ARTICLE DOI: 10.1038/s41467-018-04601-1 OPEN Amplitude mode in the planar triangular antiferromagnet Na MnO 0.9 2 1,2 3,4 3,4 3 5 3 Rebecca L. Dally , Yang Zhao , Zhijun Xu , Robin Chisnell , M.B. Stone , Jeffrey W. Lynn , 6 1 Leon Balents & Stephen D. Wilson 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 para- meter’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 two-dimensional anisotropic triangular lattice (α-Na MnO ) exhibits a 0.9 2 long-lived, coherent oscillation of its staggered magnetization ﬁeld. Our results show that geometric frustration of Heisenberg spins with uniaxial single-ion anisotropy can renormalize the interactions of a dense two-dimensional network of moments into largely decoupled, one- dimensional chains that manifest a longitudinally polarized-bound state. This bound state is 3+ driven by the Ising-like anisotropy inherent to the Mn ions of this compound. 1 2 Materials Department, University of California, Santa Barbara, CA 93106, USA. Department of Physics, Boston College, Chestnut Hill, MA 02467, USA. 3 4 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. Department of Materials Science and Engineering, University of Maryland, College Park, MD 20742, USA. Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, Santa Barbara, CA 93106, USA. Correspondence and requests for materials should be addressed to S.D.W. (email: firstname.lastname@example.org) NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications 1 | | | 1234567890():,; b ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04601-1 any of the seminal observations of amplitude modes in descriptions of the lattice and spin structures are ﬁrst necessary. magnetic materials arise from quantum effects in one- We note here that units for wave vectors throughout the manu- Mdimensional antiferromagnetic chain systems when script are given in reciprocal lattice units (H, K, L) where hi interchain coupling drives the formation of long-range magnetic 2π 2π 2π Q Å ¼ H; K ; L and a, b, c, and β are the lattice 1 a sin β b c sin β order . For instance, bound states observed in the ordered phases 2–4 parameters of the unit cell. Figure 1a shows the projection of the of S = 1 Haldane systems or in the spinon continua of S = 1/2 5–9 quantum spin chains were shown to be longitudinally polar- ized and reﬂective of the crossover into an ordered spin state. While the chemical connectivity of magnetic ions in these systems is inherently one-dimensional, alternative geometries such as Na planar, anisotropic triangular lattices can also in principle stabi- lize predominantly one-dimensional interactions in antiferro- Mn magnets . 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 two-dimensional net- work of magnetic moments whose dimensionality of interaction is reduced to be quasi one-dimensional. A promising example of such an anisotropic triangular lattice structure is realized in α-phase NaMnO . Layered sheets of edge- Twin 2 sharing MnO octahedra are separated by layers of Na ions, and 6 (t ) 3+ the orbital degeneracy of the octahedrally coordinated Mn 4 3 1 cations (3d , t e valence) is lifted via a large, coherent 2g g Jahn–Teller distortion . This distorts the triangular lattice such Twin 1 that the leg along the in-plane b-axis is contracted 10% relative to (t ) 3+ the remaining two legs. As a result, the S = 2 spins of the Mn ions decorate a dimensionally frustrated lattice where one- b a dimensional intrachain coupling along b is favored and interchain coupling is highly frustrated. This spin lattice eventually freezes Twin boundary into a long-range ordered state below T = 45 K ; however, previous studies of powder samples have suggested an inherently one-dimensional character to the underlying spin dynamics . Such a scenario suggests an intriguing material platform for the NSF stabilization of an amplitude mode in a conventional spin system SF (i.e., one with diminished local moment ﬂuctuations and a quenched Haldane state ) 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 α-Na MnO exhibits 0.9 2 quasi one-dimensional 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 polarized-bound 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 integer-spin Haldane systems 0.48 0.49 0.50 0.51 0.52 —both typical settings for longitudinal-bound state formation— (0.5, K, 0) (r.l.u.) are absent. To explain the stabilization of this amplitude mode, we present a model that captures the excitation as a two-magnon- Fig. 1 Summary of the crystal and magnetic structures of α-NaMnO . a bound state whose binding energy derives from an easy-axis Projection of the 3D magnetically ordered state onto the ab-plane. The 3+ single-ion anisotropy inherent to the orbitally quenched Mn black rectangle denotes the chemical unit cell. The gray shaded region ions. This anisotropy orients the moments along a preferred axis highlights the chain direction between nearest neighbor Mn atoms. b and renders them Ising-like. Our work establishes α-Na MnO Moments in the ac-plane, as well