Two-dimensional flat band on the (011) surface of UTe$_2$: Implication for STM measurements with a superconducting tip

This paper theoretically demonstrates that a two-dimensional nearly flat band on the (011) surface of UTe$_2$, arising from nontrivial Berry phases and weak spin conservation in the $B_{3u}$ state, explains the zero-bias peak observed in STM experiments with superconducting tips and offers crucial insights into the material's superconducting pairing symmetry.

The Gist

Imagine a superconductor as a bustling city where electrons usually move chaotically. In a normal metal, they bump into each other like pedestrians in a crowded market. But in a superconductor, they pair up and move in perfect, frictionless harmony, like a synchronized dance troupe gliding across an ice rink.

The material in this paper, UTe2, is a mysterious new "dance hall" that scientists are trying to understand. It's famous for being a potential candidate for a very exotic type of superconductivity called spin-triplet, where the electron pairs spin in a specific, coordinated way.

Here is the story of what the researchers discovered, explained through simple analogies:

1. The Mystery of the "Zero-Bias Peak"

Scientists recently used a super-sensitive microscope (called an STM) to look at the surface of UTe2. They used a special tip made of a superconductor to probe the surface.

• The Observation: They saw a giant, sharp spike in their data right at zero energy. Think of this like looking at a city's traffic report and seeing a massive, sudden jam of cars right at the starting line (zero speed), even though the road is supposed to be clear.
• The Puzzle: This "Zero-Bias Peak" (ZBP) is a hallmark of topological superconductivity, but usually, these peaks are small or broad. This one was huge and sharp. The scientists wanted to know: Why is there such a massive crowd of electrons sitting right at zero energy?

2. The "Flat Band" Analogy

To solve the mystery, the researchers looked at the "energy map" of the electrons on the surface of UTe2.

• Normal Hills and Valleys: Usually, electron energy levels are like a hilly landscape. Electrons roll down into valleys (low energy) or climb up hills (high energy).
• The Flat Plateau: The researchers found that in one specific configuration (called the B3u state), the landscape changes dramatically. Instead of hills, there is a massive, perfectly flat plateau stretching across the entire surface.
• The Crowd: On a flat plateau, everyone can stand at the exact same height (energy level) without rolling away. This creates a "flat band." Because the surface is so flat, a huge number of electrons can pile up at the exact same zero-energy level. This pile-up creates the giant "Zero-Bias Peak" the microscope saw.

3. How Was This Plateau Built? (The Two Mechanisms)

The paper explains that this flat plateau didn't just happen by accident; it was built by two specific "architectural" rules of the UTe2 city:

• Mechanism A: The Berry Phase (The Magic Loop)
  Imagine walking in a circle around a park. In most places, you end up facing the same direction. But in this specific quantum city, if you walk a loop around certain points, you end up facing the opposite direction (a "nontrivial Berry phase"). This weird twist forces the electrons to stop and gather at specific points on the map. These gathering points are the anchors for the flat band.

• Mechanism B: Weak Spin Conservation (The Loophole)
  Usually, electrons have a strict rule about how they spin (like a strict dress code). However, in this specific B3u state, the rules are a bit loose ("weak spin conservation"). This loophole allows the electrons to twist their dance moves in a way that creates a "winding number." Think of it like a spiral staircase that loops perfectly back on itself, trapping the electrons in a flat, zero-energy zone.

4. The "Superconducting Tip" Test

To prove their theory, the researchers simulated what happens when you bring a superconducting tip (the microscope probe) close to the UTe2 surface.

• The Tunneling: Electrons try to "tunnel" (jump) from the tip to the surface.
• The Result: When they simulated the B3u state (the one with the flat plateau), the tunneling current showed a massive, sharp spike at zero voltage. It matched the real-world experiments perfectly.
• The Comparison: When they simulated other possible states (like the "Au" or "B1u" states), the energy landscape was bumpy (dispersive). In those cases, the electrons scattered, and no giant spike appeared.

5. The Conclusion: Solving the Identity Crisis

The main goal of the paper was to figure out which "dance style" (pairing symmetry) UTe2 actually uses.

• The Verdict: The only scenario that explains the giant, sharp peak seen in the experiments is the B3u state.
• Why it matters: This state is special because it creates a 2D flat band, which is a rare and exciting phenomenon in physics. It suggests that UTe2 is not just a regular superconductor but a topological superconductor, a material that could be the key to building future quantum computers (which need these special "zero-energy" states to store information without errors).

Summary

Think of UTe2 as a mysterious dance floor. Scientists saw a huge crowd of dancers frozen at the starting line (the Zero-Bias Peak). By analyzing the floor's geometry, they realized the floor must be a giant, flat plateau (the 2D Flat Band) created by specific quantum rules (Berry phases and spin conservation). This plateau only exists if the dancers are following the B3u dance routine. Therefore, the paper concludes: UTe2 is dancing the B3u step.

Technical Summary

1. Problem Statement

The uranium-based superconductor UTe$_2$ is a leading candidate for spin-triplet $p$-wave pairing, yet its precise pairing symmetry remains unresolved. Recent Scanning Tunneling Microscopy (STM) experiments on the naturally cleavable (011) surface of UTe$_2$ have revealed a sharp zero-bias peak (ZBP) in the differential conductance ($dI/dV$) spectrum when using a superconducting tip.

• The Puzzle: While zero-energy Andreev Bound States (ABS) are expected on topological superconductors, typical 3D spin-triplet superconductors with point nodes exhibit 1D "Fermi arcs." These 1D arcs yield a constant density of states (DOS) rather than a sharp peak.
• The Gap: It is unclear why the (011) surface of UTe$_2$ produces such a pronounced ZBP. Furthermore, there has been no microscopic calculation of the nonequilibrium DC tunneling current between a topological superconductor and an $s$-wave superconductor to rigorously interpret these STM spectra.

