Emergence of chiral $p$-wave and $d$-wave states in $g$-wave altermagnets

This study reveals that three-dimensional $g$-wave altermagnetic metals, such as CrSb, can host emergent chiral $p$-wave and $d$-wave superconducting states depending on the strength of the altermagnetic field and electron density, offering a novel platform for realizing unconventional superconductivity.

The Gist

Imagine the world of electrons inside a solid material as a bustling city. Usually, in a magnet, the "citizens" (electrons) are either all marching in the same direction (ferromagnetism) or marching in perfect, alternating opposition (antiferromagnetism).

But recently, scientists discovered a new type of city called an Altermagnet. In this city, the electrons are arranged in a complex, alternating pattern, but the city has a special twist: the "traffic rules" depend on which street you are on. If you walk down one street, the electrons spin one way; walk down the next, and they spin the other. This creates a unique "spin-splitting" effect where the energy of the electrons changes based on their direction of travel, even without a net magnetic field.

This paper focuses on a specific, exotic version of this city called a g-wave altermagnet (found in a material called CrSb). The researchers wanted to know: If we try to make these electrons pair up to become a superconductor (a material with zero electrical resistance), what kind of "dance" will they do?

Here is the breakdown of their discovery using simple analogies:

1. The Setup: A Twisted Dance Floor

Think of the electrons as dancers. In a normal superconductor, they usually pair up in a simple, symmetrical waltz (called s-wave). They hold hands and spin gently in place.

However, in this g-wave altermagnet city, the "floor" is twisted. The alternating magnetic streets push the dancers apart based on which way they are facing. This creates a chaotic environment where the usual simple waltz is hard to maintain.

2. The Discovery: Two New, Exotic Dances

The researchers ran a massive computer simulation to see what kind of dance would survive this twisted floor. They found that depending on the "strength" of the magnetic streets and how crowded the dance floor is (electron density), the electrons spontaneously switch to two very fancy, complex dances:

• The Chiral p-wave (The Spiral): Imagine the dancers forming a giant, spinning spiral. They don't just hold hands; they rotate around a central point with a specific "handedness" (like a right-handed screw). This is a chiral state. It's topologically protected, meaning it's very robust and could be used for future quantum computers.
• The Chiral d-wave (The Double-Loop): Imagine the dancers forming a figure-eight or a double-loop pattern that also spins in a specific direction. This is another exotic, chiral state.

The Key Finding:

• When the magnetic streets are strong and the floor is crowded: The electrons prefer the Chiral p-wave (the spiral).
• When the magnetic streets are weak and the floor is moderately crowded: The electrons prefer the Chiral d-wave (the double-loop).
• When the magnetic streets are very weak: They fall back to the boring, simple waltz (s-wave).

3. The Villain: The "Ghost" Floor (Bogoliubov Fermi Surfaces)

Why do the simple dances fail? The paper introduces a concept called Bogoliubov Fermi Surfaces (BFS).

Think of the simple dance (s-wave) as trying to pair up two dancers who are moving in opposite directions. In this twisted city, the magnetic streets create "ghost zones" on the dance floor where these opposite-moving dancers simply cannot exist together. The magnetic force pushes them apart so hard that the simple pairing breaks down.

However, the fancy dances (p-wave and d-wave) are different. They pair up dancers who are moving in the same direction or in a way that the magnetic streets actually help them. Because they don't rely on the "ghost zones" that kill the simple dances, they survive and become the dominant style.

4. How Do We Know? (The Experimental Signatures)

The researchers didn't just guess; they predicted how to spot these dances in real life:

• The Energy Map (ARPES): If you shine light on the material to see the energy of the electrons, the pattern of the "Chiral p-wave" or "Chiral d-wave" will break the symmetry of the map. It will look like a pattern that doesn't look the same if you rotate it by 120 degrees (breaking the 3-fold symmetry). This is a smoking gun for these exotic states.
• The Sound of the City (Density of States): If you measure how many energy states are available at different levels, the exotic dances will show a specific "V-shape" or "U-shape" in the data, distinct from the simple waltz.

