Van Hove singularity in the magnon spectrum of the antiferromagnetic quantum honeycomb lattice

In quantum magnets, magnetic moments fluctuate heavily and are strongly entangled with each other, a fundamental distinction from classical magnetism. Here, with inelastic neutron scattering measurements, we probe the spin correlations of the honeycomb lattice quantum magnet YbCl3. A linear spin wave theory with a single Heisenberg interaction on the honeycomb lattice, including both transverse and longitudinal channels of the neutron response, reproduces all of the key features in the spectrum. In particular, we identify a Van Hove singularity, a clearly observable sharp feature within a continuum response. The demonstration of such a Van Hove singularity in a two-magnon continuum is important as a confirmation of broadly held notions of continua in quantum magnetism and additionally because analogous features in two-spinon continua could be used to distinguish quantum spin liquids from merely disordered systems. These results establish YbCl3 as a benchmark material for quantum magnetism on the honeycomb lattice.

T he honeycomb lattice decorated with interacting spins is a particularly fascinating structural motif for the generation of collective quantum behavior. This bipartite lattice geometry has the minimum coordination number of three for a lattice in two dimensions. When the interactions between the spins are strongly anisotropic, as is the case for a growing number of Kitaev materials  , the result is strongly frustrating interactions and, hence, the honeycomb lattice is presently thought of as one of the primary contenders to host quantum spin liquids. In the opposite limit of isotropic spin interactions, frustrated quantum magnetism can arise through the competition of nearest neighbor and nextnearest neighbor interactions [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] . Indeed, most honeycomb lattice materials studied thus far require the addition of further neighbor interactions to explain the underlying physical behavior 6,8,[38][39][40][41][42][43][44][45][46][47] . Such materials, with a complicated phase diagram as a function of first, second, and third nearest-neighbor interactions, have been fertile ground for exploration.
On the other hand, finding an example of honeycomb lattice magnetism where nearest-neighbor Heisenberg interactions are dominant would provide an important benchmark for studying quantum magnetism in two dimensions. In this instance, due to the bipartite geometry of the honeycomb lattice, the Heisenberg exchange interactions are not frustrated and a Néel ground state is expected 25,26,48 at zero temperature. However, long-range order at finite temperature is prohibited by the Mermin-Wagner theorem as long as there are no anisotropic or interlayer interactions. Despite the lack of frustration, in this case, the low connectivity of the honeycomb lattice indicates that collective quantum effects are likely to be observable. Experimental realizations of the ideal honeycomb lattice Heisenberg model (HLHM) are thus attractive as a means of testing fundamental concepts of collective quantum behavior.
For example, quantum effects connected to two-magnon continua, such as magnon decays and renormalizations, have been predicted for a range of two-dimensional lattices 49 , including the square 50,51 , triangular 52,53 , and honeycomb 54 lattices. However, while the Van Hove singularities of the two-magnon continua have a crucial role in these quantum effects, the Van Hove singularities themselves have not been experimentally observed despite extensive studies of square lattice [55][56][57] and triangular lattice 58-60 quantum magnets. More generally, it is important to distinguish quantum effects that arise entirely due to the honeycomb geometry from those that also require other sources, such as frustration or anisotropic interactions. A further motivation is the identification of a model system where the energy scale of the spin-spin interactions is modest enough to allow the quantum properties to be studied as a function of a relevant tuning parameter, for example, an applied magnetic field 54 .
