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Physica C Superconductivity

Spin-triplet f-wave pairing via cyclic exchange in the anisotropic triangular Hubbard model: A route to unconventional superconductivity

Satoshi Nishimoto, Y. Ohta

Published May 29, 2025• 0 citations
Publisher

Abstract

We investigate the interplay between ferromagnetism and unconventional superconductivity in a two-dimensional anisotropic triangular-lattice Hubbard model using the density-matrix renormalization group (DMRG) method. A novel mechanism for spin-triplet f -wave superconductivity is identified, driven by three-site cyclic-exchange interactions. This mechanism highlights the crucial role of lattice anisotropy and triangular hopping networks in stabilizing ferromagnetic correlations necessary for spin-triplet pairing. The slow decay of pair-correlation functions and their sign change under a π / 3 rotation confirm the dominant f -wave symmetry of the superconducting state. These findings offer a unified theoretical framework for understanding unconventional superconductivity in strongly correlated electron systems, with potential relevance to materials such as Bechgaard salts ( TMTSF ) 2 X , cobalt oxide Na 0 . 35 CoO 2 ⋅ 1 . 3 H 2 O , and layered perovskite Sr 2 RuO 4 . • Identifies a mechanism for spin-triplet superconductivity via three-site cyclic exchange. • Demonstrates f -wave pairing symmetry in a triangular-lattice Hubbard model. • Shows lattice anisotropy promotes ferromagnetism that stabilizes triplet pairing.

