Exact diagonalization study of triangular Heisenberg model with four-spin ring-exchange interaction

TL;DR

This work tackles the ground-state phase diagram of the spin-1/2 triangular-lattice Heisenberg model with $J_1$, $J_2$, and four-spin ring exchange $J_r$, motivated by frustrated magnetism and spin-liquid candidates in triangular materials. The authors apply Lanczos exact diagonalization on two 36-site tori and use level spectroscopy to locate phase boundaries via crossings of low-lying states at high-symmetry momenta, complemented by spin, dimer, and chiral structure factors. They unveil a rich tapestry of phases, including $120^{\circ}$ AFM, zigzag, tetrahedral magnetic orders and several nonmagnetic phases (I–V), with phase III and IV appearing at larger $J_r$, and find no definitive evidence for a spinon Fermi surface QSL. The results illuminate how four-spin ring exchange shapes triangular-lattice magnetism and provide benchmarks for future unbiased methods such as $SU(2)$ DMRG and tensor-network approaches.

Abstract

Using Lanczos exact diagonalization (ED), we study the spin-1/2 $J_1$-$J_2$ Heisenberg model with the four-spin ring-exchange interaction $J_r$ on triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer and chiral correlation functions. The ground state has rich phases including several magnetic ordered phases like zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction, and help to understand the relevant triangular materials.

Figures (11)

• Figure 1: (a) and (b) are two 36-site clusters (or tori) with two different periodic boundary conditions (PBC), which are called 36$A$ and 36$B$ respectively. The black arrows at the bottom left represent the PBC directions. Different colored lines represent the corresponding interactions. (c) and (d) are the first Brillouin zones of the 36$A$ and 36$B$ clusters respectively. Solid lines denote the Brillouin-zone boundaries. Symbols indicate the 36 momenta inside the first Brillouin zone. Colorful symbols are the momenta used in the ED calculations.
• Figure 2: (a) is the energy spectra of 36$A$ as functions of $J_r$ when $J_2=0$. The vertical black dashed lines indicate the phase boundaries. Symbols with different shapes and colors indicate different momenta [refer to Figs. \ref{['fig:LattBZ']}(c) and \ref{['fig:LattBZ']}(d), the rest momenta with gray color are equivalent to momenta with other colors under point-group symmetries for 36$A$.], the solid and hollow symbols indicate different spin-inversion sectors. The energies below the red dashed line form the full lowest spectra without missing any energy levels. (b) is the energy spectra of 36$B$ as functions of $J_r$ when $J_2=0$. Compared to 36$A$, for 36$B$, we only present two energy levels at each high-symmetry momentum sector ($\Gamma,K,M$). We have also calculated energy levels at other non-equivalent momenta (not shown in the figure) to ensure that the ground state energy is accurate and that there are no other level crossings between the ground state and the excited states. (c) $J_1-J_r$ phase diagram obtained on 36$A$ and 36$B$. We can identify similar phases using these two tori, but the phase boundaries are quite different for some of these phases.
• Figure 3: Structure factors for $120^{\circ}$ AFM phase, possible SL phase, phase I, phase II, zigzag phase, phase III and phase IV, respectively. Each column of figures corresponds to the different parameters (phases). The top two rows represent spin structure factors. The first row corresponds to the results obtained on 36$A$, and the second row corresponds to the results obtained on 36$B$. Similarly, the middle two rows show the dimer structure factors, and the last two rows show the chiral structure factors, with the first row in each pair corresponding to 36$A$ and the second row to 36$B$. For spin structure factors, above the boundary value $U_0 = 1.13$ labeled by a black line on the color bar, a logarithmic mapping is used, $U = U_0 + \log_{10} [S(\mathbf{q})] -\log_{10} [U_0]$, and $U = S(\mathbf{q})$ while below the boundary value.
• Figure 4: (a1)-(a3) and (b1)-(b3) show the real-space dimer correlations for $120^{\circ}$ AFM phase, phase I and phase II in 36$A$ and 36$B$, respectively. (c1)-(c3) and (d1)-(d4) show the real-space dimer correlations for zigzag phase, phase III and IV in 36$A$ and 36$B$, respectively. The corresponding $J_2$ and $J_r$ are marked at the top of corresponding figures. In each figure, the thickest horizontal blue bond in the middle represents the reference bond. Bond thickness corresponds to the magnitude of dimer correlation, while color indicates the sign: blue signifies positive, and red signifies negative. The black dash rhombus frames indicate the unit cell of possible VBS pattern.
• Figure 5: The ground state phase diagram of the $J_1$-$J_2$-$J_r$ model obtained from the 36$A$ torus. The gray dashed line indicates the $j_1-j_4$ model used in some variational Monte Carlo calculations Mishmash2013RingVMCLiJX2023RingVMCLiT2023RingVSED.