: If you twist the scarf (chiral pairing), you expect the dancers to spin wildly, creating a huge magnetic moment.
• The Paper's Finding: Surprisingly, just twisting the scarf isn't enough!
  • If the dance floor is perfectly symmetrical (like a flat, round table), the twisting creates a lot of "geometric twist" (Berry curvature), but zero net magnetic moment. The spins cancel each other out perfectly.
  • To get a magnetic moment, you need to break the symmetry of the floor itself (like having a bumpy table or a current flowing across it).

Why this matters: It's like saying that just because a dancer is spinning in a circle, it doesn't mean they are generating a magnetic field. The shape of the room matters just as much as the dance move.

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What Does This Actually Do? (The Effects)

The authors showed that this weird magnetic moment has real-world consequences:

1. The Energy Shift (The "Magnetic Push")

When you apply a real magnetic field to the superconductor, these ghost dancers feel a push.

• Analogy: Imagine the dance floor is slightly tilted by the magnetic field. The dancers on one side of the room get pushed down (lower energy), and those on the other side get pushed up (higher energy).
• Result: This changes the energy spectrum of the material. If you could measure the energy of the dancers with a microscope (like a scanning tunneling microscope), you would see these shifts.

2. The Orbital Nernst Effect (The "Thermal Wind")

This is a transport phenomenon. Imagine you heat one side of the superconductor.

• Analogy: Usually, heat makes things move randomly. But here, because the dancers have this special "magnetic spin" and are moving on a "twisted" floor, the heat creates a sideways wind.
• Result: A temperature difference creates a flow of "orbital magnetic moment" sideways. It's like a thermal wind that pushes magnetic fields to the side, which could be useful for new types of sensors or energy converters.

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The Honeycomb Model (The Test Drive)

To prove their theory, the authors built a computer model of a "honeycomb" lattice (like a beehive or graphene).

• They put a "chiral d-wave" pairing (a specific type of twisted dance) on this honeycomb.
• The Result: They mapped out where the magnetic moment was strongest.
  • The Twist: The spots where the "geometric twist" (Berry curvature) was strongest were not the same spots where the "magnetic moment" was strongest.
  • The Lesson: In regular materials, these two things usually go hand-in-hand. In superconductors, they can be completely different. It's like finding that the loudest part of a song (the beat) is in a different room than the part where the singer is standing.

Summary for the General Audience

1. Superconductors have "ghost dancers" (quasiparticles) that are a mix of electrons and holes.
2. These dancers have a unique magnetic spin (orbital magnetic moment) caused by their self-rotation.
3. The "Charge Confusion" (being half-electron, half-hole) makes this magnetic moment behave very differently than in normal metals.
4. Twisting the dance (chirality) isn't enough to create magnetism; you also need an asymmetrical dance floor.
5. This creates new effects: It shifts energy levels and creates a "thermal wind" (Orbital Nernst effect) that could be used in future technology.

In a nutshell: The authors wrote the "instruction manual" for how these ghost dancers generate magnetism, revealing that the rules are much stranger and more complex than we previously thought.

Technical Summary

1. Problem Statement

While the orbital magnetic moment (OMM) of Bloch electrons in normal metals is well-understood and plays a crucial role in various transport and magnetic phenomena, its counterpart in superconducting systems remains theoretically ambiguous.

• The Gap: In superconductors, quasiparticles are described by Bogoliubov-de Gennes (BdG) Hamiltonians. Unlike normal electrons, BdG quasiparticles are quantum superpositions of electrons and holes, leading to non-conservation of charge.
• The Conflict: Previous studies using different methods (Wannier functions, linear response) have yielded conflicting results regarding the OMM of superconducting quasiparticles. A key difficulty is the distinction between the orbital magnetic moment (related to charge distribution rotation) and the orbital angular momentum (related to probability distribution rotation).
• The Question: Does a chiral superconducting gap (which induces nontrivial Berry curvature) automatically generate a non-zero orbital magnetic moment, as it does for orbital angular momentum in normal systems?

2. Methodology

The authors employ a multi-faceted theoretical approach to derive and verify the OMM:

• Semiclassical Wavepacket Approach:

  • They construct a Bogoliubov quasiparticle wavepacket using BdG eigenstates.
  • They calculate the energy correction of this wavepacket to the linear order of an external magnetic field ($\mathbf{B}$).
  • The coefficient of the linear energy term ($-\mathbf{B} \cdot \mathbf{m}$) is defined as the orbital magnetic moment $\mathbf{m}$.
  • They derive an analytical expression involving the gauge-invariant momentum derivative and the particle-hole structure of the BdG Hamiltonian.

• Linear Response Theory (Verification):

  • To validate the semiclassical result, they perform a full quantum mechanical linear response calculation.
  • They introduce a spatially varying magnetic field (long-wavelength limit) and calculate the correction to the grand thermodynamic potential.
  • This approach confirms the semiclassical formula, ensuring the result is robust and gauge-invariant.

