Nonreciprocal impurity scattering as a probe for pairing symmetries in kagome superconductors

This paper proposes that analyzing the distinct local density of states patterns generated by two magnetic impurities in scanning tunneling microscopy experiments can effectively distinguish between conventional on-site $s$-wave and time-reversal symmetry breaking $d_{x^2-y^2}+id_{xy}$-wave pairing symmetries in kagome superconductors, thereby resolving ambiguities associated with sublattice interference and charge-density-wave entanglement.

The Gist

Imagine a superconductor as a perfectly choreographed dance floor where electrons move in perfect pairs, gliding without any friction. In certain exotic materials called kagome superconductors (named after a Japanese woven basket pattern), the dance floor itself is a tricky, triangular lattice. Scientists have been arguing for years about the exact "steps" these electron pairs take. Are they doing a simple, standard waltz (called s-wave), or are they performing a complex, time-bending tango that breaks the rules of symmetry (called TRSB pairing)?

The problem is that when you look at just one "intruder" on the dance floor (a single magnetic impurity), both types of dances look exactly the same. It's like watching a solo dancer; whether they are doing a waltz or a tango, a single observer might not be able to tell the difference.

The Solution: The "Echo" Test

The authors of this paper propose a clever new way to solve this mystery: put two intruders on the dance floor instead of one.

Think of the two magnetic impurities as two people shouting across a canyon.

• In a standard dance (s-wave): The rules of the universe (Time-Reversal Symmetry) say that if Person A shouts to Person B, the echo coming back is identical to if Person B shouted to Person A. The sound waves interfere with each other in a very predictable way. Specifically, if you stand exactly in the middle between them, the "echoes" cancel each other out so perfectly that the sound disappears. The paper shows that for this standard dance, this "silence" happens no matter where you put the two intruders.
• In the exotic dance (TRSB pairing): The rules are different. The universe is no longer symmetrical in time. If Person A shouts to Person B, the echo is not the same as if Person B shouted to Person A. It's like shouting into a canyon where the wind only blows one way. Because the "forward" and "backward" echoes are different, they don't cancel out perfectly in the middle. The silence is broken, and you can hear the distinct patterns of the dance.

The Experiment

The researchers used computer simulations to model this scenario on the kagome lattice:

1. One Intruder: They confirmed that a single magnetic impurity creates a specific energy signature (called a YSR state) that looks identical for both the simple s-wave and the complex TRSB dance. You can't tell them apart.
2. Two Intruders (Symmetrical): When they placed two intruders in a perfectly symmetrical spot (like two people standing on identical tiles), both dances looked similar again. The echoes interfered to create a predictable pattern where some signals vanished in the middle.
3. Two Intruders (Asymmetrical): This is where the magic happened. When they placed the two intruders on different types of tiles (breaking the symmetry), the two dances behaved completely differently:
   • The Simple Dance (s-wave): The "forward" and "backward" echoes remained identical. The signals in the middle still canceled out, leaving a distinct "hole" or silence in the data.
   • The Exotic Dance (TRSB): The echoes became different. The "forward" signal was strong, but the "backward" signal was weak or different. This meant the "silence" in the middle didn't happen. Instead, a unique, messy pattern of signals appeared that could only be explained by the exotic, time-breaking dance.

Why This Matters

The paper claims that by using a Scanning Tunneling Microscope (STM)—which is like a super-powerful camera that can "see" these electron energy levels—scientists can look at the space between two magnetic impurities.

• If they see a gap (silence) in the middle, the material is likely doing the standard s-wave dance.
• If they see a full pattern (noise) in the middle, the material is likely doing the exotic TRSB dance.

This method is a direct way to distinguish between the two types of superconductivity without relying on other, more confusing measurements (like critical current) that might be influenced by other factors in the material. It's a new, clear way to listen to the electron dance and finally figure out the steps.

Technical Summary

Technical Summary: Nonreciprocal Impurity Scattering as a Probe for Pairing Symmetries in Kagome Superconductors

Problem Statement
The microscopic nature of the superconducting (SC) ground state in vanadium-based kagome superconductors (specifically the $AV_3Sb_5$ family) remains unresolved. A primary challenge is determining the electron pairing symmetry and establishing its link to time-reversal symmetry breaking (TRSB). While experimental evidence suggests a conventional spin-singlet $s$-wave origin, conflicting reports indicate anisotropic gaps and TRSB signatures persisting even when charge-density-wave (CDW) order is suppressed. Two major obstacles hinder the resolution of this ambiguity:

1. Sublattice Interference: In kagome lattices, sublattice interference causes all conceivable unconventional spin-singlet pairings to exhibit excitation characteristics qualitatively similar to conventional $s$-wave superconductors, making them difficult to distinguish via single-impurity spectroscopy.
2. CDW Entanglement: It is debated whether observed TRSB signatures in the SC state are intrinsic to the pairing mechanism or a legacy of the CDW order.

