Effective Spin Model for the Spin-Liquid Phase of the Hubbard Model on the Triangular Lattice

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.

Figure 1

The most relevant exchange couplings displayed as a function of $t/U$. The 12th order bare series of all couplings appearing up to order 4 and the largest four-spin interaction $J_{\mathrm{O}6}^{4\mathrm{sp}}$ plus the largest six-spin interaction $J^{6\mathrm{sp}}$ appearing in order 6 are plotted. The shaded area marks the region where extrapolation effects become important. The vertical dashed line indicates the magnetic phase transition. Inset: various Padé approximants as a function of $t/U$ for the nearest-neighbor Heisenberg exchange $J_{1,0}$.

Figure 2

(a) Energy per site as a function of $U/t$ for the Hubbard model and the effective spin model. (b) Level crossings in the effective spin model signaled by the energy difference of the two lowest energies. (c) Double occupancy (times $U^2$) as a function of $U/t$. (d) Magnetic structure factor at the 120° AFM ordering wave vector. The dashed lines denote results for a simplified 2B-4B model evaluated up to order 10. Continuous lines with symbols are the results for the full 2B-4B-6B model considered here. The observable is up to order 2 in both cases.

Figure 3

Comparison of the low-energy spectrum of a $N_s=21$ Hubbard model at $U/t=9$ (left panel) to results obtained with the effective spin model (right panel). The $x$ axis labels the different symmetry sectors, while empty (filled) symbols denote $S^z=1/2$ ($S^z=3/2$) levels. $a$, $b$, and $c$ label momenta in the interior of the Brillouin zone.

Figure 4

ED spectra of the truncated spin model for (a) $N_s=21$, (b) $N_s=27$, and (c) $N_s=36$ as a function of $U/t$. Open (black) symbols denote nonmagnetic levels. Filled (red) symbols are energy levels corresponding to excitations with finite spin above the GS. The grey shaded area signals the extension of the SL phase.