Finite-momentum superconductivity with singlet-triplet mixing in an altermagnetic metal: A pairing instability analysis

The gist: The analysis reveals that altermagnetic spin-splitting in an altermagnetic metal gives rise to finite-momentum pairing between electrons with unlike spins, leading to a superconducting phase characterized by a multi-component order parameter featuring singlet-triplet mixing

Introduction and Motivation

Altermagnetism is a newly identified magnetic order that combines zero net magnetization with a momentum-dependent spin splitting of electronic bands enforced by crystalline symmetries rather than relativistic spin-orbit coupling It provides a fundamentally different route to finite-momentum superconductivity by connecting symmetry-related regions of the Fermi surface with compatible spin polarization, avoiding orbital pair breaking This mechanism is dictated by the underlying crystal symmetry instead of external magnetic fields, directly linking the pairing wave vector to the electronic structure

Model and Methodology

The study employs a low-energy single-band Hamiltonian for a square lattice subject to d-wave altermagnetism, defined by momentum-dependent spin splittings Jk which are either dxy-wave or dx2−y2-wave in nature The interaction Hamiltonian is modeled using the extended attractive Hubbard U-V model, incorporating on-site (U) and nearest-neighbor (V and Vσ) pairing interactions The analysis uses a pairing-instability approach based on the Thouless criterion applied to the inverse two-particle vertex function Γ−1(Q, ω = 0)

Key Findings on Pairing Instabilities

1. Leading Pairing Instability: The leading pairing instability generically occurs at finite center-of-mass momentum Q and simultaneously involves spin-singlet and spin-triplet channels with mixed parity

2. Multi-component Order Parameter: Consequently, the superconducting phase is anticipated to host an FFLO state described by a multicomponent order parameter with intrinsic singlet-triplet mixing

3. Channel Evolution: The dominant pairing channel evolves as electron filling increases, switching from extended s-wave at low filling to p-wave in the intermediate regime, and finally to d-wave for higher fillings

Altermagnetism Symmetries and Momentum Dependence

The paper examines specific altermagnetic couplings, including dxy-wave and dx2−y2-wave couplings The momentum-space form factors for different pairing channels are defined by fs(k), fes(k), fp+(k), fp-(k), and fd(k) The study shows that the global maximum of the leading eigenvalue γ1(Qmax) consistently occurs when Q is aligned with the x-direction for dxy-wave altermagnetism at low filling However, for dx2−y2-wave altermagnetism, the optimal momentum Q can reorient depending on coupling strength and filling, with a local maximum appearing around large λ

Comparison with Previous Work and Implications

The results confirm the emergence of altermagnetism-driven FFLO superconducting states, which have been examined in some detail in earlier works The singlet-triplet mixing discussed here is fundamentally distinct from the admixture realized in conventional non-centrosymmetric superconductors, as it originates from the momentum-dependent spin splitting generated by altermagnetic order rather than parity mixing associated with the absence of inversion symmetry Future work should explore multiband electronic structures and the role of competing orders like charge- and spin-densitywave instabilities

Parallel Spin Pairing Considerations

While parallel-spin pairing is not the dominant instability for sufficiently weak altermagnetic couplings when Vσ ≤ V The critical nearest-neighbor attraction Vc generally decreases as the altermagnetic coupling increases

Conclusion

The systematic analysis confirms that finite-momentum pairing naturally leads to the admixture of multiple pairing channels with different parity, resulting in multi-component FFLO states with both spin-singlet and spin-triplet contributions This intrinsic mixing is a direct consequence of the altermagnetic state's symmetry and spin texture The findings suggest that the superconducting phase boundary is associated with a nonzero peak momentum Qmax, demonstrating that the emergent superconducting phase corresponds to a finite-momentum FFLO state

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The gist: The analysis reveals that altermagnetic spin-splitting in an altermagnetic metal gives rise to finite-momentum pairing between electrons with unlike spins, leading to a superconducting phase characterized by a multi-component order parameter featuring singlet-triplet mixing

How it works

1. The system is modeled using an altermagnetic Hubbard UV model incorporating both on-site (U) and nearest-neighbor (V and Vσ) attractive interactions

