Evidence for a spinon Fermi surface in the triangular $S=1$ quantum spin liquid ${\mathrm{Ba}}_3{\mathrm{NiSb}}_2{\mathrm{O}}_9$

Inelastic neutron scattering is used to study the low-energy magnetic excitations in the spin-1 triangular lattice of the $6H$-B phase of ${\mathrm{Ba}}_3{\mathrm{NiSb}}_2{\mathrm{O}}_9$. We study two powder samples: ${\mathrm{Ba}}_3{\mathrm{NiSb}}_2{\mathrm{O}}_9$ synthesized under high pressure and ${\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}{\mathrm{NiSb}}_2{\mathrm{O}}_9$ in which chemical pressure stabilizes the $6H$-B structure. The measured excitation spectra show broad gapless and nondispersive continua at characteristic wave vectors. Our data rules out most theoretical scenarios that have previously been proposed for this phase, and we find that it is well described by an exotic quantum spin liquid with three flavors of unpaired fermionic spinons, forming a large spinon Fermi surface.

Figure 1

Dynamic susceptibility ${\chi }^{\prime \prime }(Q,E)$ on a linear intensity scale as a function of wave vector $Q$ and energy $E$ at $T\approx 1.6$ K. (a) ${\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}{\mathrm{NiSb}}_2{\mathrm{O}}_9$ with an incoming neutron energy of $E_i=3.55$ meV and (b) with $E_i=8.0$ meV. (c) ${\mathrm{Ba}}_3{\mathrm{NiSb}}_2{\mathrm{O}}_9$ with $E_i=3.27$ meV (the weak feature at 1 meV is an experimental artifact). (d) Calculated powder-averaged ${\chi }^{\prime \prime }(Q,E)$ of the U(1) Fermi surface state (A) at 1/3 spinon filling (see text).

Figure 2

(a)–(c) Energy dependence of the INS data from ${\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}{\mathrm{NiSb}}_2{\mathrm{O}}_9$. Panel (a) shows the temperature dependence of the dynamic structure factor $S(Q,E)$ at wave vectors $Q=1.9\pm 0.1$ ${\AA }^{-1}$ (magnetic signal, symbols) and $Q=3.7\pm 0.1$ ${\AA }^{-1}$ (mostly nonmagnetic signal, lines) measured with an incoming neutron energy of $E_i=14.2$ meV. The excitations extend out to an energy of about 7.5 meV. Panel (b) shows $S(Q,E)$ at $Q=0.8\pm 0.1$ ${\AA }^{-1}$ (magnetic signal, solid blue circles) and $Q=0.4\pm 0.1$ ${\AA }^{-1}$ (mostly nonmagnetic signal, open black circles) for $T=0.1$ K measured with $E_i=1.13$ meV. The excitations are gapless within the experimental energy resolution (the dashed lines indicate the extension of the elastic peak). Panel (c) shows the imaginary part of the dynamic susceptibility ${\chi }^{\prime \prime }(Q,E)$ at $Q=0.8\pm 0.1 {\AA }^{-1}$ for different temperatures obtained by combining data obtained with $E_i=3.55$ and 14.2 meV. The lines are guides to the eye. (d) Theoretical susceptibility at $Q\simeq 0.8 {\AA }^{-1}$ for the U(1) Fermi surface state at $T=0$.

Figure 3

(a) $Q$ dependence of $S(Q,E)$ of ${\mathrm{Ba}}_{2.5}{\mathrm{Sr}}_{0.5}{\mathrm{NiSb}}_2{\mathrm{O}}_9$ at $T=1.5$ K integrated over the energy range 0.3–1.5 meV at $E_i=3.55$ meV (solid black circles) and 2–4 meV at $E_i=14.2$ meV (open blue squares) showing peaks at $Q=0.83$ and 1.92 ${\AA }^{-1}$. The black line shows the $T=1.5$ K data taken at $E_i=3.55$ meV and corrected for the magnetic form factor. The open red circles show data taken with $E_i=3.55$ meV at $T=150$ K, where most of the correlations have disappeared. (b) Calculated static structure factor $S\left(Q\right)$ for the U(1) Fermi surface state (A) at $T=0$.

Figure 4

Dynamical susceptibility ${\chi }^{\prime \prime }(q,E)$, energy integrated from 0 to 5% of the spinon band width, in the U(1) Fermi surface state (A) at 1/3 spinon filling. The black hexagons are the BZ boundaries, and the radii of the red circles centered at $\mathrm{\Gamma }$ correspond to the three broad intensity maxima seen in the INS data.