Conventional methods for probing nonreciprocity, such as measuring the critical current (SC diode effect), are ambiguous because other unconventional ordered phases (e.g., loop-current charge order) can also induce nonreciprocal modulations.

Methodology
The authors employ a theoretical framework based on the Bogoliubov-de Gennes (BdG) formalism applied to a kagome lattice model. They investigate the effects of magnetic impurities on the local density of states (LDOS) using the $T$-matrix formalism, treating impurity spins as classical, static variables.

The study compares two representative SC pairing symmetries:

1. Conventional On-site $s$-wave: A time-reversal symmetric (TRS) state.
2. $d_{x^2-y^2} + i d_{xy}$-wave: A TRSB state (equivalent to the frustrated SC pairing introduced in Ref. [47]).

The analysis proceeds in two stages:

• Single Impurity: Calculating the impurity-induced in-gap states (Yu-Shiba-Rusinov or YSR states) for a single magnetic impurity to establish a baseline.
• Two Impurities: Extending the calculation to two parallel-aligned magnetic impurities. The authors analyze the interference between scattering processes, specifically distinguishing between "single contributions" (scattering off one impurity) and "loop contributions" (scattering from one impurity to the other and back). They examine various impurity configurations, including:
  • Inversion-Symmetric: Impurities on equivalent sublattice sites (e.g., both on A) with separations of even or odd multiples of the lattice constant.
  • Inversion-Symmetry-Breaking: Impurities on distinct sublattice sites (e.g., one on A, one on C).

Key Results

1. Indistinguishability of Single Impurities: For a single magnetic impurity, the spectral behavior of the LDOS is qualitatively identical for both the conventional $s$-wave and the TRSB $d_{x^2-y^2} + i d_{xy}$-wave pairings. Both exhibit symmetric YSR peaks that shift with coupling strength, rendering them indistinguishable via single-impurity spectroscopy.
2. Distinct Two-Impurity Interference Patterns: The indistinguishability is broken when considering two impurities. The interference patterns of the YSR states differ fundamentally based on the pairing symmetry and the inversion symmetry of the impurity configuration.
   • Conventional $s$-wave (TRS): Time-reversal symmetry enforces equivalence between forward ($r_1 \to r_2$) and backward ($r_2 \to r_1$) scattering processes for all impurity configurations. Consequently, along the line connecting the two impurities, a pair of YSR states (specifically the "outer" particle and hole excitations) undergoes destructive interference and nearly disappears, regardless of whether the configuration is inversion-symmetric or not.
   • TRSB $d_{x^2-y^2} + i d_{xy}$-wave: Time-reversal symmetry is broken, leading to nonreciprocal scattering. The equivalence of forward and backward scattering holds only for inversion-symmetric impurity configurations.
     • In inversion-symmetric configurations, the YSR disappearance pattern resembles the $s$-wave case.
     • In inversion-symmetry-breaking configurations (e.g., impurities on distinct A and C sublattices), the forward and backward scattering processes are no longer equivalent. This results in nonreciprocal evolution of the scattering loops ($\delta\rho_{12} \neq \delta\rho_{21}$). Specifically, the "loop contributions" for hole excitations become markedly weak, while the phase matching for inner particle excitations is partially restored. Crucially, unlike the $s$-wave case, the full set of four impurity-induced LDOS peaks (two particle-like, two hole-like) remains clearly resolved at both A and C sublattice sites.

Significance and Claims
The paper claims that these distinct spectral features provide a direct avenue to discriminate between TRSB and non-TRSB SC pairing symmetries in kagome superconductors. The key contributions are:

• Resolution of Ambiguity: The method circumvents the limitations of conventional critical current-based techniques, which are confounded by other nonreciprocal ordered phases. Since YSR states arise exclusively from impurity-induced excitations within the SC state, the observed nonreciprocal scattering patterns constitute a direct and unambiguous signature of TRSB in the pairing mechanism itself.
• Experimental Feasibility: The authors assert that these distinct spectral signatures (specifically the presence or absence of specific YSR peaks along the line connecting impurities) are resolvable in scanning tunneling microscopy (STM) experiments.
• Probe for Nonreciprocity: The study establishes an alternative approach to probe nonreciprocal responses in superconductors, offering a microscopic perspective on TRSB that is distinct from macroscopic transport measurements.

The authors remain modest regarding future implications, focusing strictly on the theoretical demonstration that two-impurity interference patterns can distinguish pairing symmetries that single-impurity spectroscopy cannot, without proposing specific new experimental setups beyond standard STM application.