2. The pairing instability is determined by the Thouless criterion applied to the inverse two-particle vertex function Γ−1(Q, ω = 0), where the transition occurs when max of Q Γ−1(Q, ω = 0) = 0

3. The leading instability is identified by diagonalizing the full matrix Γ−1(Q, ω = 0) to find the largest eigenvalue γ1(Q), and superconductivity sets in once γ1(Q) > 0 at some pairing momenta Q

Key Results and Phase Diagrams

**- The leading pairing instability generically occurs at finite center-of-mass momentum Q and simultaneously involves spin-singlet and spin-triplet channels with mixed parity This leads to an FFLO state described by a multicomponent order parameter with intrinsic singlet-triplet mixing **

**- The phase diagram in the presence of dxy-wave altermagnetism shows the critical altermagnetic coupling strength λc as a function of the lattice filling factor ν, establishing a corresponding phase diagram **

**- The dominant pairing channel evolves with increasing filling: extended s-wave at low filling, transitioning to p-wave in the intermediate regime (0.3 ≲ ν ≲ 0.6), and d-wave for higher fillings **

Symmetry and Momentum Dependence

**- The analysis distinguishes between dxy-wave and dx2−y2-wave altermagnetism, noting that for low filling (ν = 0.2), the optimal momentum Q occurs along the diagonal direction, in contrast to dxy-wave where it aligns with the x-direction **

**- The global maximum of γ1(Qmax) shows non-monotonic dependence on ν, resembling the behavior in the absence of altermagnetism, indicating a transition between dominant pairing channels **

Implications and Future Directions

The singlet-triplet mixing is fundamentally distinct from parity mixing in conventional non-centrosymmetric superconductors, as it originates from the momentum-dependent spin splitting generated by altermagnetic order Future investigations should incorporate multiband effects and higher even-parity altermagnetic orders to fully understand the stability and modulation vector of the FFLO state The role of relative phases between different components of the order parameter in the mixed singlet-triplet superconducting phase is also important for future study

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The gist: The analysis reveals that altermagnetic spin-splitting in an altermagnetic metal gives rise to finite-momentum pairing between electrons with unlike spins, leading to a superconducting phase characterized by a multi-component order parameter featuring singlet-triplet mixing

How it works

1. The system is modeled using an altermagnetic Hubbard UV model incorporating both on-site (U) and nearest-neighbor (V and Vσ) attractive interactions

2. The pairing instability is determined by the Thouless criterion applied to the inverse two-particle vertex function Γ−1(Q, ω = 0), where the transition occurs when max of Q Γ−1(Q, ω = 0) = 0

1. A system capable of predicting emergent superconducting phases in altermagnetic metals can be developed by implementing a numerical framework based on a pairing-instability analysis for two electrons near the spin-split Fermi surfaces based on the Thouless criterion. This system can determine the leading pairing instability and specify the structure of the resulting superconducting order parameter by diagonalizing the inverse vertex matrix.

2. The AI system can accurately model complex, multi-component superconducting states by incorporating singlet-triplet mixing into its order parameter prediction. This allows it to distinguish between parity-even singlet and parity-odd triplet channels, reflecting the finding that the resulting finite-momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting phase is expected to exhibit a multi-component order parameter featuring singlet-triplet mixing.

3. The system can optimize material design strategies by mapping the phase diagrams of altermagnetism, showing how the critical altermagnetic coupling strength λc is plotted as a function of the lattice filling factor ν. This enables researchers to predict whether a given material configuration will favor an extended s-wave, p-wave, or d-wave pairing symmetry.

4. The AI system can perform advanced structural analysis by evaluating the weights of the different partial-wave components Wη = Mη2 at the superconducting transition. This provides quantitative insight into which pairing channels dominate, as shown in Figure 8, helping to determine the relative weights of the distinct pairing channels at the superconducting transition.

5. The system can simulate and predict topological phenomena by analyzing the role of relative phases between different components of the order parameter and its effect on spontaneous surface currents, which is relevant for understanding systems with broken time-reversal symmetry, similar to those found in topological insulators.