• Model Calculation:

  • They apply their derived formula to a tight-binding honeycomb lattice model with chiral $d$-wave pairing.
  • They numerically calculate the momentum-space distribution of the OMM, the modified energy spectrum, the density of states (DOS), and the orbital Nernst coefficient.

3. Key Contributions and Theoretical Results

A. The Analytical Formula for OMM

The central result is the derived expression for the orbital magnetic moment of the $n$-th BdG band:
$$\mathbf{m}_n(\mathbf{k}) = \frac{e}{2\hbar} \sum_{n' \neq n} \text{Im} \left[ \frac{\langle \phi_n | \partial_{\mathbf{k}} \hat{H}_{\mathbf{k}} | \phi_{n'} \rangle \times \langle \phi_{n'} | \tau_z \partial_{\mathbf{k}} \hat{H}^d_{\mathbf{k}} | \phi_n \rangle}{E_{n',\mathbf{k}} - E_{n,\mathbf{k}}} \right]$$

• Distinctive Features:
  1. $\tau_z$ Factor: The formula includes a $\tau_z$ matrix (Pauli matrix in particle-hole space), reflecting the effective charge of the quasiparticle.
  2. Diagonal Block Dependence: Crucially, the term $\partial_{\mathbf{k}} \hat{H}^d_{\mathbf{k}}$ involves only the diagonal blocks of the BdG Hamiltonian (the electron and hole parts), not the off-diagonal pairing gap $\Delta$.
  3. Implication: The OMM cannot arise solely from the momentum dependence of the superconducting gap function (chiral pairing). It requires a specific interplay between the gap symmetry and the electron band structure (specifically, asymmetry in the electron dispersion $\xi_{\mathbf{k}} \neq \xi_{-\mathbf{k}}$ or internal band structure).

B. Distinction from Orbital Angular Momentum (OAM)

The authors highlight a fundamental difference between OMM and OAM in superconductors:

• OAM: Depends on the full Hamiltonian derivative $\partial_{\mathbf{k}} \hat{H}_{\mathbf{k}}$. A chiral gap alone can generate non-zero OAM.
• OMM: Depends on $\tau_z \partial_{\mathbf{k}} \hat{H}^d_{\mathbf{k}}$. For a simple chiral $p$-wave superconductor with symmetric dispersion, the OMM vanishes ($\mathbf{m}=0$) even though the OAM and Berry curvature are non-zero.
• Conclusion: Nontrivial pairing symmetry alone is insufficient to produce an orbital magnetic moment; the electronic band structure must also break specific symmetries (e.g., inversion or time-reversal) to yield a non-zero result.

4. Numerical Results (Honeycomb Lattice Model)

Using a honeycomb lattice with chiral $d$-wave pairing, the authors demonstrate:

• Momentum Space Distribution:

  • The Berry curvature peaks at the Fermi surface pockets ($F$ points).
  • The Orbital Magnetic Moment peaks exactly at the Dirac points ($K$ points), where the band gap between specific BdG bands is minimal.
  • This spatial separation contrasts sharply with normal Bloch electrons, where OMM and Berry curvature peaks often coincide.

• Spectroscopic Signatures:

  • Energy Shift: The OMM induces a linear energy shift $\Delta E = -\mathbf{B} \cdot \mathbf{m}$, modifying the BdG band structure.
  • Density of States (DOS): This shift creates distinct peaks and dips in the field-induced DOS correction ($\delta n(E)$). The asymmetry between positive and negative energy regions is explained by the varying weights of electron ($|u|^2$) and hole ($|v|^2$) components in the wavefunctions.

• Transport: Orbital Nernst Effect:

  • The authors calculate the Orbital Nernst Coefficient ($\eta$), which arises from the interplay between the OMM and the Berry curvature under a temperature gradient.
  • $\eta$ is non-zero only when both OMM and Berry curvature are present.
  • The effect is maximized when the chemical potential is tuned near the Dirac points (where OMM is large) but vanishes exactly at the Dirac point due to the vanishing density of states.

5. Significance and Impact

• Theoretical Clarification: The paper resolves ambiguities in the literature by providing a rigorous semiclassical derivation and verifying it with linear response theory. It establishes that the OMM in superconductors is structurally distinct from that in normal metals due to charge non-conservation.
• Experimental Predictions:
  • It predicts specific spectroscopic signatures (energy shifts and DOS modulations) that can be detected via Angle-Resolved Photoemission Spectroscopy (ARPES) or Scanning Tunneling Spectroscopy (STS).
  • It proposes the Orbital Nernst Effect as a new transport probe for chiral superconductors, distinct from the thermal Hall effect.
• Fundamental Physics: The work underscores that geometric properties (Berry curvature) and magnetic responses (OMM) in superconductors are not always correlated, challenging the intuition derived from normal electronic systems. This has implications for understanding topological superconductors and Majorana modes.