Abstract
A spin1/2 triangularlattice Heisenberg antiferromagnet (TLHAF) is a prototypical frustrated quantum magnet, which exhibits remarkable quantum manybody effects that arise from the synergy between spin frustration and quantum fluctuation. The groundstate properties of a spin1/2 TLHAF are theoretically well understood. However, the theoretical consensus regarding the magnetic excitations is limited. The experimental study of the magnetic excitations in spin1/2 TLHAFs has also been limited. Here we show the structure of magnetic excitations in the spin1/2 TLHAF Ba_{3}CoSb_{2}O_{9} investigated by inelastic neutron scattering. Significantly different from theoretical expectations, the excitation spectrum has a threestage energy structure. The lowestenergy first stage is composed of dispersion branches of singlemagnon excitations. The second and third stages are dispersive continua accompanied by a columnar continuum extending above 10 meV, which is six times larger than the exchange interaction J = 1.67 meV. Our results indicate the shortcomings of the current theoretical framework.
Introduction
Exploring quantum manybody effects has been one of the central subjects of condensed matter physics. Lowdimensional frustrated quantum magnets provide a stage to produce notable quantum manybody effects such as spin liquids^{1} and quantized magnetization^{2, 3}. The simplest and prototypical frustrated quantum magnet is a spin1/2 triangularlattice Heisenberg antiferromagnet (TLHAF) with the nearestneighbor exchange interaction. Since a resonatingvalencebond(RVB) spinliquid state without a longrange magnetic ordering was proposed as the ground state of the spin1/2 TLHAFs^{4, 5}, great effort has been made to elucidate the nature of their ground state. The present theoretical consensus is that the ground state is an ordered state of the 120° spin structure with a significantly reduced sublattice magnetization^{6,7,8,9,10}.
Although the zerofield ground state of spin1/2 TLHAFs is qualitatively the same as that for the classical spin, a pronounced quantum manybody effect emerges in magnetic fields. The quantum fluctuation stabilizes an upupdown spin state in a finite magnetic field range, giving the magnetization curve a plateau at onethird of the saturation magnetization^{11,12,13,14,15,16}. The magnetization curve, which is substantially different from that for the classical spin, has been calculated precisely using various approaches^{13,14,15,16}. The quantum magnetization process has been quantitatively verified by highfield magnetization measurements on Ba_{3}CoSb_{2}O_{9}, which is described as a spin1/2 TLHAF^{17, 18}. The entire magnetization curve, including a highfield quantum phase transition above the 1/3plateau^{18}, has been explained quantitatively by taking the weak easyplane anisotropy and interlayer exchange interaction into account^{19, 20}. Thus, the groundstate properties of a spin1/2 TLHAF with a uniform triangular lattice and the nearestneighbor exchange interaction are well understood both theoretically and experimentally.
In contrast to the groundstate properties, the magnetic excitations in a spin1/2 TLHAF are less well understood. The limited theoretical consensus for singlemagnon excitations is as follows: the dispersion relation of lowenergy singlemagnon excitations near the magnetic Bragg point (K point) is described by linear spinwave theory (LSWT). However, in a large area of the Brillouin zone, the excitation energy is significantly renormalized downward by quantum fluctuations, causing the dispersion curve to become flat^{21,22,23,24,25,26}. In addition, series expansion approach^{22, 26} and fermionizedvortex theory^{27} have demonstrated that the dispersion curve shows a rotonlike minimum at the M point, and nonlinear spinwave theory^{23, 25} has shown that spontaneous decays of magnons occur owing to the magnon interaction, which leads to line broadening of the excitation spectrum. However, there is no theoretical consensus for the excitation continuum that reflects the characteristics of magnetic quasiparticles. The experimental study of the magnetic excitations in spin1/2 TLHAFs has also been limited. Recently, magnetic excitations in Ba_{3}CoSb_{2}O_{9} were investigated by inelastic neutron scattering^{28, 29}. However, the energy range is limited below 3 meV and the excitation spectrum appears to be indistinct. Little is known about the excitation continuum in Ba_{3}CoSb_{2}O_{9}.