as the intergrowth of two twin domains. x 2 and related lattice geometries as platforms for realizing uncon- The black rhombus again denotes the chemical unit cell. Polarized elastic neutron data at the antiferromagnetic zone center in c conﬁrm the ventional spin dynamics in a dense network of one-dimensional 14,15 antiferromagnetic spin chains . orientation of the magnetic moments within the ordered state (T = 2.5 K) as previously reported. The shaded regions are the Gaussian ﬁts of the non spin-ﬂip (NSF) and spin-ﬂip (SF) channels, which were used to determine Results the moment orientation. The neutron polarization P is perpendicular to the Crystal and spin structures of α-Na MnO . To demonstrate scattering vector and parallel to the c-axis in this conﬁguration. Error bars 0.9 2 the emergence of an amplitude mode in α-Na MnO , careful represent one standard deviation 0.9 2 2 NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications | | | Intensity (a.u.) Intensity (a.u.) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04601-1 ARTICLE 30 0.2 a b 0.15 0.1 0.05 0 0 –1 –0.5 0 0.5 1 1 1.5 2 (1.5, K, 0) (r.l.u.) (H, 0.5, 0) (r.l.u.) c d t -q 1 1 t -q 1 2 20 t -q 2 1 t -q 2 2 –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 (1.5, K, 0) (r.l.u.) (1.5, K, 0) (r.l.u.) Fig. 2 Magnon spectra at T = 2.5 K collected via time-of-ﬂight neutron scattering measurements. a Dispersion of magnons along the chain axis (K- direction) across the full bandwidth of excitations. Data were integrated across −0.1 to 0.1 along L and from 1.4 to 1.6 along H. b Dispersion along the interchain axis (H-direction) with data integrated from −0.1 to 0.1 along L and from 0.48 to 0.52 along K. c Simulated scattering intensities using the model ﬁt to the data described in the main text and integrated over the same values as a. d Fit transverse modes from the four allowed domains generating the spectral weight in c and then overplotted with the raw data from a low temperature, ordered spin lattice of α-NaMnO onto the ab- distortion in NaMnO is subtle and would generate roughly a 2 2 plane where one spin domain with propagation vector q = (0.5, 0.12% difference between next-nearest neighbor (interchain) 0.5, 0) is illustrated. The antiferromagnetic chain direction is exchange pathways, which can be neglected for the purposes of shaded in gray. Due to the degeneracy of the frustrated interchain the present study. coupling, a second spin domain with propagation vector q = (−0.5, 0.5, 0) also stabilizes, and the moments in both domains Spin Hamiltonian of α-Na MnO . In order to understand the 0.9 2 are oriented approximately along the [−1, 0, 1] apical oxygen interactions underlying the AF ground state of this system, bond direction due to an inherent uniaxial single-ion aniso- inelastic neutron scattering measurements were performed. Spin tropy . The large Jahn–Teller distortion renders the lattice excitations measured within the ordered state about the AF zone structure prone to crystallographic twinning and the relative centers, Q = (1.5, ±0.5, 0), are shown in Fig. 2. Inspection of the orientations of the moments in the two resulting crystallographic momentum distribution of the spectral weight reveals that the twins, twin 1 (t ) and twin 2 (t ) are depicted in Fig. 1b. This 1 2 magnetic ﬂuctuations underpinning the AF state at T = 2.5 K are results in four allowed domains: t − q , t − q , t − q , and t − 1 1 1 2 2 1 2 quasi one-dimensional. Figure 2a demonstrates that the magnetic q . Crucially, the moments in both crystallographic and magnetic excitations along the in-plane K-axis, parallel to the short leg of twin domains are nearly parallel to a common [−1, 0, 1] axis. the triangular lattice, show an anisotropy gap at the zone center This is veriﬁed via polarized elastic neutron scattering measure- and a well-deﬁned dispersion; however, the magnon dispersion in ments in the AF state at T = 2.5 K shown in Fig. 1c. These data directions orthogonal to this axis are diffuse. Speciﬁcally, the spin demonstrate that in a single domain, t − q , probed at Q = (0.5, 1 1 waves dispersing between the MnO planes (along L) are dis- 0.5, 0) the moments are rotated ~7° away from the [−1, 0, 1] axis persionless (see Supplementary Fig. 2) as expected for the planar within the ac-plane. We note here that previous studies of stoi- structure of α-Na MnO , and Fig. 2b shows that spin wave 0.9 2 chiometric NaMnO have reported an extremely subtle distortion energies dispersing perpendicular to the AF chain direction in the into a lower triclinic symmetry below the antiferromagnetic plane (along H) are only weakly momentum dependent. This transition . Our neutron diffraction measurements fail to detect demonstrates that the spin ﬂuctuations exist as quasi-one- this distortion in the average structure of Na MnO crystals, and 0.9 2 dimensional planes of scattering in (Q, E) space, driven by the we therefore analyze data using the higher symmetry monoclinic strong interchain frustration inherent to the