Spinon Fermi surface spin liquid in a triangular lattice antiferromagnet 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, NaYbCh$_2$ (Ch=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 muon spin rotation measurements on 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-crystalline 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 NaYbSe$_2$ is a QSL with a spinon Fermi surface.

Introduction. The quantum spin liquid (QSL) is a correlated quantum state in a solid where the spins of the unpaired electrons are highly entangled over long distances, yet they do not exhibit any long-range magnetic order in the zero temperature limit. Originally proposed by Anderson as the ground state for a system of S = 1/2 spins on a two-dimensional (2D) triangular lattice that interact antiferromagnetically with their nearest neighbors [1], a QSL is a novel quantum state of matter beyond the traditional Landau's symmetry breaking paradigm [2][3][4][5], and might be relevant for our understanding of high-temperature superconductivity [6][7][8] and quantum computation in certain cases [9,10]. Beyond the simple characterization of absence of a magnetic order, one key signature of the excitations in a QSL is the presence of deconfined spinons that are fractionalized quasiparticles carrying spin-1/2, observed by inelastic neutron scattering as a spin excitation continuum fundamentally different from the integer spin-wave excitations in an ordered magnet [11][12][13][14][15][16][17][18].
Although spin excitation continuum has been observed in the geometrically frustrated S = 1/2 single crystal systems with 2D Kagomé [11], 2D triangular [12,13], three-dimensional (3D) distorted Kagomé bilayers [16], and 3D pyrochlore [17,18] lattices, there is no consensus on the microscopic origin of the observed spin-excitation continuum. In the 2D S = 1/2 Kagomé lattice ZnCu 3 (OD) 6 Cl 2 [11] and an effective S = 1/2 triangular-lattice magnet YbMgGaO 4 [12-15, 19, 20], different interpretation of the observed spin-excitation continuum includes a spin glass state of magnetic [21] and nonmagnetic Mg-Ga site disorder due to intrinsic sample issues [22][23][24], respectively, rather than the fractionalized quasiparticles of a QSL [5]. To conclusively identify the presence of deconfined spinon excitations in a QSL, one needs to search for the expected spin-excitation continuum among candidate QSL materials with high quality single crystals and establish their physical properties with clear experimental signatures and structures.
Recently, geometrically frustrated 2D triangular-lattice rare-earth-based materials with effective S = 1/2 local moments have attracted considerable attentions [25,26]. Compared with YbMgGaO 4 [19], the family of Yb dichalcogenide delafossites NaYbCh 2 (Ch=O, S, Se) does not have the issue of Mg-Ga site disorders in the non-magnetic layers and thus provides a genuine example for an interacting spin-1/2 triangular-lattice antiferromagnet [27][28][29]. Moreover, NaYbCh 2 exhibit larger magnetic anisotropy than YbMgGaO 4 [12,[30][31][32][33], suggesting that the in-plane magnetic interactions play the dominant role. The combination of the strong spin-orbit coupling (SOC) and the crystal electric field (CEF) leads to a Kramers doublet ground state for the Yb 3+ ion in NaYbCh 2 that gives rise to the effective S = 1/2 local moment at each ion site. Since the energy gaps between the ground and first excited Kramers doublets CEF levels for NaYbSe 2 [ Fig. 1(b)] [30], NaYbS 2 [28], and NaYbO 2 [29] are well above ∼12 meV, the magnetic properties below 100 K can be safely interpreted from the interaction between the effective S = 1/2 local moments. Although previous experiments on powder samples of NaYbO 2 provided some positive evidence 3 for QSL ground states [34][35][36], there are no detailed neutron scattering experiments on single crystalline samples to establish the presence of the magnetic excitation continuum and further reveal its wave vector, energy, temperature dependence, which are essential for identifying the nature of the possible QSL state.
Here we report magnetic susceptibility, heat capacity, and neutron scattering results on single crystals of NaYbSe 2 . In addition to confirming the absence of magnetic order down to 40 mK and spin freezing down to 90 mK, we show the presence of a spin excitation continuum extending from 0.1 to 2.5 meV. Since our careful structure refinement and pair-distribution function (PDF) analysis reveal only ∼ 5% of Yb on Na site and no evidence for a spin glass state at 40 mK, we conclude that the ground state of NaYbSe 2 has signatures of a QSL, consistent with the expectation of a spinon Fermi surface QSL state [38,43].