2. Methodology

The authors employed a combination of theoretical modeling and numerical simulation:

• Model Hamiltonian: They constructed a minimal tight-binding model for UTe$_2$ on a body-centered orthorhombic lattice ($D_{2h}$ point group). The normal-state Hamiltonian was tuned to reproduce the experimentally observed cylindrical Fermi surfaces (one electron-like, one hole-like) elongated along the $k_z$ axis.
• Pairing Symmetry Analysis: They investigated all four possible odd-parity irreducible representations (IR) for the spin-triplet gap function: $A_u$, $B_{1u}$, $B_{2u}$, and $B_{3u}$.
• Surface State Calculation: Using a slab geometry with open (011) surfaces, they diagonalized the Bogoliubov-de Gennes (BdG) Hamiltonian to compute quasiparticle energy spectra and surface DOS using the recursive Green's function method.
• Topological Analysis: They analyzed the emergence of ABS using topological invariants, including:
  • $\mathbb{Z}_2$ Berry phases defined at time-reversal-invariant momenta (TRIMs).
  • 1D winding numbers protected by mirror symmetry ($M_{yz}$).
  • 1D winding numbers arising from weak spin conservation in specific subspaces.
• Tunneling Current Simulation: To connect theory with experiment, they calculated the DC tunneling current between an $s$-wave superconductor (tip) and UTe$_2$ using the tunneling Hamiltonian method within the Keldysh Green's function formalism. This approach included all orders of tunneling processes (resonant transmission and multiple reflections). They also derived an analytical expression for the fourth-order Andreev current in the weak-coupling limit.

3. Key Contributions and Results

A. Emergence of a 2D Nearly Flat Band in the $B_{3u}$ State

The study identifies that a 2D nearly flat band of zero-energy states emerges exclusively in the $B_{3u}$ pairing state on the (011) surface.

• Mechanism 1 (Berry Phases): Nontrivial Berry phases at multiple high-symmetry momenta (specifically $\Gamma$ and $M$ points) guarantee the existence of zero-energy states. These states are connected smoothly by low-energy in-gap states, forming a continuous 2D structure across the surface Brillouin zone (BZ).
• Mechanism 2 (Weak Spin Conservation): The $B_{3u}$ state possesses a specific symmetry where, if the $d_x$ component of the gap vector vanishes, spin is conserved along the $x$-axis. This allows the gap function to acquire a nontrivial phase winding, generating a 1D winding number that protects additional zero-energy states along the $\Gamma-X$ and $R-M$ lines.
• Robustness: Even when the $d_x$ component is finite (breaking strict spin conservation), the flat band remains robust, staying close to zero energy as long as the spin-flip term is small compared to the energy broadening. This results in a pronounced zero-energy peak in the surface DOS.
• Contrast: Other pairing states ($A_u$, $B_{1u}$, $B_{2u}$) either lack zero-energy states or exhibit only dispersive 1D flat bands (Fermi arcs) that do not produce a sharp ZBP in the DOS.

B. Tunneling Spectroscopy and the ZBP

The authors calculated the $dI/dV$ spectra for a junction between an $s$-wave tip and UTe$_2$.

• Dominant Mechanism: In the low-bias regime, single-particle tunneling is suppressed by the $s$-wave gap, and Josephson current is suppressed by spin-space orthogonality. The dominant contribution is Andreev tunneling (fourth-order process).
• Analytical Insight: They derived an analytical expression showing that the tunneling current is proportional to the convolution of the surface DOS of UTe$_2$. Consequently, a large surface DOS at zero energy directly translates to a sharp ZBP in the conductance.
• Agreement with Experiment: Only the $B_{3u}$ state reproduces the experimentally observed sharp ZBP with a magnitude comparable to the coherence peaks of the tip. The other states fail to produce this feature.
• Phase Difference Independence: The authors addressed a discrepancy regarding the splitting of the ZBP observed in some experiments. They demonstrated that while the surface Majorana band energy levels depend on the superconducting phase difference between the tip and sample, the DC tunneling current itself is independent of this phase difference. This suggests that the observed splitting in experiments may require a different explanation than simple phase-induced gap opening.

C. Validation with Realistic Models

The authors verified their findings using a more realistic two-orbital model that includes staggered Rashba spin-orbit coupling (SOC) arising from local inversion symmetry breaking at uranium sites. The 2D nearly flat band and the resulting ZBP persisted, confirming the robustness of the mechanism against SOC.

4. Significance

• Identification of Pairing Symmetry: The work provides strong theoretical evidence that the pairing symmetry in UTe$_2$ is likely $B_{3u}$. This is the first theoretical mechanism explaining how a point-nodal or fully gapped bulk superconductor can generate a 2D flat band and a sharp ZBP on a specific surface.
• Resolution of STM Anomalies: It resolves the mystery of the large ZBP observed in STM experiments with superconducting tips, linking it directly to the topological nature of the $B_{3u}$ surface states.
• Methodological Advance: The rigorous calculation of the nonequilibrium tunneling current, incorporating all orders of tunneling, provides a reliable framework for interpreting Andreev spectroscopy in topological superconductors.
• Broader Implications: The mechanism identified (cylindrical Fermi surfaces + weak spin conservation $\to$ 2D flat bands) is generalizable to other topological superconductors and suggests new avenues for exploring interaction-induced topological phases on anomalous surfaces.

In conclusion, the paper establishes that the $B_{3u}$ pairing state in UTe$_2$ hosts a unique 2D nearly flat band on the (011) surface, which is the direct origin of the sharp zero-bias peak observed in STM experiments, thereby offering a definitive solution to the long-standing debate regarding the pairing symmetry of this enigmatic material.