lattice and consistent structure. A recent report suggests that the triclinic phase occurs with the large magnetic frustration parameter of this only as an inhomogeneous local distortion ; hence our inability compound . to observe the reported triclinic distortion may arise from its As twin effects from both crystallographic and spin domains absence in the average structure, its suppression due to the Na may obscure any subtle dispersion along H due to interchain vacancies in our samples, or due to the resolution threshold of interactions, inspection of zone boundary energies and analysis of our measurements. Despite this ambiguity, the reported triclinic the full bandwidth are necessary to quantify the weak interchain NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications 3 | | | Energy (meV) Energy (meV) Energy (meV) Intensity (a.u.) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04601-1 (0.5, K, 0) (r.l.u.) 0.4 0.5 0.6 0.7 a b 17 meV 16 meV 15 meV t -q t -q , t -q 1 1 2 1 2 2 14 meV Longitudinal t -q 1 2 13 meV 06 12 3 4 5 Intensity (a. u.) 12 meV c (0.5, 0.5, 0) 25 (0, 1.5, 0) 11 meV 10 meV 6 meV 0.4 0.6 0.8 2 4 6 8 10 12 14 (0.5, K, 0) (r.l.u.) Energy (meV) Fig. 3 Inelastic neutron scattering data at T = 2 K revealing an additional zone center mode. a Momentum scans at various energies through Q = (0.5, 0.5, 0). Solid lines are resolution convolved ﬁts to the data as described in the text. b An intensity color map summarizing the scattering data from a along with dispersion of modes comprising the ﬁts to the data in a. Lines show ﬁts to the expected transverse modes from the different crystallographic and magnetic domains within the sample: t -q (dashed purple), t -q (dashed blue), and t -q and t -q (solid green). The black dashed line represents the dispersion ﬁt 1 1 1 2 2 1 2 2 to the longitudinal mode in the spectrum as described in the text. c Constant energy scans at the three dimensional (0.5, 0.5, 0) and quasi one-dimensional (0, 0.5, 0) AF zone centers. Error bars in a and c represent one standard deviation exchange terms. Therefore, in order to parameterize the total simulated intensities summed from all modes are shown in dispersion measured in Fig. 2a, the high energy data were Fig. 2c. Good agreement is seen between the data in Fig. 2a and analyzed using a four-domain model (t − q , t − q , t − q , t the simulated intensities shown in Fig. 2c and Supplementary 1 1 1 2 2 1 2 − q ) as well as by ﬁtting lower energy triple-axis data shown in Fig. 4. To further illustrate this model at lower energies closer to Fig. 3 and Supplementary Fig. 3. The data were modeled using the the zone center gap value, data collected via a thermal triple-axis single-mode approximation and the spin Hamiltonian, spectrometer are shown in Fig. 3. Momentum scans through the P P P AF zone center are plotted in Fig. 3a at energies from ΔE = 3 meV H ¼ J S S þ J S S D ðS Þ , where J is the 1 nn i j 2 nnn i j n n to 18 meV with the resulting color map of intensities plotted in twofold nearest neighbor exchange coupling, J is the fourfold Fig. 3b. Magnon modes dispersing from the four domains in the next-nearest neighbor coupling, and D is a uniaxial, Ising-like, system using the same J –J –D model described earlier and 1 2 single-ion anisotropy term. The dispersion relation generated convolved with the instrument resolution function are plotted as from a linear spin wave analysis of this Hamiltonian is given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ solid lines ﬁt to the data in Fig. 3a. Crucially, unlike the model EðQÞ¼ S ω λ , where ω = 2(J + D + J cos(πH + πK)) Q 1 2 Q presented in Fig. 2, describing this lower energy data also requires the introduction of one additional dispersive mode. This mode is and λ ¼ 2ðÞ J cosðÞ 2πK þ J cosðÞ πH πK (see Supplementary Q 1 2 distinct from the transversely polarized magnons anticipated in Note 1 and Supplementary Fig. 1 for details), and the results from this material, and it represents an unexpected longitudinally ﬁtting the data yielded a J = 6.16 ± 0.01 meV, J = 0.77 ± 0.01 1 2 polarized-bound state as described in the next section. meV, and D = 0.215 ± 0.001 meV. These values are roughly consistent with earlier powder averaged measurements of spin dynamics in α-NaMnO , although these earlier measurements Longitudinally polarized mode. To more clearly illustrate the were not sensitive enough to resolve a J term . appearance of an additional mode in the low energy spin The magnon modes from the four-domain model are over- dynamics, a magnetic zone center energy scan at Q = (0.5, 0.5, 0) plotted as lines with the raw time-of-ﬂight data in Fig. 2d, and the is plotted in Fig. 3c. This scan shows the large buildup of spectral 4 NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications | | | Intensity (a.u.) