Abstract
Amplitude modes arising from symmetry breaking in materials are of broad interest in condensed matter physics. These modes reflect an oscillation in the amplitude of a complex order parameter, yet are typically unstable and decay into oscillations of the order parameter’s phase. This renders stable amplitude modes rare, and exotic effects in quantum antiferromagnets have historically provided a realm for their detection. Here we report an alternate route to realizing amplitude modes in magnetic materials by demonstrating that an antiferromagnet on a twodimensional anisotropic triangular lattice (αNa_{0.9}MnO_{2}) exhibits a longlived, coherent oscillation of its staggered magnetization field. Our results show that geometric frustration of Heisenberg spins with uniaxial singleion anisotropy can renormalize the interactions of a dense twodimensional network of moments into largely decoupled, onedimensional chains that manifest a longitudinally polarizedbound state. This bound state is driven by the Isinglike anisotropy inherent to the Mn^{3+} ions of this compound.
Introduction
Many of the seminal observations of amplitude modes in magnetic materials arise from quantum effects in onedimensional antiferromagnetic chain systems when interchain coupling drives the formation of longrange magnetic order^{1}. For instance, bound states observed in the ordered phases of S = 1 Haldane systems^{2,3,4} or in the spinon continua of S = 1/2 quantum spin chains^{5,6,7,8,9} were shown to be longitudinally polarized and reflective of the crossover into an ordered spin state. While the chemical connectivity of magnetic ions in these systems is inherently onedimensional, alternative geometries such as planar, anisotropic triangular lattices can also in principle stabilize predominantly onedimensional interactions in antiferromagnets^{10}. In the simplest case, geometric frustration in a lattice comprised of isosceles triangles promotes dominant magnetic exchange along the short leg of the triangle while the remaining two equivalent legs frustrate antiferromagnetic coupling between the chains. The result is a closely spaced twodimensional network of magnetic moments whose dimensionality of interaction is reduced to be quasi onedimensional.
A promising example of such an anisotropic triangular lattice structure is realized in αphase NaMnO_{2}. Layered sheets of edgesharing MnO_{6} octahedra are separated by layers of Na ions, and the orbital degeneracy of the octahedrally coordinated Mn^{3+} cations (3d^{4}, t_{2g}^{3}e_{ g }^{1} valence) is lifted via a large, coherent Jahn–Teller distortion^{11}. This distorts the triangular lattice such that the leg along the inplane baxis is contracted 10% relative to the remaining two legs. As a result, the S = 2 spins of the Mn^{3+} ions decorate a dimensionally frustrated lattice where onedimensional intrachain coupling along b is favored and interchain coupling is highly frustrated. This spin lattice eventually freezes into a longrange ordered state below T_{N} = 45 K^{11}; however, previous studies of powder samples have suggested an inherently onedimensional character to the underlying spin dynamics^{12}. Such a scenario suggests an intriguing material platform for the stabilization of an amplitude mode in a conventional spin system (i.e., one with diminished local moment fluctuations and a quenched Haldane state^{13}) as the ordered state is approached, and a static, staggered mean field is established.
In this paper, we present single crystal neutron scattering data that show that the planar antiferromagnet αNa_{0.9}MnO_{2} exhibits quasi onedimensional spin fluctuations that persist into the AF ordered state. Additionally, our data reveal that an anomalous, dispersive spin mode appears as AF order sets in, and that this new mode is longitudinally polarized with an inherent lifetime limited by the resolution of the measurement. This longitudinally polarizedbound state demonstrates the emergence of a magnetic amplitude mode in a spin system where geometric frustration lowers the dimensionality of magnetic interactions and amplifies fluctuation effects. Intriguingly, this occurs in a compound where strong quantum fluctuations inherent to S = 1/2 systems and singlet formation effects inherent to integerspin Haldane systems—both typical settings for longitudinalbound state formation—are absent. To explain the stabilization of this amplitude mode, we present a model that captures the excitation as a twomagnonbound state whose binding energy derives from an easyaxis singleion anisotropy inherent to the orbitally quenched Mn^{3+} ions. This anisotropy orients the moments along a preferred axis and renders them Isinglike. Our work establishes αNa_{ x }MnO_{2} and related lattice geometries as platforms for realizing unconventional spin dynamics in a dense network of onedimensional antiferromagnetic spin chains^{14,15}.
