Symmetry-protected nodal planes and accidental nodal surfaces in mixed odd-even wave spin-momentum locking of relativistic altermagnets

This study investigates relativistic spin-momentum locking in centrosymmetric CrSb and noncentrosymmetric MnTe, revealing that while $g$-wave symmetry is preserved only under specific Néel vector and electric field alignments, ferroelectric altermagnets can exhibit mixed angular-momentum wave symmetries featuring both symmetry-protected nodal planes and accidental nodal surfaces.

The Gist

Imagine a dance floor where electrons are the dancers. In most magnetic materials, these dancers either spin in the same direction (like a crowd of people all facing north) or in opposite pairs that cancel each other out perfectly.

This paper introduces a special, rare type of magnetic material called an altermagnet. Think of an altermagnet as a dance floor where partners are arranged in a very specific, symmetrical pattern: if you rotate the floor by a certain angle, the dancers swap places, but their "spin" (the direction they are facing) flips. Crucially, they are not just mirror images; they are connected by rotation, not by simple reflection or sliding.

The researchers studied what happens when these dancers move very fast (relativistic speeds) and when the dance floor itself is slightly tilted or distorted (breaking "inversion symmetry"). Here is the breakdown of their findings using everyday analogies:

1. The "G-Wave" Pattern (The Complex Dance)

In the slow, non-relativistic world, the dominant spin pattern in these materials is called a g-wave.

• The Analogy: Imagine a complex ripple in a pond created by dropping four stones at once. This pattern has four distinct "nodal planes." Think of these as invisible walls or lines on the dance floor where the dancers stop spinning entirely (zero spin). In a perfect, symmetrical room, these four walls are fixed by the architecture of the building.

2. The Relativistic Twist (Speed and Tilt)

The paper asks: What happens when we turn on the "relativistic" effects (like spin-orbit coupling, which is like adding a strong wind or a tilt to the floor)?

• The Finding: If the magnetic "compass" (the Néel vector) points straight up (along the z-axis), the main dancers (the dominant spin component) keep their complex g-wave pattern. They still have their four walls.
• The Twist: However, the other dancers (the sub-dominant components) change their routine.
  • In the material CrSb (a symmetrical room), these extra dancers switch to a d-wave pattern (like a ripple from two stones, with fewer walls).
  • In the material MnTe (an asymmetrical room, like a tilted floor), these extra dancers switch to a p-wave pattern (like a ripple from one stone, with just one wall).

3. The "Accidental" Walls

This is where it gets interesting. In the symmetrical room (CrSb), the walls are fixed by the building's design. But in the tilted room (MnTe), the rules change.

• The Analogy: Imagine you have a wall that was supposed to be there because of the building's design. But because the floor is tilted, that wall doesn't disappear; it just moves to a slightly different spot. It's no longer "protected" by the building's rules; it's just an accidental wall that happens to be there.
• The Result: The researchers found that in these tilted materials, you can have a mix of patterns. You might have one "protected" wall (guaranteed by symmetry) and one "accidental" wall (which appears due to the specific balance of forces but isn't guaranteed).

4. Creating "P-Wave" Magnets

The paper proposes a new way to create p-wave magnets (materials with a specific, simpler spin pattern).

• The Recipe: Instead of looking for a material that is naturally a p-wave magnet (which is hard to find), take an altermagnet (which is usually a g-wave magnet) and tilt it (break the symmetry).
• The Outcome: For certain bands of electrons (certain "groups" of dancers), the complex g-wave pattern fades away, and the simpler p-wave pattern takes over. It's like the complex ripple in the pond simplifies into a single wave because of the tilt.

Summary of the Two Main Discoveries

1. Survival of the Complex: If you keep the magnetic compass pointing straight up, the main spin pattern (g-wave) survives the relativistic speed, even in tilted materials.
2. The Birth of Simplicity: If you tilt the material (break symmetry), you can force the material to behave like a p-wave magnet for specific groups of electrons. This creates a mix of "protected" walls (nodal planes) and "accidental" walls (nodal surfaces) where the spin vanishes.

In a nutshell: The authors discovered that by tilting the "dance floor" of these special magnetic materials, they can control how the electrons spin. They can keep the complex, high-order patterns alive or simplify them into new, useful patterns, creating a mix of guaranteed and accidental "no-spin" zones. This helps scientists understand how to engineer new magnetic materials for future technologies.

Technical Summary

Technical Summary: Symmetry-protected nodal planes and accidental nodal surfaces in mixed odd-even wave spin-momentum locking of relativistic altermagnets

Problem Statement
Altermagnets are a class of magnetic materials characterized by non-relativistic spin-momentum locking (SML) with even-wave symmetries (e.g., $d$-wave, $g$-wave), resulting in zero net magnetization despite broken time-reversal symmetry. In the non-relativistic limit, these systems exhibit a specific number of symmetry-protected nodal planes determined by their crystal symmetry and Néel vector orientation. However, the introduction of relativistic effects, specifically spin-orbit coupling (SOC) and the breaking of inversion symmetry (as found in ferroelectric or noncentrosymmetric structures), complicates this picture. While SOC preserves time-reversal symmetry in non-magnetic systems, in altermagnets it can induce weak ferromagnetism and modify the SML. A critical open question is how the high-order $g$-wave SML, characteristic of non-relativistic altermagnets, evolves under relativistic conditions. Specifically, it is unclear whether the $g$-wave character survives or reduces to lower symmetries (e.g., $d$-wave or $p$-wave) and how the interplay between the Néel vector orientation and inversion-symmetry breaking affects the nodal structure (nodal planes vs. accidental nodal surfaces).