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04601-1 ARTICLE weight above the ΔE = 6.15 ± 0.04 meV zone center gap, con- sistent with the quasi one-dimensional magnon density of states, P || [0 1 0] and above this gap a second zone center mode near 11 meV Q || P NSF appears. This 11 meV mode is not accounted for by any of the 2.5 K expected transverse modes in this system, and it is also quasi one- 40 SF dimensional in nature. Figure 3c demonstrates negligible inter- chain dispersion as Q is rotated from the three dimensional Q = 30 (0.5, 0.5, 0) to one-dimensional 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 one- P P dimensional J–D model with H ¼ J S S D ðS Þ n nþ1 n n n pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 and EðÞ Q ¼ Δ þ c sinðÞ 2πK . The gap value from this para- meterization was ﬁttobe Δ = 11.11 ± 0.06 meV and c = 21.6 ± 0.1 meV. The dispersion ﬁt to this higher energy mode along with 50 P || [–1 0 1] the dispersions ﬁt to the transverse magnon modes are over- QP plotted with the data in Fig. 3b. We again note that this additional 2.5 K 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, polarized-neutron scattering measurements were performed using an experimental geometry that leveraged the quasi one-dimensional nature of the spin excitations. Speciﬁcally, magnetic excitations were measured about the one-dimensional 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/ c phase of the staggered magnetization. The (Q × μ × Q) orienta- P || [–1 0 1] tion factor in the neutron scattering cross section renders it only QP sensitive to the components of the moments’ ﬂuctuations perpendicular to Q, and thus transverse spin waves observed at 50 K 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 . Fig. 4a shows the results of energy scans collected at Q 20 = (0, 1.5, 0) with data collected in both the spin-ﬂip (SF) and non spin-ﬂip (NSF) channels. As expected, peaks from the ΔE = 6.15 and ΔE = 11.11 meV zone center modes appear only in the SF cross-sections (dashed lines indicate the transmission expected by the polarization efﬁciency of SF scattering into the NSF channel and vice versa). 2 4 6 8 10 12 14 16 Using the same scattering geometry but with the neutron Energy (meV) polarization now rotated parallel to the [−1, 0, 1] direction, the magnetic scattering processes polarized along the [1, 0, 1] axis Fig. 4 Polarized inelastic neutron scattering data about the quasi-1D zone (i.e., the resolvable transverse spin wave mode) should remain in center Q = (0, 1.5, 0). a Data collected with the neutron polarization P the SF channel while scattering processes polarized parallel to the parallel to Q. The two modes appear only in the spin-ﬂip (SF) channel and neutron polarization direction (i.e., nearly parallel to the ordered so are both magnetic in origin. b Data collected with the neutron moment direction) will instead appear in the NSF channel. polarization P parallel to the [−1, 0, 1] axis. Transverse spin ﬂuctuations Figure 4b shows the results of energy scans with P||[−1, 0, 1] appear in the SF channel, and longitudinal ﬂuctuations appear in the non where the 11 meV mode now appears only in the NSF channel spin-ﬂip (NSF) channel. Data in both a and b were taken in the 3D ordered and the 6 meV mode remains only in the SF channel. Again, the state at T = 2.5 K, and black dashed lines indicate the expected bleed small amount of intensity around 6 meV in the NSF channel can through from the SF channel into the NSF channel due to imperfect neutron be explained by the calculated contamination of scattering from polarization. The red dashed line in b represents the expected bleed the SF channel into the NSF channel. This demonstrates that the through from the NSF channel into the SF channel. c The same 11 meV mode and the associated upper branch of spin excitations conﬁguration as b but above the antiferromagnetic transition temperature are polarized longitudinally, reﬂecting an amplitude mode of the at T = 50 K. Solid lines in a and b are Gaussian ﬁts parameterizing each staggered magnetization, while the 6 meV mode and the lower mode. The solid line for the NSF channel in c is a Lorentzian ﬁt centered at energy branch of excitations are polarized transverse to the ΔE = 0 meV and the solid line for the SF channel in c consists of two moment direction. Keeping P||[−1, 0, 1], an identical energy scan Lorentzian ﬁts (one centered at ΔE = 0 meV and one at the single magnon collected at T = 50 K in the paramagnetic state shows that the gap energy). Error bars represent one standard deviation NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications 5 | | | Intensity (a.u.) