Spinon Fermi Surface Spin Liquid in a Triangular Lattice Antiferromagnet ${\mathrm{NaYbSe}}_2$

Triangular lattice of rare-earth ions with interacting effective spin-$1/2$ local moments is an ideal platform to explore the physics of quantum spin liquids (QSLs) in the presence of strong spin-orbit coupling, crystal electric fields, and geometrical frustration. The Yb delafossites, $\mathrm{NaYb}C{h}_2$ ($Ch=\mathrm{O}$, S, Se) with Yb ions forming a perfect triangular lattice, have been suggested to be candidates for QSLs. Previous thermodynamics, nuclear magnetic resonance, and powder-sample neutron scattering measurements on ${\mathrm{NaYbCh}}_2$ have supported the suggestion of the QSL ground states. The key signature of a QSL, the spin excitation continuum, arising from the spin quantum number fractionalization, has not been observed. Here we perform both elastic and inelastic neutron scattering measurements as well as detailed thermodynamic measurements on high-quality single-crystal ${\mathrm{NaYbSe}}_2$ samples to confirm the absence of long-range magnetic order down to 40 mK, and further reveal a clear signature of magnetic excitation continuum extending from 0.1 to 2.5 meV. The comparison between the structure of the magnetic excitation spectra and the theoretical expectation from the spinon continuum suggests that the ground state of ${\mathrm{NaYbSe}}_2$ is a QSL with a spinon Fermi surface.

Popular Summary

In a quantum spin liquid (QSL), electrons do not assume any regular magnetic order even at zero temperature. This sought-after state may hold secrets to high-temperature superconductivity and have properties conducive to quantum computing. However, observation of a QSL is exceedingly difficult. Here, we present strong evidence of a QSL in a 2D triangular lattice material.

The only way to positively identify a QSL is by firing neutrons at the material. In an inelastic neutron scattering experiment, the neutrons create pairs of spinlike quasiparticles called spinons that, for a given neutron momentum, have a continuous range of energies (as opposed to ordinary magnets, where the spin excitations end up with discrete energies). While there have been a few reports of such a spinon continuum, these may have alternate interpretations and could be due to disorder in the materials.

In our experiments, we scatter neutrons off of high-quality samples of ${\mathrm{NaYbSe}}_2$, a material long suggested as a QSL candidate and one that has much less disorder compared with other 2D systems. We not only confirm the absence of magnetic order down to 40 mK but also see clear evidence of a spinon continuum—the hallmark of a QSL.

Although hints of a QSL state have been reported in the same family as ${\mathrm{NaYbSe}}_2$, those experiments were on powders of the material, not crystals, which left interpretation of those results somewhat ambiguous. Therefore, we believe that our measurements provide strong evidence for a QSL signature in a 2D triangular lattice—almost 50 years after their first prediction.

Figure 1

Crystal structure and reciprocal space, CEF levels, heat capacity, and stoichiometry of ${\mathrm{NaYbSe}}_2$. (a) The structure of ${\mathrm{NaYbSe}}_2$ and corresponding reciprocal spac. The lattice parameters are $a=b\approx 4.07\AA$, $c\approx 20.77\AA$. (b) Inelastic neutron scattering spectra of CEF excitations obtained by subtracting the scattering of a nonmagnetic reference ${\mathrm{NaYSe}}_2$ from the intensity of ${\mathrm{NaYbSe}}_2$ (see the Appendix), in which the energy axis of ${\mathrm{NaYSe}}_2$ was shifted by $-1.7\mathrm{meV}$ for calibrating the difference in phonon energies [38]. Three CEF energy levels are marked by white dashed lines. (c) Temperature-dependent magnetic susceptibility along $H\perp c$ direction. The fitting $(1/\chi -{\chi }_0)=(C/T-{\mathrm{\Theta }}_{\mathrm{CW},\perp }{)}^{-1}$ for high-temperature range ($\sim 160–300\mathrm{K}$) results in a Curie-Weiss temperature ${\mathrm{\Theta }}_{\mathrm{CW},\perp }\approx -51\mathrm{K}$, and the low-temperature range ($<20\mathrm{K}$) generates a ${\mathrm{\Theta }}_{\mathrm{CW},\perp }\approx -13\mathrm{K}$, in which $C$ is Curie constant, and ${\chi }_0\sim 2\times {10}^{-4}$ emu/mol is a temperature-independent background term. The inset shows the crystal for the magnetization measurements. (d) Temperature-dependent magnetic contribution ($C_{\mathrm{mag}}$) to the specific heat (with minor contribution from nuclear Schottky anomaly $C_{\mathrm{SA}}$) of ${\mathrm{NaYbSe}}_2$ and its dependence on applied magnetic fields $H\parallel c$ [38]. Phonon contribution has been subtracted. (e) Temperature-dependent $C_{\mathrm{mag}}/T$ (black circle) with $C_{\mathrm{SA}}/T$ subtracted [38] and the magnetic entropy (black curve). The red dashed line marks the value of $R\ln 2$. The inset shows $C_p/T$ as a function of $T^2$. The red solid curve is a fitting of the phonon contribution $C_{\text{phonon}}$. (f) The Rietveld refinement results of the single-crystal x-ray diffraction (SC XRD) data at 250 K yield ${\mathrm{Na}}_{0.952(10)}{\mathrm{Yb}}_{0.048(10)}{\mathrm{YbSe}}_2$. ${\mathrm{F}}_{\mathrm{cal}}^2$ and ${\mathrm{F}}_{\mathrm{obs}}^2$ are the calculated and observed structure factors, respectively. (g) The PDF analysis of neutron data on ${\mathrm{NaYbSe}}_2$ up to 30 Å. The weighted residual value is 9.56%. (h) ac susceptibility of ${\mathrm{NaYbSe}}_2$ single crystal measured with frequencies of 3983 and 9984 Hz. The red solid curves are Curie-Weiss fits for the data.

