Higher-order topological insulators in two-dimensional antiferromagnetic and altermagnetic chromium-based group-IV chalcogenides

Based on first-principles calculations and theoretical analysis, this study identifies a family of monolayer chromium-based group-IV chalcogenides as a new class of two-dimensional magnetic higher-order topological insulators that host robust, quantized corner states protected by lattice symmetry in both antiferromagnetic and altermagnetic ground states.

The Gist

Imagine a vast, flat city made of atoms. In this city, electrons are the citizens trying to get from one side to the other. Usually, in a normal city (a regular material), these citizens can walk freely everywhere. In a standard "Topological Insulator," the citizens are blocked from walking through the middle of the city (the bulk), but they are allowed to walk freely along the streets on the very edge (the boundary).

But this paper introduces a new, even more exclusive type of city: a Higher-Order Topological Insulator (HOTI).

Here is the simple breakdown of what the researchers found, using some everyday analogies:

1. The "Corner Store" Phenomenon

In a normal city, if you block the middle and the streets, you stop traffic completely. In a standard Topological Insulator, the "streets" (edges) are open.

However, in these new Higher-Order cities, the "streets" are also blocked! The only place where the electron citizens can move freely is at the very corners of the city. It's like a building where the lobby and the hallways are locked, but the four corners of the roof are open for a party. The researchers found that these specific materials naturally create "corner states" where electrons get stuck and hang out, carrying a special, fractional electric charge.

2. The New Neighborhoods: Antiferromagnets and Altermagnets

The city in this story is made of Chromium (a magnetic metal) and other elements like Sulfur or Selenium. The researchers looked at two different types of "magnetic neighborhoods":

• The "Mirror Twins" (Antiferromagnets): Imagine a neighborhood where half the houses have red lights on and the other half have blue lights, arranged perfectly so the whole neighborhood looks neutral from the outside. This is a Conventional Antiferromagnet. The electrons here are paired up nicely, and the "corner party" happens because of the symmetry of the building.
• The "Spinning Dancers" (Altermagnets): This is the exciting new discovery. Imagine a neighborhood where the lights are still red and blue, but instead of just mirroring each other, they are arranged in a spinning pattern (like a pinwheel). This is called an Altermagnet. It's a brand-new type of magnetism that was only recently discovered. The researchers found that these "spinning" neighborhoods also host the special "corner party" for electrons.

3. The "Magic Shield" (Symmetry)

Why do these corner states exist? The paper explains that the buildings in this atomic city are shaped like triangles (or hexagons with 3-fold symmetry).

Think of it like a spinning top. If you spin a triangle, it looks the same after you turn it 120 degrees. This specific "rotational symmetry" acts like a magic shield. It forces the electrons to gather at the corners. Even if you try to push them away, the rules of the city (the symmetry) force them back to the corners.

4. The "Toughness" Test (Spin-Orbit Coupling)

In the real world, materials often get messed up by things like heat or magnetic interference. In physics, there's a force called Spin-Orbit Coupling (SOC) that can sometimes ruin delicate quantum states.

The researchers tested these materials by "turning on" this force. It's like shaking the building to see if the corner party falls apart.

• The Result: The party stayed! The corner states remained robust and stable. This is huge because it means these materials aren't just a theoretical curiosity; they are sturdy enough to potentially be used in real devices.

Why Should You Care?

The researchers are essentially saying: "We found a new family of materials that act like a fortress for electrons, trapping them only at the corners, and they do this even when they are magnetic."

This is important for the future of technology (specifically Spintronics and Quantum Computing) because:

1. Efficiency: These "corner states" are very stable and don't easily scatter, meaning they could carry information with almost no energy loss.
2. New Physics: It proves that you can mix magnetism (which usually disrupts delicate quantum states) with high-order topology to create something new and useful.
3. The "Altermagnet" Bonus: Since these materials include the newly discovered "Altermagnets," they offer a way to control electron spins without creating messy magnetic fields that interfere with nearby electronics.

In a nutshell: The researchers discovered a new class of "atomic triangles" made of Chromium. These triangles are so perfectly symmetrical that they force electrons to hide in the corners, creating a super-stable, magnetic, high-tech playground for future computers.

Technical Summary

1. Problem Statement

The field of topological quantum materials has extensively explored conventional topological insulators (TIs) and higher-order topological insulators (HOTIs). However, a significant gap exists in the identification of magnetic HOTIs, particularly in two-dimensional (2D) systems.

• Scarcity: Most known HOTIs are non-magnetic. 2D magnetic HOTIs are rare, with only a few candidates proposed (e.g., transition-metal–organic frameworks, RuCl$_2$).
• Magnetic Order Complexity: While conventional antiferromagnets (cAFM) and the recently discovered altermagnets (a third class of collinear magnets with spin-split bands but zero net magnetization) offer unique advantages for spintronics (e.g., ultrafast dynamics, no stray fields), their interplay with higher-order topology remains largely unexplored.
• Goal: The authors aim to identify new 2D magnetic materials that host HOTI phases, specifically investigating whether chromium-based group-IV chalcogenides can realize both conventional antiferromagnetic and altermagnetic HOTI states.

