Spin liquid phase of the $S=\frac{1}{2}{J}_1-J_2$ Heisenberg model on the triangular lattice

We study the $S=1/2$ Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor interactions $J_1$ and $J_2$ with the density matrix renormalization group, on long open cylinders with widths up to nine lattice spacings. In an intermediate $J_2$ region $0.06\lesssim J_2/J_1\lesssim 0.17$, we find evidence for a spin liquid (SL) state with short range spin-spin, bond-bond, and chiral correlation lengths, bordered by a classical ${120}^{\circ }$ Néel ordered state at small $J_2$ and by a two sublattice collinear magnetically ordered state at larger $J_2$. Focusing on $J_2/J_1=0.1$, we find a number of signatures of a gapped SL phase.

Figure 1

For a YC6 cylinder, we vary $J_2$ with position, from $J_2=0$ on the left edge to $J_2=0.24$ on the right edge. We also apply a pinning magnetic field along both the $x$ and $z$ directions on the left edge to favor the classical ${120}^{\circ }$ order. Two approximate phase transition lines are shown. The size of the arrow represents local measurement of $\langle S\rangle =\sqrt{{\langle S_x\rangle }^2+{\langle S_z\rangle }^2}$ with the direction of the angle given by ${tan}^{-1}[\langle S_z\rangle /\langle S_x\rangle ]$, and the widths of lines proportional to $|\langle S_i·S_j\rangle +0.18|$. The solid lines along the bonds mean the bond measurement is negative, i.e., a stronger than average bond, while dashed lines indicate bonds that are weaker than average.

Figure 2

(a) The higher-energy even and (b) the lower-energy odd sector ground states for a YC6 cylinder with $J_2=0.1$, where we subtract $-0.18$ from all the bonds. The odd and even sector systems differ primarily by the removal of a single site at each edge; in addition, we needed to make the higher-energy system shorter to avoid falling into the low-energy sector through the creation of two end spinons. In the plot the bond thickness is restricted to a maximum; otherwise, many edge bonds would be much thicker. (c) Central portion of the ground state on the XC6 cylinder. The solid (dashed) bonds have strength $\langle S_i·S_j\rangle =-0.287/-0.157$. (d) A similar central region for a YC5 cylinder. The solid (dashed) bonds have strength $\langle S_i·S_j\rangle =-0.158/-0.126$.

Figure 3

The spin triplet gap for various long cylinder geometries at $J_2=0.1$ with $L_x=20$. The inset shows the linear extrapolation of spin triplet gap for YC6 cylinders vs $1/L_x^2$. The triplet gap is roughly ${\mathrm{\Delta }}_T=0.357$ for an infinitely long YC6 cylinder.