Figure 2

Neutron scattering results in $[H,H,L]$ and $[H,K,0]$ zones. Elastic neutron scattering results ($E=0\pm 0.1\mathrm{meV}$) in (a) the $[H,H,L]$ plane and (b) $[H,K,0]$ plane measured with $E_i=3.32$ and 3.70 meV, respectively. Scattering along the vertical direction [$[-K,K,0]$ for (a) and $[0,0,L]$ for (b)] is integrated. The upper half panels of (a) and (b) are data at $T=40\mathrm{mK}$, and the lower are the differences between $T=40\mathrm{mK}$ and 10 K. (c) $L$ dependence of the spin excitations along the $[H,H]$ direction at $T=40\mathrm{mK}$ (left half panel) and $T=10\mathrm{K}$ (right half panel), with $K=[-0.05,0.05]$ and $E=0.3\pm 0.1\mathrm{meV}$. (d) Spin excitations with $E=0.3\pm 0.1$ in the $[H,K]$ plane measured at $T=40\mathrm{mK}$, 2, and 10 K. Scattering along the $[0,0,L]$ direction is integrated. The black dashed lines mark the Brillouin zones of ${\mathrm{NaYbSe}}_2$.

Figure 3

Constant-energy images of spin excitations in the $[H,K,0]$ plane. (a)–(d) Images at $T=40\mathrm{mK}$ and (e)–(h) 10 K. The intensity along the vertical $[0,0,L]$ direction is integrated. Spin excitations for (a),(e) $E=0.15\pm 0.05$, (b),(f) $0.6\pm 0.1$, (c),(g) $1.1\pm 0.1$, and (d),(h) $2.1\pm 0.1\mathrm{meV}$ are measured with $E_i=1.77$, 3.70, 12.14, and 12.14 meV, respectively. The black dashed lines mark the Brillouin zones in the reciprocal space. The data are collected in 180° range of sample rotation around the $c$ axis. The 360° circular coverage is generated by averaging the raw data and its mirror in the $[H,K,0]$ plane. The $C_2$-like anisotropy has been attributed to a trivial effect caused by sample-volume change in beam during sample rotation for neutron scattering measurements in $[H,K,0]$ plane [38].

Figure 4

Spin-excitation spectra along high symmetry momentum directions. (a),(b) Schematics of the Brillouin zones with high symmetry points $\mathrm{\Gamma }$, $K$, and $M$ denoted by green, red, and blue dots, respectively, and the high symmetry directions for the images in (c)–(f) marked by pink lines with arrow heads. Spin-excitation spectra collected at (c) $T=40\mathrm{mK}$ and (d) 10 K along the $M_2\text{−}K\text{−}\mathrm{\Gamma }\text{−}K\text{−}{M}_2$ with $E_i=3.32\mathrm{meV}$. (e),(f) Intensity color maps along the ${\mathrm{\Gamma }}_1\text{−}{M}_1\text{−}{\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_2\text{−}{M}_2\text{−}{\mathrm{\Gamma }}_3$ directions measured with $E_i=3.7\mathrm{meV}$.

Figure 5

Wave vector dependence of spin excitations along high symmetry directions. The wave vector cuts in (a),(b),(d)–(g) were measured in the $[H,H,L]$ zone with $E_i=3.32\mathrm{meV}$, while those in (c),(h),(i) were measured in the $[H,K,0]$ plane with $E_i=3.70\mathrm{meV}$. Panels (a) and (b) show the energy-dependent scattering at $K_1$ and $M_1$ points measured at $T=40\mathrm{mK}$ (black circle), 2 K (red diamond), and 10 K (green square). The inset in (a) is a schematic of the reciprocal space with the $\mathrm{\Gamma }$, $K$, and $M$ points denoted by green, red, and blue dots, respectively. The black, red, and green curves are energy cuts at ${\mathrm{\Gamma }}_1$. The inset of (b) shows the difference of ${\chi }^{\prime \prime }$ between the spectra for $T=40\mathrm{mK}$ and 10 K at the $K$ and $M$ points. The light blue and red curves are fittings of the ${\chi }^{\prime \prime }$ with a damped harmonic oscillator model. Panel (c) shows the energy cuts at the $K_1$, $M_1$, $M_2$, and ${\mathrm{\Gamma }}_2$. Solid symbols represent the data collected at $T=40\mathrm{mK}$ and the open symbols collected at 10 K. The black and blue curves are energy cuts at the ${\mathrm{\Gamma }}_2$ point measured at $T=40\mathrm{mK}$ and 10 K, respectively. (d)–(g) Constant energy cuts along the $M_2\text{−}{K}_1\text{−}{\mathrm{\Gamma }}_1$ for $T=40\mathrm{mK}$, 2 K, and 10 K, with corresponding energy transfers marked in the panels. Constant energy cuts along the ${\mathrm{\Gamma }}_3\text{−}{M}_2\text{−}{\mathrm{\Gamma }}_2$ measured at (h) $T=40\mathrm{mK}$ and (i) 10 K. The solid curves are guides to the eyes, and the error bars represent one standard deviation.

Figure 6

Spin-excitation energy spectra at $K_1$ position measured with $E_i=1.55\mathrm{meV}$ at $T=40\mathrm{mK}$ (black circles) and 10 K (green squares). The yellow shaded area marks the difference between the spectra for $T=40\mathrm{mK}$ and 10 K. The black arrow marks the lowest energy (0.06 meV) magnetic excitations.