2. Methodology

The study employs a combination of first-principles calculations and theoretical topological analysis:

• Computational Framework: Density Functional Theory (DFT) calculations were performed using the Vienna ab initio Simulation Package (VASP) with the Projector Augmented-Wave (PAW) method.
• Exchange-Correlation: Generalized Gradient Approximation (GGA) with the PBE functional was used. To account for strong electron correlations in Cr 3$d$ orbitals, the DFT+U method was applied with an effective $U$ value of 3 eV (tested range 1–4 eV).
• Structural Stability: Phonon spectra were calculated using the PHONOPY code within the framework of Density Functional Perturbation Theory (DFPT) to verify dynamical stability.
• Topological Analysis:
  • Symmetry Indicators: Irreducible representations of occupied states were extracted using irvsp to calculate topological invariants based on $C_3$ rotational symmetry.
  • Model Construction: An ab initio tight-binding model was constructed using the Wannier90 package.
  • Corner State Verification: The topological nature was explicitly verified by calculating the energy spectra and charge distributions of triangular-shaped nanodisks (approx. 3.5 nm side length) to identify zero-dimensional (0D) corner states.
• Spin-Orbit Coupling (SOC): Calculations were performed both without and with SOC to assess the robustness of the topological phase.

3. Key Contributions

• Discovery of a New Material Family: The authors identify a family of monolayer chromium-based group-IV chalcogenides ($CrCX_3$ where $X=S, Se, Te$;$CrSiS_3$; and Janus compounds $Cr_2C_2S_3Se_3$ and $Cr_2Si_2S_3Se_3$) as a new class of 2D magnetic HOTIs.
• Dual Magnetic Classification: The study distinguishes between two magnetic ground states within the same material family:
  • Conventional Antiferromagnets (cAFM): $CrCX_3$ and $CrSiS_3$ exhibit Néel-type antiferromagnetism with $PT$ symmetry, resulting in spin-degenerate bands.
  • Altermagnets: The Janus compounds ($Cr_2C_2S_3Se_3$ and $Cr_2Si_2S_3Se_3$) exhibit altermagnetic order, characterized by momentum-dependent spin splitting in the absence of SOC due to specific rotational symmetries ($M_{\bar{1}10}$) rather than inversion.
• Mechanism of Topology: The paper establishes that the nontrivial topology in these systems is protected by lattice $C_3$ rotational symmetry, leading to quantized fractional corner charges ($e/3$).

4. Key Results

A. Structural and Magnetic Properties

• Crystal Structure: The materials feature an edge-sharing honeycomb network of $CrX_6$ octahedra with $C$ (or $Si$) dimers at the center. They belong to layer groups $p\bar{3}1m$ (non-Janus) and $p31m$ (Janus), preserving $C_3$ symmetry.
• Stability: Phonon spectra show no imaginary modes, confirming dynamical stability. The structures are analogous to experimentally synthesized $CrSiTe_3$ and $CrGeTe_3$, suggesting experimental feasibility.
• Magnetic Ground State: All systems energetically favor the Néel-type antiferromagnetic (NAFM) configuration.
  • Magnetic moments are localized on Cr atoms ($\approx 3 \mu_B$).
  • In Janus compounds, the two spin-opposed sublattices are related by rotational symmetry, classifying them as altermagnets.

B. Electronic Band Structure and Topology

• Band Gaps: All materials are indirect-band-gap insulators (PBE gaps range from $\approx 1.3$ eV to $2.2$ eV).
• Spin Splitting:
  • cAFM systems: Bands are fully spin-degenerate due to $PT$ symmetry.
  • Altermagnetic systems: Exhibit spin splitting along $\Gamma-X$ and $\Gamma-X'$ directions even without SOC, a hallmark of altermagnetism.
• Topological Invariants: Using $C_3$ symmetry eigenvalues at high-symmetry points (specifically the $K$ point), the topological invariant $[K^{(3)}_1]$ was calculated to be $-2$ for both spin channels.
• Fractional Corner Charge: This invariant predicts a quantized fractional corner charge of $Q = e/3$ for each spin channel.

C. Corner States and Robustness

• Zero-Energy Modes: Calculations on triangular nanodisks reveal three degenerate in-gap states localized strictly at the three corners of the disk.
• Charge Accumulation: Real-space wavefunction analysis confirms these states carry a fractional charge of $e/3$ per spin channel.
• Effect of SOC:
  • SOC has a minor impact on the low-energy bands (derived mainly from S/Se $p$-orbitals).
  • Crucially, the bulk band gap remains open upon including SOC.
  • The system remains adiabatically connected to the SOC-free phase, preserving the HOTI nature.
  • Robustness: The corner states persist with SOC, appearing as six in-gap states (three from spin-up, three from spin-down), confirming the stability of the higher-order topological phase.

5. Significance

• Material Platform: This work provides a concrete, experimentally accessible material platform (chromium-based chalcogenides) for realizing 2D magnetic HOTIs, bridging the gap between theory and experiment.
• Unification of Concepts: It reveals an intrinsic connection between higher-order topology and magnetic order (both cAFM and altermagnetism), demonstrating that altermagnets can host robust topological corner states.
• Spintronics Applications: The coexistence of altermagnetism (spin-split bands, no stray fields) and HOTI physics (robust corner states) opens new avenues for topological spintronics. The corner states, which can be probed via Scanning Tunneling Spectroscopy (STS), offer potential for low-dissipation, spin-polarized transport and quantum information applications.
• Theoretical Advancement: The study expands the known landscape of magnetic HOTIs beyond ferromagnets, highlighting the role of rotational symmetry in protecting topological phases in zero-net-magnetization systems.