Reentrant Superconductivity in Zeeman Fields

This paper proposes a theoretical model demonstrating that when a spin-triplet $d$-vector, spin-orbit interaction, and Zeeman field are mutually perpendicular, the interplay between these vectors suppresses superconductivity at weak fields but induces reentrant superconductivity at strong fields, a phenomenon characterized by a transition from odd-frequency to even-frequency Cooper pairs.

The Gist

Imagine a superconductor as a perfectly synchronized dance troupe. In this troupe, the dancers (electrons) pair up and move in perfect unison, allowing them to flow without any friction (zero electrical resistance).

Usually, if you introduce a strong magnetic field, it's like a chaotic storm blowing through the dance floor. The storm pushes the dancers apart, breaking their pairs and ruining the dance. This is why magnets usually kill superconductivity.

However, this paper discovers a surprising twist: under very specific conditions, a strong magnetic storm can actually help the dancers get back together, causing the superconductivity to "reappear" after it had vanished. This is called Reentrant Superconductivity.

Here is the simple breakdown of how the authors figured this out, using some creative analogies:

1. The Three Key Players

The researchers built a theoretical model involving three "vectors" (think of them as invisible arrows or forces) that interact with the dancers:

• The Dancer's Style ($\mathbf{d}$): This represents the specific way the electron pairs hold hands. In this paper, they are "spin-triplet" pairs, which is a bit more complex and robust than the usual "spin-singlet" pairs.
• The Magnetic Storm ($\mathbf{H}$): This is the external Zeeman field (magnetic field). Usually, this storm pushes the dancers apart.
• The Invisible Floor Friction ($\mathbf{\alpha}$): This is Spin-Orbit Interaction (SOI). Think of this as a special, sticky floor that forces the dancers to spin in a specific direction relative to their movement.

2. The Problem: The "Bad" Dance Moves

In the world of quantum physics, there are two types of dance moves based on timing:

• Even-Frequency Moves (The Good Ones): These are stable, rhythmic steps that keep the dance going. They strengthen the superconductivity.
• Odd-Frequency Moves (The Bad Ones): These are awkward, out-of-sync steps. When these appear, they make the dancers stumble and lower the temperature at which the dance can happen.

The Scenario:
When the Magnetic Storm ($\mathbf{H}$) and the Sticky Floor ($\mathbf{\alpha}$) are arranged in a specific way (perpendicular to each other and to the dancers' style), the storm usually creates those "Bad Moves" (odd-frequency pairs). This kills the superconductivity at low magnetic fields.

3. The Twist: The "Rescue" Move

Here is the magic trick the paper describes:

When the Magnetic Storm gets very strong, something unexpected happens. The interaction between the Storm and the Sticky Floor creates a new type of dance partner: a "Spin-Singlet" pair.

Think of it like this:

• At first, the storm tries to break the main dance troupe (the spin-triplet pairs).
• But as the storm gets stronger, it accidentally triggers a rescue mechanism. The storm and the sticky floor work together to create a different kind of pair (spin-singlet) that acts like a safety net.
• This safety net stabilizes the main troupe, allowing them to dance again even in the strongest storm.

4. The "Reentrant" Effect

This leads to the phenomenon the paper is famous for:

1. Low Field: The dancers are dancing happily (Superconductivity exists).
2. Medium Field: The storm gets stronger, creates "Bad Moves," and the dancers stop (Superconductivity dies).
3. High Field: The storm gets so strong that it triggers the "Rescue Move" (the safety net). The dancers realize they can dance again, and the superconductivity returns (Reentrant).

5. Why Does This Matter?

For a long time, scientists thought the only way to save superconductivity in high magnetic fields was a very specific, rare mechanism (called Jaccarino-Peter compensation, which is like a chemical counter-force).

