Two-dimensional Systems Demonstrate Interplay Between Altermagnetism and Superconductivity

The pursuit of novel superconducting states receives a significant boost from recent investigations into the interplay between superconductivity and altermagnetism, a unique magnetic order, in two-dimensional materials. Kinga Jasiewicz, Paweł Wójcik, and Michał Nowak, alongside Michał Zegrodnik, all from AGH University of Krakow, demonstrate how this interaction generates unconventional superconducting behaviours not previously observed. Their work reveals that the combined symmetries of these two states, alongside the material’s electronic structure, lead to anisotropic pairing and the emergence of exotic, multi-component superconductivity featuring a mixing of singlet and triplet pairing states. These findings not only expand our understanding of fundamental superconductivity, but also open exciting possibilities for developing new technologies, including potentially realising superconducting diodes with directional current flow.

Altermagnetism and Superconducting Hamiltonian Derivation

This document details the mathematical foundation for understanding superconductivity in materials exhibiting altermagnetism, a recently discovered magnetic state. It explains the derivation of a Hamiltonian, a mathematical description of the system’s energy, used to model the pairing of electrons that carries the superconducting current. This work provides the theoretical basis for ongoing investigations into unconventional pairing mechanisms, crucial for researchers specializing in condensed matter physics and superconductivity. Superconductivity arises when electrons overcome their mutual repulsion to form Cooper pairs, enabling current to flow without resistance.

Altermagnetism introduces complexities to this process, influencing the formation and properties of Cooper pairs and potentially leading to novel superconducting states, such as the Fulde-Ferrell and Larkin-Ovhinnikov phases, where Cooper pairs possess a net momentum. The derivation begins with a Hamiltonian describing the system in real space, then transforms it into momentum space, a mathematical simplification that facilitates calculations. This process involves applying the Fourier transform, a standard technique in physics, resulting in a Hamiltonian that describes the interaction between electrons and the pairing that leads to superconductivity. The research defines a pairing amplitude, a crucial parameter that determines the strength and symmetry of the superconducting state.

Altermagnetism and Anisotropic Superconducting Pairing

This study investigates the interplay between altermagnetism and unconventional superconductivity, employing a model based on a single-particle Hamiltonian that accurately replicates the alternating spin splitting characteristic of altermagnetic states. Researchers analyzed how the interplay between the symmetries of the superconducting and altermagnetic order parameters, alongside the shape of the Fermi surface, leads to anisotropic pairing behaviors not previously observed in conventional states. To model these systems, the team considered both a square lattice exhibiting d-wave altermagnetic symmetry and a triangular lattice with i-wave symmetry. Calculations reveal that the combination of superconducting and altermagnetic order parameters leads to anisotropic pairing, resulting in an exotic order parameter with a mixing of singlet and triplet pairing.

Analysis of Cooper pair density in momentum space and the resulting Fermi wave vector mismatch provided insight into these novel states. The strength of the altermagnetic state was tuned, demonstrating that while the magnitude of spin splitting changes, the directions in momentum space where splitting vanishes remain constant. The model also incorporated an external magnetic field to investigate its influence on the altermagnetic behavior. Applying a mean-field approximation to the pairing term, the team derived self-consistent equations for the superconducting gaps, allowing them to analyze the stability of even-parity spin-singlet and odd-parity spin-triplet Cooper pairing in the presence of altermagnetism.

Anisotropic Superconductivity and Altermagnetic Order Mixing

This work details a study of altermagnetism and its interplay with unconventional superconductivity in two-dimensional square and triangular lattices. Researchers developed a model based on a single-particle Hamiltonian that replicates the alternating spin splitting characteristic of altermagnetic states, then supplemented it with a pairing term to explore the resulting superconducting behavior. Calculations reveal that the combination of superconducting and altermagnetic order parameters, alongside the shape of the Fermi surface, leads to anisotropic pairing, a departure from previously proposed states. Specifically, the team observed the emergence of additional pairing symmetries, resulting in an exotic order parameter with a mixing of singlet and triplet pairing.

For the square lattice, researchers demonstrated alternating d-wave spin splitting, while the triangular lattice exhibited i-wave splitting, visualized through the resulting Fermi surfaces. The strength of the altermagnetic state was tuned, with the direction of spin splitting remaining consistent regardless of its magnitude. Applying a mean-field approximation to the pairing term, the team derived self-consistent equations for the superconducting gaps, allowing for non-zero center-of-mass momentum of the Cooper pairs, potentially leading to a phase. These calculations demonstrate the potential for novel superconducting states arising from the interplay of altermagnetism and unconventional pairing mechanisms.

Altermagnetism Drives Unconventional Superconducting Phases

This research presents a comprehensive analysis of how altermagnetism and unconventional superconductivity interact in two-dimensional materials, specifically focusing on square and triangular lattices. Calculations demonstrate that a phase, characterized by Cooper pairs with non-zero momentum, can emerge in both altermagnetic states due to a mismatch in Fermi wave vectors caused by alternating spin splitting. The team found that different pairing symmetries within the superconducting state lead to variations in the phase’s behavior, with the orientation of Cooper pair momentum dictated by the symmetry of the altermagnetic ordering and the shape of the Fermi surface. Notably, the study reveals a mixing of singlet and triplet pairing symmetries, resulting in an exotic multi-component order parameter.

For the square lattice, extended s-wave pairing favors a diagonal orientation of the Cooper pair momentum, while d-wave pairing aligns the momentum horizontally and vertically. Similar singlet-triplet mixing appears in the triangular lattice, resulting in a phase conserving six-fold symmetry and a unique combination of d ± id singlet and f-wave triplet pairing. The authors acknowledge that the precise composition of these superconducting gap symmetries can.