Abstract
Quantum spin liquids (QSLs) are exotic states of matter characterized by emergent gauge structures and fractionalized elementary excitations. The recently discovered triangular lattice antiferromagnet YbMgGaO_{4} is a promising QSL candidate, and the nature of its ground state is still under debate. Here we use neutron scattering to study the spin excitations in YbMgGaO_{4} under various magnetic fields. Our data reveal a dispersive spin excitation continuum with clear upper and lower excitation edges under a weak magnetic field (H = 2.5 T). Moreover, a spectral crossing emerges at the Γ point at the Zeemansplit energy. The corresponding redistribution of the spectral weight and its fielddependent evolution are consistent with the theoretical prediction based on the interband and intraband spinon particlehole excitations associated with the Zeemansplit spinon bands, implying the presence of fractionalized excitations and spinon Fermi surfaces in the partially magnetized QSL state in YbMgGaO_{4}.
Introduction
In magnetically ordered Mott insulators, the elementary excitations are the spinwavelike magnon modes and carry integer spin quantum numbers. In quantum spin liquids (QSLs) that was first proposed by Anderson as a disordered spin state, however, the situation is drastically different^{1,2}. A QSL does not have any longrange magnetic order and is an exotic quantum state of matter with longrange quantum entanglement^{3,4,5,6,7}. The description of the QSLs goes beyond the traditional Landau’s paradigm that defines phases from their symmetrybreaking patterns. For example, the wellknown ferromagnets differ from the paramagnets by breaking the time reversal and spin rotational symmetry. In contrast, QSLs are often characterized by the emergent gauge structure with deconfined spinon excitations that carry fractionalized spin quantum numbers^{8}. Depending on the type of QSL ground states, the statistics of the spinon excitations can vary from boson to fermion, or even anyon^{4,8,9,10}. Therefore, convincingly revealing the spinon excitations in a QSL candidate not only confirms the spin quantum number fractionalization but also provides an important clue about the type of a QSL^{3,8}.
In most cases, the spin quantum number fractionalization and the spinon excitations can be tested by a combination of experimental tools that include thermodynamic, thermal transport, and spectroscopic measurements^{11,12,13}. The current experimental study of this question is sometimes constrained by various practical issues, and the experimental confirmation of QSLs remains to be controversial. Indeed, the spinonlike continuum has been observed in some of the QSL candidates^{14,15,16,17}, and more schemes are needed to provide robust evidence for the very existence of spin quantum number fractionalization^{3}. The recent discovery of the triangular lattice singlecrystalline QSL candidate material YbMgGaO_{4} provides a new testing ground for the QSL research^{18}. No ordering or symmetry breaking is observed down to about 30 mK in a variety of measurements^{16,17,18,19,20,21}. It is argued that the spinorbital entanglementinduced anisotropic interactions may trigger strong quantum fluctuations and help stabilize the QSL state consequently^{22,23,24}. More substantially, recent inelastic neutronscattering measurements have discovered a broad spin excitation continuum covering a large portion of the Brillouin zone^{16,17,25}. A variety of theoretical proposals have been made, including the QSL state with a spinon Fermi surface and nearestneighbor resonating valence bond (RVB) state^{16,25,26,27}. Meanwhile, the scenario of disorder and spinglass state have also been recently suggested^{28,29}. To further confirm the fermionic spinon excitation and the spin quantum number fractionalization in YbMgGaO_{4}, the fielddependent evolution of the spin excitations needs to be tested^{30,31}. In fact, the fermionic spinon excitation in one spatial dimension has been examined in this manner. For the spin1/2 Heisenberg chain, which is essentially a onedimensional Luttinger liquid, and is not a true QSL in the modern sense, but shows spinon excitations in the form of domain walls, the external magnetic field could lead to the splitting of the spinon band, resulting in a modulation and redistribution of the spinon continuum^{32,33}. These results provided a firm confirmation of the fractionalized spinon excitation in one spatial dimension.
