Magnetic penetration depth in topological superconductors: Effect of Majorana surface states and application for UTe$_2$

This study demonstrates that magnetic penetration depth measurements in UTe$_2$ can directly probe Majorana surface states and distinguish between different topological pairing symmetries, as surface contributions dominate the low-temperature $T^n$ scaling behavior in low-$\kappa$ superconductors while bulk quasiparticles govern the response in high-$\kappa$ regimes.

The Gist

Imagine a superconductor as a bustling city where electricity flows without any traffic jams or friction. Usually, when you bring a magnet near this city, the city creates a "force field" (a screening current) to push the magnet away, keeping the inside perfectly clear. The distance this force field reaches before fading away is called the magnetic penetration depth.

In most normal superconductors, this depth changes with temperature in a very predictable, smooth way (like a gentle curve). But the material UTe2 is a "heavy fermion" superconductor, which means it's exotic, complex, and behaves differently. Scientists have been arguing for years about exactly how it works inside.

This paper acts like a detective story, using a computer simulation to figure out what's happening inside UTe2 by looking at two main suspects: the bulk (the city center) and the surface (the city walls).

Here is the breakdown of their findings using simple analogies:

1. The City Center (The Bulk) vs. The City Walls (The Surface)

Think of the superconductor as a giant cylinder.

• The Bulk: The people living in the middle of the cylinder.
• The Surface: The people living right on the edge.

In standard physics, the "people" in the middle (bulk quasiparticles) determine how the magnet is pushed away. However, UTe2 is special because it might host Majorana surface states.

• The Analogy: Imagine the city walls are guarded by ghosts (Majorana particles). These ghosts don't exist in the city center; they only live on the surface. They are "half-particles" that are their own antiparticles. The paper asks: Do these ghosts change how the city pushes away magnets?

2. The "Orbital" Twist (The Two-Lane Highway)

The researchers realized that UTe2 isn't just a simple one-lane road; it has two lanes (two orbital degrees of freedom) for electrons to travel in.

• The Discovery: When they looked at the "city center" (the bulk) with these two lanes, they found something weird. In normal physics, if you have a "hole" in the energy map (a nodal point), the magnetic depth should change with temperature like $T^4$ (a very steep curve).
• The Result: Because of the two lanes interacting, the curve flattened out to look like $T^2$.
• The Metaphor: Imagine a car trying to drive through a tunnel. In a normal tunnel, the air resistance (resistance to current) is high. But in UTe2, the two lanes allow the cars to "slip" past each other in a way that reduces the resistance much more than expected. This explains why the material behaves differently than standard theories predict, even without the ghosts.

3. The Ghosts on the Wall (Majorana Surface States)

Now, let's look at the surface ghosts. The paper found that these ghosts leave a very specific "fingerprint" on the magnetic depth, but only if the city is small enough (or the force field is short enough).

• The "Cone" Ghosts (Au State):

  • Imagine the ghosts form a perfect ice cream cone shape on the surface.
  • The Effect: They cause the magnetic depth to change with temperature like $T^3$. It's a distinct signature, like a unique song playing only on the surface.

• The "Arc" Ghosts (B1u, B2u, B3u States):

  • In other scenarios, the ghosts don't form a cone; they form a half-circle or a line (an arc) that stretches across the surface.
  • The Effect:
    • If the arc has endpoints (like a road that starts and stops), the magnetic depth changes like $T^2$.
    • If the arc is a continuous loop with no endpoints (like a racetrack), the behavior gets tricky. Along the direction the ghosts can move, it's still $T^2$. But along the direction they are stuck (flat), the behavior changes depending on the temperature.

4. The Size Matters (The "Low-Kappa" vs. "High-Kappa" Rule)

This is the most crucial practical takeaway.