Here we focus on the nearly ideal honeycomb lattice material YbCl 3 as a potential model quantum magnet in the unfustrated limit. The arrangement of the Yb 3+ ions is illustrated in Fig. 1. While YbCl 3 is formally monoclinic (space group C12/m1), there is only a very modest distortion (<0.5% difference between Yb-Yb nearest-neighbor distances) from the ideal honeycomb lattice geometry in the ab planes 61 . YbCl 3 has been proposed as a candidate for Kitaev physics 62,63 , but other studies suggest that YbCl 3 is likely to exist in the Heisenberg limit 64 . Thus, a key question concerning the physical behavior of YbCl 3 is the nature of the spin interactions and the manifestation of collective quantum effects. Experimental studies thus far have found a broad signature in the specific heat peaked at 1.8 K that com-prises~99.8% of the entropy of Rln ð2Þ expected for the ground state doublet 62 . At T = 0.6 K, a weak anomaly in the specific heat is observed. The local crystallographic environment results in easy plane anisotropy of the Yb 3+ magnetic moments 61 . Finally, the polycrystalline averaged magnetic excitation spectrum of YbCl 3 61 is rather different from that of the prototype Kitaev material RuCl 3 17 , suggesting that a different set of interactions govern the physical behavior of YbCl 3 .
In this paper, we study YbCl 3 with high-resolution inelastic neutron scattering (INS) measurements of single crystals. In addition to a conventional spin-wave (single-magnon) mode, these measurements show a sharp Van Hove singularity (VHS) within a broad two-magnon continuum that originates from longitudinal (quantum) spin fluctuations. Linear spin-wave theory with a single Heisenberg interaction on the honeycomb lattice reproduces all features of the data, demonstrating the strong quantum and almost perfectly two-dimensional character of YbCl 3 . Additional support for these conclusions is presented through polarized neutron diffraction and specific heat capacity measurements in conjunction with microcanonical thermal pure quantum state (mTPQ) calculations. Together, these results demonstrate that YbCl 3 is an ideal model system of a twodimensional quantum magnet without frustrated or anisotropic interactions. Being considerably hard to find, such model systems are crucial in quantum magnetism as they enable the controlled experimental investigation of collective quantum behavior. Moreover, on a conceptual level, the observation of a sharp feature within a continuum response demonstrates the coherent origin of the continuum. Therefore, our results for the twomagnon continuum are proof of the principle that similar features within two-spinon continua could be utilized for more unambiguous detection of fractionalized quantum magnets, such as quantum spin liquids.

Results
Inelastic neutron scattering data. We first examine the lowenergy magnetic excitation spectra of YbCl 3 at 0.24 K.  The monoclinic crystal structure (space group C12/m1) and ordered spin configuration (magnetic space group C2 0 =m) of YbCl 3 . The monoclinic structure with a = 6.729 Å, b = 11.614 Å, c = 6.313 Å, and β = 110.6°(at T = 10 K) contains nearly ideal honeycomb lattices of Yb 3+ ions (red spheres) 61 . Within each honeycomb lattice, the Yb 3+ sites have nearest-neighbor distances of 3.884 Å and 3.867 Å for the exchanges J and J 0 , respectively, and next-nearest-neighbor distances of 6.729 Å and 6.711 Å for the exchanges J 2 and J 0 2 , respectively. The resulting three bond angles for the honeycomb lattice are 120°, 119.97°, and 119.97°. The distance between the honeycomb planes is 6.313 Å, corresponding to an interlayer exchange J c . For the ideal model (see Eq. (1)), we consider J ¼ J 0 and The lack of an observable dispersion along the (00L) direction indicates that the interactions between the honeycomb planes are very weak. Interestingly, the spin-wave mode appears to be broader than the instrumental resolution. While the reason for this broadening is uncertain, it may arise due to domain formation as a result of bond disorder 65 , interaction with the multimagnon continuum (potentially following a renormalization of the spin-wave spectrum 52 ), or even the inability to fully distinguish the overlapping continuum contribution. We also point out that the T = 3.7 K data in Fig. 3f illustrate a complete lack of well- formed magnetic excitations at higher temperatures. Finally, an important feature of the data is the lack of an appreciable spin gap (see Supplementary material for additional details). This observation suggests that the spins do not possess a significant uniaxial anisotropy, in agreement with the crystal field ground state with easy plane anisotropy determined in ref. 61 .