In this paper, we use inelastic neutronscattering technique to study the spin excitations of YbMgGaO_{4} single crystals under various external magnetic fields. In a weak magnetic field of H = 2.5 T applied along the caxis, a dispersive spin excitation continuum is revealed with clear upper and lower excitations edges, leading to a spectral crossing at the Γ point at the Zeemansplit energy. The corresponding redistribution of the spectral weight and its fielddependent evolution are inconsistent with the conventional magnon behavior, but instead are unique features for fractionalized spinon excitations. In particular, we show that the measured dynamic spin structure is consistent with the theoretical prediction based on the interband and intraband spinon particlehole excitations associated with the Zeemansplit spinon bands. Our results provide an important piece of evidence for fractionalized spinon excitations and spinon Fermi surfaces in YbMgGaO_{4} under magnetic fields.
Results
Continuous excitations under weak external fields
We start by examining the magnetization of a YbMgGaO_{4} single crystal under magnetic fields along the caxis (Fig. 1a). The magnetization increases progressively with field below ~4 T, followed by a smooth transition into the highfield regime above ~7 T, where the magnetization nearly saturates. This is consistent with previous measurements^{22}. To study the effect of magnetic field on the continuous spin excitations in YbMgGaO_{4}, we first use inelastic neutron scattering to measure the spin excitations under H = 2.5 T in the lowfield regime. As can be seen in Fig. 1b–d, the constantenergy images show ringlikeshaped diffusive magnetic excitations covering a wide region of the Brillouin zone. Such momentum structure of spin excitations resembles the continuous excitations observed at zero field, which was interpreted as the evidence for spinon excitations^{16,17}. This implies that the spinon excitations persist under a weak magnetic field, as we will discuss subsequently.
To determine the dispersion of the spin excitation continuum in the weak field regime, we measured the energy dependence of the spectral intensity along the highsymmetry directions (Fig. 2a). Unlike the zerofield data in which the spectral weight near the zone center (Γ point) is strongly suppressed, a prominent enhancement can be seen at ~0.6 meV, leading to a spectral crossing near Γ. Moreover, the continuum shows clear lower and upper excitation edges near ~0.6 meV (marked by the Xshaped cross in Fig. 2a) around Γ, which is distinct from the Vshaped upper excitation edge at zero field^{16}. As the energy is further lowered, there is another upper excitation edge below ~0.3–0.4 meV (marked by the Vshaped edge in Fig. 2a). The dispersion of the broad continuum is further confirmed by the constantenergy cuts along the highsymmetry directions in Fig. 3.
Field dependence of the spin excitations
To gain further insights into the origin of the continuum, we present in Fig. 4 the field and energy dependence of the spin excitation at Γ, M, and K points. At the zero field, it is found that the spectral weights mainly spread along the zone boundary, resulting in a suppressed intensity near the zone centers (Γ points). At low fields, however, a spectral peak occurs at a finite energy at Γ (Fig. 4a), which corresponds to the spectral crossing point in Fig. 2a. As the field is increased, the spectral peak shifted to higher energy in a linear manner as denoted in Fig. 4b. Meanwhile, the broad continuum at M and K persists except that the spectral intensity is gradually suppressed with increasing field. It seems that part of the spectral weight has been transferred from M and K to Γ point and no clear shift of the overall continuum is detected. Such behavior differs from what one would expect for the spinwave excitations in a conventional magnet, where the whole spinwave band should shift to high energy with increasing field, since magnons couple to magnetic field directly^{23,34}. Indeed, sharp spinwave excitations with a ~1.2 meV gap are observed under a high magnetic field of 9.5 T in a nearly polarized state (Fig. 2b). Moreover, the highfield spinwave spectrum shows a clearly distinct dispersion from that in the lowfield regime (Fig. 2a). This further indicates that the lowfield continuum cannot be magnon excitations.
