Possible Observation of Quadrupole Waves in Spin Nematics

Abstract

Discovery of new states of matter is a key objective in modern condensed matter physics, which often leads to revolutionary technological advancements such as superconductivity. Quantum spin nematic, a ``hidden order'' that evades conventional magnetic probes, is one such state. Na$_2$BaNi(PO$_4$)$_2$ is a potential spin nematic material, suggested by the observation of a two-magnon Bose-Einstein condensation from above the saturation field. However, direct confirmation of the spin nematicity remains elusive. This Letter presents inelastic neutron scattering spectra from the putative spin nematic phases of Na$_2$BaNi(PO$_4$)$_2$, revealing low-energy quadrupole waves that are absent in the neighboring conventional magnetic phases. A spin-one model quantitatively captures the full details of the spin excitation spectra across all low-temperature phases, providing direct evidence of the spin nematic orders. Additionally, we show evidence of the three-magnon continuum and two-magnon bound states in the $1/3$-magnetization plateau, revealing condensation of the two-magnon bound state as the origin of the low-field spin nematic supersolid phase.

Figures (15)

• Figure 1: Low-energy excitations of conventional magnetic order and SN order. (a) Goldstone mode of dipole order where the DSSF diverges at the ordering wave vector $\bm{k}_0$. (b) Goldstone mode of SN order where the DSSF is zero at $\bm{k}_0$ and scales linearly with the distance $\delta k$ away from $\bm{k}_0$ at low energy. (c) Schematics of the Goldstone mode for conventional magnetic order, where a global rotation with $\theta \rightarrow 0$ of dipole moments costs zero energy. (d) Schematics of the Goldstone mode for SN order, where a global rotation with $\theta \rightarrow 0$ of quadrupole moments costs zero energy.
• Figure 2: Field dependence of the model \ref{['eq:Ham']} at $T=0$. (a) Magnetization from DMRG (reproduced from Ref. ShengJ2025_NiP) and iPEPS calculations with bond dimension $\chi$. The experimentally measured $M/M_\text{s}$ and $\mathrm{d}M/\mathrm{d}B$ at 50mK from Ref. ShengJ2025_NiP are overlaid for comparison. (b) The SN order parameter $|Q|$ and the solid order parameter $\mathcal{S}^{zz}(\text{K})$ calculated by iPEPS.
• Figure 3: Spin excitation spectra of Na$_{2}$BaNi(PO$_4$)$_{2}$ in the SN-supersolid, UUD, and SN phases. (a)--(c) Background-subtracted INS intensities at 60mK compared to the DSSF $\mathcal{S}(\bm{k},E)$ calculated by iPEPS at $T=0$ with bond dimension $\chi=4$. INS data collected at 60mK and 4T were used as the background for subtraction. (d) DSSF $\mathcal{S}(\bm{k},E)$ by LSW calculations of the effective model \ref{['eq:Heff']} in the SN-supersolid phase ($B=\qty{0.01}{T}$) and the SN phase ($B=\qty{1.87}{T}$), where a Gaussian broadening factor $\sigma=\qty{0.001}{meV}$ is used. (e) Raw INS intensities at the $K$ point measured at $T=\qty{60}{mK}$. The dashed rectangle highlights the low-energy quadrupole waves exclusive to the SN phases. (f) Raw INS intensities measured at $T=\qty{60}{mK}$ with $B=0$ and 1.72T. The solid lines in (e) and (f) are Gaussian fits to the diffusive background.
• Figure 4: Single- and multimagnon excitations. (a) $T=0$ GLSW results of the one-magnon excitations (solid lines) and two-magnon bound states (dashed lines) at the $\Gamma$ point, and the two-boson continuum contribution to the transverse fluctuations $\mathcal{S}^{xx+yy}(\Gamma,E)$ (heat map). (b) $T=0$ iPEPS results of the one-magnon (squares) and two-magnon bound states (circles) at the $\Gamma$ point calculated with bond dimension $\chi=4$, where the lines are guides to the eye. (c) $T=0$ DMRG results of the lowest-energy two-magnon bound states calculated on a $9\times 6$ TL torus with bond dimension 3500. The two-magnon BEC critical point is indicated by the red dot at $B_\text{c1}$ in both (b) and (c). (d) $T=0$ iPEPS results of the DQSF $\mathcal{S}^{++--}(\bm{k},E)$ calculated at $B=\qty{0.8}{T}$ with bond dimension $\chi=4$.
• Figure S1: (a) Picture of a Na$_{2}$BaNi(PO$_4$)$_{2}$ single crystal. Each square in the picture represents $\qty{1}{mm}\times\qty{1}{mm}$. (b) Co-alignment of single crystals on the oxygen-free copper plate. (c)-(d) Assembly pictures of the co-aligned samples for neutron scattering experiments. (e) (H,K,L=0) scattering plane collected at $T=\qty{60}{mK}$ and $B=\qty{0}{T}$. (f) A line cut of the diffraction pattern along the (H,H,0) direction around (0,1,0) peak. The blue line represents a Gaussian fit to the experimental data, yielding a Full Width at Half Maximum (FWHM) of $\qty{0.0365(2)}{\angstrom^{-1}}$, indicating good co-alignment of the sample.