The most unusual part of the spin excitation spectrum is the sharp feature toward the top of the broad continuum. While there is precedence for the observation of spinon and multimagnon continua in one-dimensional [66][67][68][69][70] and two-dimensional square and triangular lattice 55-60 quantum magnets, the observation of a sharp feature within such a continuum has, to the best of our knowledge, not yet been reported. This sharp multimagnon feature is explored further through constant wave vector scans in Fig. 2g. At energies where the sharp feature is well separated from the most intense parts of the continuum, the width is essentially limited to the calculated energy resolution of the instrument, FWHM = 0.04 meV at ℏω = 1.19 meV. As will be described in more detail below, this result provides an experimental demonstration of how a sharp feature can be generated within a continuum response in a quantum magnet. Note that the hexagonal symmetry of the spin excitations is manifest in both these higher energy features and the transverse spin-wave modes at lower energies, as shown in the right sides of Fig. 2h and i, respectively.
Linear spin-wave theory. To understand the physics begetting the novel spin excitation spectrum of YbCl 3 , we consider a Heisenberg model on the honeycomb lattice with a single antiferromagnetic exchange interaction J between nearest-neighbor S = 1/2 spins On the bipartite honeycomb lattice, the ground state 0 j i of this Heisenberg Hamiltonian H is the antiferromagnetic Néel state 25,26,48 . Assuming without loss of generality that the spins are parallel to the z direction, the transverse and the longitudinal components of the dynamical spin structure factor are S ± ðq; ωÞ ¼ 1 4πN corresponding to the spin-wave dispersion where e iϑ q ¼ λ q =jλ q j, and r 1,2,3 are the three bond vectors connecting nearest-neighbor sites on the honeycomb lattice. For the longitudinal component, the spins must be expanded up to quadratic order to get a nontrivial inelastic contribution This two-magnon contribution gives a broad continuum over a finite energy range for each momentum q because the energy transfer, ω = ε k + ε q−k , depends on the momenta k and q − k of the individual magnons. We note that, in linear spin-wave theory, the staggered magnetic moment of the Néel state is only ≈ 48% of its classical value on the honeycomb lattice, in comparison to ≈ 61% on the square lattice 71 . Such a large reduction of the magnetic moment indicates that quantum fluctuations are strong due to the low coordination number of the honeycomb lattice 72 .

Discussion
Comparison between data and model. The HLHM in Eq. (1) reproduces the experimental data for the transverse spin-wave mode, the broad continuum, and the sharp feature toward the top of the continuum. We first note that, due to the summation over the momentum k, the contribution from the two-magnon states in Eq. (5)  We remark that the excellent agreement between the experimental data and the ideal HLHM is also in accordance with the prediction of ref. 64 . The sharp feature toward the top of the continuum is a particularly interesting aspect of the spectrum that, to the best of our knowledge, has not been previously observed in a quantum magnet. In the model, such a sharp feature appears within the two-magnon continuum due to a VHS in the joint density of states 49,50,[52][53][54] . Indeed, on the level of pure kinematics (i.e., ignoring any matrix element effects), the longitudinal twomagnon response in Eq. (5) is proportional to the joint density of states,ĝ q ðωÞ ¼ P k δðω À ε k À ε qÀk Þ, at each momentum q, which corresponds to the joint band dispersionε q ðkÞ ¼ ε k þ ε qÀk as a function of the individual magnon momentum k. Being a two-dimensional band dispersion,ε q ðkÞ has a saddle-point VHS which gives a logarithmic divergence in the density of statesĝ q ðωÞ and, thus, in the longitudinal spin response. Physically, the VHS is a specific energy transfer ω that can create many distinct magnon pairs with a fixed total momentum q but different individual momenta k and q − k. The coalescence of such distinct scattering processes is analogous to the coalescence of light rays giving rise to caustic features in ray optics (see Fig. 4). Therefore, using this analogy, the VHS can also be understood as a caustic feature in the longitudinal spin dynamics. We emphasize that the observation of a VHS is direct evidence for strong quantum fluctuations in YbCl 3 (because the VHS appears in the longitudinal spin response) as well as the two-dimensional nature of its quantum magnetism (because significant interlayer exchange would smear out the VHS).