Results
Crystal and spin structures of αNa_{0.9}MnO_{2}
To demonstrate the emergence of an amplitude mode in αNa_{0.9}MnO_{2}, careful descriptions of the lattice and spin structures are first necessary. We note here that units for wave vectors throughout the manuscript are given in reciprocal lattice units (H, K, L) where \({\mathbf{Q}}\left[{{{\mbox{\AA}}}^{  1}} \right] = \left( {\frac{{2\pi }}{{a\sin \beta }}H,\frac{{2\pi }}{b}K,\frac{{2\pi }}{{c \sin \beta }}L} \right)\) and a, b, c, and β are the lattice parameters of the unit cell. Figure 1a shows the projection of the low temperature, ordered spin lattice of αNaMnO_{2} onto the abplane where one spin domain with propagation vector q_{1} = (0.5, 0.5, 0) is illustrated. The antiferromagnetic chain direction is shaded in gray. Due to the degeneracy of the frustrated interchain coupling, a second spin domain with propagation vector q_{2} = (−0.5, 0.5, 0) also stabilizes, and the moments in both domains are oriented approximately along the [−1, 0, 1] apical oxygen bond direction due to an inherent uniaxial singleion anisotropy^{11}. The large Jahn–Teller distortion renders the lattice structure prone to crystallographic twinning^{16} and the relative orientations of the moments in the two resulting crystallographic twins, twin 1 (t_{1}) and twin 2 (t_{2}) are depicted in Fig. 1b. This results in four allowed domains: t_{1} − q_{1}, t_{1} − q_{2}, t_{2} − q_{1}, and t_{2} − q_{2}. Crucially, the moments in both crystallographic and magnetic twin domains are nearly parallel to a common [−1, 0, 1] axis. This is verified via polarized elastic neutron scattering measurements in the AF state at T = 2.5 K shown in Fig. 1c. These data demonstrate that in a single domain, t_{1} − q_{1}, probed at Q = (0.5, 0.5, 0) the moments are rotated ~7° away from the [−1, 0, 1] axis within the acplane. We note here that previous studies of stoichiometric NaMnO_{2} have reported an extremely subtle distortion into a lower triclinic symmetry below the antiferromagnetic transition^{11}. Our neutron diffraction measurements fail to detect this distortion in the average structure of Na_{0.9}MnO_{2} crystals, and we therefore analyze data using the higher symmetry monoclinic structure. A recent report suggests that the triclinic phase occurs only as an inhomogeneous local distortion^{17}; hence our inability to observe the reported triclinic distortion may arise from its absence in the average structure, its suppression due to the Na vacancies in our samples, or due to the resolution threshold of our measurements. Despite this ambiguity, the reported triclinic distortion in NaMnO_{2} is subtle and would generate roughly a 0.12% difference between nextnearest neighbor (interchain) exchange pathways, which can be neglected for the purposes of the present study.
Spin Hamiltonian of αNa_{0.9}MnO_{2}
In order to understand the interactions underlying the AF ground state of this system, inelastic neutron scattering measurements were performed. Spin excitations measured within the ordered state about the AF zone centers, Q = (1.5, ±0.5, 0), are shown in Fig. 2. Inspection of the momentum distribution of the spectral weight reveals that the magnetic fluctuations underpinning the AF state at T = 2.5 K are quasi onedimensional. Figure 2a demonstrates that the magnetic excitations along the inplane Kaxis, parallel to the short leg of the triangular lattice, show an anisotropy gap at the zone center and a welldefined dispersion; however, the magnon dispersion in directions orthogonal to this axis are diffuse. Specifically, the spin waves dispersing between the MnO_{6} planes (along L) are dispersionless (see Supplementary Fig. 2) as expected for the planar structure of αNa_{0.9}MnO_{2}, and Fig. 2b shows that spin wave energies dispersing perpendicular to the AF chain direction in the plane (along H) are only weakly momentum dependent. This demonstrates that the spin fluctuations exist as quasionedimensional planes of scattering in (Q, E) space, driven by the strong interchain frustration inherent to the lattice and consistent with the large magnetic frustration parameter of this compound^{18}.
As twin effects from both crystallographic and spin domains may obscure any subtle dispersion along H due to interchain interactions, inspection of zone boundary energies and analysis of the full bandwidth are necessary to quantify the weak interchain exchange terms. Therefore, in order to parameterize the dispersion measured in Fig. 2a, the high energy data were analyzed using a fourdomain model (t_{1} − q_{1}, t_{1} − q_{2}, t_{2} − q_{1}, t_{2} − q_{2}) as well as by fitting lower energy tripleaxis data shown in Fig. 3 and Supplementary Fig. 3. The data were modeled using the singlemode approximation and the spin Hamiltonian, \(H = J_1\mathop {\sum}\nolimits_{nn} {{\mathbf{S}}_i} \cdot {\mathbf{S}}_j + J_2\mathop {\sum}\nolimits_{nnn} {{\mathbf{S}}_i} \cdot {\mathbf{S}}_j  D\mathop {\sum}\nolimits_n {(S_n^z)} ^2\), where J_{1} is the twofold nearest neighbor exchange coupling, J_{2} is the fourfold nextnearest neighbor coupling, and D is a uniaxial, Isinglike, singleion anisotropy term. The dispersion relation generated from a linear spin wave analysis of this Hamiltonian is given by \(E({\mathbf{Q}}) = S\sqrt {\omega _{\mathbf{Q}}^2  \lambda _{\mathbf{Q}}^2}\), where ω_{ Q } = 2(J_{1} + D + J_{2}cos(πH + πK)) and \(\lambda _{\mathbf{Q}} = 2\left( {J_1\cos \left( {2{\mathrm{\pi }}K} \right) + J_2{\mathrm{cos}}\left( {{\mathrm{\pi }}H  {\mathrm{\pi }}K} \right)} \right)\) (see Supplementary Note 1 and Supplementary Fig. 1 for details), and the results from fitting the data yielded a J_{1} = 6.16 ± 0.01 meV, J_{2} = 0.77 ± 0.01 meV, and D = 0.215 ± 0.001 meV. These values are roughly consistent with earlier powder averaged measurements of spin dynamics in αNaMnO_{2}, although these earlier measurements were not sensitive enough to resolve a J_{2} term^{12}.