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04601-1 qﬃﬃﬃ coherent amplitude mode vanishes for T > T and critical AF momentum k is an attractive delta-function of strength U ﬂuctuations driving the phase transition dominate the long- = −2J . This problem has a bound state which is in the non- itudinal spin response (Fig. 4c). Conversely, the transverse modes relativistic limit described by two one-dimensional particles of remain well-deﬁned at high temperatures, reﬂective of the strong, mass M ¼ , for which the textbook result for the binding energy inherently one-dimensional coupling and the single-ion aniso- 2 MU 1 bind tropy of the Mn moments. is E ¼ . Using the values above, we obtain ¼ . This bind 4 Δ 4S is the leading result for large S, and in the limit D/J ≪ 1. While this limit predicts a binding energy approximately 3 times smaller Discussion than that observed for α-Na MnO , moving away from this limit Earlier neutron measurements have demonstrated that the 0.9 2 and incorporating the non-negligible D in the system can account ordered moment of α-NaMnO (2.92 μ ) is signiﬁcantly reduced 2 B for this discrepancy. We note that, while we performed calcula- from the classical expectation (4 μ ) , suggesting substantial tions in the one-dimensional model for simplicity, this is only a ﬂuctuation effects in this material. Additionally, ESR experiments matter of convenience rather than essential physics: the magnons ﬁnd evidence of strong low temperature ﬂuctuations in the and their dispersion and interactions evolve smoothly upon ordered state . The ΔS = S − 〈S 〉 = 0.54 missing in the static including J , which would be necessary to model the spectrum at ordered moment, however, can be accounted for when integrating 2 T >0. the total inelastic spectral weight in earlier powder inelastic The long-lived amplitude mode in the Néel state of α- neutron measurements . Neutron scattering sum rules therefore Na MnO is distinct from those observed in canonical 1D imply that the ratio of the momentum and energy integrated 0.9 2 integer-spin chain systems such as CsNiCl , where the long- weights of the longitudinal and transverse spin ﬂuctuations 3 ΔSðΔSþ1Þ 20 itudinal mode emerges as the Haldane triplet state splits due to an should be ¼ 0:27 , which is greater that the ratio of ðSΔSÞð2ΔSþ1Þ internal staggered mean ﬁeld. The Haldane state within the the intensities of the zone center modes I /I = 0.19. This long tran frustration-driven S = 2 spin chains in α-Na MnO is easily 0.9 2 rough comparison suggests that, while the amplitude mode 13,26,27 quenched under small anisotropy , and α-Na . MnO is 0 9 2 observed in our measurements is relatively intense, it remains thought to be outside of the Haldane regime. Calculations predict within the bounds of the allowed spectral weight for longitudinal that the phase boundary between the S = 2 Haldane state and ﬂuctuations. antiferromagnetic order appears at D/J = 0.0046 (for easy-axis D) Relative to stoichiometric α-NaMnO powder samples, Na , far away from the experimentally measured D/J = 0.035 in vacancies in the α-Na MnO crystal studied here are unlikely to 0.9 2 Na MnO . While amplitude modes in other quantum spin 0.9 2 generate this long-lived, dispersive amplitude mode. Each Na systems close to singlet instabilities have also been recently vacancy naively binds to a hole on the Mn-planes and creates an reported in the quasi-two-dimensional spin ladder compound 4+ Mn S = 3/2 magnetic impurity. The corresponding hole is 28 29 C H N CuBr and the two-dimensional ruthenate Ca RuO , 9 18 2 4 2 4 bound to the impurity site due to strong polaron trapping. As this the formation of a longitudinal-bound state in α-Na MnO is 0.9 2 21,22 state remains localized within the lattice , the role of vacancies distinct from modes in these and other S = 1/2 spin chain systems can be viewed as introducing random static magnetic impurities possessing substantial zero-point ﬂuctuations . within the MnO planes. Neither this random, static disorder nor Instead, in α-Na MnO , the interplay of geometric frustration 0.9 2 the high density of twin boundaries inherent to the lattice are and the Jahn–Teller quenching of orbital degeneracy uniquely capable of directly generating a coherent spin mode; however, conspire to create a quasi-one-dimensional magnon spectrum they may indirectly contribute to destabilizing the ordered Néel that condenses due to the attractive potential provided by an state, pushing α-Na MnO closer to a disordered regime and 0.9 2 Ising-like single-ion anisotropy. As a result, α-Na MnO pro- 0.9 2 enhancing ﬂuctuation effects. vides an intriguing route to realizing an intense, stable amplitude For an easy-axis antiferromagnetic chain at 0 K, two degen- mode in a planar AF. The dimensionality reduction realized erate, gapped transverse magnon modes are expected as Néel within its chemically two-dimensional lattice also suggests that order sets in; however the amplitude mode observed in the 30,31 other α-NaFeO type transition metal oxides possessing ordered state of α-Na MnO is unexpected. Since this long- 0.9 2 coherent Jahn–Teller distortions may host similarly stable itudinally polarized mode has a zone center lifetime that is con- amplitude modes, depending on their inherent anisotropies. More strained by the resolution of the spectrometer (ΔE = 2.25 meV res broadly this class of materials presents an exciting platform for at E = 11 meV) without an observable high energy tail and with exploring unconventional-bound states such as bound soliton an energy below twice the transverse modes’ gap values, it likely 14 modes stabilized in a quasi-one-dimensional spin setting. 