Results. High-quality single crystals of NaYbSe 2 were grown by using flux method with Te as the flux (see Methods for further synthesis and experimental details). Figure 1(a) displays schematics of crystal structure and reciprocal space of NaYbSe 2 , where Yb ions form a perfect triangular lattice layer. Inelastic neutron scattering spectra of CEF excitations obtained by subtracting the scattering of a non-magnetic reference NaYSe 2 from NaYbSe 2 are shown in Fig. 1(b) [38]. Consistent with previous work [30], the CEF levels of Yb 3+ have a Kramers doublet ground state and three excited Kramers doublets at E = 15.7, 24.5, and 30.2 meV at T = 7 K, thus ensuring that all measurements below ∼ 100 K can be safely considered as an effective S = 1/2 ground state [30]. Note that the CEF levels with E = 24. 5 and 30.2 meV are instrumental energy resolution limited [38] while the lowest CEF excitation is broader than the energy resolution indicative of the internal structure of this mode. The broadening could be attributed to the Yb 3+ -Yb 3+ exchange effects on CEF excitations that can split the lowest CEF excitation into two levels [36,38].
To characterize the behavior of the local moments of Yb and their exchange interactions, we measured the magnetic susceptibility of single-crystalline NaYbSe 2 . The temperature dependence of magnetization and the in-plane magnetic susceptibility χ ⊥ (T ) is depicted in Fig. 1(c), and a simple fit to the Curie-Weiss law yields Θ CW,⊥ −13 K in the low-temperature region (< 20 K), whose absolute value is larger than |Θ CW,⊥ | 7 K when the Van Vleck contribution is subtracted [44], indicating the predominantly antiferromagnetic spin interactions in NaYbSe 2 . Heat capacity measurements were also performed to characterize the thermodynamics of NaYbSe 2 , and the magnetic contribution C mag (T ) to the specific heat of NaYbSe 2 and its dependence on applied magnetic fields from 0 T to 8 T are presented in Fig. 1(d). The data shows a broad peak that shifts upward in temperature as a function of increasing magnetic field for H c, no sharp anomaly indicative of the onset of long-range order, consistent with the susceptibility result and earlier work [44]. Figure 1(e) also shows the estimated temperature dependence of C mag (T )/T (left axis) and the corresponding magnetic entropy S mag (right axis). It is noted that C mag (T )/T in the low-temperature regime (< 0.5 K) is almost a constant, well compatible with the fact that the spinon Fermi surface alone has a constant density of states and would give a heat capacity depending linearly on temperature. Moreover, the temperature dependence of the magnetic entropy saturates to a value close to S mag ≈ R ln 2 (where R is the ideal gas constant) around 15 K, consistent with an effective spin-1/2 description of the Yb 3+ local moment [44].
Although stoichiometric NaYbSe 2 has no intrinsic structural disorder in the Na + intercalating layer [27][28][29], real crystal could still have structural defects in Na + and Se 2− sites, and these vacant sites could be replaced by Yb 3+ and Te 2− , respectively (see Methods). To accurately determine the stoichiometry of our NaYbSe 2 , we carried out single crystal X-ray structural refinement by recording 1334 Bragg reflections, corresponding to 238 non-equivalent reflections. The Rietveld refinement results of the single-crystal Xray diffraction data collected at T = 250 K are shown in Fig. 1(f) and the fitting outcome reveals full occupancy of the Yb 3+ (3a) and Se 2− (6c) sites in the YbSe 2 layers and ∼ 4.8% ± 1% of the Na (3b) sites occupied by the Yb ions. A small amount of Yb occupying the Na site is not surprising because Yb and Na have the same chemical environment for bonding, where both cations have six Se coordinates and are located at the center of NaSe 6 /YbSe 6 octahedra [32]. To further characterize the structural character of the sample, we have also performed PDF analysis on neutron diffraction data measured on 2.7 grams of NaYbSe 2 powder ground from large amount of single crystals obtained from the same batches as the spin-excitation measurements. As shown in Fig. 1(g), the local PDF peaks are well reproduced by fitting with the refined average structure using the X-ray diffraction data, indicating the absence of substantial local distortions. The average structure includes a Yb substitution at the Na site and possible excess Te at the Se site. The PDF analysis suggests an upper limit of 10% of Yb at the Na site and 0% Te at the Se site. While this value is larger than that obtained by single crystal X-ray refinements, single crystal refinement results are more accurate as more Bragg peaks are measured in the X-ray refinements. These results are consistent with the inductively-coupled plasma measurements of chemical composition of the sample (see Methods for details). Although Yb ions in the Na layers may be magnetic, our frequency-dependent ac susceptibility measurements down to 90 mK can be well described with a Curie-Weiss fit and show no evidence of spin freezing [ Fig. 1(h)].
In the previous inelastic neutron scattering measurements on single crystals of CsYbSe 2 (Θ CW −13 K), spin excitations were found to be centered around the K point in reciprocal space [ Fig. 1(a)], with no intensity modulation along the c-axis, and extending up to 1 meV [45]. To determine what happens in NaYbSe 2 , we must first determine if the system has long/short-range magnetic order. For this purpose,  Fig. 4(a) at 40 mK and 10 K, respectively. The excitation continuum here is analogous to that calculated from the free spinon theory. The excitation bandwidth (∼ 2 meV), together with the Curie-Weiss temperatures, characterize the scale of the magnetic interactions [12,43]. At both 40 mK and 10 K, the spectral intensity is broadly distributed in the energy-momentum plane, and the excitation intensity gradually decreases with increasing energy and finally vanishes above ∼2.2 meV. The broad neutron-scattering spectral intensity at 40 mK persists to the lowest energy that we measured implying a high density of spinon scattering states at low energies. Moreover, the spectral weight 6 around Γ point is suppressed to form a V-shaped upper bound [38]. Combining these two facts, it strongly suggests a spinon Fermi surface QSL since this scenario not only provides a high density of spinon states near the Fermi surface, but also well explains the V-shaped upper bound on the excitation energy near the Γ point [43].