Ba_{3}CoSb_{2}O_{9} crystallizes in a highly symmetric hexagonal structure, P6_{3}/mmc ^{30}. Magnetic Co^{2+} ions form a uniform triangular lattice parallel to the ab plane. Because the triangular layers are separated by nonmagnetic layers consisting of Sb_{2}O_{9} double octahedra and Ba^{2+} ions, the interlayer exchange interaction is much smaller than the intralayer exchange interaction^{18}. The effective magnetic moment of Co^{2+} ions with an octahedral environment can be described by the pseudospin1/2 at low temperatures sufficiently below λk _{B} 250K (λ: spinorbit coupling constant)^{17}. Because the octahedral environment of Co^{2+} is close to a cubic environment in Ba_{3}CoSb_{2}O_{9}, the anisotropy of the exchange interaction is small^{18}. Because of the highly symmetric crystal structure, the antisymmetric DzyaloshinskiiMoriya interaction is absent between neighboring spins in the triangular lattice. Ba_{3}CoSb_{2}O_{9} undergoes a magnetic phase transition at T _{N} = 3.8 K owing to the weak interlayer interaction^{30}. In the ordered phase, spins lie in the ab plane and form a 120° structure^{18} ^{,} ^{19} ^{,} ^{29}.
The effective exchange interaction between pseudospins S _{ i } is described by the spin1/2 XXZ model with small easyplane anisotropy as
with 0 < Δ << 1. Here, the first and second terms are the exchange interactions in the triangular layer and between layers, respectively. From the analyses of the saturation field and the collective modes observed by elecron spin resonance (ESR) measurements, the exchange parameters were evaluated to be J = 1.67 meV, Δ = 0.046 and J′ ≈ 0.12 meV^{18}. Because of the small value of Δ, the exchange interaction can approximate the Heisenberg model. For simplification, the small anisotropy in the interlayer exchange interaction is neglected.
Here, we present the results of inelastic neutron scattering experiments on Ba_{3}CoSb_{2}O_{9}, which provide the whole picture of magnetic excitations in a spin1/2 TLHAF. It is revealed that the excitation spectrum has a threestage structure composed of singlemagnon branches and two strong dispersive continua, and that the excitation continuum extends to over 10 meV that is six times larger than the exchange constant J.
Results
Twodimensional excitation spectrum
Figures 1a–d show energymomentum maps of the scattering intensity along two highsymmetry directions parallel to Q = (H, H) and (−K, K) in the twodimensional (2D) reciprocal lattice. The scattering data were collected at 1.0 K, well below T _{N} = 3.8 K, with incident neutron energies of E _{i} = 3.14 and 7.74 meV. These two highsymmetry directions in the 2D reciprocal lattice are illustrated in Fig. 1e. The scattering intensities were integrated over L (the wave vector along the c ^{*} direction) to map the scattering intensity in the 2D reciprocal lattice, assuming good twodimensionality, as shown below. Two weak Qindependent spectra between 5 and 6 meV in Fig. 1b, d are extrinsic spectra, which stem from γrays emitted by the collision of neutrons with E _{i} = 4.68 meV to objects made of cadmium or boron in the beam line.
Figures 2a–c show energymomentum maps of the scattering intensity along Q = (1/3, 1/3, L) and (1/2, 1/2, L) measured with E _{i} = 3.14 and 7.74 meV. The lowenergy excitations for Q = (1/3, 1/3, L) are dispersive, while all the excitations for Q = (1/2, 1/2, L) are almost independent of L. This indicates that the interlayer exchange interaction is small and does not affect the excitations above 1 meV. Because the lowenergy excitations in the vicinity of the K point can be described by LSWT, as shown below, we evaluate the interlayer exchange interaction \(J'\) by applying LSWT to the dispersion curves of the singlemagnon excitations for Q = (1/3, 1/3, L). The solid lines in Fig. 2a are fits with \(J'\) = 0.080 meV, with J and Δ fixed at J = 1.67 meV and Δ = 0.046^{18}, which were determined from the analysis of the saturation field H _{s} = 32.5 T with the gfactor of 3.85 and the zerofield ESR gap of 0.68 meV that corresponds to the excitation gap at Q = (1/3, 1/3, ± 1)^{17, 18}. Lowenergy singlemagnon excitations for Q = (1/3, 1/3, L) are well described by LSWT with these exchange parameters. Because both the interlayer exchange interaction and the anisotropy of the exchange interaction are less than 5% of J and magnetic excitations above 1 meV are almost dispersionless along the c ^{*} direction as shown in Fig. 2b, c, we can deduce that all the excitations except the lowenergy excitations near the K point can be attributed to the 2D spin1/2 TLHAF.
