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arXiv:0711.0244 · Physical Review Letters

Hubbard Model on the Triangular Lattice: Spiral Order and Spin Liquid

P Sahebsara, David Sénéchal

Published Mar 31, 2008• 139 citations
arXivHTMLPDF

Abstract

We investigate the half-filled Hubbard model on an isotropic triangular lattice with the variational cluster approximation. By decreasing the on-site repulsion U (or equivalently increasing pressure) we go from a phase with long-range, three-sublattice, spiral magnetic order, to a nonmagnetic Mott insulating phase--a spin liquid--and then, for U less or similar to 6.7t, to a metallic phase. Clusters of sizes 3, 6, and 15 with open boundary conditions are used in these calculations, and an extrapolation to infinite size is argued to lead to a disordered phase at U = 8t, but to a spiral order at U greater or similar to 12.

DOI
10.1103/physrevlett.100.136402

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We investigate the half-filled Hubbard model on an isotropic triangular lattice with the variational cluster approximation. By decreasing the on-site repulsion U (or equivalently increasing pressure) we go from a phase with long-range, three-sublattice, spiral magnetic order, to a nonmagnetic Mott insulating phase--a spin liquid--and then, for U less or similar to 6.7t, to a metallic phase. Clusters of sizes 3, 6, and 15 with open boundary conditions are used in these calculations, and an extrapolation to infinite size is argued to lead to a disordered phase at U = 8t, but to a spiral order at U greater or similar to 12.
We investigate the metal-insulator transition in the half-filled Hubbard model on the two-dimensional triangular lattice using both the Kotliar-Ruckenstein slave-boson technique and an exact numerical diagonalization of finite clusters. Contrary to the case of a square lattice, where a perfect nesting of the Fermi surface leads to a metal-insulator transition at arbitrarily small values of U, always accompanied by antiferromagnetic ordering, on a triangular lattice, due to the lack of perfect nesting, the transition takes place at a finite value of U, and frustration induces a nontrivial competition among different magnetic phases. Indeed, within the mean-field approximation in the slave-boson approach, as the interaction grows the paramagnetic metal turns into a metallic phase with incommensurate spiral ordering. Increasing the interaction further, a linear spin-density wave is stabilized, and finally for strong coupling the latter phase undergoes a first-order transition toward an antiferromagnetic insulator. No trace of the intermediate phases is seen in the exact diagonalization results, indicating a transition between a paramagnetic metal and an antiferromagnetic insulator.
We show that the spin-liquid phase of the half-filled Hubbard model on the triangular lattice can be described by a pure spin model. This is based on a high-order strong coupling expansion (up to order 12) using perturbative continuous unitary transformations. The resulting spin model is consistent with a transition from three-sublattice long-range magnetic order to an insulating spin-liquid phase, and with a jump of the double occupancy at the transition. Exact diagonalizations of both models show that the effective spin model is quantitatively accurate well into the spin-liquid phase, and a comparison with the Gutzwiller projected Fermi sea suggests a gapless spectrum and a spinon Fermi surface.
We investigate the quantum phase transitions in the half-filled Hubbard model on the triangular lattice by means of the path-integral renormalization group method with a new iteration and truncation scheme proposed recently. It is found for a cluster of 36 sites that as the Hubbard interaction U increases, the paramagnetic metallic state undergoes a first-order phase transition to a nonmagnetic insulating (NMI) state at Uc1 approximately 7.4t, which is followed by another first-order transition to a 120 degrees Néel ordered state at Uc2 approximately 9.2t, where t is the transfer integral. The size dependence of the results is also addressed. Our results suggest the existence of the intermediate NMI phase and resolve some controversial arguments on the nature of the previously proposed quantum phase transitions.
We investigate the Hubbard model on the anisotropic triangular lattice with two hopping parameters ttt and ${t}^{\ensuremath{'}}$ in different spatial directions, interpolating between decoupled chains (t=0t=0t=0) and the isotropic triangular lattice ($t={t}^{\ensuremath{'}}$). Variational wave functions that include both Jastrow and backflow terms are used to compare spin-liquid and magnetic phases with different pitch vectors describing both collinear and coplanar (spiral) order. For relatively large values of the on-site interaction $U/{t}^{\ensuremath{'}}\ensuremath{\gtrsim}10$ and substantial frustration, i.e., $0.3\ensuremath{\lesssim}t/{t}^{\ensuremath{'}}\ensuremath{\lesssim}0.8$, the spin-liquid state is clearly favored over magnetic states. Spiral magnetic order is only stable in the vicinity of the isotropic point, while collinear order is obtained in a wide range of interchain hoppings from small to intermediate frustration.
We study Mott transition in the two-dimensional Hubbard model on an anisotropic triangular lattice. We use the Lanczos exact diagonalization of finite-size clusters up to eighteen sites, and calculate Drude weight, charge gap, double occupancy and spin structure factor. We average these physical quantities over twisted boundary conditions in order to reduce finite-size effects. We find a signature of the Mott transition in the dependence of the Drude weight and/or charge gap on the system size. We also examine the possibility of antiferromagnetic order from the spin structure factor. Combining these information, we propose a ground-state phase diagram which has a nonmagnetic insulating phase between a metallic phase and an insulating phase with antiferromagnetic order. Finally, we compare our results with those reported in the previous theoretical studies, and discuss the possibility of an unconventional insulating state.
The Mott transition is observed experimentally in materials that are magnetically frustrated so that long-range order does not hide the Mott transition at finite temperature. The Hubbard model on the triangular lattice at half filling is a paradigmatic model to study the interplay of interactions and frustration on the normal-state phase diagram. We use the dynamical cluster approximation with continuous-time auxiliary-field quantum Monte Carlo to solve this model for 1-, 4-, 6-, 12-, and 16-site clusters with detailed analysis performed for the 6-site cluster. We show that (a) for every cluster there is an inflection point in the double occupancy as a function of interaction, defining a Widom line that extends above the critical point of the first-order Mott transition; (b) the presence of this line and the cluster size dependence argue for the observability of the Mott transition at finite temperature in the thermodynamic limit; and (c) the loss of spectral weight in the metal-to-Mott-insulator transition as a function of temperature and for strong interactions is momentum dependent, the hallmark of a pseudogap. That pseudogap spans a large region of the phase diagram near the Mott transition.