Competing spin-liquid states in the spin-$\frac{1}{2}$ Heisenberg model on the triangular lattice

We study the spin-$\frac{1}{2}$ Heisenberg model on the triangular lattice with antiferromagnetic first- $\left(J_1\right)$ and second- $\left(J_2\right)$ nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\circ }$ magnetic order phase for $J_2\lesssim 0.07J_1$ and a stripe antiferromagnetic phase for $J_2\gtrsim 0.15J_1$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped $Z_2$ spin liquids.

Figure 1

Quantum phase diagram of the isotropic spin-$\frac{1}{2} J_1-J_2$ Heisenberg model on a triangular lattice ($J_1=J_1^{\prime }$). With growing $J_2$, the system has a ${120}^{\circ }$ AF phase for $J_2\lesssim 0.07$, a stripe AF phase for $J_2\gtrsim 0.15$, and a SL phase in between. The schematic figures of the different phases also show the (a), (b) YC and (c) XC cylinder geometries. (d)–(f) are the contour plots of the spin structure factor for each phase.

Figure 2

The NN bond energy $\langle S_i·S_j\rangle$ for $J_2=0.1$ on the YC8-24 cylinder in (a) the even and (b) the odd sector. The left 16 columns are shown here. The odd sector is obtained by removing one site in each boundary of the cylinder. In both figures, all the bond energies have subtracted the average value of $-0.18$. The red solid and blue dashed bonds denote the negative and positive bond energies after subtraction (with some numbers shown for clarity).

Figure 3

(a) and (b) are the spin and dimer correlation functions on different YC cylinders for $J_2=0.1$. $\xi$ denotes the corresponding correlation lengths on the YC10 cylinder in the odd sector. (c) Chiral correlations for $J_2=0.1$ on different cylinders. All the data are obtained from real number DMRG calculations except “YC10, even (c),” which denotes the data obtained from complex wave function DMRG calculations on the YC10 cylinder in the even sector. (d) Chiral order from the boundary to the bulk for the bond anisotropic system in the even sector, which is obtained from the complex DMRG. (e) Schematic phase diagram for the bond anisotropic system at $J_2=0.1$.

Figure 4

Bulk energy per site $e$ vs DMRG truncation error $\varepsilon$ for ground states in the even and odd sectors for (a) $J_2=0.1$ and (b) $J_2=0.125$ on the YC10-24 cylinder. The numbers denote the kept $SU\left(2\right)$ states. The energy data are fitted using the formula $e\left(\varepsilon \right)=e\left(0\right)+a\varepsilon +b{\varepsilon }^2$.

Figure 5

(a) Real-space configuration of spin magnetization $〈S_i^z〉$ after adiabatically inserting a flux $\theta =2\pi$. The red and blue circles denote the positive and negative $\langle S_i^z\rangle$ with the area of the circle proportional to the amplitude of $\langle S_i^z\rangle$. $\mathrm{\Delta }{Q}_{\mathrm{R}}$ is the net spin accumulation on the right edge $\mathrm{\Delta }{Q}_{\mathrm{R}}={\sum }_i\langle S_i^z\rangle$, where $i$ denotes the site on the right edge of the cylinder. (b) Real-space configuration of the accumulated spin magnetization in each column with increasing flux. Low-lying ES at the flux (c) $\theta =0$ and (d) $\theta =2\pi$. The numbers in ES denote the near-degenerate eigenvalues in large weight levels. At $\theta =2\pi$ in (d), all the eigenvalues are double degenerate, as explicitly shown by the circles for the low-lying levels.