This paper proposes a new, more general explanation:

• It's not just about chemical counters; it's about the geometry of the forces.
• If you arrange the magnetic field, the material's internal spin-orbit interaction, and the electron pairs at right angles to each other (like the three axes of a 3D graph: X, Y, and Z), you can create this "reappearance" effect.

The Big Picture Analogy

Imagine a group of people trying to hold hands in a strong wind.

• Normal Wind: They hold hands tight.
• Stronger Wind: The wind blows them apart, and they let go.
• Hurricane: Suddenly, the wind is so strong that it pushes them into a corner where they can lock arms in a completely different, tighter formation that the wind actually helps them maintain.

The authors show that by understanding the "dance moves" (odd vs. even frequency pairs) and the "geometry" of the forces, we can predict and potentially design materials (like the exotic material UTe2) that stay superconducting even in incredibly strong magnetic fields. This could be a game-changer for building powerful magnets for MRI machines or future fusion reactors.

Technical Summary

1. Problem Statement

External magnetic fields typically suppress superconductivity through two mechanisms: the orbital effect (fluctuating the phase of the condensate) and the Zeeman effect (aligning electron spins, breaking Cooper pairs).

• Spin-singlet superconductors: The Zeeman effect imposes a strict upper limit on the critical field, known as the Pauli limit ($H_P$).
• Spin-triplet superconductors: While a Zeeman field parallel to the pairing vector ($\mathbf{d}$) suppresses superconductivity, a perpendicular field does not.
• The Phenomenon: Experimental observations in materials like $\lambda-(BETS)_2FeCl_4$, UCoGe, and UTe2 have revealed Zeeman-field-induced superconductivity (ZFIS) or reentrant superconductivity, where superconductivity reappears or is enhanced at high magnetic fields after being suppressed at lower fields.
• The Gap: The widely accepted mechanism for ZFIS is the Jaccarino-Peter compensation (cancellation of internal exchange fields by external fields). However, this paper argues that this is not the only explanation and proposes a new theoretical scenario based on the interplay between Spin-Orbit Interactions (SOI) and Zeeman fields in the absence of internal magnetic ordering.

2. Methodology

The authors propose a theoretical model based on the Bogoliubov-de Gennes (BdG) formalism to investigate the stability of superconducting states under Zeeman fields.

• Model System: A two-dimensional electronic structure with two valleys (momentum space points $\mathbf{K}$ and $-\mathbf{K}$).
• Hamiltonian Components: The model incorporates three vectors in spin space:
  1. $\mathbf{d}$: The vector describing the spin-triplet pairing state.
  2. $\mathbf{\alpha}$: A vector representing Spin-Orbit Interaction (SOI), which changes sign between valleys.
  3. $\mathbf{V}$: The Zeeman field vector ($\mathbf{V} = \mu_B \mathbf{H}$).
• Key Assumption: The SOI ($\mathbf{\alpha}$) and the pairing vector ($\mathbf{d}$) are treated as $k$-independent potentials that change sign under valley interchange, allowing the magnetic properties to be derived purely from the relative geometric configuration of $\mathbf{\alpha}$, $\mathbf{d}$, and $\mathbf{H}$.
• Analytical Tools:
  • Solution of the Gor'kov equation to derive the anomalous Green's function.
  • Calculation of the Ginzburg-Landau free energy to determine phase boundaries and transition order (first vs. second).
  • Analysis of the Superfluid Weight ($Q_F$): The authors use the superfluid weight as a stability metric. They distinguish between even-frequency pairs (which increase $Q_F$ and stabilize superconductivity) and odd-frequency pairs (which decrease $Q_F$ and destabilize superconductivity).

3. Key Contributions and Theoretical Framework

The paper's primary contribution is identifying a mechanism where spin-singlet Cooper pairs, induced by the interplay of SOI and Zeeman fields, stabilize a spin-triplet superconducting state at high magnetic fields.

• Symmetry Classification: The authors classify induced pairing correlations based on frequency (even/odd) and spin symmetry (singlet/triplet).

