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arXiv:2506.15516 · Physical review. B./Physical review. B

Unconventional spin dynamics and supersolid excitations in the triangular-lattice XXZ model

R. Flores-Calderón, Roderich Moessner, Frank Pollmann

Published Nov 20, 2025• 3 citations

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 at T = 0 . 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 K 2 Co ( SeO 3 ) 2 . The low-energy spectrum reveals a rotonlike 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 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 wave function and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photonlike 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.

DOI: 10.1103/xdwz-8k94

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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 at T = 0 . 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 K 2 Co ( SeO 3 ) 2 . The low-energy spectrum reveals a rotonlike 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 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 wave function and the DMRG ground state exhibit nearly identical structure factors with pronounced transverse photonlike 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.

Motivated by a recent experimental study on the quantum Ising magnet

{\text{K}}_{2}\text{Co}{({\text{SeO}}_{3})}_{2}

that presented spectroscopic evidence of zero-field supersolidity (Chen et al., arXiv:2402.15869), we simulate the excitation spectrum of the corresponding microscopic

XXZ

model for the compound using the recently developed excitation ansatz for infinite projected entangled-pair states. We map out the ground state phase diagram and compute the dynamical spin structure factors across a range of magnetic field strengths, focusing especially on the two supersolid phases found near zero and saturation fields. Our simulated excitation spectra for the zero-field supersolid ``Y'' phase are in excellent agreement with the experimental data, recovering the low-energy branches and integer quantized excited energy levels ${\ensuremath{\omega}}_{n}=n{J}_{zz}$. Furthermore, we demonstrate the nonlocal multi-spin-flip features for modes at ${\ensuremath{\omega}}_{2}$, indicative of their multimagnon nature. Additionally, we identify characteristics of the high-field supersolid ``$\mathrm{\ensuremath{\Psi}}$'' phase in the simulated spectra, which should be compared with future experimental results.

We investigate the dynamical signatures of the supersolid phase in the triangular-lattice hardcore Bose-Hubbard model via large-scale quantum Monte Carlo simulations and linear spin-wave theory. We reveal a distinct momentum-space separation in the spectral weights. The transverse spectrum shows gapless Goldstone modes at the

Γ

and

K

points, while the longitudinal spectrum features a gapless mode at

K

and a roton-like minimum at

M

. Directly in the imaginary-time domain, the supersolid is unambiguously characterized by the coexistence of transverse spectral lines at

Γ

and

K

. Furthermore, we show that strong quantum fluctuations induce a breakdown of the sharp quasiparticle picture, leading to asymmetric line shapes and broad continua. Our work establishes the definitive dynamical signatures of the hardcore-boson supersolid, providing key insights into analogous spin supersolids in triangular quantum magnets.

We present a systematic numerical study of the spin-1 antiferromagnetic Heisenberg model on the triangular lattice in an out-of-plane magnetic field, using Density Matrix Renormalization Group (DMRG) methods. By mapping out the quantum phase diagram as a function of the single-ion anisotropy

D_z

and magnetic field, we identify distinct dipolar and quadrupolar spin supersolid states, characterized by spontaneous U(1) symmetry breaking with finite spin superfluid stiffness coexisting with longitudinal translational symmetry breaking. At zero field, the dipolar spin supersolid with a 'Y'-type spin configuration persists down to

D_z = 0

, whereas the quadrupolar spin supersolid prevails at large

D_z

. At intermediate fields, the phase diagram is dominated by an up-up-down phase. At high fields below saturation, a quadrupolar spin superfluid emerges in the large-

D_z

regime, whereas a dipolar spin supersolid with a 'V'-type spin configuration dominates at small

D_z

. These phases are characterized through their order parameters and spin superfluid stiffness using calculations on various system sizes. Furthermore, the dynamical spin structure factor is obtained across the phase diagram, where characteristic spectral signatures of different phases are observed, including the gapless Goldstone mode and the roton-like minima. These features are directly accessible to inelastic neutron scattering experiments. Our results provide a theoretical understanding of the interplay between frustrations, anisotropy, and Zeeman interactions in driving distinct spin supersolid phases in the spin-1 system, which are relevant to various triangular-lattice antiferromagnets such as Na

_2

BaNi(PO

_4

)

_2

and K

_2

Ni(SeO

_3

)

_2

.