• Small City (Low $\kappa$): If the superconductor is small or the "force field" (penetration depth) is short, the ghosts on the wall dominate. You can hear their song ($T^3$ or $T^2$) clearly. This is a direct way to prove the ghosts exist!
• Huge City (High $\kappa$): UTe2 is actually a "Huge City." The force field is very long (about 1 micron), while the ghosts only live in a tiny zone (about 10 nanometers).
  • The Result: In a huge city, the ghosts are too few to change the overall traffic flow. The "city center" (bulk) rules again. The magnetic depth follows the bulk rules (the $T^2$ caused by the two-lane highway interaction), and the ghosts become invisible.

The Big Conclusion

The paper solves a mystery about UTe2:

1. Why does it act weird? It's partly because of the "two-lane highway" effect in the bulk, which changes the standard rules from $T^4$ to $T^2$.
2. Can we see the ghosts? Yes, but only in materials where the "force field" is short (low $\kappa$). In UTe2, the field is too long, so the ghosts are drowned out by the bulk behavior.
3. The Takeaway: If you want to find these magical Majorana ghosts, you can't just look at any superconductor. You need to look at low-$\kappa$ materials where the surface ghosts are big enough to be heard over the noise of the city center.

In short: The paper tells us that UTe2 is a complex city where the "traffic laws" (physics) are different because of two-lane roads, and while magical ghosts live on the walls, they are too small to be seen in this specific city. However, the method they developed is a perfect map for finding these ghosts in other, smaller cities.

Technical Summary

Here is a detailed technical summary of the paper "Magnetic penetration depth in topological superconductors: Effect of Majorana surface states and application for UTe2" by Akuzawa et al.

1. Problem Statement

The superconducting pairing symmetry of the heavy-fermion compound UTe$_2$ remains a subject of intense debate. While experimental evidence points toward spin-triplet unconventional superconductivity, the specific gap symmetry (fully gapped vs. nodal) and the irreducible representation (among $A_u$, $B_{1u}$, $B_{2u}$, and $B_{3u}$ of the $D_{2h}$ point group) are unresolved.

• The Conflict: Some experiments suggest a fully gapped state, while others indicate point nodes. Recent measurements of magnetic penetration depth show a $T^2$ dependence in all directions, which contradicts conventional theory for point-nodal superconductors (which predicts $T^4$ for currents along antinodal directions).
• The Gap in Knowledge: Conventional theories often neglect orbital degrees of freedom and the contribution of Majorana surface states to the electromagnetic response. In topological superconductors, surface states can dominate low-temperature properties, potentially masking bulk signatures. The authors aim to clarify how orbital effects and Majorana surface states (cones and arcs) influence the magnetic penetration depth ($\lambda$) to resolve the ambiguity in UTe$_2$.

2. Methodology

The authors employ a comprehensive theoretical approach combining effective modeling, linear response theory, and numerical simulations:

• Model Hamiltonian: They utilize a two-orbital tight-binding model focusing on the $5f$electrons of uranium dimers. This model captures the quasi-2D cylindrical Fermi surfaces observed in UTe$_2$ and incorporates inter-orbital hopping terms essential for describing the orbital degrees of freedom.
• Pairing States: They analyze four candidate odd-parity spin-triplet pairing states belonging to the $D_{2h}$ irreducible representations: $A_u$, $B_{1u}$, $B_{2u}$, and $B_{3u}$.
• Geometry: Calculations are performed on a slab geometry with open boundary conditions to explicitly include surface states.
• Formalism:
  • They calculate the current response to a static magnetic field using linear response theory, separating the response into paramagnetic (quasiparticle) and diamagnetic (Meissner) kernels.
  • They solve Maxwell's equations for the vector potential in the slab to derive the spatial profile of the magnetic field and extract the penetration depth $\lambda(T)$.
  • They perform power-counting analysis on the paramagnetic kernel to derive analytical scaling laws for Majorana cones and arcs, distinguishing between cases with and without arc endpoints.