  • Principal Correlation: Spin-triplet, odd-valley parity (forms the main pair potential).
  • Induced Odd-Frequency Pairs: Generated by SOI or Zeeman fields; these are destabilizing (negative superfluid weight).
  • Induced Even-Frequency Spin-Singlet Pairs: Generated specifically when the scalar chiral product $\mathbf{\alpha} \times \mathbf{d} \cdot \mathbf{H} \neq 0$. These are stabilizing (positive superfluid weight).

• The Mechanism of Reentrance:

  • In weak Zeeman fields, the SOI induces odd-frequency pairs that suppress superconductivity.
  • In strong Zeeman fields, the Zeeman field screens the destabilizing effect of the SOI. Crucially, the interplay generates even-frequency spin-singlet pairs (proportional to $\mathbf{\alpha} \times \mathbf{d} \cdot \mathbf{H}$).
  • These induced singlet pairs compensate for the negative superfluid weight of the odd-frequency pairs, thereby enhancing the total superfluid weight and stabilizing the superconducting state at high fields.

4. Results

The authors analyzed three geometric configurations of the vectors $\mathbf{\alpha}$, $\mathbf{d}$, and $\mathbf{H}$:

1. Parallel Configuration ($\mathbf{\alpha} \parallel \mathbf{d} \parallel \mathbf{H}$):

   • The Zeeman field suppresses superconductivity.
   • Odd-frequency pairs appear, reducing the superfluid weight.
   • The transition becomes first-order at low temperatures, but no reentrance occurs.

2. Perpendicular Configuration ($\mathbf{\alpha} \perp \mathbf{d} \perp \mathbf{H}$):

   • At $H=0$: Strong SOI ($\alpha > \mu_B H_P$) suppresses superconductivity entirely due to odd-frequency pairing.
   • At High $H$: As the Zeeman field increases, it screens the SOI effect. The induced even-frequency spin-singlet pairs ($q_\perp$) compensate for the negative weight of odd-frequency pairs ($q_{odd}$).
   • Outcome: The critical temperature $T_c$ increases with the magnetic field, leading to reentrant superconductivity. The phase boundary shows a region where the normal state exists at low fields, followed by a superconducting phase at high fields.

3. Mixed Configuration:

   • When vectors have mixed components, the system exhibits complex phase diagrams.
   • The model predicts a transition from a low-field superconducting phase (dominated by one component of $\mathbf{d}$) to a high-field phase (dominated by another component), separated by a normal state region, confirming the reentrant behavior.

• Single-Band Validation: The authors confirmed in the Supplemental Material that this reentrant behavior also occurs in standard single-band superconductors with Rashba SOI, proving the mechanism is not limited to their specific two-valley model.

5. Significance

• Alternative to Jaccarino-Peter: This work provides a novel physical picture for ZFIS that does not rely on the compensation of internal magnetic moments. Instead, it relies on the symmetry-induced generation of stabilizing singlet pairs in a triplet superconductor.
• Role of Odd-Frequency Pairs: It rigorously demonstrates how odd-frequency pairs act as a destabilizing agent and how their suppression (or compensation by even-frequency pairs) dictates the stability of the superconducting state.
• Experimental Relevance: The findings offer a theoretical framework to interpret recent experimental data on UTe2, a candidate for spin-triplet superconductivity, where signs of spin-singlet pairing have been observed in high-field phases.
• General Applicability: The theory suggests that reentrant superconductivity is a generic phenomenon in systems with broken inversion symmetry (SOI) and spin-triplet pairing, provided the scalar chiral product $\mathbf{\alpha} \times \mathbf{d} \cdot \mathbf{H}$ is non-zero.

In conclusion, the paper establishes that the interplay between Spin-Orbit Interaction and Zeeman fields can induce even-frequency spin-singlet Cooper pairs that stabilize spin-triplet superconductivity at high magnetic fields, offering a robust explanation for reentrant superconductivity in various quantum materials.