Probing pairing symmetries through quasiparticle interference in chiral Bloch bands

This paper presents a theoretical framework for analyzing quasiparticle interference in superconductors with chiral Bloch bands, demonstrating how the interplay between quantum geometry and the order parameter's phase enables the distinction of various pairing symmetries through spatial variations in the local spectral function.

The Gist

Imagine a superconductor not as a smooth, featureless sheet of metal, but as a complex, multi-layered dance floor where electrons (the dancers) move in very specific, choreographed patterns. In some exotic materials, like a special type of stacked graphene, these dancers don't just move in circles; they spin in a specific direction, creating a "chiral" (handed) state. This is like a dance where everyone is spinning clockwise, never counter-clockwise.

The scientists in this paper are trying to figure out the exact "dance steps" (pairing symmetry) the electrons take when they become superconductors. The problem is, if you just look at the energy of the dancers, many different dance routines look exactly the same. It's like trying to guess a song just by listening to the volume; a loud rock song and a loud classical piece sound the same if you only measure the volume, not the melody.

The Detective Tool: Quasiparticle Interference (QPI)
To solve this mystery, the researchers use a technique called "Quasiparticle Interference" (QPI). Think of this like dropping a pebble into a calm pond. The pebble is an impurity (a tiny defect) in the material. As the electron waves hit this pebble, they scatter and create ripples. By studying the pattern of these ripples, you can figure out the shape of the pond and the nature of the water.

In this paper, the "ripples" are measured using a super-sensitive microscope (Scanning Tunneling Microscopy) that can peek at the electrons on the very top layer or the very bottom layer of the material.

The Twist: Quantum Geometry
Here is where the paper gets interesting. In normal materials, the ripples from a pebble look the same whether you measure them on the top or bottom of the water. But in these special "chiral" materials, the water itself has a weird, twisted geometry.

The authors discovered a surprising effect:

• Same Layer: If you drop a pebble on the top layer and measure the ripples on the top layer, you see a standard pattern of ripples.
• Cross Layer: If you drop a pebble on the top layer but measure the ripples on the bottom layer, something magical happens. Right at the spot where the pebble is, the ripples completely cancel out. The signal vanishes.

The Analogy: Imagine two people holding opposite ends of a long, twisted rope. If one person shakes the rope (the impurity), the other person feels a wave. But because the rope is twisted in a specific way, if you stand directly opposite the shaker, the waves from the twist cancel each other out perfectly, leaving you feeling nothing. This "destructive interference" is a unique fingerprint of the material's twisted geometry.

Solving the Mystery of the Dance
The main goal of the paper is to use these ripple patterns to tell the difference between two types of superconducting dances:

1. Achiral (Non-chiral): A simple, symmetric dance.
2. Chiral: A complex, spinning dance.

The researchers found that by looking at the ripples on the top layer (where both the pebble and the measurement are on the same side), they could clearly distinguish between the two dances.

• For the Achiral dance, the ripples look like a simple, smooth ring.
• For the Chiral dance, the ripples look different because the "twist" of the electrons interacts with the "twist" of the dance steps, creating a unique, distorted pattern.

What About Moving Ponds?
The paper also looked at what happens if the whole system is moving (finite momentum). In this case, the circular ripples get squashed into an oval shape, like a ripple in a flowing river. However, even with this distortion, the unique difference between the "simple dance" and the "spinning dance" remains visible in the top-layer measurements.

The Bottom Line
The paper concludes that by carefully watching how electron waves scatter off tiny defects—specifically by checking if the signal cancels out on opposite layers or how the ripples look on the same layer—scientists can finally identify the exact "pairing symmetry" of these exotic superconductors. It's a new way to read the "melody" of the electrons by listening to the ripples they make when they hit a bump in the road.

Technical Summary

Problem Statement
Recent experiments in van der Waals multi-layer systems, such as rhombohedral tetralayer graphene (RTG) and twisted MoTe$_2$, have revealed superconductivity emerging from symmetry-reduced, chiral normal states. These systems lack intravalley anti-unitary or reflection symmetries, resulting in chiral Bloch states with finite net Berry curvature. A primary challenge in characterizing these systems is identifying the pairing symmetry of the superconducting state. Standard scanning tunneling microscopy (STM) measures the local density of states (LDOS), which is sensitive only to the magnitude of the superconducting gap $|\Delta_k|$, rendering it unable to distinguish between order parameters with different symmetries (e.g., $s$-wave vs. chiral $p$-wave) if they share the same spectral function. While quasiparticle interference (QPI) via Fourier-transform STM (FT-STS) offers a pathway to momentum-resolved information, interpreting QPI patterns in low-symmetry, multi-band systems is complicated by "quantum geometry" effects—the non-trivial momentum dependence of Bloch state wavefunctions. There is a need for a theoretical framework to understand how these quantum geometric effects, combined with the momentum-dependent phase of the order parameter, influence QPI in chiral Bloch bands to distinguish between candidate pairing states.

