Interplay of Zeeman field, Rashba spin-orbit interaction, and superconductivity: spin susceptibility

This paper presents a self-consistent theoretical framework using the Bogoliubov-de Gennes Hamiltonian and Kubo formula to analyze how Zeeman fields and Rashba spin-orbit coupling jointly influence the spin susceptibility of $s$-wave and $p$-wave superconductors, providing quantitative benchmarks for identifying pairing symmetries and interaction strengths in non-centrosymmetric materials.

The Gist

Imagine a superconductor as a grand ballroom where electrons are the dancers. In a normal state, they dance chaotically. But when they become superconductors, they pair up into perfect couples (Cooper pairs) and move in a synchronized, frictionless waltz. This is the magic of superconductivity.

This paper is like a choreographer's guidebook, investigating what happens when you mess with this perfect dance in two specific ways:

1. The Zeeman Field (The Magnetic Push): Imagine a strong wind blowing across the ballroom, trying to push all the dancers' heads in one direction.
2. Rashba Spin-Orbit Coupling (The Tangled Shoes): Imagine the dancers are wearing shoes that are magically tied to their feet in a way that makes them spin whenever they move forward. Their direction of travel dictates how they spin.

The authors, Chen Pang and Yi Zhou, wanted to know: How does the "spin susceptibility" (how easily the dancers can be nudged to spin) change when you have both the wind and the tangled shoes?

Here is the breakdown of their findings using simple analogies:

1. The Two Types of Dancers (Pairing States)

The paper looks at two main styles of dancing:

• The "Opposite-Spin" Dancers (s-wave): These are like a classic couple where one dancer spins left and the other spins right. They are very sensitive to the "wind" (magnetic field). If the wind blows too hard, it forces them to spin the same way, breaking their partnership.
• The "Equal-Spin" Dancers (p-wave): These are more adventurous. Sometimes both spin left, sometimes both spin right. They are more complex and react differently depending on which way the wind blows.

2. The Rules of the Dance (The Findings)

The "s-wave" (Opposite-Spin) Dancers

• Just the Wind: If you only blow the wind, it breaks up the couples. The dance stops (superconductivity dies) if the wind gets too strong.
• Just the Tangled Shoes: If you only have the tangled shoes (no wind), the couples stay together! The dance continues. However, even though they are dancing, they can't spin perfectly anymore. They retain a "residual" ability to spin (about 2/3 of their normal ability) because the shoes force them to keep moving.
• Wind + Tangled Shoes: When you have both, it gets weird. The wind tries to break them up, but the shoes keep them moving. The authors found a "kink" in the data—a sudden change in behavior—where a new type of "ghost dance floor" (called a Bogoliubov Fermi surface) appears. It's like the dancers are half-dancing, half-floating.

The "p-wave" (Equal-Spin) Dancers

These dancers are much more sensitive to direction.

• The Direction Matters: If the wind blows parallel to their spin axis, they act like the s-wave dancers (sensitive). If the wind blows perpendicular (from the side), they act like they are in a normal state and don't care at all.
• The Tangled Shoes Effect: When you add the tangled shoes to these dancers, things get chaotic.
  • For some styles, the shoes make the dance floor slippery, and the dancers can spin more than usual (susceptibility goes up).
  • For other styles, the shoes create "holes" in the dance floor (nodal lines). If the wind and shoes hit a specific sweet spot, the dancers get stuck in these holes, and the spin susceptibility goes infinite (diverges). Imagine the dancers spinning so fast they become a blur that can't be stopped.

3. Why Does This Matter? (The Real-World Connection)

The authors mention a specific family of materials called $A_2Cr_3As_3$ (where A is Sodium, Potassium, etc.). These are real-world materials that scientists are currently studying.

Think of this paper as a diagnostic tool.

• Scientists can measure how these materials react to magnetic fields (the Knight shift experiment).
• By comparing the real-world measurements to the "dance choreography" predicted in this paper, they can figure out:
  • Are the electrons dancing in the "Opposite" style or the "Equal" style?
  • How strong are the "tangled shoes" (spin-orbit coupling)?
  • Is the "wind" (magnetic field) strong enough to break the dance?

The Big Takeaway

This paper provides a universal rulebook for how superconductors behave when you push them with magnets and spin-orbit coupling. It tells us that:

1. At absolute zero: Only the "particle-particle" interactions matter (the dancers interacting with their future selves).
2. At the melting point ($T_c$): Only the "particle-hole" interactions matter (the dancers interacting with empty spots).
3. The "2/3 Rule": In many cases with strong spin-orbit coupling, the ability to spin settles at exactly 2/3 of what it was in a normal metal.

In short: The authors built a sophisticated simulation to predict how different types of superconducting "dancers" react to a stormy, tangled dance floor. This helps experimentalists decode the secret language of new, exotic materials that could one day power our future technology.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of calculating the static, uniform spin susceptibility ($\chi$) in superconductors subjected to simultaneous strong Zeeman magnetic fields and Rashba-type spin-orbit coupling (SOC).

• Context: While perturbation theory is often used for weak fields, modern experiments (e.g., on non-centrosymmetric superconductors like $A_2Cr_3As_3$) involve magnetic fields with energy scales comparable to or larger than the superconducting gap.
• Gap: Existing linear-response theories fail to capture the non-perturbative interplay between Zeeman splitting, SOC-induced band splitting, and Cooper pairing. There is a need for a self-consistent framework to predict Knight shift behavior (spin susceptibility) across various pairing symmetries (s-wave and p-wave) under these strong conditions.

2. Methodology

The authors employ a self-consistent mean-field theory based on the Bogoliubov-de Gennes (BdG) formalism.

