Effect of symmetry breaking on altermagnetism in CrSb and Formation of fragmented nodal curves

By employing DFT calculations and symmetry analysis on CrSb and related models, this study reveals that reducing six-fold to two-fold rotational symmetry via vacancy engineering, doping, or strain induces band-specific fragmented nodal curves and enables tunable anomalous Hall conductivity, thereby expanding the potential of altermagnets for future quantum devices.

The Gist

Imagine a bustling city where the traffic lights are perfectly synchronized. In this city, there are two types of cars: "Red" cars and "Blue" cars. In a normal city (a standard magnet), all the Red cars might go one way, and all the Blue cars go the other, creating a net flow of traffic in one direction. In a standard anti-magnet, the Red and Blue cars are perfectly balanced, canceling each other out so there is no net flow at all.

This paper introduces a third, very strange type of city called an Altermagnet. Here, the Red and Blue cars are still perfectly balanced overall (no net flow), but if you look at specific streets (directions in space), the Red cars zoom fast while the Blue cars crawl, or vice versa. It's like a dance where the partners move in opposite directions depending on exactly where they are on the dance floor.

The researchers studied a specific material called CrSb (Chromium Antimony) to understand how this dance works and what happens if we mess with the city's layout.

The Perfect City: Pristine CrSb

In its natural, perfect state, CrSb is like a city built on a hexagonal grid (like a honeycomb). It has a high degree of symmetry, meaning if you rotate the city by 60 degrees, it looks exactly the same.

Because of this perfect symmetry, the "dance" of the electrons follows strict rules. There are invisible walls in the city called Nodal Planes. On these walls, the Red and Blue cars move at the exact same speed (they are "degenerate"). Everywhere else, they split apart. In this perfect city, there are four of these walls: one flat floor and three diagonal walls cutting through the city.

Breaking the Rules: Vacancies and Doping

The researchers asked: "What happens if we break the symmetry of this city?" To find out, they created five "Model Cities" (Model Structures) by playing with the atoms:

1. Removing atoms (Vacancies): Taking out some Antimony (Sb) atoms.
2. Adding atoms (Doping): Stuffing extra Antimony atoms into empty spaces.

The Result:

• Small changes: When they removed or added just a few atoms, the city still kept its 6-fold rotational symmetry (or a slightly twisted version of it). The dance still had those four straight walls (nodal planes). The "split" between Red and Blue cars got weaker, but the pattern remained the same.
• Big changes (The Discovery): When they arranged the atoms in a specific way (Model V) or squashed the city with uniaxial strain (squeezing it from one side), they broke the symmetry down to just a 2-fold rotation (like flipping a coin).

The Big Surprise: Fragmented Nodal Curves (FNCs)

This is the paper's main discovery. When the symmetry dropped from 6-fold to 2-fold, the straight, infinite walls (nodal planes) vanished.

Instead of straight walls, the researchers found Fragmented Nodal Curves (FNCs).

• The Analogy: Imagine the straight walls of the city were replaced by a series of floating, broken rings or loops scattered randomly throughout the 3D space.
• The Rule: These loops are "band-specific." This means that for one pair of dancing electrons, the loop might look like a circle. For a different pair of electrons, the loop might look like a figure-eight or a squiggly line. They are not the same shape for everyone.
• Why it matters: In the perfect city, the rules were the same everywhere. In this broken city, the "meeting points" where Red and Blue cars move at the same speed are now scattered, unique, and specific to each pair of dancers.

Validating the Discovery

To prove this wasn't just a fluke of their computer models, they looked at two other things:

1. Squeezing CrSb: They simulated squeezing the perfect CrSb crystal. Just like their model, the straight walls broke apart into these scattered loops (FNCs).
2. RbMnPO4: They looked at a different material, RbMnPO4, which naturally has this lower symmetry. They found the same scattered loops there, confirming that this "Fragmented Nodal Curve" phenomenon is real and happens in other materials too.

The Traffic Flow: Anomalous Hall Conductivity (AHC)

The paper also looked at how this affects the "traffic flow" (electrical current).

• In the perfect city: If the "Néel vector" (the direction the magnetic dance is oriented) points up (out of the floor), the traffic flow cancels out completely. No current flows sideways.
• In the broken city (with FNCs): Because the symmetry is lower, the rules change. Now, even if the magnetic dance points up (out-of-plane), a sideways current can flow.
• The Analogy: In the perfect city, the traffic lights forced all sideways movement to cancel out. In the broken city, the lights are different, allowing a "sideways drift" of electrons even when the main magnetic direction is vertical.

Summary

The paper shows that by breaking the symmetry of a magnetic material (CrSb) through defects or strain, you can destroy the straight "walls" where electrons behave identically. In their place, you get scattered, unique "loops" (Fragmented Nodal Curves). This change unlocks a new ability: the material can now generate a sideways electrical current (Anomalous Hall Effect) even when its magnetic direction is pointing straight up, a feat impossible in the perfect, symmetric version of the material.

