Inspired by recent advances in the fabrication of surface superlattices, and in particular the triangular lattice made of tin (Sn) atoms on silicon, we study an extended Hubbard model on a triangular lattice. The observations of magnetism in these systems justify the inclusion of a strong on-site repulsion, and the observation of superconductivity suggests including an effective, nearest-neighbor attractive interaction. The attractive interaction mimics the effect of strong on-site repulsion near half filling, which can be seen in strong-coupling vertex calculations such as the Eliashberg method. With this extended Hubbard model on a triangular lattice with its geometrical frustration, we find a rich phase diagram of various magnetic orders and pairing functions, within the framework of self-consistent mean field theory. We uncover the competition between magnetism and unconventional superconductivity, and their coexistence for triplet pairings. We follow the Fermi surface of the system as the system is doped away from half filling and find nesting vectors and a Lifshitz transition, which provide an intuitive understanding of the phase transitions between the many orders we consider.
Two-dimensional density-matrix renormalization group method is employed to examine the ground-state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction
U at ${U}_{\mathrm{c}1}\ensuremath{\sim}7.8t$ and ${U}_{\mathrm{c}2}\ensuremath{\sim}9.9t$ (
t being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\ensuremath{\le}U<{U}_{\mathrm{c}1}$ implies a metallic phase in this regime. For
U>Uc2, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\ensuremath{\circ}}$ N\'eel-ordered state realized in the limit of $U\ensuremath{\rightarrow}\ensuremath{\infty}$. For
Uc1<U<Uc2, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the
K and ${\mathrm{K}}^{\ensuremath{'}}$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing
U and thus the superconductivity is unlikely in the intermediate phase.
We study the Hubbard model on the triangular lattice for general fillings by means of the slave-boson technique. In particular, we consider the stability regions of the paramagnetic, antiferromagnetic and linearly polarized spin-density-wave phases that are present at half-filling. The model presents a large anisotropy under particle or hole doping of the half-filled band, a consequence of the non-bipartite nature of the lattice. We also compute the helicity modulus in the antiferromagnetic phase, which characterizes the stiffness of the magnetic order.
Abstract Using a mean-field theory based upon Hartree—Fock approximation, we theoretically investigate the competition between the metallic conductivity, spin order and charge order phases in a two-dimensional half-filled extended Hubbard model on anisotropic triangular lattice. Bond order, double occupancy, spin and charge structure factor are calculated, and the phase diagram of the extended Hubbard model is presented. It is found that the interplay of strong interaction and geometric frustration leads to exotic phases, the charge fluctuation is enhanced and three kinds of charge orders appear with the introduction of the nearest-neighbor interaction. Moreover, for different frustrations, it is also found that the antiferromagnetic insulating phase and nonmagnetic insulating phase are rapidly suppressed, and eventually disappeared as the ratio between the nearest-neighbor interaction and on-site interaction increases. This indicates that spin order is also sensitive to the nearest-neighbor interaction. Finally, the single-site entanglement is calculated and it is found that a clear discontinuous of the single-site entanglement appears at the critical points of the phase transition.
An extended Hubbard model for a single band, including Coulomb repulsion and correlated hopping between nearest neighbors, is studied using a generalized mean-field approach. Antiferromagnetism and superconductivity are probed for arbitrary occupation number, near and away from half filling. Binding of pairs in the superconducting state of this purely repulsive model is mediated by the correlated hopping in the form of a covalent-bond configuration, with partial intrasite and intersite pairings. A region of coexistence is conjectured, the superconductivity being suppressed by the saturation of the staggered magnetic moment. Singlet superconducting nonmagnetic states are obtained for the almost-empty- or full-band cases. On the other hand, antiferromagnetism induces mixed s- and p-type superconductivities in the neighborhood of half filling.
The superconducting pairing symmetry is significantly constrained by lattice symmetry, yet how this pairing symmetry evolves with changes in lattice symmetry remains unclear. In this study, through variational quantum Monte Carlo simulations on the Hubbard model, we investigate the evolution of the superconducting ground state on an anisotropic triangular lattice with smoothly tunable frustration. At half-filling, we identify a hidden
C6-like
s+d+id-wave superconductivity, which breaks time-reversal symmetry, emerging for moderate frustrations. This occurs as the lattice symmetry transitions from
C4 to
C2, where the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave and
s+dxy-wave superconducting ground states dominate, respectively. Notably, we find an isotropic-like
d+id-wave superconducting phase at ${t}^{\ensuremath{'}}/t\ensuremath{\approx}0.7$, much lower than the expected value ${t}^{\ensuremath{'}}/t=1$ predicted by most previous studies. At high electron doping levels, we further discover an unexpected quasi-one-dimensional ${d}_{xy}^{\ensuremath{'}}$-wave superconductivity characterized by strong pairing on the next-nearest-neighbor bond, despite the absence of direct hopping. The numerical data also reveal that pairing symmetries, belonging to distinct irreducible representations of the given point group, can mix together, beyond the conventional theories based on the linear gap equations.