Spin-liquid versus spiral-order phases in the anisotropic triangular lattice

We study the competition between magnetic and spin-liquid phases in the Hubbard model on the anisotropic triangular lattice, which is described by two hopping parameters $t$ and $t^{\prime }$ in different spatial directions and is relevant for layered organic charge-transfer salts. By using a variational approach that includes spiral magnetic order, we provide solid evidence that a spin-liquid phase is stabilized in the strongly correlated regime and close to the isotropic limit $t^{\prime }/t=1$. Otherwise, a magnetically ordered spiral state is found, connecting the (collinear) Néel and the (coplanar) ${120}^{\circ }$ phases. The pitch vector of the spiral phase obtained from the unrestricted Hartree-Fock approximation is substantially renormalized in the presence of electronic correlations, and the Néel phase is stabilized in a wide regime of the phase diagram, i.e., for $t^{\prime }/t<0.75$. We discuss these results in the context of organic charge-transfer salts.

Figure 1

Illustration of the anisotropic triangular lattice in the square topology (left), used in this work, and in the equivalent triangular topology (right). Solid and dashed lines denote hopping amplitudes $t$ and $t^{\prime }$, respectively.

Figure 2

HF (left) and VMC (right) energies of the spiral state, for $U/t=16$, as a function of the pitch angle $\theta$ (in unit of $\pi$), for $t^{\prime }/t=0.85$ (top), $0.9$ (middle), and $0.95$ (bottom). Cluster of sizes $l\times l$ have been used, with $l=12,14,16,18,20$.

Figure 3

Upper panel: HF (red circles) and VMC (blue squares) energies of the optimal spiral state as a function of $t^{\prime }/t$ for $U/t=16$. Lower panel: the pitch angle $\theta$ (in unit of $\pi$) of the optimal spiral state as a function of $t^{\prime }/t$ for $U/t=16$. The dotted horizontal line corresponds to $\theta =2\pi /3$, suitable for the isotropic point $t^{\prime }/t=1$. The error bars in the VMC calculations are due to the finite-size limitations in the accessible pitch angles.

Figure 4

Upper panel: VMC energy of the optimal spiral (blue squares) and spin-liquid (black diamonds) states as a function of the inverse system size $1/L$, with $L$ ranging from $10\times 10$ to $20\times 20$. Data refer to the case $t^{\prime }/t=0.85$ and $U/t=16$. Lower panel: the same as in the upper panel but for $t^{\prime }/t=0.95$.

Figure 5

The energy (in unit of $J=4t^2/U$) as a function of $t/U$ for the HF approximation (red circles), the optimal spiral state in VMC (blue squares), and the spin-liquid state (black diamonds) for $t^{\prime }/t=0.8$ (top), $0.9$ (middle), and $0.95$ (bottom).

Figure 6

Schematic phase diagram of the Hubbard model on the anisotropic triangular lattice, as obtained by VMC: metal (blue), insulator with magnetic Néel order (red), insulator with spiral magnetic order (gradient red-yellow-green), and spin liquid (cyan). The ${120}^{\circ }$ ordered state with $\theta =2\pi /3$ (vertical green line) is stable only for $t^{\prime }/t=1$. The spiral state illustrated with the red-yellow gradient has a pitch angle $\theta$ ranging from $\pi$ to $0.9\pi$. The border between the spin-liquid and the magnetic state for $t^{\prime }/t<0.8$ (dashed cyan-white line) is only inferred, since pitch angles too close to $\pi$ could not be resolved.