Technical Summary

Technical Summary: Effect of Symmetry Breaking on Altermagnetism in CrSb and Formation of Fragmented Nodal Curves

Problem Statement
Altermagnetism represents a distinct magnetic phase characterized by momentum-space spin polarization (MSSP) where antiferromagnetic sub-bands split in a manner resembling time-reversal symmetry breaking, yet maintaining zero net magnetization. While high-symmetry altermagnets like CrSb (hexagonal $P6_3/mmc$) exhibit robust altermagnetic spin splitting (AMSS) and nodal planes, the mechanisms governing the evolution of these properties under symmetry reduction remain to be fully explored. Specifically, there is a need to understand how vacancy engineering, interstitial doping, and strain affect the nodal structures (planes vs. curves) and the resulting anomalous Hall conductivity (AHC), particularly regarding the orientation of the Néel vector.

Methodology
The authors employed first-principles Density Functional Theory (DFT) calculations using the Vienna ab-initio Simulation Package (VASP) with the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation. For the correlated system RbMnPO$_4$, the DFT+$U$ method was utilized.

• Model Construction: Five hypothetical model structures (MS-I to MS-V) were derived from pristine CrSb by manipulating non-magnetic Sb atoms at interstitial voids (vacancy creation and interstitial doping).
• Symmetry Analysis: The study utilized Spin Space Group (SSG) notation to analyze the symmetry operations connecting opposite spin sublattices. The authors examined how reducing rotational symmetry from six-fold ($C_{6z}$ or $S_{6z}$) to two-fold ($C_{2z}$) alters the band topology.
• Transport Properties: Anomalous Hall Conductivity (AHC) was calculated using the Kubo formula within the Quantum ESPRESSO framework, employing maximally localized Wannier functions (Wannier90) and the WannierBerri code. Calculations were performed for various Néel vector orientations (out-of-plane and in-plane).
• Validation: The findings were validated by applying uniaxial strain to pristine CrSb and by analyzing the electronic structure of the experimentally synthesized antiferromagnet RbMnPO$_4$.

Key Contributions and Results

1. Symmetry Reduction and Nodal Topology:

   • Pristine and High-Symmetry Models (MS-I, MS-II): In pristine CrSb and models retaining six-fold symmetry (via $C_{6z}$ or improper rotation $S_{6z}$), the system exhibits four nodal planes (one basal and three diagonal). The study mathematically demonstrates that in the non-relativistic limit, proper rotation ($C_{6z}$) and improper rotation ($S_{6z}$) yield identical altermagnetic characteristics and nodal plane structures.
   • Low-Symmetry Models (MS-III, MS-IV): Structures possessing an inversion center were found to be conventional antiferromagnets with no altermagnetic splitting, as the inversion symmetry enforces band degeneracy across the entire Brillouin zone.
   • Fragmented Nodal Curves (FNCs): In Model Structure-V (MS-V), where symmetry is reduced to two-fold rotation ($C_{2z}$) and mirror symmetry ($M_z$) without an inversion center, the three diagonal nodal planes vanish. Instead, Fragmented Nodal Curves (FNCs) emerge across the Brillouin zone. Unlike nodal planes, these FNCs are band-specific; different pairs of spin-opposite sublattice bands exhibit distinct curve shapes and locations. Along these curves, bands are degenerate; elsewhere, they split.

2. Realization via Strain and Other Materials:

   • The formation of FNCs was confirmed to be achievable in pristine CrSb by applying 5% uniaxial strain along the $a$-axis, which breaks the $C_{6z}$ symmetry down to $C_{2z}$.
   • The phenomenon was further validated in RbMnPO$_4$, a known antiferromagnet with $P2_1$ symmetry, which also exhibits FNCs in its constant energy surfaces.

3. Anomalous Hall Conductivity (AHC) Engineering:

   • Pristine/High-Symmetry: In pristine CrSb and MS-I/II, AHC is zero when the Néel vector is out-of-plane (along $c$-axis) due to symmetry-enforced cancellation of Berry curvature. A finite AHC is only permitted when the Néel vector is oriented in-plane.
   • Low-Symmetry (MS-V and Strained CrSb): The reduction to $C_{2z}$ symmetry fundamentally alters the AHC landscape. In MS-V and strained CrSb, a finite AHC is permitted for both out-of-plane and in-plane Néel vector orientations.
   • Specifically, for the out-of-plane orientation in strained CrSb, the AHC tensor components $\sigma_{xy}^z$ and $\sigma_{yz}^x$ become non-zero. The magnitude of AHC near the Fermi level in strained CrSb was observed to be higher than in the pristine case.

Significance and Claims
The paper claims to identify a new topological feature in altermagnets: Fragmented Nodal Curves (FNCs), which arise specifically when the rotational symmetry of an altermagnet is restricted to two-fold. The authors posit that this symmetry lowering offers a tunable mechanism to manipulate the momentum space spin polarization (MSSP) and the Néel vector orientation.

Crucially, the work demonstrates that symmetry breaking (via strain or doping) can unlock finite AHC in configurations where it was previously forbidden (specifically for out-of-plane Néel vectors). The authors suggest that this flexibility in generating AHC, combined with the tunability of FNCs, provides a pathway for utilizing altermagnets in future quantum devices and spintronic applications. The study emphasizes that these findings are derived from theoretical modeling and DFT calculations, offering a framework for understanding how chemical and structural modifications can tailor the transport properties of altermagnetic materials.