Methodology
The authors employ first-principles calculations based on Density Functional Theory (DFT) using the Vienna Ab initio Simulation Package (VASP) with the Projector Augmented Wave (PAW) method and the Generalized Gradient Approximation (GGA-PBE). To account for localized $d$-orbital correlations, the DFT+$U$ formalism is applied ($U=2$ eV for Cr, $U=4$ eV for Mn). The study focuses on two hexagonal altermagnetic systems:

1. Centrosymmetric CrSb: Space group $P6_3/mmc$ (No. 194), with the Néel vector oriented along the $z$-axis.
2. Noncentrosymmetric Wurtzite MnTe: Space group $P6_3mc$ (No. 186), exhibiting ferroelectric properties, with Néel vectors oriented along the $x$, $y$, and $z$ axes.

The analysis involves calculating spin-resolved Fermi surfaces at specific $k_z$ planes ($0, \pm 0.25 \pi/c$) to map the momentum-space spin texture. The authors decompose the relativistic spin-momentum locking (RSML) into multipole expansions (monopole, dipole, quadrupole) to identify the dominant wave symmetries ($s, p, d, g$). They specifically analyze the superposition of Rashba-type contributions (arising from inversion-symmetry breaking) and the intrinsic magnetic quadrupole terms. The degree of collinearity is assessed by examining the relativistic spin density of states projected onto the two magnetic sublattices.

Key Contributions and Results

• Survival of $g$-wave Symmetry: The study establishes that the dominant spin component retains its $g$-wave character in the relativistic regime, but only under specific symmetry conditions. In both centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe, the dominant component ($S_z$) preserves the $g$-wave SML (four nodal planes) only when the Néel vector is aligned along the $z$-axis. If the Néel vector lies in the $xy$-plane, the symmetry reduces, and the $g$-wave character is lost or significantly diminished.
• Emergence of Mixed Symmetries and Subdominant Components: While the dominant component may retain high-order symmetry, the subdominant components ($S_x, S_y$) acquire lower angular-momentum symmetries. In centrosymmetric CrSb, subdominant components exhibit $d$-wave symmetry. In noncentrosymmetric wurtzite MnTe, the interplay between the Rashba effect and the magnetic order induces $p$-wave symmetry in the subdominant components.
• Nodal Plane Evolution and Accidental Nodal Surfaces (ANS):
  • In centrosymmetric systems, nodal planes are symmetry-protected.
  • In noncentrosymmetric systems (wurtzite MnTe), the breaking of inversion symmetry transforms some symmetry-protected nodal planes into accidental nodal surfaces (ANS).
  • The authors identify two distinct scenarios for the mixing of dipole ($p$-wave) and quadrupole ($d/g$-wave) terms:
    1. Common Nodal Plane: When the dipole and quadrupole share a nodal plane (e.g., $Q_{yz} + Q_y$), the result is a system with one symmetry-protected nodal plane and one ANS.
    2. No Common Nodal Plane: When they do not share a nodal plane (e.g., $Q_{yz} + Q_x$), the system exhibits no symmetry-protected nodal planes but possesses two ANS and two nodal lines, while maintaining zero net magnetization.
• Robust $p$-wave Magnetism in Specific Bands: A significant finding is that wurtzite MnTe with a Néel vector along the $x$-axis can host bands with robust $p$-wave magnetism (broken time-reversal symmetry). While the system generally exhibits mixed $p$- and $d$-wave character, specific energy bands (e.g., at $-2$ eV) are dominated by the $p$-wave component, whereas others (e.g., at $-1$ eV) remain dominated by even-wave ($g/d$) components.
• Collinearity: The study confirms that for Néel vectors along the $x$ or $z$ axes, the wurtzite MnTe phase remains a pure collinear altermagnet without spin canting, despite the presence of SOC and inversion-symmetry breaking. Weak ferromagnetism and spin canting only emerge when the Néel vector is aligned along the $y$-axis.

Significance and Claims
The paper claims to establish a framework for understanding the evolution of spin textures in relativistic altermagnets, specifically addressing the fate of high-order $g$-wave symmetries. The authors assert that their results provide a general guideline for identifying altermagnetic materials where higher-order wave symmetries remain robust against SOC.

Furthermore, the work proposes a "recipe" for realizing $p$-wave magnets with broken time-reversal symmetry by starting from altermagnets with broken inversion symmetry. The authors highlight that searching for $p$-wave states in systems with mixed odd- and even-wave components offers simpler symmetry requirements compared to previous proposals for pure odd-wave materials, thereby broadening the range of candidate materials to include noncentrosymmetric systems, surfaces, and engineered heterostructures.

The study emphasizes that while the $g$-wave character is restricted to the dominant component under specific conditions, the subdominant components can exhibit significant $p$-wave or $d$-wave character in momentum space, which is crucial for understanding phenomena such as the spin Hall conductivity, non-linear Hall effect, and spin photocurrents. The paper concludes that altermagnets can host distinct spin components that realize a mixture of angular-momentum wave symmetries in the relativistic limit.