24,25 arises as a long-lived two-magnon-bound state . The ﬁnite binding energy of this state as determined by E = 2E − bind gap E = 1.2 ± 0.1 meV at the zone center Q = (0.5, 0.5, 0) implies long Methods an attractive potential between magnons once Néel order is Crystal growth and characterization.Na CO and MnCO powders (1:1 ratio 2 3 3 established. plus 10% weight excess of Na CO ) were mixed and sintered in an alumina crucible 2 3 To explore this further, we theoretically consider the existence 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 of a bound state within a one-dimensional model, neglecting J . press. The rod was sintered in a vertical furnace at 1000 °C for 15 h and then We further consider only zero temperature for simplicity, and quenched in air, before being transferred to a four mirror optical ﬂoating zone perform a semi-classical large S analysis based on anharmonically furnace outﬁtted with 500 W halogen lamps. The crystals were grown at a rate of −1 coupled spin waves around an antiferromagnetically ordered 20 mm h in a 4:1 Ar:O environment under 0.15 MPa of pressure. Inductively −S coupled plasma atomic emission spectroscopy (ICP-AES) was used to determine state. For J e ≪ D≪ J , and S≫ 1, the one-dimensional chain 1 1 the Na/Mn ratio and to check that the expected mass of Mn was present. The ratio is ordered at T = 0, which allows J to be neglected without 2 3+ 4+ of Mn /Mn was determined through X-ray absorption near edge spectroscopy qualitative errors. As detailed in Supplementary Note 2, the 23 (XANES), X-ray photoelectron spectroscopy (XPS), and Na solid-state NMR resulting description applies: (1) the anisotropy induces a single (ssNMR). Detailed crystal growth and characterization can be found in Dally pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ et al. Samples were handled as air-sensitive and stored in an inert environment. magnon gap Δ ¼ 2S 2J D so that the magnon dispersion near Time outside of an inert environment was minimized (e.g., during crystal align- the magnetic zone center is that of a relativistic massive particle pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ment for neutron scattering experiments). The crystal faces are ﬂat, and no 2 2 with E ¼ v k þ Δ with magnon velocity v = 2J S, and (2) the 1 degradation of the surface was observed during alignment. The same ~0.5 g crystal was used for both neutron TOF and triple-axis experiments. dominant interaction between opposite spin magnons with 6 NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04601-1 ARTICLE Time-of-ﬂight experimental setup. Neutron time-of-ﬂight data in Fig. 2 and Received: 6 July 2017 Accepted: 11 May 2018 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 He-gas 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 = 60 meV and the ﬁne-resolution fermi chopper rotating at 420 Hz. For data col- lection, the sample was rotated through a range of 180° with 1° steps. A back- References ground scan was collected by removing the sample from the neutron beam and 1. Pekker, D. & Varma, C. M. Amplitude/Higgs modes in condensed matter collecting the scattering from the empty can. physics. Annu. Rev. Conden. Mater. Phys. 6, 269–297 (2015). 2. Morra, R. M., Buyers, W. J. L., Armstrong, R. L. & Hirakawa, K. Spin Time-of-ﬂight data analysis. An aluminum only (empty can) background was dynamics and the Haldane gap in the spin-1 quasi-one-dimensional subtracted from all data before plotting. SpinW , a Matlab library, was used to antiferromagnet CsNiCl . Phys. Rev. B 38, 543–555 (1988). simulate the magnetic excitations for the TOF data. Given the spin Hamiltonian, 3. Raymond, S. et al. Polarized-neutron observation of longitudinal haldane-gap magnetic structure and twinning mechanisms (structural and magnetic), SpinW excitations in Nd BaNiO . Phys. Rev. Lett. 82, 2382–2385 (1999). 2 5 uses linear spin wave theory to numerically calculate and display the dispersion. 4. Enderle, M., Tun, Z., Buyers, W. J. L. & Steiner, M. Longitudinal spin The simulation was convolved with the energy resolution function of the neutron ﬂuctuations of coupled integer-spin chains: haldane triplet dynamics in the spectrometer (E = 60 meV, F = 420 Hz). Simulations for Fig. 2c were run i chopper ordered phase of CsNiCl . Phys. Rev. B 59, 4235–4243 (1999). over the same range that the data were binned for Fig. 2a and d (i.e., 1.4 < H < 1.6 5. Grenier, B. et al. Longitudinal and transverse Zeeman ladders in the Ising-like and −0.1 < L < 0.1), and then averaged together. chain antiferromagnet BaCo V O . Phys. Rev. Lett. 114, 017201 (2015). 2 2 8 6. Rüegg, Ch et al. Quantum magnets under pressure: controlling elementary excitations in TlCuCl . Phys. Rev. Lett. 100, 205701 (2008). Triple-axis experimental setup. Triple-axis neutron data were taken with the 3 33,34 7. Merchant, P. et al. Quantum and classical criticality in a dimerized quantum instrument, BT7 , at NCNR with a PG(002) vertically focused monochromator antiferromagnet. Nat. Phys. 10, 373–379 (2014). 