Discussion and Conclusion
Overall, the magnetic and heat capacity measurements, combined with the neutron scattering results on single crystals of NaYbSe 2 demonstrate the absence of long/short-range magnetic order even down to 40 mK, implying a quantum disordered QSL state. In particular, besides the naive disorder and the simple spectral continuum of spin excitation, the almost linear temperature dependence of magnetic heat capacity C mag (T ) at the low temperature regime, the enormous low energy gapless excitations and the V-shaped upper bound around the Γ point in inelastic neutron scattering spectrum all strongly indicate the existence of a spinon Fermi surface. Theoretically, although the pure compact U (1) gauge theory in two spatial dimensions is always confined due to the non-perturbative instanton events [47], it has been shown and understood that in the presence of spinon Fermi surface and gapless excitations, the QSL phase could be stable against gauge fluctuations, and a noncompact U (1) gauge theory remains to be a good low energy description [8,49]. Therefore, our experimental results and conclusion about spinon Fermi surface QSL can be compatible with theory. The scenario of spinon Fermi surface QSL could further be verified by low-temperature thermal-transport measurement, which has an advantage to unveil the nature of low-energy itinerant excitations.
Very recently, the pressure-induced insulator to metal transition followed by an emergence of superconductivity in NaYbSe 2 was observed in experiments [50,51]. This is quite remarkable since the QSL has long been thought to be a parent state of the high-temperature superconductivity [6][7][8]. It was suggested that doping a QSL could naturally result in superconductivity [6][7][8] due to the intimate relationship between high-temperature superconductivity and QSL, but the definitive experimental evidence showing that doping QSLs give rise to superconductivity is still lacking. Instead of doping, Ref. [50,51] obtained the superconductivity by pressure, which opens up a promising way to study the superconductivity in QSL candidates and sheds light on the mechanism of high temperature superconductivity.

Acknowledgments
We thank M. Stone for suggestions of appropriate neutron scattering instrumentation, Feng Ye (ORNL) for the assistance with the single-crystal x-ray diffraction measurements. We also thank Ling Wang and the magnetic field applied along the c-axis at Fudan University and Rice University. The total specific heat is described as a sum of magnetic and lattice contributions: C p = C mag + C phonon . We fit the phonon contribution with C phonon = βT 3 + αT 5 .

Magnetic Susceptibility
The magnetic susceptibility of a rare-earth magnetic system with strong spinorbit coupling can be determined by CEF excitations, particularly the first CEF excitation level. In this case, the Curie-Weiss analysis is applicable only in the limited temperature range much smaller than the first CEF excitation level. In NaYbSe 2 , spin-orbit coupling is quite a large energy scale and generates the local moment J = 7/2 for Yb 3+ ion. The crystal field further splits the eight J = 7/2 states into 4 Kramers doublets, and the ground state doublet contributes to the effective spin-1/2 description that is responsible for the low temperature magnetism. Since the lowest CEF excitation is ∼ 15.7meV (∼ 180 K), the Curie-Weiss fitting of the T < 20 K range is not affected by spin-orbit coupling and CEF levels. for high-temperature range (∼ 160 − 300K) results in a Curie-Weiss temperature Θ CW,⊥ ≈ −51K, and the low temperature range (< 20K) generates a Θ CW,⊥ ≈ −13 K, in which C is Curie constant, and χ 0 ∼ 2 × 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 mag ) to the specific heat (with minor contribution from nuclear Schottky anomaly C SA ) of NaYbSe 2 and its dependence on applied magnetic fields H c [38]. Phonon contribution has been subtracted. (e) Temperature dependent C mag /T (black circle) with C 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 phonon . (f) The Rietveld refinement results of the single-crystal X-ray diffraction data at 250 K yield Na 0.952(10) Yb 0.048(10) YbSe 2 . F 2 cal and F 2 obs are the calculated and observed structure factors, respectively. (g) The PDF analysis of neutron data on NaYbSe 2 up to 30 Å. The weighted residual value is 9.56%. (h) AC susceptibility of NaYbSe 2 single crystal measured with frequencies of 3983 Hz and 9984 Hz.
The red solid curves are Curie-Weiss fits for the data. 10 K (green squares). The yellow shaded area marks the difference between the spectra for T = 40 mK and 10 K.
The black arrow marks the lowest energy (0.06 meV) magnetic excitations.