Discussion
We propose that the modulation of the spectral weights of the continuum in the lowfield regime is consistent with the previously predicted behavior of the spinon Fermi surface QSL state under magnetic fields^{30}. In the weak field regime, the proposed zerofield spinon Fermi surface QSL state is expected to persist and the spinon remains to be a valid description of the magnetic excitation^{30}, which is confirmed by our data that continuum excitations are observed at all energy measured. It was previously shown in ref. ^{30} that the degenerate spinon bands are split and the splitting is given by the Zeeman energy. The meanfield results for the specific parameter choice of the present experiment are given in details in Methods. In an inelastic neutronscattering measurement, the neutron energymomentum loss creates the spin excitation that at the meanfield level corresponds to both the interband and intraband particlehole excitation of the spinons. The particlehole excitation continuum of the spinons persists into the weak field regime. In particular, for zero momentum transfer of the neutron, the relevant particlehole excitation would simply be the vertical interband excitation between the spinup and spindown spinon bands and leads to the spectral peak at the Γ point and the Zeemansplit energy (Figs. 2a, d and 4a). For momenta away from the Γ point, depending on how the momenta are connected to the two split spinon bands, the interband particlehole continuum is bounded by the upper and lower excitation edges that cross each other at the Zeemansplit energy and the Γ point (marked by the Xshaped excitation edges in Fig. 2a). As for the intraband particlehole excitations below the Zeemansplit energy, a minimal momentum transfer, p_{min} ≈ E/v_{F}, is needed to create the intraband particlehole excitation across each spinon Fermi surface for a small and finite neutron energy loss E, where v_{F} refers to the corresponding Fermi velocity. Thus, the spinon continuum near the Γ point is bounded by an upper excitation edge (marked by the Vshaped edge below the Xshaped edge in Fig. 2a).
It is the spinon Fermi surface and the fractionalized nature of the spinons that give the excitation continuum in the zero field^{16} and the particular spectral structure of the continuum in the weak field regime. The magnetic field shifts the spinup and spindown spinon bands in an opposite fashion. Being fractionalized spin excitations, the spinup and spindown spinons should combine together and contribute to the magnetic excitations measured by the inelastic neutron scattering. This opposite behavior of the two spinon bands in magnetic field is manifested by the spectral peak and the spectral crossing at the Γ point as well as the lower and upper excitation edges near the Γ point. Indeed, the calculated spinon excitation spectrum associated with the split spinon bands (Supplementary Fig. 2c) captures such features of the measured excitation continuum (Fig. 2a).
Despite a reasonable agreement between the experimental data and the theoretical prediction, we would also point out the discrepancy for a future improvement on the theory side. The spectral peak at the Γ point in Fig. 4a shows a wider broadening and a reduced intensity compared to the theoretical expectation^{30}. This spectral broadening and reduced intensity could arise from two sources. One is the intrinsic broadening due to the gauge fluctuation. The spinongauge coupling not only blurs the quasiparticle nature of the spinon but also gives extra scattering processes with gauge photons involved in the inelastic neutron scattering. Both effects are beyond the meanfield analysis for noninteracting spinons and will be discussed in future work. The other is the extrinsic broadening due to the gfactor randomness. It was argued that Mg/Ga site disorder may create crystal electric field variations and thus induce gfactor randomness^{20}. The gfactor randomness will give the Zeeman energy disorder and is thus responsible for the broadening of the spectral peak. Finally we emphasize that the spin quantum number fractionalization with spinon excitations is one of the key properties of QSL and could survive even with weak local perturbations such as weak disorder.
We would also like to discuss about other scenarios. Recently, ref. ^{25} suggested the nearestneighbor RVB scenario and claimed that the excitation continuum in the inelastic neutronscattering measurement bears no obvious relation to spinons. In fact, it is well known that the nearestneighbor RVB state on frustrated lattices such as the triangular lattice is a fully gapped \({\Bbb Z}_2\) QSL^{35,36,37}. As a result, even in the nearestneighbor RVB scenario, the excitation continuum should be the spinon continuum. For such a \({\Bbb Z}_2\) QSL, all the excitations, both spinons and visons, are gapped, and the spinon gap would be of the order of the exchange coupling. There does not seem to be any signature of gapped visons and spinons in the heat capacity and the spin susceptibility data. As was explained in our previous work, the usual gapped \({\Bbb Z}_2\) QSL is a bit difficult to reconcile with the dynamic spin structure factor in this system^{16}.
