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Disorder-tuned crossing of monopole and conventional pairing instabilities in multi-Weyl semimetals

This paper demonstrates that non-magnetic impurity scattering in multi-Weyl semimetals can tune the leading pairing instability from a topologically nontrivial monopole channel to a conventional s-wave channel, establishing a specific criterion for this crossover based on the interplay between disorder strength and the relative critical temperatures of the competing pairing states.

Original authors: Enrique Muñoz, Rodrigo Soto-Garrido

Published 2026-09-15

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CC BY 4.0

Original authors: Enrique Muñoz, Rodrigo Soto-Garrido

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the hidden world of solid materials, electrons do not always behave like simple particles bouncing around a room. In certain exotic crystals known as semimetals, these electrons can organize themselves into a state where they act as if they are massless, moving at a constant speed regardless of their energy. Within these materials, the paths the electrons take are not just lines on a map; they are twisted by a fundamental property of quantum mechanics called topology. Imagine the surface of a sphere where the magnetic field lines all point outward from a single center; in these crystals, the electron waves carry a similar kind of "charge" at specific points where energy bands touch, acting like tiny magnetic monopoles. When scientists dope these materials with extra electrons, they can sometimes force the electrons to pair up and flow without resistance, creating a superconductor. However, the nature of this superconducting state is delicate. It can take one of two forms: a standard, robust state that is immune to many types of impurities, or a more fragile, topologically complex state that carries the unique "monopole" charge of the underlying crystal. The question of which state wins, and whether one can be forced to switch to the other, has remained a puzzle, especially when the material is not perfectly pure.

A team of researchers from the Pontificia Universidad Católica de Chile has now solved a key part of this puzzle by studying how non-magnetic impurities—tiny, harmless defects scattered throughout the crystal—affect the competition between these two types of superconductivity. They focused on a specific class of materials called multi-Weyl semimetals, where the topological charge at the electron crossing points can be stronger than usual, carrying a value of one, two, or three. In a perfectly clean crystal, the topological "monopole" pairing is often the strongest, but the researchers wanted to know what happens when disorder is introduced. Using a detailed theoretical model, they simulated the behavior of electrons in these materials as they encountered random, non-magnetic defects. Their calculations revealed a surprising and tunable switch: as the amount of disorder increases, the material does not simply lose its superconductivity. Instead, the leading form of superconductivity changes. The disorder acts as a dial that suppresses the fragile, topological monopole state more quickly than the standard, conventional state. At a specific, moderate level of impurity, the two states cross paths, and the material flips from the exotic, topological superconductor to the ordinary, fully gapped one.

The researchers found that this transition depends heavily on the strength of the topological charge. In materials where the charge is higher, the topological state is more sensitive to disorder and gives way to the conventional state at a lower level of impurity. For the simplest case, where the charge is one, the topological state survives up to a higher level of disorder before switching. The team calculated that for the materials they modeled, this switch happens when the disorder is strong enough to reduce the electron's lifetime to a few hundredths of a picosecond, a regime where the material is still a good metal but far from the point where the crystal structure itself would dissolve. This is a crucial distinction; the transition occurs while the material remains a functioning metal, not in a chaotic, disordered mess. The study also highlighted that the two states are fundamentally different in their response to defects. The conventional state is protected by a principle known as Anderson's theorem, which allows it to ignore certain types of scattering, while the topological state lacks this shield and is gradually worn down by the same impurities.

Beyond the switching mechanism, the paper provides a clear way to identify which state a material is in by looking at its heat capacity and how it conducts electricity at very low temperatures. In the topological state, the electrons form a pattern with specific points where the energy gap vanishes, creating a unique signature in how the material stores heat. This signature changes in a predictable way depending on the topological charge, following a specific power law that distinguishes a charge of one from a charge of two or three. In contrast, the conventional state has a full energy gap with no such points, behaving like a standard superconductor. The researchers also discovered that the way disorder affects the material depends on the charge. For the simplest charge, there is a clear threshold of disorder below which the material remains perfectly clean of residual electrons, but for higher charges, even a tiny amount of disorder creates a small, lingering background of conducting electrons. This difference offers a potential experimental fingerprint to tell the states apart.

The work suggests that by carefully controlling the level of impurities in a multi-Weyl semimetal, scientists could deliberately tune the material to switch between these two distinct quantum phases. This is not just a theoretical curiosity; it points toward a way to engineer materials with specific superconducting properties by managing their disorder. The researchers emphasize that their findings are based on a specific type of disorder—smooth, non-magnetic impurities that do not jump between different parts of the crystal structure. If the disorder were different, such as magnetic or short-range, the rules might change. However, within the conditions they studied, the crossing of the two instabilities is a robust prediction. The study leaves open the question of exactly how the material behaves right at the moment of the switch, whether it jumps abruptly or passes through a mixed state, but the existence of the switch itself is firmly established. This provides a new roadmap for experimentalists looking to observe these exotic topological transitions in real materials, offering a clear set of parameters to look for, such as specific quantum lifetimes and heat capacity signatures, to confirm when the topological state has yielded to the conventional one.

Technical Summary: Disorder-tuned crossing of monopole and conventional pairing instabilities in multi-Weyl semimetals

Problem Statement
The paper investigates how non-magnetic impurity scattering influences the competition between two distinct superconducting pairing channels in doped multi-Weyl semimetals: a topologically nontrivial "monopole" channel and a conventional fully gapped $s$ -wave channel. While previous work established the competition in clean systems, this study addresses the disordered regime. The central question is whether scalar disorder can act as a tuning parameter to select the pairing topology, specifically by driving a crossing of the linearized pairing instabilities where the leading superconducting state switches from the monopole channel to the conventional one. The authors emphasize the necessity of ensuring this transition occurs while the host material remains a good metal, well below the disorder strength that would drive the normal state into a diffusive metal phase.