The magnon modes from the fourdomain model are overplotted as lines with the raw timeofflight data in Fig. 2d, and the total simulated intensities summed from all modes are shown in Fig. 2c. Good agreement is seen between the data in Fig. 2a and the simulated intensities shown in Fig. 2c and Supplementary Fig. 4. To further illustrate this model at lower energies closer to the zone center gap value, data collected via a thermal tripleaxis spectrometer are shown in Fig. 3. Momentum scans through the AF zone center are plotted in Fig. 3a at energies from ΔE = 3 meV to 18 meV with the resulting color map of intensities plotted in Fig. 3b. Magnon modes dispersing from the four domains in the system using the same J_{1}–J_{2}–D model described earlier and convolved with the instrument resolution function are plotted as solid lines fit to the data in Fig. 3a. Crucially, unlike the model presented in Fig. 2, describing this lower energy data also requires the introduction of one additional dispersive mode. This mode is distinct from the transversely polarized magnons anticipated in this material, and it represents an unexpected longitudinally polarizedbound state as described in the next section.
Longitudinally polarized mode
To more clearly illustrate the appearance of an additional mode in the low energy spin dynamics, a magnetic zone center energy scan at Q = (0.5, 0.5, 0) is plotted in Fig. 3c. This scan shows the large buildup of spectral weight above the ΔE = 6.15 ± 0.04 meV zone center gap, consistent with the quasi onedimensional magnon density of states, and above this gap a second zone center mode near 11 meV appears. This 11 meV mode is not accounted for by any of the expected transverse modes in this system, and it is also quasi onedimensional in nature. Figure 3c demonstrates negligible interchain dispersion as Q is rotated from the three dimensional Q = (0.5, 0.5, 0) to onedimensional Q = (0, 0.5, 0) AF zone center, and the 11 meV mode’s dispersion along the chain direction is plotted in Fig. 3b. While there is a limited bandwidth (ΔE = 11–15 meV) where this new mode remains resolvable inside of the dispersing transverse magnon branches, the narrow region of dispersion was empirically parameterized using a onedimensional J–D model with \(H = J\mathop {\sum}\nolimits_n {{\mathbf{S}}_n} \cdot {\mathbf{S}}_{n + 1}  D\mathop {\sum}\nolimits_n {(S_n^z)} ^2\) and \(E\left( {\mathbf{Q}} \right) = \sqrt {\Delta ^2 + c^2{\mathrm{sin}}^2\left( {2{\mathrm{\pi }}K} \right)}\). The gap value from this parameterization was fit to be Δ = 11.11 ± 0.06 meV and c = 21.6 ± 0.1 meV. The dispersion fit to this higher energy mode along with the dispersions fit to the transverse magnon modes are overplotted with the data in Fig. 3b. We again note that this additional 11 meV dispersive mode was incorporated within the fits shown in Fig. 3a.
To further explore the origin of the anomalous 11 meV branch of excitations near the AF zone center, polarizedneutron scattering measurements were performed using an experimental geometry that leveraged the quasi onedimensional nature of the spin excitations. Specifically, magnetic excitations were measured about the onedimensional zone center, Q = (0, 1.5, 0). As the magnetic moment μ is oriented nearly parallel to the [−1, 0, 1] crystallographic axis, two transversely polarized magnon modes along the [1, 0, 1] and the [0, 1, 0] directions are expected in the ordered state, each carrying an oscillation of the orientation/phase of the staggered magnetization. The (Q × μ × Q) orientation factor in the neutron scattering cross section renders it only sensitive to the components of the moments’ fluctuations perpendicular to Q, and thus transverse spin waves observed at the Q = (0, 1.5, 0) are dominated by [1, 0, 1] polarized modes. By further orienting the neutron’s spin polarization, P, parallel to Q, all allowed magnetic scattering is guaranteed to appear in the channel where the neutron’s spin is flipped during the scattering process^{19}. Fig. 4a shows the results of energy scans collected at Q = (0, 1.5, 0) with data collected in both the spinflip (SF) and non spinflip (NSF) channels. As expected, peaks from the ΔE = 6.15 and ΔE = 11.11 meV zone center modes appear only in the SF crosssections (dashed lines indicate the transmission expected by the polarization efficiency of SF scattering into the NSF channel and vice versa).
