Finite-momentum bound pairs of two electrons in an altermagnetic metal

The gist: The altermagnetic spin-splitting in electronic bands naturally gives rise to a ground state of two-electron bound pairs with nonzero center-of-mass momentum, suggesting that altermagnetic spin splitting may favor pairing at nonzero center-of-mass momentum, which could be relevant for proposed Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconductivity in altermagnetic systems<ref:2601.12905#pg2>.

Model and Hamiltonian

The study solves the two-electron problem on a square lattice with d-wave altermagnetism, considering both on-site and nearest-neighbor attractive interactions<ref:2601.12905#pg4>. The model Hamiltonian is written as H = H0+Hint, where the non-interacting kinetic part H0 in momentum space takes the simplest one-band form with a single-particle dispersion relation εk ≡ −2t(cos kx + cos ky) and the momentum-dependent spin-splitting Jk arising from altermagnetism<ref:2601.12905#pg3>. For dxy-wave altermagnetism, Jk is given by Jk = λ sin (kx) sin (ky)<ref:2601.12905#pg4>. The interaction Hamiltonian is described by the extended Hubbard U-V model, where the on-site interaction occurs only between spin-up and spin-down electrons, while the nearest-neighbor interaction can happen between electrons with same spins<ref:2601.12905#pg5>. In momentum space, the nearest-neighbor term can be decomposed into distinct channels (i.e., the extended s-wave, the two p-wave, and the d-wave channels)<ref:2601.12905#pg5>.

Theoretical Framework for Bound States

The theoretical framework involves solving a Schrödinger equation for the two-electron wave-function by introducing a center-of-mass momentum Q<ref:2601.12905#pg4>. For an infinite lattice system, the two-electron wave-function is formulated as Φi = Xp Φp pi = Xp Φpc† p+Q/2↑ c† -p+Q/2↓vaci<ref:2601.12905#pg4>. The resulting two-electron Schrödinger equation, when inserted with the separable form of the interaction potential, leads to a secular equation det Lηη′ (E) − δη′ Vη = 0<ref:2601.12905#pg5>. This secular equation determines all the two-electron energy levels E, including bound states below the threshold of the two-particle continuum E.

Results for On-site Attraction

When considering only attractive on-site interaction (U < 0), a dxy-wave altermagnetic coupling λ = 3t leads to the ground-state of the bound pairs occurring at finite momentum along either the Qx or Qy axis<ref:2601.12905#pg6>. This emergence of finite-momentum bound pairs appears to directly reflect the non-trivial two-electron dispersion induced by altermagnetism<ref:2601.12905#pg6>. The analysis of the two-electron continuum shows that when λ > 2t, the minimum shifts continuously to a finite momentum Q = arccos(2t/λ), signaling a tendency toward finitemomentum pairing<ref:2601.12905#pg7>. This shift demonstrates that the altermagnetic coupling drives a transition of the continuum minimum from zero to finite momentum at the critical value λc = 2t<ref:2601.12905#pg7>.

Results for Nearest-Neighbor Attraction

In the case with nearest-neighbor attractive interaction (V < 0), a strong attraction can support up to four distinct bound states<ref:2601.12905#pg8>. The inclusion of an altermagnetic coupling λ = 3t leads to several notable changes in the bound-pair dispersion, where the lowest bound-pair energy with dominant develops a global minimum at a finite center-of-mass momentum along the Qx or Qy axis<ref:2601.12905#pg8>. Furthermore, the second lowest bound-pair energy also acquires a global minimum at nonzero momentum, but along the diagonal direction (i.e., Qx = Qy along the Γ-M line)<ref:2601.12905#pg8>. This suggests that in a many-electron system, one would anticipate an FFLO superconducting state with a finite center-of-mass momentum Q<ref:2601.12905#pg8>.

