🔬 mesoscale physics

A relationship between nonunitary mixed parity superconductivity and magnetism with spin-orbit coupling

This paper demonstrates that nonunitary mixed parity superconductivity is physically equivalent to magnetism with spin-orbit coupling via the Schrieffer-Wolff transformation, revealing that phenomena such as Dzyaloshinskii-Moriya interactions, magnetoelectric effects, and altermagnetism can emerge purely from superconductivity without intrinsic magnetic or spin-orbit coupling.

Original authors: Takehito Yokoyama

Published 2026-07-30

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CC BY 4.0

Original authors: Takehito Yokoyama

Original paper licensed under CC BY 4.0 (http://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

Imagine a world where the tiny particles that make up everything around us, like electrons, have a secret double life. On one hand, they are charged particles that flow as electricity to power your phone. On the other, they possess a hidden property called "spin," which acts like a tiny internal compass needle. Usually, these two lives are separate, but in the exotic realm of condensed matter physics, scientists love to mix them up. When an electron's movement (its orbit) gets tangled with its spin, it creates a phenomenon called "spin-orbit coupling." Think of it like a dancer who, when they spin, their arms automatically fling out in a specific direction. This dance is the engine behind some of the coolest tech we have today, from faster computer chips to materials that conduct electricity without losing any energy.

For a long time, scientists believed that to get this special "tangled" dance going, you needed a specific ingredient: a crystal structure that lacks a mirror image (no inversion symmetry), which naturally forces the spin and orbit to interact. However, there is another mysterious state of matter called superconductivity, where electricity flows with zero resistance. In some superconductors, the electrons pair up in a weird way called "mixed parity," meaning their partnership has a jumbled mix of symmetries. The big question has been: Can this jumbled superconducting dance itself create the same effects as the spin-orbit coupling dance, even if the crystal structure is perfectly symmetrical and doesn't force the interaction? If so, it would mean superconductors could mimic magnetic materials and spin-orbit effects all on their own, opening up a whole new playground for physics.

In this paper, Takehito Yokoyama from the Institute of Science Tokyo proposes a fascinating connection: he suggests that the complex math describing these jumbled superconductors can be rewritten to look exactly like the math for magnets with spin-orbit coupling. It's as if he found a secret translation dictionary that turns a story about superconducting electrons into a story about magnetic spins. By using a mathematical tool called the Schrieffer-Wolff transformation, the author shows that the "mixed parity" superconducting state naturally creates its own version of spin-orbit coupling and magnetic fields, without needing any external magnets or special crystal structures.

The paper demonstrates this by looking at four specific effects that usually require spin-orbit coupling or magnetism, and showing they can happen purely because of the superconductivity itself. First, there are "Dzyaloshinskii-Moriya interactions," which are like a twist in the magnetic alignment of neighbors; the paper suggests these can arise from the superconducting mix, but with a twist: they would vanish if you warmed the material up past its superconducting temperature, unlike the standard magnetic version. Second, there is the "Edelstein effect," where a flowing electric current creates a spin polarization (a crowd of electrons all pointing their compass needles in one direction). The author calculates that in these mixed superconductors, a supercurrent can generate a spin polarization with an energy scale comparable to what you'd get from a topological insulator, all without any external spin-orbit coupling.

Third, the paper explores "supercurrent-induced spin current." Usually, you need a magnetic field to turn a flow of charge into a flow of spin. Here, the author shows that the internal structure of the superconducting pairs acts like that magnetic field, driving a spin current. Finally, the paper points to "altermagnetism," a state where spin polarization depends on the direction of motion in a specific, alternating pattern (like a checkerboard). The math shows that the momentum-dependent "magnetic field" created by the superconductivity can produce this exact pattern.

The author is careful to note that these effects are purely derived from the superconductivity itself, meaning no external magnetism or spin-orbit coupling is required to start the show. While the calculations are theoretical and rely on specific assumptions (like ignoring strong electron-electron interactions), the results suggest that noncentrosymmetric superconductors might be far more versatile than we thought. They could host these exotic magnetic-like phenomena simply by virtue of how their electron pairs are arranged, offering a new way to think about how we might control spin and charge in future quantum devices.

Technical Summary: Nonunitary Mixed Parity Superconductivity and Magnetism with Spin-Orbit Coupling

Problem Statement
The paper addresses the physical relationship between nonunitary mixed parity superconductivity and magnetism in the presence of spin-orbit coupling (SOC). While SOC is a well-established driver for phenomena such as topological phases, spintronics, and specific magnetic orderings (e.g., Dzyaloshinskii-Moriya interactions), its role in noncentrosymmetric superconductors is often treated as an external perturbation mixing singlet and triplet pairing states. The author investigates whether the intrinsic physics of nonunitary mixed parity superconductivity can, in the absence of explicit SOC and magnetic fields in the normal state, mimic the effective Hamiltonian of a magnetic system with SOC.