Using the same scattering geometry but with the neutron polarization now rotated parallel to the [−1, 0, 1] direction, the magnetic scattering processes polarized along the [1, 0, 1] axis (i.e., the resolvable transverse spin wave mode) should remain in the SF channel while scattering processes polarized parallel to the neutron polarization direction (i.e., nearly parallel to the ordered moment direction) will instead appear in the NSF channel. Figure 4b shows the results of energy scans with P[−1, 0, 1] where the 11 meV mode now appears only in the NSF channel and the 6 meV mode remains only in the SF channel. Again, the small amount of intensity around 6 meV in the NSF channel can be explained by the calculated contamination of scattering from the SF channel into the NSF channel. This demonstrates that the 11 meV mode and the associated upper branch of spin excitations are polarized longitudinally, reflecting an amplitude mode of the staggered magnetization, while the 6 meV mode and the lower energy branch of excitations are polarized transverse to the moment direction. Keeping P[−1, 0, 1], an identical energy scan collected at T = 50 K in the paramagnetic state shows that the coherent amplitude mode vanishes for T > T_{AF} and critical fluctuations driving the phase transition dominate the longitudinal spin response (Fig. 4c). Conversely, the transverse modes remain welldefined at high temperatures, reflective of the strong, inherently onedimensional coupling and the singleion anisotropy of the Mn moments.
Discussion
Earlier neutron measurements have demonstrated that the ordered moment of αNaMnO_{2} (2.92 μ_{B}) is significantly reduced from the classical expectation (4 μ_{B})^{11}, suggesting substantial fluctuation effects in this material. Additionally, ESR experiments find evidence of strong low temperature fluctuations in the ordered state^{18}. The ΔS = S − 〈S_{ z }〉 = 0.54 missing in the static ordered moment, however, can be accounted for when integrating the total inelastic spectral weight in earlier powder inelastic neutron measurements^{12}. Neutron scattering sum rules therefore imply that the ratio of the momentum and energy integrated weights of the longitudinal and transverse spin fluctuations should be \(\frac{{{\mathrm{\Delta }}S({\mathrm{\Delta }}S + 1)}}{{(S  {\mathrm{\Delta }}S)(2{\mathrm{\Delta }}S + 1)}} = 0.27\)^{20}, which is greater that the ratio of the intensities of the zone center modes I_{long}/I_{tran} = 0.19. This rough comparison suggests that, while the amplitude mode observed in our measurements is relatively intense, it remains within the bounds of the allowed spectral weight for longitudinal fluctuations.
Relative to stoichiometric αNaMnO_{2} powder samples, Na vacancies in the αNa_{0.9}MnO_{2} crystal studied here are unlikely to generate this longlived, dispersive amplitude mode. Each Na vacancy naively binds to a hole on the Mnplanes and creates an Mn^{4+} S = 3/2 magnetic impurity. The corresponding hole is bound to the impurity site due to strong polaron trapping. As this state remains localized within the lattice^{21,22}, the role of vacancies can be viewed as introducing random static magnetic impurities within the MnO_{6} planes. Neither this random, static disorder nor the high density of twin boundaries inherent to the lattice^{23} are capable of directly generating a coherent spin mode; however, they may indirectly contribute to destabilizing the ordered Néel state, pushing αNa_{0.9}MnO_{2} closer to a disordered regime and enhancing fluctuation effects.
For an easyaxis antiferromagnetic chain at 0 K, two degenerate, gapped transverse magnon modes are expected as Néel order sets in; however the amplitude mode observed in the ordered state of αNa_{0.9}MnO_{2} is unexpected. Since this longitudinally polarized mode has a zone center lifetime that is constrained by the resolution of the spectrometer (ΔE_{res} = 2.25 meV at E = 11 meV) without an observable high energy tail and with an energy below twice the transverse modes’ gap values, it likely arises as a longlived twomagnonbound state^{24,25}. The finite binding energy of this state as determined by E_{bind} = 2E_{gap} − E_{long} = 1.2 ± 0.1 meV at the zone center Q = (0.5, 0.5, 0) implies an attractive potential between magnons once Néel order is established.