Why Does This Matter?

This is a big deal for two reasons:

1. Quantum Computing: These "chiral" states are the holy grail for building topological quantum computers. They are stable and can store information in a way that is immune to small errors (like noise in a room).
2. New Materials: The paper points to CrSb (Chromium Antimonide) as a real-world material where this might already be happening. It gives experimentalists a roadmap: "Look at CrSb, tune the electron density, and you might find these superconducting spirals."

Summary

In short, this paper says: "We found a new type of magnetic material (g-wave altermagnet) that acts like a twisted dance floor. On this floor, the usual simple electron pairing fails, forcing the electrons to perform exotic, spinning 'chiral' dances (p-wave and d-wave). These dances are not only stable but are the perfect candidates for the next generation of quantum technology."

Technical Summary

1. Problem Statement

Altermagnetism is a newly identified magnetic phase characterized by antiferromagnetic-like spin arrangements combined with crystallographic rotational symmetries. Unlike conventional ferromagnets or antiferromagnets, altermagnets exhibit momentum-dependent spin splitting (altermagnetic spin splitting) without net magnetization or spin-orbit coupling.

While previous research has extensively explored superconductivity in d-wave altermagnets, the potential for unconventional superconductivity in g-wave altermagnets (recently observed in materials like CrSb) remains largely unexplored. The central scientific question addressed in this work is:

• How does the unique g-wave spin-splitting of the Fermi surfaces in 3D hexagonal lattices influence the dominant superconducting instability?
• Can these systems host chiral superconducting states (carrying non-zero angular momentum and potentially hosting Majorana edge modes), which are crucial for topological quantum computing but difficult to realize?

2. Methodology

The authors employ a theoretical framework combining tight-binding modeling and mean-field analysis:

• Model System: A three-dimensional hexagonal lattice with two sublattice layers (space group 194, point group $D_{6h}$), representative of the material CrSb.
• Hamiltonian Construction:
  • Kinetic Term ($\hat{H}_0$): A tight-binding model derived for CrSb, incorporating nearest-neighbor hopping ($t_1$), inter-layer hopping ($t_2$), and a local symmetry-breaking term ($t_z$). Crucially, it includes an exchange field ($J$) representing the g-wave altermagnetic order, which induces momentum-dependent spin splitting.
  • Interaction Term ($\hat{H}_{int}$): An extended attractive Hubbard model including onsite interaction ($U$), nearest-neighbor intralayer interaction ($V_1$), and interlayer interaction ($V_2$).
• Theoretical Approach:
  • Bogoliubov-de Gennes (BdG) Formalism: The authors perform a self-consistent mean-field analysis to solve for the superconducting gap equations.
  • Pairing Channels: They systematically investigate all possible intralayer and interlayer pairing symmetries, including:
    • Spin-singlet: s-wave, extended s-wave (es), chiral d-wave ($d+id$), and $s_z$-wave.
    • Spin-triplet: f-wave, chiral p-wave ($p+ip$), and $p_z$-wave.
  • Ground State Determination: By numerically solving the self-consistent gap equations at zero temperature, they calculate the condensation energy for each state to identify the energetically favored ground state across a parameter space defined by chemical potential ($\mu$), altermagnetic field strength ($J$), and interaction strength ($V_1$).

3. Key Contributions

1. Discovery of Chiral States in g-wave Altermagnets: The study establishes that g-wave altermagnetic metals can stabilize chiral p-wave ($p+ip$) and chiral d-wave ($d+id$) superconducting states, which were previously difficult to realize.
2. Mechanism of Suppression (BFS Formation): The paper elucidates the physical mechanism driving the transition from singlet to triplet pairing. It demonstrates that strong altermagnetic fields induce the formation of Bogoliubov Fermi Surfaces (BFSs) in spin-singlet states. These BFSs (regions of zero-energy quasiparticles) suppress the spin-singlet pairing amplitude by excluding specific momentum regions from the pairing integration, thereby destabilizing singlet phases.
3. Robustness of Triplet States: In contrast, spin-triplet states are shown to be immune to the formation of BFSs because the altermagnetic field does not split the energy levels of electrons with the same spin. This allows triplet states to remain stable and energetically favorable in high-field regimes.
4. Experimental Signatures: The authors propose specific experimental probes to distinguish these pairing symmetries, focusing on quasiparticle energy dispersions and the Density of States (DOS).