Threestage energy structure
The most noteworthy feature of the excitation spectrum is its threestage energy structure. The lowest stage (ħω < 1.6 meV) is composed of two distinct branches of singlemagnon excitations, which rise up from the K point. The middle (1.1 < ħω < 2.4 meV) and highest (ħω > 2.4 meV) stages are dispersive continua. In the spin3/2 TLHAF CuCrO_{2}, an excitation spectrum with such a threestage energy structure is not observed^{31}. Because the quantum fluctuation in spin1/2 case is considerably stronger than that in spin3/2 case, we infer that the threestage energy structure arises from the quantum manybody effect characteristic of a spin1/2 TLHAF.
As shown in Fig. 1a–d, a significant feature of the magnetic excitations is the two strong dispersive continua that form the middle and third stages of the excitation spectrum. The highest third stage is accompanied by a columnar continuum extending to at least 6 meV. Figures 3a–f show constantenergy slices of the scattering intensity in the continuum range plotted in 2D reciprocal lattice space. The evolution of the scattering intensity with increasing energy is clearly observed in these figures. At intermediate energies of ħω ~ 2.0 meV, strong scattering occurs around the K point. With increasing energy, the position of the strong scattering shifts to the M point and the intensity at the K point decreases. We can see that the excitation continuum extends to over 8 meV.
Highenergy excitation continuum
Figure 4a, b show the energy dependence of the excitation spectrum at the M point for Q = (1/2, 1/2) measured with E _{i} = 7.74 and 15.16 meV, respectively, where the scattering intensity was integrated over L. Horizontal bars in Fig. 4a are the energy resolution. The inset of Fig. 4b shows the energy dependence of the excitation spectrum for Q = (1/2, 1/2) measured with E _{i} = 3.14 meV. Horizontal blue lines in Fig. 4a, b are background level, which was estimated from the scattering intensity between 0.12 and 0.5 meV measured with E _{i} = 3.14 meV. Small peaks between 5 and 6 meV in Fig. 4a are extrinsic peaks originating from γrays emitted in the beam line. A threestage energy structure composed of two singlemagnon excitations and two excitation continua is clearly observed. For E _{i} = 7.74 meV, the widths of two singlemagnon peaks are approximately the same as the energy resolution. The energy ranges of two excitation continua are much larger than the energy resolution. The third stage has a long energy tail of excitation continuum. The energy tail continues to over 10 meV, which is six times larger than the exchange interaction J.
Discussion
The solid lines in Fig. 1a, c are dispersion curves calculated by LSWT with J = 1.67 meV and Δ = 0.046^{18} on the basis of the 2D model described by the first term of Eq. 1. For the lowenergy singlemagnon excitations near the K point, the spectrum becomes visually broad owing to the finite dispersion along the c* direction. In the vicinity of the K point, the lower bound of the spectrum, which corresponds to that for odd L and closely approximates the spectrum at the 2D limit, coincides with the LSWT result. However, the further the wave vector moves away from the K point, the more rapidly the excitation energy deviates downward from the LSWT dispersion. At the M point, the energies of lower and higher singlemagnon excitations are renormalized downward by a factor of 0.69 and 0.61, respectively. This result is qualitatively consistent with the theory^{21,22,23,24,25,26}. Both singlemagnon branches show distinct rotonlike minima at the M point, although the theory predicts that only the lowest branch shows a minimum^{22, 26, 27}. The rotonlike minimum is theoretically interpreted in terms of pairs of spinons characteristic of the RVB state^{22, 26} or vortex excitations with fermionic character^{27}. The experimental dispersion curve for the lowest branch in Ba_{3}CoSb_{2}O_{9} is qualitatively in agreement with the result of the series expansion approach^{22, 26}. In the present experiment, we confirmed that the dispersion of singlemagnon excitations is largely renormalized downward at high energies by quantum fluctuations, while for low energies, the renormalization is small. Note that this quantum renormalization is in contrast with that observed in the spin1/2 kagome antiferromagnet Cs_{2}Cu_{3}SnF_{12} with magnetic ordering, where a uniform quantum renormalization with a Qindependent renormalization factor takes place^{32}.