To explore this further, we theoretically consider the existence of a bound state within a onedimensional model, neglecting J_{2}. We further consider only zero temperature for simplicity, and perform a semiclassical large S analysis based on anharmonically coupled spin waves around an antiferromagnetically ordered state. For J_{1} e^{−S} ≪ D ≪ J_{1}, and S ≫ 1, the onedimensional chain is ordered at T = 0, which allows J_{2} to be neglected without qualitative errors. As detailed in Supplementary Note 2, the resulting description applies: (1) the anisotropy induces a single magnon gap \({\mathrm{\Delta }} = 2S\sqrt {2J_1D}\) so that the magnon dispersion near the magnetic zone center is that of a relativistic massive particle with \(E = \sqrt {v^2k^2 + {\mathrm{\Delta }}^2}\) with magnon velocity v = 2J_{1}S, and (2) the dominant interaction between opposite spin magnons with momentum \(k \ll \sqrt {\frac{D}{{J_1}}}\) is an attractive deltafunction of strength U = −2J_{1}. This problem has a bound state which is in the nonrelativistic limit described by two onedimensional particles of mass \(M = \frac{\Delta }{{v^2}}\), for which the textbook result for the binding energy is \(E_{{\mathrm{bind}}} = \frac{{MU^2}}{4}\) . Using the values above, we obtain \(\frac{{E_{{\mathrm{bind}}}}}{\Delta } = \frac{1}{{4S^2}}\). This is the leading result for large S, and in the limit D/J_{1} ≪ 1. While this limit predicts a binding energy approximately 3 times smaller than that observed for αNa_{0.9}MnO_{2}, moving away from this limit and incorporating the nonnegligible D in the system can account for this discrepancy. We note that, while we performed calculations in the onedimensional model for simplicity, this is only a matter of convenience rather than essential physics: the magnons and their dispersion and interactions evolve smoothly upon including J_{2}, which would be necessary to model the spectrum at T > 0.
The longlived amplitude mode in the Néel state of αNa_{0.9}MnO_{2} is distinct from those observed in canonical 1D integerspin chain systems such as CsNiCl_{3}^{4}, where the longitudinal mode emerges as the Haldane triplet state splits due to an internal staggered mean field. The Haldane state within the frustrationdriven S = 2 spin chains in αNa_{0.9}MnO_{2} is easily quenched under small anisotropy^{13,26,27}, and αNa_{0}._{9}MnO_{2} is thought to be outside of the Haldane regime. Calculations predict that the phase boundary between the S = 2 Haldane state and antiferromagnetic order appears at D/J = 0.0046 (for easyaxis D)^{27}, far away from the experimentally measured D/J = 0.035 in Na_{0.9}MnO_{2}. While amplitude modes in other quantum spin systems close to singlet instabilities have also been recently reported in the quasitwodimensional spin ladder compound C_{9}H_{18}N_{2}CuBr_{4}^{28} and the twodimensional ruthenate Ca_{2}RuO_{4}^{29}, the formation of a longitudinalbound state in αNa_{0.9}MnO_{2} is distinct from modes in these and other S = 1/2 spin chain systems possessing substantial zeropoint fluctuations^{8}.
Instead, in αNa_{0.9}MnO_{2}, the interplay of geometric frustration and the Jahn–Teller quenching of orbital degeneracy uniquely conspire to create a quasionedimensional magnon spectrum that condenses due to the attractive potential provided by an Isinglike singleion anisotropy. As a result, αNa_{0.9}MnO_{2} provides an intriguing route to realizing an intense, stable amplitude mode in a planar AF. The dimensionality reduction realized within its chemically twodimensional lattice also suggests that other αNaFeO_{2} type transition metal oxides^{30,31} possessing coherent Jahn–Teller distortions may host similarly stable amplitude modes, depending on their inherent anisotropies. More broadly this class of materials presents an exciting platform for exploring unconventionalbound states such as bound soliton modes^{14} stabilized in a quasionedimensional spin setting.
Methods
Crystal growth and characterization
Na_{2}CO_{3} and MnCO_{3} powders (1:1 ratio plus 10% weight excess of Na_{2}CO_{3}) were mixed and sintered in an alumina crucible at 350 °C for 15 h, reground and sintered for an additional 15 h at 750 °C. Dense polycrystalline rods were made by pressing the powder at 50,000 psi in an isostatic press. The rod was sintered in a vertical furnace at 1000 °C for 15 h and then quenched in air, before being transferred to a four mirror optical floating zone furnace outfitted with 500 W halogen lamps. The crystals were grown at a rate of 20 mm h^{−1} in a 4:1 Ar:O_{2} environment under 0.15 MPa of pressure. Inductively coupled plasma atomic emission spectroscopy (ICPAES) was used to determine the Na/Mn ratio and to check that the expected mass of Mn was present. The ratio of Mn^{3+}/Mn^{4+} was determined through Xray absorption near edge spectroscopy (XANES), Xray photoelectron spectroscopy (XPS), and ^{23}Na solidstate NMR (ssNMR). Detailed crystal growth and characterization can be found in Dally et al.^{23} Samples were handled as airsensitive and stored in an inert environment. Time outside of an inert environment was minimized (e.g., during crystal alignment for neutron scattering experiments). The crystal faces are flat, and no degradation of the surface was observed during alignment. The same ~0.5 g crystal was used for both neutron TOF and tripleaxis experiments.
