Unconventional Spin Dynamics and Supersolid Excitations in the Triangular-Lattice XXZ Model

TL;DR

This work investigates the spin-1/2 XXZ model on the triangular lattice with strong Ising anisotropy, combining large-scale DMRG calculations of the dynamical spin structure factor with two complementary analytical frameworks (Schwinger-boson and hard-core boson approaches, and a variational quantum dimer model). The key findings are a roton-like minimum at the $M$ point and approximately linear dispersion near $\Gamma$, consistent with neutron-scattering data for a layered material, and the demonstration that both quantum-dimer and Schwinger-boson descriptions capture the observed spectra and ground-state correlations, including transverse photon-like excitations and three-sublattice supersolid order. The work establishes a microscopic origin for supersolid excitations in this frustrated magnet and highlights the proximity of the supersolid to a spin-liquid phase, with emergent gauge-field-like dynamics and strong interplay between lattice geometry, anisotropy, and quantum fluctuations. These insights advance understanding of supersolid behavior in quantum magnets and guide interpretation of current and future experiments in frustrated spin systems.

Abstract

Motivated by recent experiments, we investigate the spin-1/2 XXZ model on the triangular lattice with strong Ising anisotropy, combining large-scale numerical simulations and analytical methods to uncover unconventional spin dynamics. First, we compute the dynamical spin structure factor using density matrix renormalization group (DMRG) simulations and find excellent agreement with inelastic neutron scattering data on the layered compound $\text{K}_2\text{Co}(\text{SeO}_3)_2$. The low-energy spectrum reveals a roton-like minimum at the $M$ point, absent in linear spin-wave theory, accompanied by peak intensity and a broad continuum above it. Near the $Γ$ point, we observe an approximately linear dispersion with vanishing spectral weight. Second, we compare two analytical frameworks that reproduce the observed features. The first is a hard-core boson approach, which includes: (i) an effective staggered boson model (ESBM) at zero magnetic field, (ii) perturbation theory applied to the one-third magnetization plateau, and (iii) a self-consistent mean-field Schwinger boson theory (SBT). The second framework is based on a variational supersolid quantum dimer model (QDM) ansatz, combined with a single-mode approximation. The SBT captures the broad continuum, the $M$-point minimum, and linear dispersion at $Γ$, whereas the QDM reproduces the roton minimum and linear dispersion at finite momentum near $Γ$. Remarkably, both the QDM wavefunction and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photon-like excitations. Together, our comprehensive theoretical and numerical analysis elucidates the microscopic origin of supersolid excitations in the XXZ triangular lattice model and their proximity to a spin liquid phase observed experimentally.

Figures (23)

• Figure 1: Dynamical structure factor for $J_z = 3\ \text{meV}$, $J_\perp = 0.2 \ \text{meV}$ of the spin-1/2 XXZ model zenodo_data. (a) DMRG results shown as a density plot for a $24\times 6$ cylinder geometry with circumference $L = 6$ and bond dimensions up to $\chi = 800$. The lines overlaid on the heatmap show predictions from Schwinger boson theory (SBT), linear spin-wave theory (LSWT), the variational quantum dimer model wavefunction (QDM), the effective staggered boson model (ESBM) and in pink a fit $\omega = 0.0849 \abs{\textbf{k}}$ to the maximum dynamical structure factor $S_{\text{max}}$ computed from DMRG with representative points. . (b) A zoom into the low-energy region with the same theoretical curves overlaid. The left inset shows the Brillouin zone with the relevant momenta and the path used for the plots. The right inset shows the energy bound near the $K$ point computed using the single-mode approximation of the QDM. Panels (c) and (d) display inelastic neutron scattering data for $\text{K}_2\text{Co}(\text{SeO}_3)_2$, taken from Refs. zhu_continuum_2024 and chen_phase_2024, respectively. Note that the experimental plots have been cropped and rescaled to match the energy scales of the DMRG data shown in (b).
• Figure 2: Static horizontal dimer structure factor calculated (a) for the ground state of the XXZ model Eq. \ref{['XXZmodel']} from DMRG simulations of a simulate a $36\times 6$ cylinder geometry with circumference $L = 6$, with bond dimension $\chi=800$, normalized and interpolated in momentum space. (b) Dimer structure factor calculated exactly from the supersolid variational wavefunction of sen_variational_2008 with $z=0.925$ and the mapping to free Kasteleyn fermions. We have added in white the points where the structure factor diverges with system size.
• Figure 3: Static spin structure factor obtained from DMRG for different anisotropy values. We simulate a $18 \times 6$ cylinder geometry with circumference $L = 6$, with bond dimensions up to $\chi = 512$ for (a),(b) (c) and circumference $L = 6$ length $24$ for (d) with bond dimension $\chi=1000$.
• Figure 4: Static structure factor comparison between our DMRG ground state for a $24\times 6$ cylinder geometry with circumference $L = 6$ and bond dimension up to $\chi=800$. A white rectangle was selected to indicate the DMRG resolution for $J_z = 3\ \text{meV}$, $J_\perp = 0.2 \ \text{meV}$. In panel b) we show the results of elastic neutron scattering of Ref. chen_phase_2024.
• Figure 5: Dynamical spin structure factor of the spin-1/2 XXZ model calculated from the DMRG simulations with parameters as in Fig. \ref{['fig:DSF_exp']} with $J_z = 3\ \text{meV}$, $J_\perp = 0.2 \ \text{meV}$ separated into the a) $S^z_iS^z_j$ and b) $S^x_iS^x_j+S^y_iS^y_j$ contributions.