Unveiling the superconducting scenario in multiphase superconductor CeRh$_2$As$_2$ from space-group symmetry analysis and DFT calculations

This paper investigates the unique superconducting scenario in CeRh$_2$As$_2$ by combining space-group symmetry analysis and DFT calculations to propose that the observed field-induced transition between low- and high-field phases arises from a change in the internal structure of triplet pairing states ($E_{1u}^{\prime +}$) rather than a conventional singlet-triplet transition, potentially driven by non-symmorphic lattice effects at the Brillouin zone boundary.

The Gist

Imagine you have a very special, tiny city made of atoms called CeRh₂As₂. In this city, electrons (the tiny messengers of electricity) usually behave like shy introverts, pairing up to move in perfect lockstep. This is superconductivity: a state where electricity flows with zero resistance.

Usually, scientists think there are only two ways these electron pairs can dance:

1. The "Singlet" Dance: The partners hold hands tightly, spinning in opposite directions (one up, one down). They are very sensitive to magnetic fields.
2. The "Triplet" Dance: The partners spin in the same direction. They are tough and can survive strong magnetic fields.

The Mystery of CeRh₂As₂
For a long time, scientists were puzzled by CeRh₂As₂. It has two different "superconducting phases" (let's call them Phase 1 and Phase 2).

• Phase 1 acts like a shy "Singlet" dancer.
• Phase 2 acts like a tough "Triplet" dancer and can survive incredibly strong magnetic fields (much stronger than physics usually allows for a Singlet).

The big question was: How can one single material switch from a shy dancer to a tough dancer just by turning on a magnet? Usually, this would require a complete change in the material's internal rules, which seemed impossible.

The New Discovery: The "Non-Symmorphic" Twist
This paper proposes a clever solution using a concept called space-group symmetry. Think of the crystal structure of CeRh₂As₂ not as a simple grid, but as a M.C. Escher staircase.

In a normal building, if you walk up a flight of stairs, you end up exactly above where you started. But in a "non-symmorphic" structure (like this crystal), if you take a step forward, you also slide slightly to the side or flip upside down. It's a glide-reflection.

The authors used advanced math (group theory) and computer simulations (DFT) to look at how electrons move in this "Escher city." They found something amazing:

1. The Rules Change at the Edges: In most places in the city, the rules are strict: "If you spin the same way, you must be a Triplet." But at the very edges of the electron's world (called the Brillouin Zone boundaries), the "Escher slide" changes the rules.
2. The Same Dance, Different Costumes: The authors realized that the electrons in Phase 1 and Phase 2 are actually doing the same type of dance (both are Triplets, spinning the same way). They aren't changing their fundamental nature. Instead, they are just wearing different "costumes" (spatial symmetries) because of the weird geometry of the crystal.
   • Phase 1 (Low Field): The electrons pair up in a way that looks like a "Singlet" to the outside world, but it's actually a special kind of Triplet that fits the crystal's glide-rules.
   • Phase 2 (High Field): When the magnet gets strong, the electrons switch to a different "costume" (a different symmetry pattern) that allows them to survive the magnetic pressure.

The Analogy: The Magic Mirror Room
Imagine a room with a special mirror on the floor.

• Normal Room: If you walk forward, your reflection walks forward. If you raise your right hand, the reflection raises its left.
• CeRh₂As₂ Room: If you walk forward, your reflection slides to the side and flips upside down.

The paper suggests that the electrons in this material are like people walking in this magic room. Because the room itself is twisted, the "reflection" (the electron pair) can look different depending on where you are in the room.

• In the center of the room, the reflection looks like a standard "Singlet."
• At the edges of the room (where the crystal's weird geometry kicks in), the reflection looks like a "Triplet."

Why This Matters
The authors found that the electrons responsible for this magic are the Cerium (Ce) 4f electrons. These are heavy, moody electrons that interact strongly with each other.