Methodology
The study employs a theoretical framework combining the Born approximation for disorder self-energies with the generalized Abrikosov–Gor'kov (AG) theory for pair-breaking.

Model: The system is modeled as a two-node multi-Weyl semimetal with monopole charge $J \in \{1, 2, 3\}$ . The disorder is treated as a smooth, non-magnetic, intra-node scalar potential. A specific "projected impurity kernel" is defined, valid in a two-scale window where the disorder correlation length is large compared to intra-pocket momentum transfers but small compared to the inter-node separation ( $q_{\text{intra}}^{\text{max}} \xi_{\text{dis}} \ll 1 \ll |2Q| \xi_{\text{dis}}$ ).
Self-Energy Calculation: The authors compute the Born self-energy in the chiral band basis. They distinguish between the normal scattering rate (frequency renormalization) and the anomalous rate (gap renormalization).
Pair-Breaking Analysis: Using the AG formalism, they derive the critical pair-breaking rates for both channels. The conventional channel is assumed to be Anderson-protected ( $\eta_s = 1$ ) within the projected kernel, while the monopole channel experiences pair-breaking governed by a protection factor $\eta_m(J)$ .
Thermodynamics: The study calculates specific heat jumps, low-energy density of states (DOS) scaling laws, and residual DOS under disorder to provide thermodynamic fingerprints of the transition.

Key Contributions and Results

Exact Protection Factors:
- The authors derive that for the projected intra-node scalar disorder, the conventional channel is exactly Anderson-protected ( $\eta_s = 1$ ) because the Berry phase dressing the normal scattering rate cancels identically with the anomalous self-energy.
- For the monopole channel, the pair-breaking rate is softened by chiral suppression of backscattering. The protection factor is found to be an exact eigenvalue of the projected kernel: $\eta_m(J) = 1/(J + 2)$ . Consequently, the pair-breaking rate is $\Gamma_{\text{pb}}^m = (1 - \eta_m)\Gamma_N = \frac{J+1}{J+2}\Gamma_N$ .
Disorder-Tuned Crossing Criterion:
- The competition between the two channels is resolved by a crossing of their critical temperatures ( $T_c$ ) as disorder strength ( $\Gamma_N$ ) increases.
- A crossing occurs if the conventional channel is pair-broken sufficiently more slowly than the monopole channel. The general criterion for a crossing at finite $T_c$ is derived as:
  $\frac{1 - \eta_s}{1 - \eta_m} < r \equiv \frac{T_{c0}^{(s)}}{T_{c0}^{(m)}}$
  where $r$ is the ratio of clean critical temperatures.
- Under the assumption $\eta_s = 1$ , the crossing location in the dimensionless disorder strength $\Gamma_N / T_{c0}^{(m)}$ decreases as the monopole charge $J$ increases (due to the decreasing $\eta_m$ ).
Topological Distinction of Pure States:
- The two competing pure states possess distinct nodal topology. The monopole solution features point nodes on the Fermi surface carrying Bogoliubov-de Gennes (BdG) Berry charges of $\pm J$ .
- The conventional solution is fully gapped with zero BdG nodal charge.
- The authors establish that the local BdG nodal charge is a rigorous property of the clean projected Hamiltonian, independent of inter-node gauge conventions.
Thermodynamic Signatures:
- Specific Heat: The clean specific heat jump $\Delta C/C_n$ is reduced for the monopole channel compared to the BCS value (1.43), taking values of 1.19, 0.95, and 0.79 for $J=1, 2, 3$ respectively.
- Low-Energy DOS: The quasiparticle DOS follows a charge-dependent power law $N_{\text{SC}}(E) \propto E^{2/J}$ , leading to a specific heat $C \propto T^{1+2/J}$ .
- Residual DOS: Under disorder, the emergence of a residual zero-energy DOS depends on $J$ . For $J=1$ , a finite disorder threshold exists below which the DOS remains zero. For $J \ge 2$ , the threshold vanishes; arbitrarily weak disorder generates a residual DOS (exponentially small for $J=2$ , algebraically small for $J=3$ ).
Regime of Validity:
- The calculated crossing occurs in a "moderately metallic" regime. For the illustrative parameters used ( $T_{c0}^{(m)}/\mu = 0.133$ ), the ratio of chemical potential to scattering rate at the crossing is $\mu/\Gamma_N^* \approx 11\text{--}14$ . This ensures the system is far from the disorder-driven semimetal-to-diffusive-metal transition where the Born expansion fails.

Significance and Claims
The paper claims to provide a "proof-of-principle" prediction that scalar disorder can tune the pairing symmetry in multi-Weyl superconductors. The primary significance lies in demonstrating that the topological monopole channel is not robust against non-magnetic disorder in the same way the conventional channel is; instead, it is pair-broken at a rate determined by the monopole charge $J$ .

The authors explicitly state that their results are conditional on the specific projected kernel (valley-diagonal, intra-pocket momentum-independent) and the assumption of Anderson protection for the conventional channel. They do not claim to have solved the full node-resolved anomalous impurity vertex for generic short-range disorder, nor do they determine the precise order of the phase transition (e.g., first-order vs. coexistence) at the crossing, noting that a full Ginzburg-Landau analysis with disorder-dressed vertices is required for that.

The work serves as a theoretical extension of clean-system analyses into the disordered regime, offering specific, charge-dependent thermodynamic fingerprints (such as the specific heat jump and the nature of the residual DOS) that could distinguish between monopole and conventional pairing in future experimental studies of candidate materials like HgCr $_2$ Se $_4$ or SrSi $_2$ , provided the disorder is non-magnetic and smooth on the scale of the node separation.

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