Remarkable features of magnetic excitations in Ba_{3}CoSb_{2}O_{9} are the two strong dispersive excitation continua, in which the higher energy excitation continuum extends to over the energy of 6 J. Because the highest energy of a singlemagnon excitation is approximately equal to J = 1.67 meV as shown in Fig. 1a, c, the observed excitation continuum cannot be explained in terms of conventional twomagnon excitations. Recently, magnetic excitations in a spin1/2 TLHAF were discussed from the standpoint of spin1/2 fractionalized excitations, spinons, using a mean field Schwinger boson approach^{26}. The excitation continuum in a spin1/2 antiferromagnetic Heisenberg chain that arises from independently propagating spinons is well established^{33,34,35,36}. However, it is difficult to describe the highenergy excitation continuum observed in Ba_{3}CoSb_{2}O_{9} in terms of a twospinon continuum in a spin1/2 TLHAF, because the highest upper bound of continuum is approximately 2 J at most^{26}. At present, no theory describes the structure of the excitation continua observed in this experiment, and thus, a new theoretical framework is required. Our results show that the magnetic excitations in a spin1/2 TLHAF include rich quantum manybody effects yet to be fully explained.
Methods
Sample preparation
Ba_{3}CoSb_{2}O_{9} powder was prepared via the chemical reaction 3BaCO_{3} + CoO + Sb_{2}O_{5}→ Ba_{3}CoSb_{2}O_{9} + 3CO_{2}. Reagentgrade materials were mixed in stoichiometric quantities and calcined at 1100°C for 20 h in air. Ba_{3}CoSb_{2}O_{9} was sintered at 1200 and 1600°C for more than 20 h after being pressed into a pellet. Single crystals were grown from the melts, using a Pt crucible. The temperature at the center of the furnace was decreased from 1700 to 1600°C over 3 days. A single crystal of 10 × 8 × 4 mm^{3}size was used in the neutron inelastic scattering experiments. The mosaicity of crystal was found to be 0.6°.
Measurements of magnetic excitations
Magnetic excitations in a wide momentumenergy range were measured using the coldneutron disk chopper spectrometer AMATERAS^{37} installed in the Materials and Life Science Experimental Facility (MLF) at JPARC, Japan. The sample was mounted in a cryostat with its (1, 1, 0) and (0, 0, 1) directions in the horizontal plane. The sample was cooled to 1.0 K using a ^{3}He refrigerator. Scattering data were collected by rotating the sample around the (−1, 1, 0) direction with a set of incident neutron energies, E _{ i } = 3.14, 4.68, 7.74 and 15.16 meV. All the data were analyzed using the software suite Utsusemi^{38}. At 10 K, excitation spectra shown in Fig. 1 are considerably smeared and the intensities decrease. From this result, their origin was verified to be magnetic.
Data Availability
All relevant data are available from the corresponding author on request.
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Acknowledgements
We express our sincere thanks to T. Masuda and M. Soda for their support in determining crystal orientations by the Xray diffraction. This work was supported by GrantsinAid for Scientific Research (A) (No. 26247058 and 17H01142) and (C) (No. 16K05414) from Japan Society for the Promotion of Science.
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S.I. and H.T. grew the single crystal. S.I., N.K., H.T., S.O.K., K.N., Sh.I. and Ke.K. performed the inelastic neutron scattering experiments. Ka.K. supervised the project. H.T. wrote the manuscript. All the authors discussed the results and contents of the manuscript.
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Ito, S., Kurita, N., Tanaka, H. et al. Structure of the magnetic excitations in the spin1/2 triangularlattice Heisenberg antiferromagnet Ba_{3}CoSb_{2}O_{9} . Nat Commun 8, 235 (2017). https://doi.org/10.1038/s4146701700316x
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DOI: https://doi.org/10.1038/s4146701700316x
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