Magnetic and spin-liquid phases in the frustrated $t-t^{\prime }$ Hubbard model on the triangular lattice

The Hubbard model and its strong-coupling version, the Heisenberg one, have been widely studied on the triangular lattice to capture the essential low-temperature properties of different materials. One example is given by transition metal dichalcogenides, as $1\mathrm{T}-{\mathrm{TaS}}_2$, where a large unit cell with 13 Ta atoms forms weakly coupled layers with an isotropic triangular lattice. By using accurate variational Monte Carlo calculations, we report the phase diagram of the $t-t^{\prime }$ Hubbard model on the triangular lattice, highlighting the differences between positive and negative values of $t^{\prime }/t$; this result can be captured only by including the charge fluctuations that are always present for a finite electron-electron repulsion. Two spin-liquid regions are detected: one for $t^{\prime }/t<0$, which persists down to intermediate values of the electron-electron repulsion, and a narrower one for $t^{\prime }/t>0$. The spin-liquid phase appears to be gapless, though the variational wave function has a nematic character, in contrast to the Heisenberg limit. We do not find any evidence for nonmagnetic Mott phases in the proximity of the metal-insulator transition, at variance with the predictions (mainly based upon strong-coupling expansions in $t/U$) that suggest the existence of a weak-Mott phase that intrudes between the metal and the magnetically ordered insulator.

Figure 1

Energy (per site) in units of $J=4t^2/U$, as a function of $t/U$ for $t^{\prime }/t=+0.3$ (upper panel) and $t^{\prime }/t=-0.3$ (lower panel). Data are shown for four different trial wave functions: The spin liquids “SL $d$-wave” (red empty squares) and “SL complex” (red empty circles), the magnetic state with the three-sublattice ${120}^{\circ }$ order (blue circles), and the magnetic state with the stripe collinear order (blue squares). Black arrows denote the metal-insulator transitions. Data are shown for a $L=18\times 18$ lattice size. Error bars are smaller than the symbol size.

Figure 2

Static density-density structure factor $N\left(\mathbf{q}\right)$, divided by $\left|\mathbf{q}\right|$, over the optimal wave function at different values of $U/t$, for $t^{\prime }/t=+0.3$ (upper panel) and $t^{\prime }/t=-0.3$ (lower panel). Data are shown for the $L=18\times 18$ lattice size, along the line connecting $\mathrm{\Gamma }=(0,0)$ to $\mathbf{M}=(\pi ,\frac{\pi }{\sqrt{3}})$. Error bars are smaller than the symbol size.

Figure 3

Ground-state phase diagram of the $t-t^{\prime }$ Hubbard model on the triangular lattice at half-filling. The magnetic phases are denoted by blue (for ${120}^{\circ }$ order) and green (for stripe collinear order) regions; the spin-liquid phase (with $d$-wave symmetry) is denoted by the red region; finally, the white part denotes the metallic phase. Points (with error bars) indicate the places where phase transitions have been located by our calculations.

Figure 4

Fermi surface at $U=0$, for different values of $t^{\prime }/t$, in the $(k_x,k_y)$ plane. The first Brillouin zone is denoted by black lines, while the Fermi surface is drawn in blue.

Figure 5

Static spin-spin structure factor $S\left(\mathbf{q}\right)$, divided by $\left|\mathbf{q}\right|$, over the optimal wave function at $t^{\prime }/t=-0.3, U/t=10$ (red empty circles) and at $t^{\prime }/t=-0.3, U/t=20$ (red empty squares), shown along the line connecting $\mathrm{\Gamma }=(0,0)$ to $\mathbf{M}=(\pi ,\frac{\pi }{\sqrt{3}})$. $S\left(\mathbf{q}\right)/\left|\mathbf{q}\right|$ is also shown on the frustrated square lattice at $U/t=16$ from $\mathrm{\Gamma }=(0,0)$ to $\mathbf{M}=(\pi ,\pi )$. All data are presented on a $L=18\times 18$ lattice size. Error bars are smaller than the symbol size.