Upon close examination, the analytic model does not fully capture the intensity and dispersion of the VHS over the entire zone, as can be seen in Fig. 2a-f. By plotting the fitted peak positions of the VHS from Fig. 2g on the calculated spectra in Fig. 2d-f, we notice that there are differences between the calculated and the observed VHS energies near the (100) and ð 1 2 20Þ wave vectors (≈0.2 meV). These energy differences may arise from a renormalization of the two-magnon continuum due to magnon interactions neglected in linear spin-wave theory. Also, we point out that domain formation due to bond disorder 65 could, in principle, play a role in partially smearing out the VHS and thus explaining why certain portions of the predicted VHS are absent from the experimental data.
We now explore the potential importance of additional exchange interactions through comparison of the experimental spin-wave dispersion, including points measured along the L direction, to calculations using the SpinW software 73 . Since there are very small differences in the bond lengths within the honeycomb layers of YbCl 3 , as described in Fig. 1, we label two of the three nearestneighbor exchanges as J for the d = 3.884 Å bonds and the third one as J 0 for the d = 3.867 Å bond. The numerical comparison yields J = 0.42(1) meV and J 0 ¼ 0:43ð1Þ meV, while J c refines to a value below the detection limit of 0.016 meV for the measurements reported here (see Supplementary material for additional details). The resulting cross-section is shown in Fig. 3g-k. The numerically determined J and J 0 are indistinguishable from each other and close to the value determined by a comparison to the analytical model. The spin-wave modes from linear spin-wave theory accurately reproduce the dispersion and intensity distribution of this portion of the measured spectrum (Fig. 3g-k). We also attempted to include next-nearest-neighbor exchange interactions within the plane of the honeycomb lattice; the best fit values of J 2 and J 0 2 are three orders of magnitude smaller than J and zero within the error bars of the refined value (see Supplementary material for additional details).
Magnetic moment and Néel temperature. To further check the validity of the ideal HLHM, we have performed polarized neutron diffraction measurements at T = 0.3 K (see Supplementary material for additional details). We observe an antiferromagnetic spin arrangement with a well-defined staggered magnetic moment. By symmetry, this moment can either lie in the ac plane or point along the b axis, i.e., a linear combination of the two directions is not allowed. Our measurements reveal that the magnitude of the ordered moment is 1.06(4) μ B and that the moment points primarily along the a axis with only a small deviation of 5(3)°along the c axis. The expected moment from the ground state crystal field doublet is 2.24 (5) μ B 61 . The value of the ordered moment is thus ≈47% of the fully polarized moment, which is in excellent agreement with the value of ≈48% obtained from linear spin-wave theory and the value of ≈54% determined by more accurate techniques 72,[74][75][76][77] . We emphasize, however, that these theoretical values are extremely sensitive to any interlayer exchange J c . For example, if we include |J c /J| ≈ 4%, corresponding to the detection limit from the INS response (see above), the ordered moment in linear spin wave theory already increases from ≈48% to ≈63%. Therefore, the small value of the experimental moment indicates that the actual value of J c is significantly below the INS detection limit and further confirms our conclusion that the magnetism in YbCl 3 is two dimensional to a very good approximation.
The same conclusion is demonstrated even more strikingly by the very small value of the Néel temperature, T N ≈ 0.6 K 62 . Theoretically, for a quasi-two-dimensional Heisenberg magnet, the Néel temperature is approximately given by T N ¼ 4πρ s = log jJ=J c j, where ρ s is the spin stiffness of the purely twodimensional system. At this temperature, T = T N , the individual interlayer exchanges J c add constructively within an area of linear size ξ $ expð2πρ s =TÞ 78,79 to produce an effective interlayer exchange on the order of the intralayer exchange J. Taking ρ s ≈ 0.101J from the literature 76,77 and J ≈ 0.42 meV from the neutron scattering data, the relative interlayer exchange is then estimated to be jJ c =Jj $ expðÀ4πρ s =T N Þ % 3 10 À5 . This value is three orders of magnitude smaller than the detection limit from the INS response and indicates that the magnetism in YbCl 3 is as close to two-dimensional as in the canonical cuprate antiferromagnet La 2 CuO 4 80 . Moreover, the specific ordering pattern below T N signals the presence of very small anisotropic interactions on top of the dominant Heisenberg exchange J. Since these anisotropic spin interactions should also contribute to T N , the interlayer exchange may be even smaller than our estimate. We finally remark that the application of a small magnetic field should, in principle, cause the ordering transition to sharpen and the Néel temperature to increase 79 . While there are experimental indications for such behavior in the specific heat 62 , more detailed studies of the field dependence would be very important to confirm not only this prediction but also further predictions of interesting quantum effects at larger magnetic fields 54 .
