Magnetic Moment Fragmentation in an All-in-All-out Pyrochlore $\mathrm{Nd_2Sn_2O_7}$

TL;DR

This work demonstrates magnetic moment fragmentation in Nd$_2$Sn$_2$O$_7$, revealing a nearly flat, pinch-point–like band coexisting with dispersive modes within an all-in–all-out ordered state. Single-crystal neutron scattering data are quantitatively captured by a minimal dipolar–octupolar DO XYZ spin Hamiltonian, with a four-sublattice LSWT+ extit{Bogoliubov} treatment yielding two flat and two dispersive magnon branches in excellent agreement with experiment. The static AIAO component and dynamic, fragmentation-driven fluctuations arise from a rotated DO doublet and are evidenced by the observed structure factors and the absence of photon-like excitations within the measured window, thereby constraining possible Coulombic antiferromagnet scenarios. Overall, Nd$_2$Sn$_2$O$_7$ provides a clean quantitative platform to study emergent gauge fields and multipolar spin dynamics in frustrated magnets, extending fragmentation physics to a Nd-based pyrochlore and clarifying its energetic landscape and excitations.

Abstract

We report single crystal neutron spectroscopy and bulk characterization on hydrothermally grown $\mathrm{Nd_2Sn_2O_7}$, revealing magnetic moment fragmentation embedded within the all-in-all-out ordered state. The spectra show a nearly flat band with pinch-point-like momentum dependence, accompanied by dispersive branches that generate half-moon features across multiple Brillouin zones. These defining signatures are captured quantitatively by a minimal dipolar-octupolar spin Hamiltonian, demonstrating excellent agreement between experiment and theory. The higher flat-mode energy helps account for the absence of dynamical interference in prior $\mathrm{μSR}$ studies, while the lack of any photon-like excitation imposes strict constraints on the proposed Coulombic antiferromagnet scenario. Our results extend moment fragmentation to $\mathrm{Nd_2Sn_2O_7}$ and identify it as a clean, tractable platform for quantitative exploration of emergent gauge field physics and multipolar spin-wave dynamics in frustrated magnets.

Figures (10)

• Figure 1: (a) Magnetization $M(H)$ at $T = 2$ K for $\text{Nd}_2\text{Sn}_2\text{O}_7$ measured along $[100]$, $[111]$, and $[110]$. Inset (i): schematic of the all-in–all-out (AIAO) order below $T_{\mathrm N}$. (b) Heat capacity $C_p(T)$ after subtracting the La$_2$Sn$_2$O$_7$ lattice contribution; the vertical dashed line marks $T_{\mathrm N} = 0.89(3)$ K; the red curve marks the calculated spin-wave contribution $C_{\mathrm{SW}}(T)$. (c) Magnetic entropy $S_{\mathrm{mag}}(T) = \int_0^T (C_p/T')\,\mathrm{d}T'$ recovering $R\ln2$ by $\sim$5 K. (d) DC susceptibility $\chi(T)$ of pulverized single crystals showing a peak at $T_{\mathrm N}$. Inset (ii): Curie–Weiss fit to $\chi^{-1}(T)$ over 2–10 K, yielding $\theta_{\mathrm{CW}} = -0.08(1)$ K and $\mu_{\mathrm{eff}} = 2.397(2)\,\mu_{\mathrm B}$/Nd$^{3+}$.
• Figure 2: (a) False color map of inelastic scattering intensity $S(\mathbf{Q},\omega)$ versus energy transfer $\hbar\omega$ and momentum transfer $Q$ along a path connecting the labeled high symmetry points. Data were taken at $T=0.25$ K with incident energy $E_i=2.19$ meV on the LET spectrometer (elastic resolution FWHM $\approx0.07$ meV), integrated over a perpendicular momentum window of $\pm 0.1$ Å$^{-1}$, and not symmetrized. (b) Simulated intensity from the spin-wave model of Ref. benton2016quantum with parameters $(\tilde{J}_x,\tilde{J}_y,\tilde{J}_z)=(0.10(2),\,0.00(1),\,-0.075(2))~\mathrm{meV}$, convolved with the LET energy resolution and scaled by a single global factor to match the data.
• Figure 3: (a–d) Constant-$Q$ energy cuts at the indicated points, integrated over $\pm0.1$ rlu along each principal direction and measured with $E_i=2.19$ meV. Blue/black: $T=0.25$ K/$10$ K. Green/red: simulations for Solution 1/Solution 2, averaged over the same $Q$-windows as the data, convolved with the LET energy resolution, and scaled by a single global factor (per solution) to match the total intensity across panels. Experimental spectra are averaged over symmetry-equivalent points in the $(HHL)$ plane using space group $Fd\bar{3}m$. Gray dashed lines mark the calculated spin-wave energies.
• Figure 4: (a,c,e,g,i,k,m) Constant energy $Q$ maps collected at $T=0.25$ K with $E_i=2.19$ meV in the $(HHL)$ plane. Data are integrated along $[K\bar{K}0]$ with $K\in[-0.07,0.07]$ rlu, symmetrized in the $(HHL)$ plane using space group $Fd\bar{3}m$. (b,d,f,h,j,l,n) Simulated $Q$ maps from the spin wave model of Ref. benton2016quantum with parameters $(\tilde{J}_x,\tilde{J}_y,\tilde{J}_z)=(0.10(2)),0.00(1),-0.075(2)))$ meV, convolved with the LET resolution and scaled by a single global factor to match the data. (o,p) $Q$-dependence of the simulated intensity for the flat band (o) and the dispersive branches (p).
• Figure S1: (a) Photograph of a portion of the co-aligned single-crystal $\text{Nd}_2\text{Sn}_2\text{O}_7$ array (total mass $\sim$0.93 g), with the vertical axis approximately aligned along the $[1\bar{1}0]$ crystallographic direction. (b) Optical microscopy image of a representative single crystal. (c) Optical microscopy image of crystals affixed to the copper plate within the co-aligned array shown in (a). (d) False-color intensity map of the $(HHL)$ plane of the co-aligned $\text{Nd}_2\text{Sn}_2\text{O}_7$ crystal array, collected at $T=0.25$ K on the cold neutron multi-chopper spectrometer LET with incident energy $E_i = 22.7$ meV. The map displays the elastic channel integrated over energy transfer $[-2.1,2.1]$ meV and within $\pm0.1$ rlu along the $[K\bar{K}0]$ direction. The data reveal a single, slightly broadened crystal grain with an estimated mosaicity of $\sim$5$^\circ$ (FWHM), further illustrated in (e--h). (e--h) Transverse cuts through the Bragg peaks $\mathbf{Q}=(\bar{4}\bar{4}0)$ and $\mathbf{Q}=(008)$ along the in-plane directions $[001]$ and $[110]$, respectively, as well as the out-of-plane direction $[K\bar{K}0]$, measured at $T=0.3$ K. Horizontal axes are expressed in angular units $\psi_{\mathrm{in}}$ (in-plane) and $\phi_{\mathrm{out}}$ (out-of-plane), normalized to $|\mathbf{Q}|$. Data are integrated within $\pm0.09$ Å$^{-1}$ in the perpendicular $\mathbf{Q}$ directions.