• Hamiltonian: A single-band BdG Hamiltonian is constructed including:
  • Kinetic energy ($\xi_k$).
  • Zeeman term ($\mu_B \mathbf{H} \cdot \hat{\sigma}$).
  • Rashba SOC term ($g \mathbf{k} \cdot \hat{\sigma}$).
  • Pairing potential ($\Delta(k)$) for both spin-singlet (s-wave) and spin-triplet (p-wave) states.
• Pairing States: The study covers:
  • s-wave: Conventional spin-singlet.
  • p-wave: Six representative unitary spin-triplet states, categorized into Opposite-Spin Pairing (OSP) and Equal-Spin Pairing (ESP) classes.
• Calculation of Susceptibility:
  • The spin susceptibility $\chi_{\mu\nu}(T)$ is calculated using the Kubo formula.
  • The formula is decomposed into particle-hole (ph) and particle-particle (pp) channels, further split into intra-band and inter-band contributions.
  • The gap equation is solved self-consistently to determine the order parameter $\Delta(T, H, g)$ before calculating $\chi$.

3. Key Contributions & Theoretical Framework

The paper establishes two universal constraints on spin susceptibility derived from the Kubo formula decomposition:

1. At Zero Temperature ($T=0$): Only particle-particle (pp) terms contribute to the susceptibility. The particle-hole terms vanish because the derivative of the Fermi distribution function goes to zero.
2. At Critical Temperature ($T \to T_c$): Only particle-hole (ph) terms remain. This ensures continuity, where $\chi(T_c^-) = \chi(T_c^+) = \chi_N$ (the normal-state susceptibility) for continuous phase transitions.

These constraints serve as rigorous benchmarks for numerical calculations and experimental interpretations.

4. Key Results

A. s-Wave (Spin-Singlet) Pairing

• Zeeman Field Only: Reduces $T_c$. For strong fields ($H > H_P$, the Pauli limit), a first-order phase transition occurs. $\chi(T=0)$ vanishes.
• Rashba SOC Only: Preserves $T_c$ (due to time-reversal symmetry) but induces a residual zero-temperature susceptibility $\chi(0)$. In the strong SOC limit ($g k_F \gg \Delta$), $\chi(0) \to \frac{2}{3}\chi_N$.
• Combined Fields:
  • The competition creates a Bogoliubov Fermi surface (gapless excitations) at intermediate fields.
  • This results in a kink in the $\chi(0)$ vs. $H$ curve at the critical field where the Bogoliubov Fermi surface emerges.
  • $\chi(0)$ remains non-zero and approaches $\frac{2}{3}\chi_N$ in the strong SOC limit.

B. p-Wave (Spin-Triplet) Pairing

The response is highly anisotropic and depends on the pairing class (OSP vs. ESP) and field direction.

• Opposite-Spin Pairing (OSP) States (e.g., $(k_x+ik_y)\hat{z}$, $k_z\hat{z}$):

  • Parallel Field ($H \parallel \hat{z}$): Behaves like a spin-singlet; $\chi_{zz}(T)$ decreases below $T_c$. A finite Zeeman field induces a residual $\chi_{zz}(0)$ due to nodal points/lines.
  • Perpendicular Field ($H \perp \hat{z}$): Behaves like an ESP state; $\chi_{xx}(T)$ remains constant at $\chi_N$ for $T < T_c$.
  • Rashba SOC: Can suppress $T_c$ and alter the nodal structure. In the strong SOC limit, $\chi(0) \to \frac{2}{3}\chi_N$.

• Equal-Spin Pairing (ESP) States (e.g., $k_x\hat{x} \pm k_y\hat{y}$):

  • Parallel Field: $\chi_{zz}(T)$ remains constant at $\chi_N$ (unaffected by the field).
  • Perpendicular Field: $\chi_{xx}(T)$ decreases below $T_c$.
  • Rashba SOC Effects:
    • Generally lowers $T_c$.
    • Critical Finding: For specific ESP states ($k_x\hat{x} - k_y\hat{y}$ and $k_y\hat{x} + k_x\hat{y}$), Rashba SOC induces nodal lines in the quasiparticle spectrum.
    • Divergence: When the SOC strength satisfies $g k_F = \Delta(T=0)$, the zero-temperature longitudinal susceptibility $\chi_{zz}(0)$ diverges logarithmically. This is a unique signature of SOC-driven topological changes in the nodal structure.

5. Significance and Applications

• Diagnostic Tool for Pairing Symmetry: The paper provides quantitative benchmarks for Knight shift experiments in non-centrosymmetric superconductors.
  • The field-direction dependence of $\chi(T)$ can distinguish between OSP and ESP states.
  • The presence of a residual $\chi(0)$ or specific anisotropy patterns helps identify the pairing symmetry in materials like $A_2Cr_3As_3$ ($A$ = Na, K, Rb, Cs).
• Experimental Signatures:
  • Kinks in $\chi(0)$: Indicate the formation of Bogoliubov Fermi surfaces in s-wave systems.
  • Divergence in $\chi_{zz}(0)$: A direct signature of SOC-induced nodal lines in specific p-wave ESP states.
  • Residual Plateau ($\approx 2/3 \chi_N$): Indicates the dominance of Van Vleck-like contributions from band splitting in the strong SOC limit.
• Future Directions: The framework sets the stage for investigating multi-orbital effects, mixed-parity pairing, and non-unitary triplet states, offering a path to understand the interplay of symmetry, topology, and correlations in unconventional superconductors.

In summary, this work moves beyond perturbative approaches to provide a comprehensive, non-perturbative theory of spin susceptibility in superconductors under strong magnetic and spin-orbit fields, offering critical predictions for interpreting NMR data in emerging quantum materials.