By proving that the material doesn't need to change its fundamental "spin" (multiplicity) to switch phases, but only needs to change its spatial arrangement (symmetry), the paper solves a major puzzle. It explains how CeRh₂As₂ can be a "shape-shifter" superconductor, surviving extreme magnetic fields without breaking the laws of physics.

In a Nutshell:
CeRh₂As₂ isn't two different materials fighting for dominance. It's one material wearing two different hats. The "hat" it wears depends on how strong the magnetic field is, thanks to the crystal's unique, twisted architecture. This discovery helps us understand how to design future superconductors that can work in powerful magnets, which is crucial for things like MRI machines and fusion energy.

Technical Summary

1. Problem Statement

The superconductor CeRh$_2$As$_2$ exhibits a unique phase transition between two superconducting states (SC1 and SC2) under an applied magnetic field.

• SC1 (Low-field): An even-parity state strongly limited by the Pauli-Clogston paramagnetic limit.
• SC2 (High-field): An odd-parity state with an upper critical field ($H_{c2} \approx 14$ T) that exceeds the Pauli-Clogston limit, suggesting triplet pairing.
• The Paradox: Conventional wisdom suggests a transition from SC1 to SC2 implies a change in spin multiplicity (singlet $\to$ triplet). However, having two distinct pairing scenarios (singlet and triplet) within a single compound is theoretically unlikely.
• The Goal: The authors investigate whether the symmetry change between SC1 and SC2 can occur without changing the spin multiplicity (i.e., a triplet-to-triplet transition with different spatial parities), leveraging the material's non-symmorphic crystal structure ($P4/nmm$) and the behavior of Cooper pairs at the Brillouin Zone (BZ) boundaries.

2. Methodology

The study employs a dual approach combining rigorous group-theoretical analysis with advanced electronic structure calculations.

A. Group-Theoretical Symmetry Analysis

• Framework: The authors utilize the space-group approach to the Cooper pair wavefunction, extending Anderson's ansatz to non-symmorphic space groups.
• Key Concepts:
  • Non-Symmorphic Symmetry: The space group $P4/nmm$ (No. 129, $D_{4h}^7$) contains glide planes and screw axes. This disrupts the direct relationship between spin multiplicity and spatial parity at high-symmetry points (BZ faces) where the Fermi surface intersects the zone boundary.
  • Magnetic Groups: The analysis incorporates time-reversal symmetry and magnetic groups ($4/mm'm'$) to account for phase winding and the effect of magnetic fields.
  • Pair Density Waves (PDW): The authors consider scenarios where Cooper pairs have non-zero total momentum ($K = b/2$), leading to PDW states, particularly at the X point.
• Order Parameter (SOP) Construction:
  • Constructed using Anderson pair functions for singlet and triplet states.
  • Incorporated phase winding ($e^{im\theta}$) to satisfy the symmetry of the point group $D_{4h}$ and the magnetic group.
  • Analyzed the nodal structure (nodes vs. nodeless states) based on irreducible corepresentations (ICRs).

B. Density Functional Theory (DFT) Calculations

• Software/Methods: Performed using VASP with Projector Augmented Wave (PAW) potentials.
• Functionals:
  • HSE06: Used for accurate treatment of correlation effects and Ce 4f electron positioning.
  • GGA+U: Used for Fermi surface calculations, with Hubbard $U$ and Hund's $J$ parameters tuned to match HSE06 results.
• Focus: Specifically examined the contribution of Ce 4f electrons at the Fermi level, particularly at high-symmetry points (X, M, Z) and the intersection of the Fermi surface (FS) with BZ boundaries.
• Structural Relaxation: Investigated the impact of internal atomic relaxation on the band structure, noting that while lattice parameters changed minimally, interlayer distances shifted significantly, altering the band spectrum.

