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arXiv:2609.05181

Dipolar and quadrupolar spin supersolid states in a spin-1 triangular antiferromagnet

Yixuan Huang, Yuan Gao, Wei Li, Seiji Yunoki, Sadamichi Maekawa

Published Sep 4, 2026• 0 citations
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Abstract

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$_3$)$_2$.

DOI
10.48550/arxiv.2609.05181

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The antiferromagnetic Heisenberg model on the triangular lattice is perhaps the best known example of frustrated magnets, but it orders at low temperatures. Recent density matrix renormalization group (DMRG) calculations find that the next nearest neighbor interaction J_{2} enhances the frustration, and it leads to a spin liquid for J_{2}/J_{1}∈(0.08,0.15). In addition, a DMRG study of a dipolar Heisenberg model with longer range interactions gives evidence for a spin liquid at a small dipole tilting angle θ∈[0,10°). In both cases, the putative spin liquid region appears to be small. Here, we show that for the triangular lattice dipolar Heisenberg model, a robust quantum paramagnetic phase exists in a surprisingly wide region, θ∈[0,54°), for dipoles tilted along the lattice diagonal direction. We obtain the phase diagram of the model by functional renormalization group (RG), which treats all magnetic instabilities on equal footing. The quantum paramagnetic phase is characterized by a smooth continuous flow of vertex functions and spin susceptibility down to the lowest RG scale, in contrast to the apparent breakdown of RG flow in phases with stripe or spiral order. Our finding points to a promising direction to search for quantum spin liquids in ultracold dipolar molecules.
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