Superconductivity in a Two-Orbital Hatsugai-Kohmoto Model at Half Filling

The gist: This study provides a systematic framework for analyzing superconductivity in two-orbital extensions of the Hatsugai-Kohmoto model, classifying symmetry-allowed superconducting gap structures and computing critical temperature and order parameter as functions of interaction and pairing strength within a mean-field treatment.

Model Framework

The research focuses on the multiband extension of the HK model, referred to as the orbital HK model, which is an exactly solvable model of correlated electrons with momentum-local interactions that provides a minimal framework to explore the interplay of strong correlations, orbital structure and pairing symmetry The general form of the orbital HK model is given by HˆOHK = ∑ k,α,β,σ Hαβ (k)− µδαβ cˆ† kασ cˆ kβ σ,(2) where an orbital index α is added to the electronic operators The model assumes a two-band system for the Bloch Hamiltonian H(k), considering nearest- and second-nearest-neighbour hopping between p-orbitals on the square lattice, which leads to the Bloch Hamiltonian H(k) =h 4s cos kx cos ky + t (1) σ +t (1) π (cos kx +cos ky) i τ0 −4rsinkx sinkyτx + t(1) σ −t(1) π (cos kx −cos ky) τz,(3)> The parameters are determined by the overlap integrals [45] t(1) σ = (ppσ)1, t(1) π = (ppπ)1, s = ((ppσ)2 + (ppπ)2)/2, r = ((ppσ)2−(ppπ)2)/2

Symmetry Classification

The classification of pairing symmetries is significantly more extensive due to the inclusion of the orbital degree of freedom alongside spin and momentum The analysis restricts itself to two-orbital systems, where the symmetry classification becomes significantly more extensive due to the inclusion of the orbital degree of freedom alongside spin and momentum The pairing can thus be a spin-singlet or triplet and an orbital-singlet or triplet under the overall antisymmetry constraint The resulting basis functions of the corresponding irreducible representations are the symmetry-allowed candidates for the gap functions For selected pairing channels, the critical temperature and the superconducting order parameter as functions of interaction and pairing strength are computed The point group D4h constrains the possible gap functions W(k) and provides the basis for the symmetry classification in the following section The resulting basis functions are summarized in Tab. II for the spin-singlet and orbital-singlet sector, Tab. III for the spinsinglet and orbital-triplet sector, Tab. IV for the spin-triplet and orbital-singlet sector and Tab. V for the spin-triplet and orbital-triplet sector For a crystal lattice, these functions are replaced by the corresponding lowest-order lattice harmonics, obtained by the substitutions x → sin kx, x squared → cos kx etc., to ensure that the functions respect the periodicity of the Brillouin zone while retaining their transformation properties under the point group D4h The internal matrix structure decomposes as (1⊕3)spin ⊗(1⊕3)orbital, corresponding to the four sectors with dimensions 1, 3, 3, and 9, respectively

Mean-Field Analysis and Results

A mean-field decoupling of the pairing term preserves the momentum-sector decomposition of the HK model such that the resulting local Hamiltonians can be diagonalized exactly for each sector The mean-field Hamiltonian is Hˆ = HˆOHK −∆∗Aˆ −∆Aˆ† + N g ∆2,(13) where the order parameter is given by the thermal average ∆ = g/N D Aˆ E, ⟨·⟩ = 1/Z tr· e−βH. (14) The mean-field Hamiltonian couples opposite momenta k and −k and can thus be written as Hˆ = N g ∆2 + ∑ k∈HBZ Hˆk,(15) where the sum runs over the half Brillouin zone containing one representative of each pair and the local Hamiltonian Hˆk = ∑ s=±,α,β,σ Hαβ (sk)− µδαβ cˆ† skασ cˆ skβ σ +U ∑s=±,α nˆ skα↑ nˆ skα↓ −∆∗ ∑s=± ψ T−skW(sk)ψ T-sk -∆ ∑s=± ψ†skW†(sk)ψ†T−sk (16) The critical temperature TC is the maximal temperature for which f as a function of ∆ has a global minimum at ∆ ≠ 0 In the Mott regime, the global minimum remains at ∆ ≠ 0 upon increasing the temperature and is separated from the normal-state solution at ∆ = 0 by a local maximum