Timeofflight experimental setup
Neutron timeofflight data in Fig. 2 and Supplementary Figs. 2b and 4 were taken at the Spallation Neutron Source at Oak Ridge National Laboratory using the instrument SEQUOIA. The sample was sealed in a Hegas environment, mounted in a cryostat, and aligned in the (H, K, 0) horizontal scattering plane. All data were taken at 4 K with an incident energy E_{i} = 60 meV and the fineresolution fermi chopper rotating at 420 Hz. For data collection, the sample was rotated through a range of 180° with 1° steps. A background scan was collected by removing the sample from the neutron beam and collecting the scattering from the empty can.
Timeofflight data analysis
An aluminum only (empty can) background was subtracted from all data before plotting. SpinW^{32}, a Matlab library, was used to simulate the magnetic excitations for the TOF data. Given the spin Hamiltonian, magnetic structure and twinning mechanisms (structural and magnetic), SpinW uses linear spin wave theory to numerically calculate and display the dispersion. The simulation was convolved with the energy resolution function of the neutron spectrometer (E_{i} = 60 meV, F_{chopper} = 420 Hz). Simulations for Fig. 2c were run over the same range that the data were binned for Fig. 2a and d (i.e., 1.4 < H < 1.6 and −0.1 < L < 0.1), and then averaged together.
Tripleaxis experimental setup
Tripleaxis neutron data were taken with the instrument, BT7^{33,34}, at NCNR with a PG(002) vertically focused monochromator and the horizontally flat focus mode of the PG analyzer system. PG filters before and after the sample were used during collection of elastic data (E_{i} = 14.7 meV), and only a PG filter after the sample was used during inelastic operation (fixed E_{f} = 14.7 meV). Unpolarized data from Fig. 3 and Supplementary Fig. 3 were taken with open−25′−50′−120′ collimations (denoting the collimation before the monochromator, sample, analyzer, and detector, respectively), and the sample was aligned in the (H, K, 0) scattering plane. Supplementary Fig. 2a data were unpolarized and taken with open−50′−50′−120′ collimators in the (H, H, L) plane. Polarized data in Fig. 1c were taken with open−25’−25′−120′ collimation in the (H, K, 0) scattering plane. Polarized data in Fig. 4 were taken in the (H, K, H) plane with open−80′−80′−120′ collimations.
Tripleaxis polarization efficiency corrections
It was determined that only two (one NSF and one SF) of the available four neutron scattering crosssections were needed for polarization analysis. Flipping ratios were taken throughout the experiment at the (2, 0, 0) nuclear Bragg peak and at all temperatures probed. These flipping ratios were used to correct for the polarization efficiency.
Tripleaxis data analysis
All data were normalized to the neutron monitor counts, M. Error bars represent one standard deviation of the data. For unpolarized data, this was calculated by the square root of the number of counts, \(\sqrt N\), where N is the number of counts. The lower monitor counts in polarized data were considered by propagating the error in the monitor counts, \(\sqrt M\), such that \(\sigma ^2 = \frac{N}{{M^2}}\left( {1 + \frac{N}{M}} \right)\). The determination of the moment angle utilized the polarized elastic data shown in Fig. 1c. After correcting for the polarization efficiency, the integrated intensities of the NSF and SF peaks were found by fitting the data to Gaussian functions. These intensities were used to find the moment angle following the technique in Moon et al.^{19}
Fits to the constant energy scans (unpolarized inelastic neutron scattering data) in Fig. 2d, 3a, b and Supplementary Fig. 3 used the Cooper–Nathans approximation^{35} in ResLib^{36}, a program that calculates the convolution of the spectrometer resolution function with a user supplied cross section. The cross section used for the transverse excitation was the singlemode approximation of a twodimensional spin lattice with singleion anisotropy, as described in Supplementary Note 1. Crosssections for t_{1} − q_{1}, t_{1} − q_{2}, t_{2} − q_{1}, and t_{2} − q_{2} were all included during the fitting routine using the relation between the first moment sum rule and the dynamical structure factor, \({\int}_{  \infty }^\infty {\left( {\hbar \omega } \right)} S^{\alpha \alpha }\left( {{\mathbf{q}},\hbar \omega } \right){\mathrm d}\left( {\hbar \omega } \right) =  \mathop {\sum}\nolimits_{n,\beta } {J_n} \left[ {1  \cos \left( {{\mathbf{q}} \cdot {\mathbf{a}}_n} \right)} \right]\left( {1  \delta _{\alpha \beta }} \right)\langle\langle S_{{\mathbf{R}}_j}^\beta ,S_{{\mathbf{R}}_j + {\mathbf{a}}_n}^\beta \rangle\rangle\). The contribution to the scaling factor from the singleion term is small^{37}, and therefore, was not included. The longitudinal excitation was empirically fit using the singlemode approximation for a onedimensional chain, given its unresolvable dispersion along H. The longitudinal mode gap was determined by fitting the data in the range where it was resolvable (below 18 meV). This gap value was fixed and the fitting routine was run again, allowing all other parameters to vary. A single intrinsic HWHM for all excitations was refined during fitting and refined to be negligibly small. Additionally, a scaling prefactor was also refined for each crystallographic twin, where the two magnetic domains within a crystallographic twin were assumed to have the same weight (i.e., t_{1} − q_{1} and t_{1} − q_{2} had the same prefactor).