3. Key Contributions

1. Orbital-Dependent Bulk Anomaly: The authors demonstrate that in bulk nodal superconductors with cylindrical Fermi surfaces and orbital degrees of freedom, the penetration depth along the antinodal direction scales as $T^2$ rather than the conventional $T^4$. This deviation is caused by quasiparticles near point nodes contributing significantly to the inter-orbital paramagnetic current.
2. Majorana Surface State Signatures: They establish that Majorana surface states can dominate the low-temperature penetration depth in low-$\kappa$ superconductors (where the penetration depth $\lambda$ is comparable to the coherence length $\xi$).
   • Majorana Cones (e.g., $A_u$ state): Produce a $T^3$ dependence.
   • Majorana Arcs (e.g., $B_{1u}, B_{2u}, B_{3u}$ states): Generally produce a $T^2$ dependence.
3. Role of Arc Endpoints: A critical theoretical finding is that the power-law exponent for currents flowing along the dispersionless direction of a Majorana arc depends on whether the arc terminates at endpoints within the surface Brillouin zone.
   • With endpoints: Robust $T^2$ behavior.
   • Without endpoints (spanning the zone): The exponent remains $T^2$ along the dispersive direction but deviates (becoming larger, e.g., $T^{2+\eta}$) along the dispersionless direction due to thermally activated contributions.
4. $\kappa$-Dependence: They show that as the Ginzburg-Landau parameter $\kappa$ increases (approaching the type-II limit where $\lambda \gg \xi$), the surface contribution becomes negligible, and the temperature dependence reverts to being governed by bulk quasiparticles.

4. Key Results

• Bulk Nodal States ($B_{2u}, B_{3u}$):
  • For currents along the antinodal direction, the penetration depth scales as $T^2$.
  • This is attributed to the inter-orbital paramagnetic current, which is non-vanishing at the nodal points, unlike the intra-orbital current which vanishes (leading to the conventional $T^4$).
• Fully Gapped States ($A_u$):
  • In the bulk, $\lambda(T)$ shows exponential decay.
  • With surface states (Majorana cones), $\lambda(T)$ follows a $T^3$ power law for currents flowing along the cone's dispersive directions.
• Nodal States with Surface Arcs ($B_{1u}, B_{2u}, B_{3u}$):
  • $B_{1u}$: Exhibits Majorana arcs without endpoints. The penetration depth shows anisotropic power laws: $T^2$ along the dispersive direction and a higher exponent ($>2$) along the dispersionless direction.
  • $B_{2u}$ & $B_{3u}$: Exhibit Majorana arcs with endpoints. Both dispersive and dispersionless directions show a robust $T^2$ dependence.
• Comparison with Experiment:
  • The experimental observation of $T^2$ dependence in UTe$_2$ in all directions is consistent with the inter-orbital bulk effect in nodal states (like $B_{2u}$) or the surface arc effect in low-$\kappa$ regimes.
  • However, since UTe$_2$ is a type-II superconductor with $\lambda(0) \gg \xi$, the authors argue that the observed $T^2$ behavior is likely dominated by the bulk inter-orbital effect rather than surface states, as surface contributions are suppressed in the large-$\kappa$ limit.

5. Significance

• Diagnostic Tool: The study provides a direct method to probe Majorana surface states via magnetic penetration depth measurements, specifically in low-$\kappa$ superconductors where surface effects are prominent.
• Resolution of Controversy: It offers a theoretical explanation for the anomalous $T^2$ penetration depth in UTe$_2$ that does not require assuming a non-unitary pairing state or broken time-reversal symmetry, attributing it instead to orbital physics and nodal quasiparticles.
• Fundamental Physics: The work clarifies the distinct electromagnetic signatures of Majorana cones versus arcs and highlights the crucial role of orbital degrees of freedom in modifying standard power-law behaviors in topological superconductors.
• Future Direction: It suggests that to definitively identify the pairing symmetry of UTe$_2$, one must account for both bulk orbital effects and the specific surface termination, as the temperature dependence of $\lambda$ is highly sensitive to these factors.