Apart from the scenario of the nearestneighbor RVB state, the scenario with disorders was also proposed^{28}. Disorders might play some role in the verylowenergy magnetic properties and the thermal transport^{21,38}. The important questions are whether the disorder is the driving force of the possible QSL state in YbMgGaO_{4}, and to what extent the disorder affects the ground state in YbMgGaO_{4}. Spin quantum number fractionalization, which is one of the key properties of QSL, could persist even with weak disorder. In addition, a spinglass scenario has been proposed in ref. ^{29}, where the a.c. susceptibility shows a broad peak at an extremely low temperature of ~80–100 mK. The authors attribute this behavior to a spinglass transition. However, it is known that a spin liquid can also display a peak structure in a.c. susceptibility because of the presence of slow dynamics^{15}. In fact, the spinglass scenario is also in conflict with μSR measurements where no spin freezing was observed down to 48 mK^{19}. Moreover, this a.c. susceptibility peak appears at such a low temperature where the Rln2 magnetic entropy has been already released by more than 99%. Furthermore, the redistribution of the spin excitation continuum under a weak field observed in our current work is a bit difficult to be explained by the conventional magnon behavior of a spin glass, but can be explained straightforwardly by fractionalized spinon excitations.
Methods
Inelastic neutronscattering experiments
Inelastic neutronscattering measurements were performed on the ThALES cold tripleaxis spectrometer at the Institut LaueLangevin, Grenoble, France. Highquality YbMgGaO_{4} single crystals were synthesized using the optical floating zone technique^{16}. PG(002) was used as a monochromator. In the singledetectormode measurements (Fig. 4), PG(002) was used as an analyzer and the final neutron energy was fixed at E_{f} = 3.5 meV, resulting in an energy resolution of ~0.11 meV. For the measurements with the Flatcone detector (Figs. 1–3, Supplementary Fig. 1), Si (111) was used as an analyzer and the final energy was fixed at E_{f} = 4 meV, resulting in an energy resolution of ~0.16 meV. A velocity selector was installed in front of the monochromator to remove the contamination from higherorder neutrons. A dilution insert was used to reach temperatures down to ~70 mK in the vertical magnet.
In order to reduce the influence of the neutronbeam selfattenuation (by the sample), same correction method is used as that in ref. ^{16} for data shown in Figs. 1–3. The selfattenuation effect can be presented as anisotropic intensity distribution in the elastic incoherent scattering image measured at 20 K (Supplementary Fig. 1a). Similar anisotropy is also observed in the raw constantenergy images in the inelastic channel (Supplementary Fig. 1b–d). The correction can be done by normalizing the inelastic data with a linear attenuation correction factor converted from the elastic incoherent scattering intensity, which is dependent on the sample position (ω) and scattering angle (2θ). The normalized constantenergy images are presented in Fig. 1b–d. All the data in the manuscript are presented without symmetrization/folding.
Spinon Fermi surface in a weak magnetic field
Here we explore the coupling of the candidate spinon Fermi surface state for YbMgGaO_{4} to the external magnetic field. This spinon Fermi surface QSL state was originally proposed for the triangular lattice organic materials κ(ET)_{2}Cu_{2}(CN)_{3} and EtMe_{3}Sb[Pd(dmit)_{2}]_{2}^{39,40,41,42}. For the organics, due to the small Mott gap and proximity to the Mott transition, the coupling to the magnetic field may involve a significant Lorentz coupling^{43}. For YbMgGaO_{4} that is in the strong Mott regime, however, only Zeeman coupling is necessary^{30}.
In previous works, spinon meanfield theory and a systematic projective symmetry analysis have suggested a SU(2) rotational invariant spinon meanfield Hamiltonian with shortrange spinon hoppings to describe YbMgGaO_{4}^{16,27}. A more recent theoretical work by two of us has extensively studied the effect of weak magnetic field on the spinon continuum based on meanfield theory^{30}. Here we adjust the early theoretical formulation into the parameter choice of the current experiment. In the current experiment, the field is applied along the caxis (normal to the triangular plane). From the SU(2) symmetry of the spinon meanfield theory, the direction of the magnetic field will probably not induce any qualitatively different behavior on the spinon continuum from the cdirection magnetic field at the meanfield level.