Pairing Symmetry and Coexistence

The analysis reveals that the two-electron states exhibit pairing in both the spin-singlet and spin-triplet channels when altermagnetism is present, as the dispersion relation E2p(k, Q) contains both even and odd components of k<ref:2601.12905#pg5>. This mixed pairing character can be easily identified through the quantities Mη<ref:2601.12905#pg5>. Specifically, for nearest-neighbor attraction, the ground-state bound pair clearly forms a coherent superposition of spin-singlet (extended s-wave) and spin-triplet (p+-wave) components<ref:2601.12905#pg9>. This mixed character is expected to have significant consequences for the finite-momentum FFLO superconducting state that may emerge in a many-electron system, suggesting an FFLO order parameter with a mixed singlet-triplet character<ref:2601.12905#pg10>.

Outlook

Future research should focus on elucidating the relationship between the symmetry of bound states and the symmetry of the underlying lattice, as this issue may be closely connected to the pairing symmetry of superconducting states<ref:2601.12905#pg11>. A promising direction is to pursue a comprehensive investigation of the pairing instability in many-electron systems using established methods such as the Thouless criterion under the ladder approximation within the many-body T-matrix approach<ref:2601.12905#pg10>. The paper also notes that for standard zero-momentum pairing, classification can be carried out using group-theoretical methods<ref:2601.12905#pg5>. This suggests that extending such analyses to finite-momentum pairing states requires a more rigorous and comprehensive theoretical framework<ref:2601.12905#pg10>. The paper also mentions that for the dx2−y squared-wave altermagnetism, once λ > 4t, a global minimum develops at the M point in the lower edge of the two-electron continuum, supporting finitemomentum bound pairs at the same M point<ref:2601.12905#pg11>. This indicates that in a many-electron system, one would anticipate the possible formation of a finite-momentum FFLO-type order parameter<ref:2601.12905#pg11>. The center-of-mass momentum is not necessarily pinned to the M point Q = (π, π) due to many-particle correlation<ref:2601.12905#pg11>. The paper concludes by stating that a comprehensive investigation of the pairing instability in many-electron systems could be pursued in future work using the well-established Thouless criterion under the ladder approximation within the many-body T-matrix approach<ref:2601.12905#pg10>. The final sentence states that a comprehensive investigation of the pairing instability in many-electron systems could be pursued in future work using the well-established Thouless criterion under the ladder approximation within the many-body T-matrix approach<ref:2601.12905#pg10>.

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The gist: The altermagnetic spin-splitting in electronic bands naturally gives rise to a ground state of two-electron bound pairs with nonzero center-of-mass momentum, suggesting that altermagnetic spin splitting may favor pairing at nonzero center-of-mass momentum, which could be relevant for proposed Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconductivity in altermagnetic systems<ref:2601.12905#pg2>.

How it works

The altermagnetic spin-splitting in the single-particle dispersion naturally gives rise to a ground state of two-electron bound pairs with nonzero center-of-mass momentum<ref:2601.12905#pg2>. The emergence of finite-momentum bound states suggests that altermagnetic spin splitting may favor pairing at nonzero center-of-mass momentum, which could be relevant for proposed Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconductivity in altermagnetic systems<ref:2601.12905#pg2>.

1. The AI system can accurately predict FFLO superconducting states in altermagnetic systems by calculating the two-electron analog of many-electron FFLO pairing observed in altermagnetic systems as suggested by the paper. This enables the design of materials with predicted finite-momentum pairing tendencies based on intrinsic spin splitting rather than external fields.

2. The system can classify superconducting order parameters as a coherent superposition of spin-singlet and spin-triplet pairings by analyzing partial-wave components, such as how the groundstate bound pair clearly forms a coherent superposition of spin-singlet (extended s-wave) and spin-triplet (p+-wave) components. This allows for the prediction of mixed singlet-triplet character in many-electron FFLO order parameters.

3. The AI can determine the critical coupling strength for finite momentum pairing by analyzing the continuum edge behavior, specifically identifying the transition of the continuum minimum from zero to finite momentum at the critical value λc = 2t. This capability allows for targeted material design where altermagnetic coupling is tuned to favor a specific center-of-mass momentum Q.

4. The system can predict which partial-wave components dominate pairing based on momentum, as shown by analyzing the weight of the partial-wave components (M˜η, as explicitly indicated) in Fig. 5 and Fig. 4(b)-4(e). This provides insight into how lattice symmetry and altermagnetic coupling dictate the specific pairing symmetry in a many-electron system.