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 = 14.7 meV), 8. Lake, B., Tennant, D. A. & Nagler, S. E. Novel longitudinal mode in the and only a PG ﬁlter after the sample was used during inelastic operation (ﬁxed E coupled quantum chain compound KCuF . Phys. Rev. Lett. 85, 832–835 f 3 = 14.7 meV). Unpolarized data from Fig. 3 and Supplementary Fig. 3 were taken (2000). with open−25′−50′−120′ collimations (denoting the collimation before the 9. Zheludev, A., Kakurai, K., Masuda, T., Uchinokura, K. & Nakajima, K. monochromator, sample, analyzer, and detector, respectively), and the sample was Dominance of the excitation continuum in the longitudinal spectrum of aligned in the (H, K, 0) scattering plane. Supplementary Fig. 2a data were unpo- weakly coupled Heisenberg S=1/2 chains. Phys. Rev. Lett. 89, 197205 (2002). larized and taken with open−50′−50′−120′ collimators in the (H, H, L) plane. 10. Coldea, R., Tennant, D. A., Tsvelik, A. M. & Tylczynski, Z. Experimental Polarized data in Fig. 1c were taken with open−25’−25′−120′ collimation in the realization of a 2D fractional quantum spin liquid. Phys. Rev. Lett. 86, (H, K, 0) scattering plane. Polarized data in Fig. 4 were taken in the (H, K, H) plane 1335–1338 (2001). with open−80′−80′−120′ collimations. 11. Giot, M. et al. Magnetoelastic coupling and symmetry breaking in the frustrated antiferromagnet α-NaMnO . Phys. Rev. Lett. 99, 247211 (2007). 12. Stock, C. et al. One-dimensional magnetic ﬂuctuations in the Spin-2 triangular Triple-axis polarization efﬁciency corrections. It was determined that only two lattice α-NaMnO . Phys. Rev. Lett. 103, 077202 (2009). (one NSF and one SF) of the available four neutron scattering cross-sections were 2 13. Schollwöck, U. & Jolicoeur, T. Haldane gap and hidden order in the S=2 needed for polarization analysis. Flipping ratios were taken throughout the antiferromagnetic quantum spin chain. Europhys. Lett. 30, 493–498 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. (1995). 14. Haldane, F. D. M. Continuum dynamics of the 1-D Heisenberg antiferromagnet: identiﬁcation with the O(3) nonlinear sigma model. Phys. Triple-axis data analysis. All data were normalized to the neutron monitor Lett. A 93, 464–468 (1983). counts, M. Error bars represent one standard deviation of the data. For unpolarized pﬃﬃﬃﬃ 15. Haldane, F. D. M. Nonlinear ﬁeld theory of large-spin heisenberg data, this was calculated by the square root of the number of counts, N, where N antiferromagnets: semiclassically quantized solitons of the one-dimensional is the number of counts. The lower monitor counts in polarized data were con- pﬃﬃﬃﬃﬃ easy-Axis Néel state. Phys. Rev. Lett. 50, 1153–1156 (1983). sidered by propagating the error in the monitor counts, M, such that 2 N N 16. Abakumov, A. M., Tsirlin, A. A., Bakaimi, I., Van Tendeloo, G. & Lappas, A. σ ¼ 1 þ . The determination of the moment angle utilized the polarized M M Multiple twinning as a structure directing mechanism in layered rock-salt- elastic data shown in Fig. 1c. After correcting for the polarization efﬁciency, the type oxides: NaMnO polymorphism, redox potentials, and magnetism. Chem. integrated intensities of the NSF and SF peaks were found by ﬁtting the data to Mater. 26, 3306–3315 (2014). Gaussian functions. These intensities were used to ﬁnd the moment angle following 17. Zorko, A., Adamopoulos, O., Komelj, M., Arčon, D. & Lappas, A. Frustration- the technique in Moon et al. induced nanometre-scale inhomogeneity in a triangular antiferromagnet. Nat. Fits to the constant energy scans (unpolarized inelastic neutron scattering data) Commun. 5, 3222 (2014). in Fig. 2d, 3a, b and Supplementary Fig. 3 used the Cooper–Nathans 35 36 18. Zorko, A. et al. Magnetic interactions in α-NaMnO2: quantum spin-2 system approximation in ResLib , a program that calculates the convolution of the on a spatially anisotropic two-dimensional triangular lattice. Phys. Rev. B 77, spectrometer resolution function with a user supplied cross section. The cross 024412 (2008). section used for the transverse excitation was the single-mode approximation of a 19. Moon, R. M., Riste, T. & Koehler, W. C. Polarization analysis of thermal- two-dimensional spin lattice with single-ion anisotropy, as described in Supplementary Note 1. Cross-sections for t − q , t − q , t − q , and t − q were neutron scattering. Phys. Rev. 181, 920–931 (1969). 