Interplay between Relativistic Spin-Momentum Locking and Breaking of Inversion Symmetry: conditions for p-wave magnetism

This paper investigates how the interplay between relativistic spin-momentum locking and various forms of inversion symmetry breaking in the altermagnet Ca2RuO4 leads to diverse magnetic phases, including Rashba and Weyl-type spin-orbit couplings and exotic states with weak ferromagnetism, thereby explaining experimental observations and predicting conditions for p-wave magnetism.

The Gist

The Big Picture: A Magnetic Dance Floor

Imagine a crowded dance floor where the dancers are electrons. In most magnets, the dancers all spin in the same direction (like a line of soldiers). In this specific material, Ca₂RuO₄, the dancers are arranged in a very specific, alternating pattern: some spin up, some spin down, but the total number of dancers spinning up equals the total spinning down. The floor has no net "spin" to it.

Scientists call this altermagnetism. It's a special state where the dancers are locked in a pattern based on where they are standing on the floor (their momentum). If you move to the left, you spin one way; if you move to the right, you spin the other. This is called spin-momentum locking.

This paper asks a simple question: What happens if we tilt the dance floor or change the rules of the room? Specifically, what happens if we break the perfect symmetry of the room (inversion symmetry) using electric fields or structural shifts?

The Main Characters

1. The Material (Ca₂RuO₄): Think of this as a layered cake made of Ruthenium and Oxygen. It's a "testbed" material, meaning scientists use it to test theories because it's complex and interesting.
2. The "Locking" (Spin-Momentum): Imagine that every dancer has a rule: "If I step forward, I must spin clockwise. If I step backward, I must spin counter-clockwise." This rule is the spin-momentum locking.
3. The Symmetry Breaking: Imagine the dance floor is perfectly square and balanced. Now, imagine someone pushes the floor so it tilts, or shifts the tiles so the pattern changes. This is breaking inversion symmetry.

The Three Scenarios Explored

The researchers tested three different ways to "tilt" or "shift" this magnetic dance floor to see how the dancers' rules change.

1. The "Rashba" Tilt (The One-Way Street)

When they applied a specific type of shift (like a ferroelectric distortion), they created a Rashba effect.

• The Analogy: Imagine the dance floor has a strong wind blowing in one direction.
• The Result: The dancers who were following the original "step forward = spin clockwise" rule found that rule broken for two of their directions. However, the rule for the direction parallel to the wind remained intact.
• The Twist: The dancers who lost their old rule didn't just stop spinning; they adopted a new, simpler rule (like a "p-wave" pattern). It's like they switched from a complex dance to a simple march, but only in the direction the wind was blowing.
• Key Finding: The material still had zero net spin (no weak ferromagnetism), but the complex locking pattern was simplified for some dancers.

2. The "Weyl" Tilt (The Maze)

When they applied a different shift (antiferroelectric distortion along the x-axis), they created a Weyl effect.

• The Analogy: Imagine the dance floor turns into a maze where the walls are constantly moving.
• The Result: This was the most chaotic scenario. The original "step forward = spin clockwise" rule was completely destroyed for all dancers in all directions.
• The Twist: Instead of flat "nodal planes" (areas where the spin rule was zero), the dancers now only had "nodal lines" (thin lines where the rule was zero). It's like the dance floor lost its flat spots and became a series of ridges.
• Key Finding: Even though the complex locking was shattered, the material still had zero net spin. The "Weyl" effect broke the locking for everyone, but didn't make the whole group spin in one direction.

3. The "Stripe" Shift (The Patchwork Quilt)

Finally, they simulated a "stripe" phase, where only one layer of the cake was shifted, while the others stayed put.

• The Analogy: Imagine a patchwork quilt where one square has a different pattern than the rest.
• The Result: This created a unique situation where two different sets of rules existed at the same time. Some dancers followed the "3D bulk" rule, while others followed a "2D surface" rule that was usually hidden.
• The Twist: This mixing of two different locking patterns created a new, exotic state. Crucially, this specific mix caused the material to develop a tiny bit of weak ferromagnetism (a tiny net spin), which didn't happen in the other scenarios. It's like the patchwork quilt finally tipped over just enough to have a slight lean.

The "Spin Canting" (The Lean)

The paper also looked at how the dancers lean. In the perfect, balanced room, the dancers lean slightly but cancel each other out perfectly, resulting in no net lean.

• When the room is tilted (symmetry broken), the dancers lean differently.
• However, the researchers found that in most cases, the material still managed to keep its "zero net spin" status, even with the tilt. It only developed a net lean (weak ferromagnetism) in the specific "stripe" scenario where two different patterns mixed.

Summary of Findings

• Symmetry is Key: The way the material locks spin to movement depends entirely on the symmetry of the crystal structure.
• Different Tilts, Different Rules: Breaking symmetry doesn't just "break" the rules; it often replaces complex rules with simpler ones (like switching from a d-wave dance to a p-wave march) or destroys the rules entirely (Weyl effect).
• No Net Spin (Usually): Even when the complex patterns are destroyed by electric fields or structural shifts, the material usually remains a "pure" altermagnet with zero net magnetization.
• The Exception: Only when you create a "stripe" pattern (mixing two different internal rules) does the material develop a tiny, detectable magnetic lean (weak ferromagnetism).

In short, the paper maps out how a specific magnetic material behaves when you poke, prod, and tilt it. It shows that while the material is robust and usually keeps its "zero net spin" promise, the internal dance of the electrons changes dramatically depending on how you break its symmetry.

