[Paper Review] Low-temperature anomaly and anisotropy of critical magnetic fields in transition-metal dichalcogenide superconductors

Tomoya Sano, Kota Tabata+2 more|arXiv (Cornell University)|Jan 28, 2026

2D Materials and Applications0 citations

TL;DR

The paper explains why spin-singlet superconductivity persists in monolayer TMDs under high parallel magnetic fields by identifying Ising protection through even- and odd-frequency spin-triplet pairing, and derives the associated superfluid weight to account for low-temperature anomaly and anisotropy.

ABSTRACT

We clarify why spin-singlet superconductivity persists in monolayer transition-metal dichalcogenides even in high magnetic fields beyond the Pauli limit. The phenomenon called Ising protection is caused by two magnetically active potentials: a Zeeman field and an Ising spin-orbit interaction. These potentials induce two spin-triplet pairing correlations in a spin-singlet superconductor. One belonging to odd-frequency symmetry class arises solely from a Zeeman field and always makes the superconducting state unstable. The other belonging to even-frequency symmetry class arise from the interaction between the two magnetic potentials and eliminate the instability caused by odd-frequency pairs. The presence or absence of even-frequency spin-triplet pairs explains the anisotropy of the Ising protection. The analytical expression of the superfluid weight enables us to conclude that even-frequency spin-triplet Cooper pairs support spin-singlet superconductivity in high Zeeman fields.

Motivation & Objective

• Explain how Ising-type spin-orbit interaction protects superconductivity against parallel Zeeman fields in monolayer TMDs.
• Clarify the origin of the low-temperature anomaly in the Ising protection.
• Elucidate the anisotropy of the protection with respect to the orientation of the magnetic field.
• Provide analytical and numerical insights into the superfluid density contributions from different pairing symmetries.

Proposed method

• Solve the Gor’kov equation for the Bogoliubov-de Gennes Hamiltonian of a two-valley system with Ising spin-orbit interaction and Zeeman field.
• Compute the anomalous Green’s function to classify Cooper-pair symmetries by frequency, spin, and valley parity.
• Derive the gap equation to obtain the H–T phase diagram and Tc(H).
• Decompose the superfluid density into even- and odd-frequency pairing contributions and analyze their temperature dependence.
• Analyze configurations with β parallel and perpendicular to H, and discuss Rashba SOI and impurities effects.

Figure 1: The critical magnetic field $H_{c}$ is plotted as a function of temperature $T$ for several values of the Ising spin-orbit interactions $\beta$ in (a). The arrow on the vertical axis indicates the Pauli limit $\mu_{\mathrm{B}}H_{p}=\Delta_{0}/\sqrt{2}$ . The superfluid weights along the tr

Experimental results

Research questions

• RQ1 How does Ising spin-orbit interaction modify the stability of spin-singlet superconductivity under in-plane Zeeman fields?
• RQ2 What are the roles of even-frequency and odd-frequency spin-triplet pairings in determining the critical field and its temperature dependence?
• RQ3 How does the orientation of β relative to the magnetic field influence the Ising protection and the anisotropy in Hc?
• RQ4 Can analytical expressions for the superfluid weight explain the low-temperature anomaly and anisotropy observed in experiments?

Key findings

• Even- and odd-frequency spin-triplet Cooper pairs arise due to the interplay of Zeeman field and Ising SOI; even-frequency pairs stabilize the superconducting state while odd-frequency pairs tend to destabilize it.
• For β ⟂ H, the Ising protection greatly enhances the critical field at low temperatures, with q_even dominating over q_odd and yielding a large Hc.
• For β ∥ H, the critical field remains essentially unchanged from the β = 0 case, explaining anisotropy in Ising protection.
• Analytical low-temperature expressions show q_even ∝ log(√(β^2+μ_B^2H_c^2)/(2πT)) and a power-law vs. logarithmic behavior for q_odd, accounting for the low-T anomaly.
• The presence of β × H–induced even-frequency triplet pairs (with valley-odd parity) is key to eliminating odd-frequency-induced instability and sustaining superconductivity at high fields.
• Rashba SOI can counteract Ising protection by introducing other pairing channels, reducing the low-temperature enhancement of Hc.

Figure 2: The critical Zeeman field $H_{c}$ at $T=0.1T_{0}$ is plotted as a function of the angle $\theta$ between $\bm{\beta}$ and $\bm{H}$ for two values of the Ising spin-orbit interactions $\beta$ . The angles $\theta=0$ and $\theta=\pi/2$ correspond to the out-of-plane and in-plane magnetic fie