Polarized inelastic neutron data in Fig. 4 are plotted as raw data, uncorrected for polarization efficiency. The dashed lines representing the expected bleed through from the SF channel into the NSF channel in Fig. 4a and b were determined from the measured flipping ratio.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
 1.
Pekker, D. & Varma, C. M. Amplitude/Higgs modes in condensed matter physics. Annu. Rev. Conden. Mater. Phys. 6, 269–297 (2015).
 2.
Morra, R. M., Buyers, W. J. L., Armstrong, R. L. & Hirakawa, K. Spin dynamics and the Haldane gap in the spin1 quasionedimensional antiferromagnet CsNiCl_{3}. Phys. Rev. B 38, 543–555 (1988).
 3.
Raymond, S. et al. Polarizedneutron observation of longitudinal haldanegap excitations in Nd_{2}BaNiO_{5}. Phys. Rev. Lett. 82, 2382–2385 (1999).
 4.
Enderle, M., Tun, Z., Buyers, W. J. L. & Steiner, M. Longitudinal spin fluctuations of coupled integerspin chains: haldane triplet dynamics in the ordered phase of CsNiCl_{3}. Phys. Rev. B 59, 4235–4243 (1999).
 5.
Grenier, B. et al. Longitudinal and transverse Zeeman ladders in the Isinglike chain antiferromagnet BaCo_{2}V_{2}O_{8}. Phys. Rev. Lett. 114, 017201 (2015).
 6.
Rüegg, Ch et al. Quantum magnets under pressure: controlling elementary excitations in TlCuCl_{3}. Phys. Rev. Lett. 100, 205701 (2008).
 7.
Merchant, P. et al. Quantum and classical criticality in a dimerized quantum antiferromagnet. Nat. Phys. 10, 373–379 (2014).
 8.
Lake, B., Tennant, D. A. & Nagler, S. E. Novel longitudinal mode in the coupled quantum chain compound KCuF_{3}. Phys. Rev. Lett. 85, 832–835 (2000).
 9.
Zheludev, A., Kakurai, K., Masuda, T., Uchinokura, K. & Nakajima, K. Dominance of the excitation continuum in the longitudinal spectrum of weakly coupled Heisenberg S=1/2 chains. Phys. Rev. Lett. 89, 197205 (2002).
 10.
Coldea, R., Tennant, D. A., Tsvelik, A. M. & Tylczynski, Z. Experimental realization of a 2D fractional quantum spin liquid. Phys. Rev. Lett. 86, 1335–1338 (2001).
 11.
Giot, M. et al. Magnetoelastic coupling and symmetry breaking in the frustrated antiferromagnet αNaMnO_{2}. Phys. Rev. Lett. 99, 247211 (2007).
 12.
Stock, C. et al. Onedimensional magnetic fluctuations in the Spin2 triangular lattice αNaMnO_{2}. Phys. Rev. Lett. 103, 077202 (2009).
 13.
Schollwöck, U. & Jolicoeur, T. Haldane gap and hidden order in the S=2 antiferromagnetic quantum spin chain. Europhys. Lett. 30, 493–498 (1995).
 14.
Haldane, F. D. M. Continuum dynamics of the 1D Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model. Phys. Lett. A 93, 464–468 (1983).
 15.
Haldane, F. D. M. Nonlinear field theory of largespin heisenberg antiferromagnets: semiclassically quantized solitons of the onedimensional easyAxis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983).
 16.
Abakumov, A. M., Tsirlin, A. A., Bakaimi, I., Van Tendeloo, G. & Lappas, A. Multiple twinning as a structure directing mechanism in layered rocksalttype oxides: NaMnO_{2} polymorphism, redox potentials, and magnetism. Chem. Mater. 26, 3306–3315 (2014).
 17.
Zorko, A., Adamopoulos, O., Komelj, M., Arčon, D. & Lappas, A. Frustrationinduced nanometrescale inhomogeneity in a triangular antiferromagnet. Nat. Commun. 5, 3222 (2014).
 18.
Zorko, A. et al. Magnetic interactions in αNaMnO2: quantum spin2 system on a spatially anisotropic twodimensional triangular lattice. Phys. Rev. B 77, 024412 (2008).
 19.
Moon, R. M., Riste, T. & Koehler, W. C. Polarization analysis of thermalneutron scattering. Phys. Rev. 181, 920–931 (1969).
 20.