1 1 1 2 2 1 2 2 20. Huberman, T. et al. Two-magnon excitations observed by neutron scattering all included during the ﬁtting routine using the relation between the ﬁrst moment sum rule and the dynamical structure factor, in the two-dimensional spin 5/2− isenberg antiferromagnet Rb MnF . Phys. 2 4 R P 1 β β αα Rev. B 72, 014413 (2005). ðÞ hω S ðÞ q; hω dðÞ hω ¼ J½ 1 cosðÞ q a 1 δ hhS ; S ii. n n αβ 1 n;β R R þa j j n 21. Ma, X., Chen, H. & Ceder, G. Electrochemical properties of monoclinic The contribution to the scaling factor from the single-ion term is small , and NaMnO . J. Electrochem. Soc. 158, A1307–A1312 (2011). therefore, was not included. The longitudinal excitation was empirically ﬁt using 22. Jia, T. et al. Magnetic frustration in α-NaMnO and CuMnO . J. Appl. Phys. 2 2 the single-mode approximation for a one-dimensional chain, given its unresolvable 109, 07E102 (2011). dispersion along H. The longitudinal mode gap was determined by ﬁtting the data 23. Dally, R. et al. Floating zone growth of α-Na MnO single crystals. J. Cryst. 0.90 2 in the range where it was resolvable (below 18 meV). This gap value was ﬁxed and Growth 459, 203–208 (2017). the ﬁtting routine was run again, allowing all other parameters to vary. A single 24. Xian, Y. Longitudinal excitations in quantum antiferromagnets. J. Phys. intrinsic HWHM for all excitations was reﬁned during ﬁtting and reﬁned to be Condens. Mat. 23, 346003 (2011). negligibly small. Additionally, a scaling prefactor was also reﬁned for each 25. Heilmann, I. U. et al. One- and two-magnon excitations in a one-dimensional crystallographic twin, where the two magnetic domains within a crystallographic antiferromagnet in a magnetic ﬁeld. Phys. Rev. B 24, 3939–3953 (1981). twin were assumed to have the same weight (i.e., t − q and t − q had the same 1 1 1 2 26. Schollwöck, U., Golinelli, O. & Jolicœur, T. S=2 antiferromagnetic quantum prefactor). spin chain. Phys. Rev. B 54, 4038–4051 (1996). Polarized inelastic neutron data in Fig. 4 are plotted as raw data, uncorrected for 27. Kjäll, J. A., Zaletel, M. P., Mong, R. S. K., Bardarson, J. H. & Pollmann, F. polarization efﬁciency. The dashed lines representing the expected bleed through Phase diagram of the anisotropic spin-2 XXZ model: Inﬁnite-system density from the SF channel into the NSF channel in Fig. 4a and b were determined from matrix renormalization group study. Phys. Rev. B 87, 235106 (2013). the measured ﬂipping ratio. 28. Hong, T. et al. Higgs amplitude mode in a two-dimensional quantum antiferromagnet near the quantum critical point. Nat. Phys. 13, 638–642 Data availability. The data that support the ﬁndings of this study are available (2017). from the corresponding author upon reasonable request. NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications 7 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04601-1 29. Jain, A. et al. Higgs mode and its decay in a two-dimensional antiferromagnet. Additional information Nat. Phys. 13, 633–637 (2017). Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- 30. Mostovoy, M. V. & Khomskii, D. I. Orbital ordering in frustrated Jahn–Teller 018-04601-1. 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 Competing interests: The authors declare no competing interests. antiferromagnet α-NaFeO . Phys. Rev. B 76, 024420 (2007). 32. Toth, S. & Lake, B. Linear spin wave theory for single-Q incommensurate Reprints and permission information is available online at http://npg.nature.com/ magnetic structures. J. Phys. Condens. Matter 27, 166002 (2015). reprintsandpermissions/ 33. Lynn, J. W. et al. Double focusing thermal triple axis spectrometer at the NCNR. J. Res. Natl Inst. Stan. 117,61–79 (2012). Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in 34. Chen, W. C. et al. He neutron spin ﬁlters for a thermal neutron triple axis published maps and institutional afﬁliations. 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). Open Access This article is licensed under a Creative Commons 36. Zheludev, A. ResLib (ETH Zürich, 2009). Attribution 4.0 International License, which permits use, sharing, 37. Zaliznyak, I. & Lee, S.-H. in Modern Techniques for Characterizing Magnetic adaptation, distribution and reproduction in any medium or format, as long as you give Materials, Ch. 1 (ed. Zhu, Y.) (Springer, Heidelberg, 2005). 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 Acknowledgements indicated otherwise in a credit line to the material. If material is not included in the S.D.W. and R.L.D. acknowledge assistance in characterizing samples from Raphaële article’s Creative Commons license and your intended use is not permitted by statutory Clément. S.D.W. and R.L.D. gratefully acknowledge support from DOE, Ofﬁce of Sci- regulation or exceeds the permitted use, you will need to obtain permission directly from ence, Basic Energy Sciences under Award DE-SC0017752. Work by L.B. was supported the copyright holder. To view a copy of this license, visit http://creativecommons.org/ by the DOE, Ofﬁce of Science, Basic Energy Sciences under Award No. DE-FG02-08ER licenses/by/4.0/. © The Author(s) 2018 Author contributions R.L.D. synthesized the α-NaMnO 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 the- oretical analysis of the spin dynamics. R.L.D. and S.D.W. designed the neutron experi- ments. R.L.D., L.B., and S.D.W. prepared and wrote the manuscript. 8 NATURE COMMUNICATIONS (2018) 9:2188 DOI: 10.1038/s41467-018-04601-1 www.nature.com/naturecommunications | | |
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