4. Key Results

A. Phase Diagrams and Stability

The phase diagrams reveal a rich landscape of superconducting states dependent on $J$, $V_1$, and $\mu$:

• Strong Altermagnetic Fields ($J \gtrsim 0.6$) & High Electron Density: The chiral p-wave ($p+ip$) state becomes the dominant ground state. This is particularly relevant for the candidate material CrSb (where $J \approx 0.6, \mu \approx 2$).
• Weak Altermagnetic Fields & Intermediate Density: The chiral d-wave ($d+id$) state can be stabilized, particularly when $V_1$ is large and $\mu$ is near zero ($-1.1 \le \mu \le 0.3$).
• Weak Fields & Typical Density: Conventional non-chiral states (s-wave, extended s-wave, or f-wave) are stabilized.
• Mixed Phases: Regions exist where mixed phases, such as $(f, p+ip)$, occur, driven by the competition between different pairing channels.

B. The Role of Bogoliubov Fermi Surfaces (BFSs)

• Singlet Collapse: In the intermediate-to-strong $J$ regime, the altermagnetic spin splitting causes the lower energy bands of spin-singlet quasiparticles to cross zero energy, forming BFSs. This leads to a vanishing of the spin-singlet pair correlation functions in those momentum regions, effectively suppressing singlet pairing.
• Triplet Survival: Spin-triplet quasiparticle bands remain fully gapped (or nodal without BFSs) because the energy splitting $E_{\alpha+}(k) - E_{\alpha+}(-k) = 0$ for same-spin electrons. Consequently, triplet pairing amplitudes remain robust.

C. Experimental Signatures

The authors propose two primary methods to detect these states experimentally:

1. Symmetry Breaking in Energy Dispersion:
   • Pure non-chiral states (s, es) and pure chiral states ($d+id$, $p+ip$) preserve the three-fold rotational symmetry ($C_{3z}$) in their energy dispersions due to the $|g(k)|^2$ dependence in the quasiparticle energy.
   • However, mixed states (e.g., $f + p+ip$) break $C_{3z}$ symmetry due to interference terms between the form factors. This symmetry breaking serves as a definitive signature of chiral pairing mixing with non-chiral components.
2. Density of States (DOS):
   • Hard Gap (U-shaped): Observed in s-wave, chiral d-wave, and chiral p-wave states (sharp coherence peaks).
   • V-shaped Gap: Observed in f-wave and mixed $(f, p+ip)$ states due to nodes in the pairing amplitude along high-symmetry lines or at the $\Gamma$ point.
   • Absence of Coherence Peaks: Mixed states lack the sharp coherence peaks seen in pure states.

5. Significance

• New Platform for Topological Superconductivity: This work identifies g-wave altermagnets (like CrSb and MnTe) as a promising platform for realizing chiral superconductivity without the need for external magnetic fields or strong spin-orbit coupling.
• Quantum Computing: The realization of chiral p-wave and d-wave states implies the potential existence of Majorana edge modes, which are essential for fault-tolerant topological quantum computing.
• Theoretical Insight: The study provides a fundamental understanding of how altermagnetic spin splitting selectively suppresses singlet pairing via BFS formation while stabilizing triplet pairing, offering a new design principle for engineering unconventional superconductors.
• Experimental Guidance: By predicting specific signatures in ARPES (Angle-Resolved Photoemission Spectroscopy) and STS (Scanning Tunneling Spectroscopy), the paper provides a roadmap for experimentalists to verify these theoretical predictions in real materials.

In conclusion, the paper demonstrates that the unique momentum-dependent spin splitting of g-wave altermagnets acts as a powerful tuning knob, suppressing conventional singlet superconductivity and stabilizing topologically non-trivial chiral triplet states, thereby opening a new frontier in the search for topological superconductivity.