DOI
10.1016/j.physc.2025.1354742

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We investigate the interplay between ferromagnetism and unconventional superconductivity in a two-dimensional anisotropic triangular-lattice Hubbard model using the density-matrix renormalization group (DMRG) method. A novel mechanism for spin-triplet f -wave superconductivity is identified, driven by three-site cyclic-exchange interactions. This mechanism highlights the crucial role of lattice anisotropy and triangular hopping networks in stabilizing ferromagnetic correlations necessary for spin-triplet pairing. The slow decay of pair-correlation functions and their sign change under a π / 3 rotation confirm the dominant f -wave symmetry of the superconducting state. These findings offer a unified theoretical framework for understanding unconventional superconductivity in strongly correlated electron systems, with potential relevance to materials such as Bechgaard salts ( TMTSF ) 2 X , cobalt oxide Na 0 . 35 CoO 2 ⋅ 1 . 3 H 2 O , and layered perovskite Sr 2 RuO 4 . • Identifies a mechanism for spin-triplet superconductivity via three-site cyclic exchange. • Demonstrates f -wave pairing symmetry in a triangular-lattice Hubbard model. • Shows lattice anisotropy promotes ferromagnetism that stabilizes triplet pairing.
Ferromagnetiam and superconductivity in a two-dimensional triangular-lattice Hubbard model are studied using the density-matrix renormalization group method. We propose a mechanism of the {\it f}-wave spin-triplet pairing derived from the three-site cyclic-exchange ferromagnetic interactions. We point out that a triangular network of hopping integrals, which is required for the three-site cyclic hopping processes, is contained in the (possibly) spin-triplet superconducting systems, such as Bechgaard salts (TMTSF)2_22​X, cobalt oxide Na0.35_{0.35}0.35​CoO2_22​$\cdot1.3H1.3H1.3H_2O,andlayeredperovskiteSrO, and layered perovskite SrO,andlayeredperovskiteSr_2RuORuORuO_4$.
Spin-triplet superconductivity is actively pursued in condensed matter physics due to the potential applications in topological quantum computations. The related pairing mechanism involving the interaction remains an important research topic. Here we propose a universal approach to obtain p-wave triplet superconductivity in the Hubbard models by simply changing the sign of the hopping amplitudes of the spin-down electrons, and apply it to three prototype two-dimensional lattices (honeycomb, square, and triangular). The parent Hamiltonian at half filling has long-range magnetic order, which is ferromagnetic in all three directions for the frustrated triangular lattices, and ferromagnetic (antiferromagnetic) in the xy plane (z direction) for the bipartite honeycomb and square lattices. The magnetic transitions occur at some critical interactions on honeycomb and triangular lattices, which are estimated by finite-size scalings. When the systems are doped, we find the triplet p-wave pairing is a dominating superconducting instability. We demonstrate its emergence is closely related to the strong ferromagnetic spin fluctuations induced by the doping. Our results provide an understanding of the microscopical triplet-pairing mechanism, and will be helpful in the search for spin-triplet superconducting materials.
Altermagnets - collinear, zero-net-moment magnets with momentum-odd spin splitting protected by crystalline symmetries - offer a tunable route to suppress long-range antiferromagnetism while preserving strong short-range spin fluctuations. We show that this environment robustly stabilizes unconventional superconductivity and naturally produces mixed-symmetry pairing. Through a strong-coupling analysis of a spin-anisotropic Hubbard model, we derive an anisotropic t-J model where exchange interactions cooperatively enhance singlet d-wave pairing and promote triplet p-wave pairing. Our mean-field analysis reveals a pairing evolution driven by altermagnetic anisotropy: for small spin anisotropy, the d-wave channel is enhanced, closely resembling the dominant pairing symmetry in cuprate superconductors, which suggests that weak spin anisotropy may be an essential ingredient in realistic models of these materials. Constrained-path quantum Monte Carlo simulations confirm this picture, showing a regime where dominant d-wave correlations coexist with an emergent p-wave component near optimal doping. As spin anisotropy increases, strong C2 anisotropy and spin splitting activate the triplet channel, leading to a stable d+p mixed-pairing state. This synergistic state exhibits significantly enhanced overall pairing strength, suggesting the possibility of a higher superconducting transition temperature.
Recent experimental progress has established the twisted bilayer transition metal dichalcogenide (TMD) as a highly tunable platform for studying many-body physics. Particularly, the homobilayer TMDs under displacement field are believed to be described by a generalized triangular-lattice Hubbard model with a spin-dependent hopping phase $\ensuremath{\theta}$. To explore the effects of $\ensuremath{\theta}$ on the system, we perform density matrix renormalization group calculations for the relevant triangular lattice t-J model. By changing $\ensuremath{\theta}$ at small hole doping, we obtain a region of quasi-long-range superconducting order coexisting with charge and spin density wave within $0<\ensuremath{\theta}<\ensuremath{\pi}/3$. The superconductivity is composed of a dominant spin singlet ddd-wave and a subdominant triplet ppp-wave pairing. Intriguingly, the ${S}_{z}=\ifmmode\pm\else\textpm\fi{}1$ triplet pairing components feature pair-density waves. In addition, we find a region of triplet superconductivity coexisting with charge-density wave and ferromagnetism within $\ensuremath{\pi}/3<\ensuremath{\theta}<2\ensuremath{\pi}/3$, which is related to the former phase at smaller $\ensuremath{\theta}$ by a combined operation of spin-flip and gauge transformation. Our findings provide insights and directions for experimental search for exotic superconductivity in twisted TMD systems.
We study the effect of the nearest neighbor anisotropic exchange interaction on superconductivity in extended Hubbard model in a square lattice. This Hamiltonian is one of a base model to describe various effects on triplet superconductors, e.g. Sr 2 RuO 4 , because the ferromagnetic exchange is advantagous for triplet pairing. Within a one-band meanfield approximation, we calculate the critical temperatures of spin singlet and triplet superconductivity. We obtain a phase diagram in the longitudinal vs transverse exchange couplings and we show how the stable pairing state is controlled by the anisotropy of the exchange interaction.
With the discovery of strong-coupling physics and superconductivity in moir\'e superlattices, it is essential to have an understanding of strong-coupling driven superconductivity in systems with trigonal symmetry. The simplest lattice model with trigonal symmetry is the triangular lattice Hubbard model. Although the triangular lattice spin model is a heavily studied model in the context of frustration, studies of the hole-doped triangular lattice Hubbard model are rare. Here, we use density matrix renormalization group to investigate the dominant superconducting channels in the hole-doped triangular lattice Hubbard model over a range of repulsive interaction strengths. We find a clear transition from ppp-wave superconductivity at a moderate on-site repulsion strength (U/t=2U/t=2U/t=2) at filling above 1/4 ($n\ensuremath{\sim}0.65$) to ddd-wave superconductivity at a strong on-site repulsion strength (U/t=10U/t=10U/t=10) at filling below 1/4 ($n\ensuremath{\sim}0.4$). The unusual tunability that moir\'e superlattices offer in controlling U/tU/tU/t would open up the opportunity to realize this transition between ddd-wave and ppp-wave superconductivity.