Here we explain the basic idea and the underlying physics, and also point out the difference from the zerofield results^{16,27}. We introduce the Abrikosov fermion representation for the spin operator such that \({\bf{S}}_i = \mathop {\sum}\nolimits_{\alpha \beta } {\kern 1pt} f_{i\alpha }^\dagger \frac{{{\boldsymbol{\sigma }}_{\alpha \beta }}}{2}f_{i\beta }\) with the Hilbert space constraint \(\mathop {\sum}\nolimits_\alpha {\kern 1pt} f_{i\alpha }^\dagger f_{i\alpha } = 1\). We start with the meanfield Hamiltonian for the spinons,
where t_{1} and t_{2} are the nearest and nextnearestneighbor spinon hoppings, respectively. The chemical potential μ is introduced to impose the Hilbert space constraint, and the last Zeeman terms accounts for effects of the external magnetic field along the caxis. Since the system is in the strong Mott regime, the charge fluctuation is strongly suppressed, the Lorentz coupling due to charge fluctuation in the weak Mott regime does not apply here^{43}. We only need to consider the Zeeman coupling to the magnetic field^{30}. We choose the hopping term in \({\cal H}_{{\mathrm{MFT}}}\) to be spatially uniform, since it was shown to be the only symmetric meanfield state that is compatible with the existing experiments^{27}. The fractionalized nature of the spin excitations is already captured by this simple spinon meanfield Hamiltonian, and the further neighbor spinon hopping is introduced to improve the comparison with experiments. We remark on the SU(2) spin rotational symmetry of the spinon meanfield Hamiltonian \({\cal H}_{{\mathrm{MFT}}}\). This SU(2) spin symmetry at the meanfield level is protected by the projective symmetry group^{27}. This symmetry is clearly absent in the microscopic spin model^{23}. It is then pointed out^{27,30} that the anisotropic spin interaction enters as SU(2) symmetrybreaking interactions between the spinons. A random phase approximation was then introduced to capture the anisotropic interaction and compute the dynamic spin structure factor. It was found that the spectral weight of the spinon continuum is redistributed and the qualitative features of the continuum persist. More detailed meanfield theory and the random phase approximation have been discussed in the previous theoretical works^{27,30}.
Without the magnetic field, the ground state of Eq. (1) is a filled Fermi sea of degenerate spinup and spindown spinons with a large Fermi surface. It has already been shown that the particlehole continuum of the spinon Fermi surface gives a consistent explanation for the excitation continuum in the inelastic neutronscattering measurement with the zero field^{16,26}. Moreover, due to the spin rotational invariance and the degenerate spinup and spindown spinon bands, the spinflipping process and the spinpreserving process in the neutron scattering, which correspond to the interband particlehole excitation and the intraband particlehole excitation respectively, give the same momentumenergy relation for the inelastic neutronscattering spectrum. Therefore, in the previous calculations^{16,27}, considering the interband particlehole excitation is sufficient.
In the presence of a weak magnetic field H, such that the Zeeman coupling would only have a perturbative effect on the QSL ground state and the spinon remains to be a valid description of the magnetic excitation, the previously degenerate spinup and spindown bands are now split by an energy separation set by the Zeeman energy Δ ≡ g_{z}μ_{B}H^{30}. The inelastic neutron scattering measures the correlation function of the spin component that is transverse to the momentum transfer. The dynamic spin structure factor, which is detected by the inelastic neutron scattering, is given by
where \(\left {\mathrm{\Omega }} \right\rangle\) is the filled Fermi sea ground state of the splitted spinon bands, and \(\widehat {\bf{p}}\) is a unit vector that defines the direction of the momentum p. Both the S^{+}S^{−} correlation and the S^{z}S^{z} correlation are involved in the above equation. The correlation between S^{z} and S^{+} or S^{−} is vanishing at the meanfield level because the spinon meanfield Hamiltonian still preserves the U(1) spin rotational symmetry around the zaxis. To understand these two contributions, we explain their spectroscopic signatures in turns in the following discussion.