Heat capacity calculations. Specific heat capacity measurements provide an additional means of examining the HLHM in YbCl 3 . The experimental specific heat and the entropy of YbCl 3 for 0.5 K < T < 8 K are shown in Fig. 5a and b. Between T = 0.5 K and T = 8 K, nearly all of the entropy, Rln ð2Þ, for the ground state doublet has been recovered by the system with only a very small contribution in the region of the transition to long-range magnetic order 62 . We use mTPQ calculations 81 , as implemented in the HΦ library 82 , for a cluster size of 32 spin 1 2 elements to calculate the heat capacity as a function of the reduced temperature T/J (see Supplementary material for additional details). The results for the HLHM with J = 0.42 meV, obtained by fitting the INS data, are shown in Fig. 5a and b. The overall shape is in reasonable agreement with the data, but a somewhat improved comparison is found by using J = 0.32 meV. The discrepancy may be due to quantum corrections, such as renormalizations due to magnon interactions 51,52 , that are captured by the mTPQ calculations but neglected in linear spin wave theory.
We have used INS to investigate the collective magnetic excitation spectrum of YbCl 3 . In addition to a conventional transverse spin-wave (single-magnon) mode, there is a longitudinal two-magnon continuum harboring a sharp VHS. These components are all reproduced by linear spin-wave theory with a single nearest-neighbor Heisenberg interaction on the honeycomb lattice. The sharp VHS is observable due to the almost perfectly two-dimensional quantum magnetism in YbCl 3 , which is further reflected in a strongly reduced ordered moment and a very small ordering temperature. Our results establish YbCl 3 as an ideal model system to investigate the collective quantum behavior of the honeycomb antiferromagnet in the unfrustrated limit. Finally, we point out that the observation of a VHS in a two-magnon continuum provides a strong indication that a similar observation in the two-spinon spectrum of a twodimensional quantum spin liquid 83,84 is experimentally feasible. Such an observation in a quantum spin liquid would be important in ruling out competing sources of a continuum response, such as quenched disorder or overdamped magnons, and could provide a more unambiguous signature of a long-range-entangled quantum state. Fig. 4 Alternative understanding of Van Hove singularities as caustic features in the collective spin dynamics. a Caustic features in ray optics. Parallel light rays (black lines) enter an optical system at different positions. When these light rays reflecting from a circular mirror (red line) coalesce, they give rise to caustic features in real space. b Caustic features in a spin response. The two-magnon continuum can be understood as a sum of sharp contributions, ω = ε k + ε q−k , each corresponding to a fixed momentum k of the first magnon. When these sharp contributions (black lines) coalesce, they give rise to caustic features in the two-magnon continuum. Note that the spin response shown here is for a one-dimensional model system; for the two-dimensional system in consideration, the caustic features appear inside the continuum (not at its edge) and are weaker as they correspond to logarithmic (rather than square-root) singularities. Fig. 5 Magnetic specific heat capacity, entropy, and mTPQ calculations for YbCl 3 . Specific heat capacity (a) and entropy (b) for YbCl 3 . The solid red line is the best fit calculation using mTPQ as described in the text. The dashed blue line is the calculated specific heat using mTPQ with the value J = 0.42 meV obtained by fitting the INS data.