Methodology
The study employs a theoretical framework based on the Bogoliubov-de Gennes Hamiltonian for a superconductor containing both singlet ( $\Delta_s$ ) and triplet ( $\mathbf{d} \cdot \boldsymbol{\sigma}$ ) pairing components. The Hamiltonian is formulated as $H = H_0 + H_\Delta$ , where $H_0$ represents the normal state kinetic energy and $H_\Delta$ the pairing potential. Crucially, the normal part $H_0$ is assumed to lack spin-orbit coupling to isolate the effects arising purely from the mixed parity pairing.

The core methodological tool is the Schrieffer-Wolff transformation. By treating the pairing term $H_\Delta$ as a perturbation relative to the kinetic energy, the author derives an effective low-energy Hamiltonian ( $H'$ ). This transformation integrates out the high-energy degrees of freedom to reveal the effective interactions in the quasiparticle sector. Additionally, the paper utilizes the full Green's function formalism to calculate response functions (such as spin polarization and spin currents) via the Kubo formula, verifying that the derived correspondences hold beyond the perturbative limit.

Key Contributions and Results

Effective Hamiltonian Equivalence:
The primary result is the demonstration that the effective Hamiltonian for nonunitary mixed parity superconductivity can be recast into the form of a magnetic system with spin-orbit coupling. The transformation yields:
$H' = H_0 + \tau_3 (\varepsilon_k \sigma_0 + \mathbf{a}(\mathbf{k}) \cdot \boldsymbol{\sigma}) + \tau_0 \mathbf{b}(\mathbf{k}) \cdot \boldsymbol{\sigma} + \dots$
Here, two emergent vectors are identified:
- $\mathbf{a}(\mathbf{k}) = \frac{1}{\xi} \text{Re}(\Delta_s^* \mathbf{d})$ : An even-parity vector in momentum space that breaks inversion symmetry, acting as an effective spin-orbit coupling.
- $\mathbf{b}(\mathbf{k}) = \frac{i}{2\xi} \mathbf{d} \times \mathbf{d}^*$ : An odd-parity vector that breaks time-reversal symmetry, acting as an effective magnetic field.
  This establishes a direct correspondence: nonunitary mixed parity superconductivity shares the same physics as magnetism with SOC, even if the underlying material possesses neither intrinsic SOC nor magnetic order.
Demonstration of Four Physical Phenomena:
The paper applies this correspondence to explain four specific effects that originate purely from the superconducting state:
- Dzyaloshinskii-Moriya (DM) Type Interactions: The paper confirms that DM interactions arise from the mixed parity pairing (specifically determined by $\mathbf{a}(\mathbf{k})$ ). It distinguishes this mechanism from SOC-driven DM interactions by their temperature dependence: the superconductivity-induced effect vanishes at the critical temperature ( $T_c$ ), whereas SOC-driven effects persist.
- Magnetoelectric Effect (Edelstein Effect): The study demonstrates that a supercurrent can induce spin polarization in mixed parity superconductors without intrinsic SOC. The induced spin polarization is proportional to the product of the singlet and triplet gap functions ( $\Delta_s \Delta_t$ ). Calculations show this effect is comparable in magnitude to the Edelstein effect in Rashba systems.
- Supercurrent-Induced Spin Current: In the presence of the effective magnetic field $\mathbf{b}(\mathbf{k})$ (requiring nonunitary triplet pairing where $\mathbf{d} \times \mathbf{d}^* \neq 0$ ), a supercurrent drives a spin supercurrent. This is analogous to spin currents generated in ferromagnets by charge currents.
- Altermagnetism: The momentum-dependent effective magnetic field leads to momentum-resolved spin polarization. For specific $\mathbf{d}$ -vector configurations (e.g., $d \propto (k_x, ik_y, 0)$ ), the spin polarization exhibits $d$ -wave symmetry in momentum space, a hallmark of altermagnetism.

Significance and Claims
The paper claims that these results reveal a new perspective on noncentrosymmetric superconductors. Traditionally, the breaking of spatial inversion symmetry in these materials is incorporated via Rashba-type SOC. This work argues that broken inversion symmetry can manifest in a variety of physical phenomena even without Rashba-type SOC, provided the superconductivity is nonunitary and of mixed parity.

The author posits that the "spin-orbit physics" and "magnetic physics" observed in these systems may not require external relativistic interactions or magnetic order but can emerge intrinsically from the superconducting order parameter itself. The paper concludes that this correspondence allows for the exploration of orbitronics and altermagnetism in superconducting systems by replacing spin Pauli matrices with orbital matrices, suggesting a pathway to orbital physics without orbital Rashba interactions.

The work is supported by JSPS KAKENHI grants and explicitly excludes strong correlation effects, noting them as a subject for future investigation.

Technical Summary: Nonunitary Mixed Parity Superconductivity and Magnetism with Spin-Orbit Coupling

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