Revisiting the $J_1$-$J_2$ Heisenberg Model on a Triangular Lattice: Quasi-Degenerate Ground States and Phase Competition

Abstract

It is generally believed that the spin-$\tfrac{1}{2}$ triangular-lattice $J_1$-$J_2$ Heisenberg model hosts a quantum spin liquid in the intermediate regime between the $120^\circ$ and stripe ordered phases. Density matrix renormalization group studies on cylinders have consistently found two nearly degenerate ground states, commonly interpreted as distinct topological sectors. Using state-of-the-art matrix product state simulations on YC6 cylinders, we compare the static and dynamical properties of these two sectors at $J_2/J_1 = 0.125$. Noticeable differences appear already in static correlations; moreover, high-resolution dynamical structure factors reveal qualitatively distinct low-energy excitations. These results suggest that the two ground states cannot be understood as merely topologically distinct sectors of a gapped $\mathbb{Z}_2$ spin liquid.

Figures (9)

• Figure 1: A schematic phase diagram of the $J_1$-$J_2$ Heisenberg model on a triangular lattice as a function of $J_2/J_1$, as conjectured from previous numerical studies Manuel1999Mishmash2013Kaneko2014Li2015Zhu2015Hu2015Bishop2015Iqbal2016Hu2019Ferrari2019Gong2019Tang2022Jiang2023Wietek2024. For $J_2 = 0$ the model exhibits coplanar $120\degree$ order, transitioning into the QSL candidate region around $J_2/J_1 \approx 0.07$, followed by a stripe-ordered phase at larger values of coupling.
• Figure 2: Nearest-neighbor spin-spin correlation $S_{\langle i,j\rangle}$ on the YC6-36 cylinder computed for (a) $120\degree$ ordered state at $J_2 = 0$, (b) even- and (c) odd-sector states at $J_2/J_1 = 0.125$, and (d) stripe-ordered state at $J_2/J_1 = 0.2$. Bond strengths are represented via a colormap, with the same scale applied globally across all panels to allow direct comparison.
• Figure 3: The equal-time structure factor $\chi({\boldsymbol{\mathrm{k}}})$ extracted from ground-state MPSs of the YC6-36 cylinder. Panels (a) and (d) correspond to $J_2 = 0$ ($120\degree$ order) and $J_2/J_1 = 0.2$ (stripe order), respectively. Results for even (b) and odd (c) ground states in the QSL candidate phase are plotted side-by-side for comparison. Each panel is normalized independently to its own maximum intensity to highlight the relative distribution of spectral weight across different coupling regimes. The dashed white lines indicate the boundaries of the first Brillouin zone, which we also plot separately in panel (e), marking high symmetry points and the cuts considered in our DSF computations (see Fig. \ref{['fig:QSL_DSF']}).
• Figure 4: The dynamical structure factor computed for (a) the $120\degree$ ordered phase at $J_2 = 0$ and (b,c) the putative QSL phase at $J_2/J_1 = 0.125$ along the momentum cuts indicated in Fig. \ref{['fig:QSL_static']}(e). The normalization factor $1/S_{\mathrm{max}}({\boldsymbol{\mathrm{k}}},\omega)$ is provided in the top left corner of each panel.
• Figure S-1: DMRG sweeping protocol for the TLHAF at $J_2/J_1 = 0.125$. The two insets show equal-time structure factors for the MPS before and after the drop observed around the fifth sweep at $D^* = 2048$.