Methodology
The authors develop a theoretical framework for QPI in superconductors with chiral Bloch bands, using a minimal two-band continuum model of RTG as a prototype. The model focuses on the valley-polarized regime, retaining only the dominant low-energy degrees of freedom (A1 and B4 sublattices).

1. Model Construction: The normal-state Hamiltonian is defined with layer-asymmetry energy ($u_0$) and interlayer hopping ($w_0$). The Bloch eigenstates are chosen to transform trivially under $C_{3z}$ symmetry, revealing a non-trivial momentum-dependent phase winding ($e^{-4i\phi_k}$) between the top and bottom sublattice components.
2. QPI Formalism: The impurity-induced change in the local spectral function is calculated using the Born approximation. The authors project the impurity scattering potential onto the active low-energy band, explicitly accounting for the momentum-dependent form factors of the Bloch states.
3. Superconducting Response: The formalism is extended to the superconducting state using a Nambu basis. The study considers both zero-momentum ($q=0$) and finite-momentum ($q \neq 0$) pairing. Pairing symmetries are classified by their irreducible representations (IR): achiral (IR A) and chiral (IR E and E*).
4. Analysis: The authors compute the spatial modulation of the LDOS for different configurations of the STM tip and impurity locations (top-top, bottom-bottom, and cross-layer). They analyze the Fourier transform of these spatial modulations to determine the QPI patterns, focusing on the interplay between the "normal" (particle-particle) and "anomalous" (particle-hole) scattering contributions.

Key Contributions and Results

• Normal State Quantum Geometry: In the normal state, the non-trivial quantum geometry of the Bloch states induces significant sublattice dependence. Specifically, for cross-layer configurations (impurity on one layer, STM tip on the other), the authors find a complete destructive interference at the impurity site ($r=0$), causing the spectral function change to vanish. This is a direct consequence of the relative momentum-dependent winding of the Bloch state components. Same-layer configurations (top-top or bottom-bottom) exhibit qualitatively similar Friedel oscillations but differ in magnitude.
• Distinguishing Pairing Symmetries in Superconductors:
  • Zero Momentum ($q=0$): The QPI response depends critically on the "angular winding number" ($l$) of the scattering terms. For the top-top configuration, the normal scattering contribution is always isotropic ($l=0$). However, the anomalous contribution inherits the angular winding directly from the superconducting order parameter.
    • For achiral pairing (IR A), the anomalous term remains isotropic ($l=0$), resulting in a QPI pattern similar to the normal state.
    • For chiral pairing (IR E), the anomalous term acquires a non-zero winding ($l=-1$), leading to a qualitatively distinct radial profile in the real-space LDOS and a different QPI pattern.
    • The top-top configuration is identified as the most robust channel for distinguishing between achiral and chiral pairing, as the winding mismatch produces clear signatures. In contrast, the bottom-bottom configuration involves complex interference between form factors and order parameter phases, making the distinction subtler.
  • Finite Momentum ($q \neq 0$): Introducing finite center-of-mass momentum breaks the continuous rotational symmetry of the continuum model, imprinting a universal directional anisotropy (dumbbell-shaped envelope) on all QPI patterns, dictated by the constant-energy contour of Bogoliubov quasiparticles. Despite this anisotropy, the qualitative distinctions between chiral and achiral pairing observed in the top-top configuration at $q=0$ persist.
• Role of Quantum Geometry: The paper demonstrates that the intensity distribution within the allowed momentum-transfer region is not determined solely by the energy spectrum but is heavily influenced by the non-trivial quantum geometry of the Bloch wavefunctions and their interference with the order parameter phase.

Significance
The authors claim that their work provides a theoretical guide for interpreting QPI patterns in materials with chiral bands. They establish that QPI measurements, particularly in specific sublattice configurations (top-top), can serve as a sensitive probe to uniquely identify or narrow down candidate pairing states in systems where the normal state breaks time-reversal symmetry and possesses low symmetry. The study highlights that the interplay between the quantum geometry of the Bloch states and the momentum dependence of the order parameter is central to the QPI response, offering a mechanism to distinguish between topologically trivial and non-trivial superconducting states that might otherwise appear identical in standard spectroscopic measurements.