Technical Summary

Technical Summary: Interplay between Relativistic Spin-Momentum Locking and Breaking of Inversion Symmetry: Conditions for p-wave Magnetism

Problem Statement
Recent advancements in the field of magnetism have established the classification of non-relativistic spin-momentum locking in altermagnets. However, a comprehensive understanding of how these non-relativistic lockings evolve when relativistic effects (spin-orbit coupling, SOC) are introduced, particularly in the presence of broken inversion symmetry, remains incomplete. While previous studies have addressed Rashba-type spin-splitting in altermagnets using model Hamiltonians, a thorough investigation of the interplay between intrinsic altermagnetic spin-splitting and Rashba- or Weyl-type relativistic spin-splitting is lacking. Specifically, it is unclear under which conditions nodal planes persist, how different spin components behave under symmetry breaking, and whether exotic states such as p-wave magnetism or the coexistence of multiple spin-momentum lockings can emerge.

Methodology
The authors utilize the quasi-two-dimensional material $\text{Ca}_2\text{RuO}_4$ as a testbed system due to its complex magnetic structure (featuring four magnetic atoms per unit cell) and its status as an orbital-selective altermagnet. The study employs a combination of:

1. Density Functional Theory (DFT) Calculations: Performed on the centrosymmetric bulk phase (space group 61) and various non-centrosymmetric distortions. The calculations analyze electronic band structures, magnetic moments, and spin canting under different magnetic configurations (A-centered and B-centered) and Néel vector orientations.
2. Analytical Modeling: Development of tight-binding models and effective Hamiltonians to describe the non-relativistic and relativistic spin-momentum locking. These models incorporate inter- and intralayer hybridization terms and Rashba/Weyl Hamiltonians to derive analytical expressions for eigenvalues and spin textures.
3. Symmetry Analysis: Systematic investigation of ferroelectric (uniform displacement) and antiferroelectric (alternating displacement) distortions, as well as modulated electric fields simulating stripe phases, to determine the resulting space groups and symmetry-protected nodal features.

Key Contributions and Results

• Centrosymmetric Ground State: The study confirms that the experimental ground state of bulk $\text{Ca}_2\text{RuO}_4$ is an A-centered magnetic order with the Néel vector along the $b$-axis. This state is a pure altermagnet with zero net magnetization, despite exhibiting significant spin canting (up to 10% of the total moment) driven by staggered Dzyaloshinskii–Moriya interaction (DMI) and strong SOC. The non-relativistic spin-momentum locking is identified as a planar $d_{xz}$-wave, with nodal planes at $k_x=0$ and $k_z=0$.
• Relativistic Spin-Momentum Locking: Upon including SOC, the system exhibits relativistic spin-momentum locking for all three spin components ($S_x, S_y, S_z$). The dominant $S_y$ component retains the non-relativistic $d_{xz}$ character, while the subdominant $S_x$ and $S_z$ components acquire $d_{yz}$ and $d_{xy}$ characters, respectively, due to antisymmetric exchange.
• Impact of Inversion Symmetry Breaking:
  • Ferroelectric and Antiferroelectric Distortions: Breaking inversion symmetry via uniform or alternating displacements of Ru atoms introduces Rashba-type or Weyl-type SOC.
  • Rashba Regime: In most distortion scenarios (e.g., ferroelectric distortions along $x, y, z$ or specific antiferroelectric distortions), the system exhibits Rashba-type SOC. The authors find that the Rashba interaction destroys the nodal planes for spin components perpendicular to the electric field, transforming their spin-momentum locking from $d$-wave to $p$-wave (e.g., $S_x$ and $S_y$ components adopting $p_y$ and $p_x$ characters). Crucially, the spin component parallel to the electric field preserves its original nodal plane and $d$-wave character. Importantly, Rashba-type SOC does not induce weak ferromagnetism; the net magnetization remains zero.
  • Weyl Regime: In the case of antiferroelectric distortion along the $x$-axis (space group 19), the system exhibits Weyl-type SOC. This form disrupts the nodal planes for all three spin components, leaving only symmetry-protected nodal lines (NLs) coincident with the $k_y$ and $k_z$ axes. Like the Rashba case, the Weyl interaction preserves zero net magnetization.
• Stripe Phase and Hidden Altermagnetism: The authors simulate a stripe phase by applying a modulated electric field that displaces Ru atoms in only one layer. This distortion lowers the crystal symmetry sufficiently to activate "hidden" altermagnetism. The resulting state hosts the coexistence of two distinct non-relativistic spin-momentum lockings: the bulk $d_{xz}$-wave and a $d_{xy}$-wave (typically associated with 2D or single-layer limits). This coexistence leads to a single nodal plane ($k_x=0$) and induces a weak ferromagnetic moment, representing a transition from a pure altermagnetic phase to a weak ferromagnetic phase.

Significance
The paper provides a comprehensive analysis of how relativistic effects and inversion symmetry breaking modify the topological and magnetic properties of altermagnets. It establishes that the interplay between intrinsic altermagnetic locking and Rashba/Weyl SOC is highly dependent on the specific symmetry breaking and the orientation of the Néel vector. The work identifies specific conditions under which $p$-wave magnetic orders emerge (via Rashba coupling) and when nodal lines replace nodal planes (via Weyl coupling). Furthermore, it demonstrates that structural distortions, such as those found in stripe phases, can unlock hidden altermagnetic channels and induce weak ferromagnetism without destroying the zero-net-magnetization character of the underlying altermagnetic order. These findings offer a theoretical framework for understanding and potentially manipulating spin-momentum locking in altermagnetic materials for spintronic applications, particularly in systems where electric fields or strain are used to tune symmetry.