Huberman, T. et al. Twomagnon excitations observed by neutron scattering in the twodimensional spin 5/2− isenberg antiferromagnet Rb_{2}MnF_{4}. Phys. Rev. B 72, 014413 (2005).
 21.
Ma, X., Chen, H. & Ceder, G. Electrochemical properties of monoclinic NaMnO_{2}. J. Electrochem. Soc. 158, A1307–A1312 (2011).
 22.
Jia, T. et al. Magnetic frustration in αNaMnO_{2} and CuMnO_{2}. J. Appl. Phys. 109, 07E102 (2011).
 23.
Dally, R. et al. Floating zone growth of αNa_{0.90}MnO_{2} single crystals. J. Cryst. Growth 459, 203–208 (2017).
 24.
Xian, Y. Longitudinal excitations in quantum antiferromagnets. J. Phys. Condens. Mat. 23, 346003 (2011).
 25.
Heilmann, I. U. et al. One and twomagnon excitations in a onedimensional antiferromagnet in a magnetic field. Phys. Rev. B 24, 3939–3953 (1981).
 26.
Schollwöck, U., Golinelli, O. & Jolicœur, T. S=2 antiferromagnetic quantum spin chain. Phys. Rev. B 54, 4038–4051 (1996).
 27.
Kjäll, J. A., Zaletel, M. P., Mong, R. S. K., Bardarson, J. H. & Pollmann, F. Phase diagram of the anisotropic spin2 XXZ model: Infinitesystem density matrix renormalization group study. Phys. Rev. B 87, 235106 (2013).
 28.
Hong, T. et al. Higgs amplitude mode in a twodimensional quantum antiferromagnet near the quantum critical point. Nat. Phys. 13, 638–642 (2017).
 29.
Jain, A. et al. Higgs mode and its decay in a twodimensional antiferromagnet. Nat. Phys. 13, 633–637 (2017).
 30.
Mostovoy, M. V. & Khomskii, D. I. Orbital ordering in frustrated Jahn–Teller systems with 90° exchange. Phys. Rev. Lett. 89, 227203 (2002).
 31.
McQueen, T. et al. Magnetic structure and properties of the S=5/2 triangular antiferromagnet αNaFeO_{2}. Phys. Rev. B 76, 024420 (2007).
 32.
Toth, S. & Lake, B. Linear spin wave theory for singleQ incommensurate magnetic structures. J. Phys. Condens. Matter 27, 166002 (2015).
 33.
Lynn, J. W. et al. Double focusing thermal triple axis spectrometer at the NCNR. J. Res. Natl Inst. Stan. 117, 61–79 (2012).
 34.
Chen, W. C. et al. ^{3}He neutron spin filters for a thermal neutron triple axis spectrometer. Phys. B 397, 168–171 (2007).
 35.
Cooper, M. J. & Nathans, R. The resolution function in neutron diffractometry. I. The resolution function of a neutron diffractometer and its application to phonon measurements. Acta Crystallogr. 23, 357–367 (1967).
 36.
Zheludev, A. ResLib (ETH Zürich, 2009).
 37.
Zaliznyak, I. & Lee, S.H. in Modern Techniques for Characterizing Magnetic Materials, Ch. 1 (ed. Zhu, Y.) (Springer, Heidelberg, 2005).
Acknowledgements
S.D.W. and R.L.D. acknowledge assistance in characterizing samples from Raphaële Clément. S.D.W. and R.L.D. gratefully acknowledge support from DOE, Office of Science, Basic Energy Sciences under Award DESC0017752. Work by L.B. was supported by the DOE, Office of Science, Basic Energy Sciences under Award No. DEFG0208ER 46524.
Author information
Affiliations
Contributions
R.L.D. synthesized the αNaMnO_{2} single crystals and analyzed the data. R.L.D., Y.Z., Z.X., R.C., M.B.S., and J.W.L. helped perform neutron experiments. L.B. performed theoretical analysis of the spin dynamics. R.L.D. and S.D.W. designed the neutron experiments. R.L.D., L.B., and S.D.W. prepared and wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Dally, R.L., Zhao, Y., Xu, Z. et al. Amplitude mode in the planar triangular antiferromagnet Na_{0.9}MnO_{2}. Nat Commun 9, 2188 (2018). https://doi.org/10.1038/s41467018046011
Received:
Accepted:
Published:
Further reading

ThreeMagnon Bound State in the QuasiOneDimensional Antiferromagnet α  NaMnO2
Physical Review Letters (2020)

Nanoscale degeneracy lifting in a geometrically frustrated antiferromagnet
Physical Review B (2020)

Amplitude modes in threedimensional spin dimers away from quantum critical point
Physical Review Research (2019)

Cluster glass behavior of the frustrated birnessites AxMnO2·yH2O(A=Na,K)
Physical Review B (2019)

Thermal evolution of quasionedimensional spin correlations within the anisotropic triangular lattice of α−NaMnO2
Physical Review B (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.