In free spinon meanfield theory, the S^{+}S^{−} correlation detects the interband spinon particlehole excitation. We have \({\cal S}_{ +  }({\bf{p}},E)\),
where n refers to the intermediate particlehole excited state. Since this is a spinflipping process, it naturally probes the interband spinon particlehole continuum. The Γ point, with zero momentum transfer of the neutron, simply corresponds to the vertical transition between the spinon bands (Supplementary Fig. 2a). At the meanfield level, the spectral intensity would be proportional to the density of states available for this vertical transition. Due to the large density of states for this vertical transition, there would be a spectral peak at the Γ point and the Zeemansplit energy Δ. We, however, expect that the interaction between the spinons would suppress the meanfield spectral intensity. The actual spectral peak may not be quite significant. In any case, a spectral peak at the Γ point and the Zeemansplit energy Δ is observed in Fig. 4a.
A finite momentum transfer, p, would probe the tilted particlehole process between the two spinon bands. For a fixed and small momentum near the Γ point, there exists a range of energies that connect two bands. This indicates the presence of the lower and upper excitation edges that define the energy range of the continuous excitations near the Γ point. Moreover, these two edges cross each other right at the Γ point and the Zeemansplit energy. These features are observed in Supplementary Fig. 2a.
The S^{z}S^{z} correlation detects the intraband spinon particlehole excitation, and we have
where we exclude the static p = 0 contribution from the finite magnetization along the z direction in the actual calculation. Like the zerofield case, the particle and the hole in the intraband process can be excited right next to the Fermi surface (Supplementary Fig. 2b), thus the intraband particlehole excitation can have an arbitrarily small energy. The lowenergy particlehole continuum of the intraband process, however, is bounded by an upper excitation edge near the Γ point. This is actually analogous to the one in the zerofield result. For a small and finite energy transfer E of the neutron, a minimal momentum transfer p_{min} ≈ E/v_{F} is needed to excite the spinon particlehole pair, where the Fermi velocity v_{F} depends on the momentum direction and the spin flavor of the spinon Fermi surface.
Having explained the physics of the interband and intraband scattering, we here include both the interband process and the intraband process and compute the dynamic spin structure in Eq. (2). The result is shown in Supplementary Fig. 2c and is reasonably consistent with the experimental one in Fig. 2a. Note here the calculated spectra show lower intensity at low energy compared to the experimental results. The observed high intensity at low energies is mainly due to background contamination from the incoherent elastic scattering at E = 0 meV, because the contamination is less significant when the energy resolution is improved from 0.16 (Fig. 2a) to 0.11 meV (Fig. 4). Another possible cause is owing to the simplicity of the meanfield theory that neglects the U(1) gauge fluctuation. It is well known that the gauge fluctuation would enhance the lowenergy density of spinon excitations and thus increase the spectral weights at low energies.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank C. Broholm, F. C. Zhang, M. Mourigal, and A. L. Chernyshev for discussions. This work was supported by the Innovation Program of Shanghai Municipal Education Commission (grant number 201701070007E00018), the Ministry of Science and Technology of China (Program 973: 2015CB921302), the National Key R&D Program of the MOST of China (grant number 2016YFA0300203), and the National Natural Science Foundation of China (grant number 91421106). Y.D.L. and G.C. were supported by the Ministry of Science and Technology of China with the Grant No.2016YFA0301001, the StartUp Funds and the Program of FirstClass Construction of Fudan University, and the ThousandYouthTalent Program of China.
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J.Z. and G.C. planned the project. Y.S., H.W., and S.S. synthesized the sample. Y.S., X.Z., H.C.W., P.S., and J.Z. carried out the neutron experiments with experimental assistance from M.B. J.Z. and Y.S. analyzed the experimental data. Y.D.L. and G.C. provided the theoretical explanation and calculation. J.Z., G.C., Y.S., and Y.D.L. wrote the paper. All authors provided comments on the paper.
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Shen, Y., Li, YD., Walker, H.C. et al. Fractionalized excitations in the partially magnetized spin liquid candidate YbMgGaO_{4}. Nat Commun 9, 4138 (2018). https://doi.org/10.1038/s41467018065881
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