Key Findings on Phase Transitions

The HK interaction affects not only the critical temperature, but also the topology of the free energy Below the Mott transition, channels A(ST)1g, B(TS)2g and B(ST)1g exhibit a conventional continuous transition, while in the Mott regime the phase transition becomes first order with additional metastable states In the channel E(SS)u, a local maximum persists down to T = 0 in the metallic regime The critical temperature shows a pronounced channel dependence and is maximized at finite interaction strength with the maximum occurring well below the Mott transition

Conclusion

The HK interaction affects not only the critical temperature, but also the topology of the free energy Below the Mott transition, channels A(ST)1g, B(TS)2g and B(ST)1g exhibit a conventional continuous transition, while in the Mott regime the phase transition becomes first order with additional metastable states In the channel E(SS)u, a local maximum persists down to T = 0 in the metallic regime The critical temperature shows a pronounced channel dependence and is maximized at finite interaction strength with the maximum occurring well below the Mott transition While the classification derived here is specific to the porbital model on the square lattice, the HK construction itself is more general Momentum-local interactions can be implemented for other band structures and lattice geometries, including systems where the relevant degrees of freedom are not described by the same D4h symmetry The present work therefore provides a concrete example of how superconducting order can be classified and analyzed in an orbital HK model, while the broader framework can be adapted to other correlated multiband systems

Acknowledgments

All authors acknowledge support from the Knut and Alice Wallenberg Foundation (Grant No. 2023.0087), the Swedish Research Council (VR starting Grant No. 2022-03350), the Olle Engkvist Foundation (Grant No. 229-0443), and Chalmers University of Technology, via the department of physics and the Areas of Advance Nano and Materials Science The computations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS) at C3SE partially funded by the Swedish Research Council through grant agreement no. 2022-06725

References

[1] S. Souma, Y. Machida, T. Sato, T. Takahashi, H. Matsui, S.-C. Wang, H. Ding, A. Kaminski, J. C. Campuzano, S.

1. The AI system can perform automated symmetry classification for complex correlated electron systems by leveraging the basis functions of the corresponding irreducible representations from tables like Table II and III, which are derived from the point group D4h. This allows the system to rigorously determine the symmetry-allowed candidates for the gap functions across spin, orbital, and momentum degrees of freedom.

2. The AI system can predict superconducting phase transitions in strongly correlated materials by analyzing the free-energy landscapes described in Section IV. Specifically, it can identify when the global minimum remains at ∆ ≠ 0 upon increasing the temperature and is separated from the normal-state solution at ∆ = 0 by a local maximum in the Mott regime, indicating a first-order phase transition.

3. The AI system can optimize pairing mechanisms for specific materials by correlating interaction strength with critical temperature predictions. It can calculate the critical temperature TC as a function of interaction strength U and pairing strength g for various channels, identifying that the critical temperature is maximized at finite interaction strength, with the maximum occurring well below the Mott transition.

Multiband superconductivity gives rise to a rich landscape of possible pairing states. Here we study superconductivity emerging from a half-filled normal state in a two-orbital extension of the Hatsugai-Kohmoto model, an exactly solvable model of correlated electrons with momentum-local interactions, which provides a minimal framework to explore the interplay of strong correlations, orbital structure and pairing symmetry. For a system with point-group symmetry D 4h, we classify the symmetry-allowed superconducting gap structures, taking into account spin, orbital and momentum degrees of freedom. We further compute the critical temperature and the superconducting order parameter for selected pairing channels as functions of interaction and pairing strength within a mean-field treatment. Our results provide a systematic framework for analyzing superconductivity in the orbital Hatsugai-Kohmoto model and extend symmetry-based approaches to correlated multiband settings.