3. Key Contributions & Results

A. Breaking the Multiplicity-Parity Link

The study confirms that in non-symmorphic systems, the direct link between spin multiplicity (singlet/triplet) and spatial parity (even/odd) is violated at BZ faces.

• At the X Point: Triplet pairs can be even-parity, and singlet pairs can be either even or odd.
• Implication: This allows for a transition between SC1 and SC2 where both phases are triplet states, but with different spatial parities (Even $\to$ Odd), avoiding the need for a singlet-triplet transition.

B. Identification of Nodeless Triplet States

Through the construction of the SOP with phase winding in the magnetic group $4/mm'm'$:

• SC1 (Low-field): Identified as an Opposite Spin Pairing (OSP) triplet state with symmetry $E'_{1u}$ (spatially even at X, odd in $\Gamma-X$ direction).
• SC2 (High-field): Identified as an Equal Spin Pairing (ESP) triplet state, also with symmetry $E'_{1u}$ but with a different internal structure (spatially odd).
• Nodal Structure: The analysis reveals that while nodes in vertical planes can be removed by phase winding, robust nodes exist in the basal plane for specific angular momentum projections ($m=1$ for even pairs, $m=2$ for odd pairs). However, fully gapped nodeless states are possible for specific ICRs ($E'_{1u}$).

C. DFT Validation of Ce 4f Contributions

• Fermi Surface Topology: Calculations confirm that the Fermi surface intersects the BZ boundaries (specifically near the X and M points).
• Orbital Character:
  • The X point features a Van Hove Singularity (VHS) with significant contributions from Ce 4f and Rh 4d orbitals.
  • The M point is dominated by Rh 4d orbitals.
• Relaxation Effects: While structural relaxation significantly alters interlayer distances, the HSE06 functional correctly places the Ce 4f bands relative to the Fermi level, validating the importance of Ce 4f electrons in the pairing mechanism.

D. Mechanism of Transition

The authors propose that the SC1 $\to$ SC2 transition is driven by the magnetic field stabilizing the ESP (Equal Spin Pairing) triplet state over the OSP (Opposite Spin Pairing) triplet state.

• SC1: OSP triplet (spatially even at X).
• SC2: ESP triplet (spatially odd).
• This transition occurs within a single pairing potential (nearest-neighbor exchange interaction between Ce 4f electrons connected by the non-symmorphic translation $\{I|\tau\}$), explaining the high critical field of SC2 without invoking a change in spin multiplicity.

4. Significance

• Resolution of the Multiphase Paradox: The paper provides a robust theoretical framework explaining how CeRh$_2$As$_2$ can host two superconducting phases with different parities without requiring a rare singlet-to-triplet transition.
• Role of Non-Symmorphic Symmetry: It highlights the critical role of non-symmorphic space groups in enabling unconventional superconductivity, specifically by allowing triplet pairs to exhibit even parity at BZ boundaries.
• Experimental Consistency: The proposed scenario (triplet OSP $\to$ triplet ESP) aligns with experimental observations:
  • SC1 is fully gapped (consistent with nodeless OSP triplet).
  • SC2 exhibits odd-parity characteristics and high $H_{c2}$ (consistent with ESP triplet).
  • The presence of antiferromagnetic fluctuations in SC1 and their suppression in SC2 is naturally explained by the change in pairing symmetry.
• General Applicability: The methodology (combining space-group symmetry analysis with DFT) offers a template for investigating other heavy-fermion and non-centrosymmetric superconductors (e.g., UTe$_2$, UPt$_3$).

Conclusion

The authors successfully demonstrate that the superconducting phase transition in CeRh$_2$As$_2$ is a triplet-to-triplet transition involving a change in spatial parity (even $\to$ odd) and spin configuration (OSP $\to$ ESP), facilitated by the material's non-symmorphic crystal structure and the specific behavior of Ce 4f electrons at the Brillouin zone boundaries. This resolves the theoretical conflict regarding the coexistence of singlet and triplet scenarios in a single material.