insight - Computational Complexity - # d-wave Superconductivity in the Hubbard Model on the Isotropic Triangular Lattice

Core Concepts

The ground state of the Hubbard model on the isotropic triangular lattice is the dxy-wave superconductor below the Mott insulator phase, and the chiral d+id pairing is a quasi-stable state slightly higher in energy.

Abstract

The author studied d-wave superconductivity in the Hubbard model on the isotropic triangular lattice using the variational cluster approximation (VCA) at zero temperature and half-filling. The key findings are:

- The ground state is the dxy-wave superconductor below the Mott insulator phase at U/t ≲ 6, where U is the on-site Coulomb repulsion and t is the hopping parameter.
- The energy of the chiral d+id pairing state is slightly higher, about 0.01t to 0.03t, than the dxy-wave ground state for U/t ≃ 5.
- The d-wave superconductivity is not realized above the Mott transition point at U/t ≳ 7.
- The prediction of the dxy pairing symmetry agrees with experimental observations on the organic superconductor κ-(BEDT-TTF)2Cu2(CN)3, and the estimated energy difference between the normal and superconducting states is semi-quantitatively consistent with the observed transition temperature.
- The author discusses the importance of using reference clusters that capture the symmetry properties of the system, as the discrepancies with previous studies using smaller clusters are attributed to the inability of those clusters to properly describe the mixing of the dx2-y2 and dxy order parameters.

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Stats

The energy difference between the superconducting and paramagnetic states is about 0.02t to 0.06t for U/t ≃ 5.
The energy difference between the d+id and dxy states is about 0.01t to 0.03t for U/t ≃ 5.

Quotes

"The ground state is the dxy-wave superconductor below the Mott insulator phase at U/t ≲ 6."
"The energy of the chiral d+id pairing state is slightly higher, about 0.01t to 0.03t, than the dxy-wave ground state for U/t ≃ 5."
"The d-wave superconductivity is not realized above the Mott transition point at U/t ≳ 7."

Deeper Inquiries

To fully understand the superconductivity in materials such as κ-(BEDT-TTF)2Cu2(CN)3, it is essential to consider several factors beyond the Hubbard model. Firstly, electron-phonon interactions could play a significant role in mediating superconductivity, particularly in organic materials where lattice vibrations can influence electronic properties. Additionally, spin fluctuations and magnetic interactions are crucial, as they can lead to unconventional pairing mechanisms that are not captured by the simple Hubbard model.
Moreover, the geometric frustration inherent in the triangular lattice structure can lead to complex ground states, such as spin liquids, which may affect the superconducting phase. The presence of disorder and impurities in real materials can also alter the electronic states and pairing symmetries, necessitating a more comprehensive model that includes these effects. Finally, multi-band effects should be considered, as the presence of multiple electronic bands can lead to competition between different pairing symmetries, influencing the overall superconducting behavior.

Doping the system away from half-filling would significantly alter the electronic structure and the resulting superconducting properties. In the context of the Hubbard model on the isotropic triangular lattice, moving away from half-filling introduces additional charge carriers, which can lead to a change in the effective interaction strength between electrons. This can enhance or suppress superconductivity depending on the level of doping.
For instance, electron doping could lead to an increase in the density of states at the Fermi level, potentially enhancing superconductivity. Conversely, hole doping might destabilize the dxy pairing symmetry observed at half-filling, possibly favoring other pairing states or leading to a transition to a different phase, such as a Fermi liquid or a different type of superconducting state. The energy differences between various pairing symmetries, such as dxy and chiral d+id, would also be affected, potentially making the chiral d+id state more favorable under certain doping conditions.

Yes, the chiral d+id pairing state can potentially be stabilized in other strongly correlated electron systems on the triangular lattice. The key factors that contribute to the stabilization of this pairing state include the symmetry of the lattice, the strength of electron correlations, and the presence of frustration.
In systems with similar triangular lattice geometries, such as certain organic superconductors or transition metal oxides, the conditions for chiral d+id pairing can be met, especially when the system is close to a Mott transition. The interplay between spin and charge fluctuations in these materials can lead to the emergence of chiral superconductivity, particularly when the system exhibits strong electron correlations.
Furthermore, the topological nature of the chiral d+id state, which is characterized by its non-trivial phase structure, makes it a candidate for exploration in various materials that exhibit strong correlation effects. Experimental realizations of such states have been observed in systems like heavy fermion compounds and topological insulators, suggesting that the principles governing the stabilization of chiral d+id pairing can be applicable across a range of strongly correlated electron systems on triangular lattices.

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