Recent experiments have sparked interest on the ground-state properties and on the excitation spectrum of the spin-

1/2

triangular XXZ model with strong Ising-like anisotropy. In this work, we introduce and analyze a ``projected'' coherent-state approximation to the model, constructed by projecting spin-product states onto the low-energy configurations of the triangular Ising antiferromagnet. Within this approximation, we discuss the structure of the ground-state ordering and the magnetization curve under a longitudinal field. We then discuss the excitation spectrum by linearizing equations of motion deriving from a time-dependent variational principle. This leads to a natural description of spin-wave-like excitations within the strongly constrained Hilbert space. At zero field, the variational approximation leads to a very flat landscape of quasi-degenerate states, which results in a low-energy pseudo-Goldstone mode in the excitation spectrum. We compare our results to experimental measurements on the cobaltite K

_{2}

Co(SeO

_{3}

)

_{2}

(KCSO). Our approximation leads to a dramatic improvement as compared to linear spin-wave theory. However, there remain deviations in the value of the pseudo-Goldstone gap and in the dispersion near

M

points at low field.

Based on the recent experiments on the triangular lattice antiferromagnet Na 2 BaCo ( PO 4 ) 2 , we propose the easy-axis XXZ spin-1/2 model on the triangular lattice, that exhibits a quantum spin supersolid, to be a precursory Dirac spin liquid. Despite the presence of a three-sublattice magnetic order as a spin supersolid, we suggest that this system is close to a Dirac spin liquid by exploring its spectroscopic response. The physical consequence is examined from the spectroscopic response, and we establish the continuous spectra near the M point in addition to the K point excitation from the spinon continuum on top of the three-sublattice order. Moreover, the satellite peaks were predicted at the mid-points connecting the Γ and K points. This proposal offers a plausible understanding of the recent inelastic neutron scattering measurement in Na 2 BaCo ( PO 4 ) 2 and could inspire further research in relevant models and materials, such as K 2 Co ( SeO 3 ) 2 and Rb 2 Co ( SeO 3 ) 2 , and even more anisotropic magnets like PrMgAl 11 O 19 . Published by the American Physical Society 2024

Field-Induced Spin Excitations in the Spin-1/2 Triangular-Lattice Antiferromagnet CsYbSe

_2

A layered triangular lattice with spin-1/2 ions is an ideal platform to explore highly entangled exotic states like quantum spin liquid (QSL). Here, we report a systematic in-field neutron scattering study on a perfect two-dimensional triangular-lattice antiferromagnet, CsYbSe

_2

, a member of the large QSL candidate family rare-earth chalcogenides. The elastic neutron scattering measured down to 70 mK shows that there is a short-range 120

^{\circ}

magnetic order at zero field. In the field-induced ordered states, the spin-spin correlation lengths along the

c

axis are relatively short, although the heat capacity results indicate long-range magnetic orders at 3 T

-

5 T. The inelastic neutron scattering spectra evolve from highly damped continuum-like excitations at zero field to relatively sharp spin wave modes at the plateau phase. Our extensive large-cluster density-matrix renormalization group calculations with a Heisenberg triangular-lattice nearest-neighbor antiferromagnetic model reproduce the essential features of the experimental spectra, including continuum-like excitations at zero field, series of sharp magnons at the plateau phase as well as two-magnon excitations at high energy. This work presents comprehensive experimental and theoretical overview of the unconventional field-induced spin dynamics in triangular-lattice Heisenberg antiferromagnet and thus provides valuable insight into quantum many-body phenomena.