High order magnon bound states in the quasi-one-dimensional antiferromagnet α -NaMnO 2
Rebecca L. Dally, Alvin J. R. Heng, Anna Keselman, Mitchell M. Bordelon, Matthew B. Stone, Leon Balents, Stephen D. Wilson
HHigh order magnon bound states in the quasi-one-dimensional antiferromagnet α -NaMnO Rebecca L. Dally,
1, 2, ∗ Alvin J.R. Heng,
3, 4, ∗ Anna Keselman, Mitchell M.Bordelon, Matthew B. Stone, Leon Balents, † and Stephen D. Wilson ‡ NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Materials Department, University of California, Santa Barbara, CA 93106, USA Kavli Institute for Theoretical Physics, University of California,Santa Barbara, Santa Barbara, CA 93106, USA Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 637371, Singapore Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA (Dated: January 22, 2020)Here we report on the formation of two and three magnon bound states in the quasi-one-dimensional antiferromagnet α -NaMnO , where the single-ion, uniaxial anisotropy inherent to theMn ions in this material provides a binding mechanism capable of stabilizing higher order magnonbound states. While such states have long remained elusive in studies of antiferromagnetic chains,neutron scattering data presented here demonstrate that higher order n > . We corroborate our findings with exact numerical simulations of aone-dimensional Heisenberg chain with easy-axis anisotropy using matrix-product state techniques,finding a good quantitative agreement with the experiment. These results establish α -NaMnO asa unique platform for exploring the dynamics of composite magnon states inherent to a classicalantiferromagnetic spin chain with Ising-like single ion anisotropy. INTRODUCTION
One dimensional systems are renown for their abilityto host ground states and phases markedly different fromtheir higher dimensional counterparts. In the realm ofmagnetism, the prototypical S = 1 / S = 1 chainalso avoids order, and hosts instead a topologically non-trivial Haldane-gapped state with protected S = 1 / S in one dimension in the ideal case, they become in-creasingly fragile to perturbations as S increases and thesemi-classical limit is achieved. In this regime, describedby the non-linear sigma model (NLSM) with a weak cou-pling of order 1 /S [5], quantum fluctuations are confinedto very low energies. A small anisotropy restricts thosefluctuations to the “easy” directions, and for an Ising-likesituation, this is sufficient to induce order.Thus, large spin (e.g. S ≥
2) quantum spin chainsseem an unlikely place to observe strongly quantum phe-nomena. Indeed this is true for their ground states, how-ever, their excitations can still be highly quantum andrealize interesting and paradigmatic few-body problems.Recent interest in few-body problems – i.e. the quan-tum mechanics of a finite number n > J - J models [9]. Inthe latter case, the particles binding are magnons above atrivial field-induced polarized state [10], and lattice-scalecompeting interactions play an important role.The cleanest, arguably most beautiful example of afew-body problem is the droplet mentioned above, whichis a collection of one-dimensional Bosons with attractivezero-range delta-function interactions. The fundamen-tal physics of this problem is that increasing numbers ofbosons bind more strongly, due to Bose statistics. More-over, the ground state is, in this case, exactly solubleanalytically via a simple application of the Bethe ansatz.In prior work, we uncovered a surprising connection ofthis canonical model problem to the large S nearest-neighbor antiferromagnetic chain with weak easy-axisIsing anisotropy [11]. The theory developed there pre-dicts an approximate mapping of the quantum mechan-ics of magnons to the droplet problem of bosons. Thismapping is non-trivial because the unperturbed NLSMis a gapless theory and, in it, binding would mean insta-bility. Binding occurs as a subtle balance between fluc-tuations, weak anisotropy, and the intrinsic interactionsof the NLSM.In Ref. 11, the two-body bound state predicted bythis theory was observed in the material α -NaMnO .Fundamentally, the crystal lattice of α -NaMnO is builtfrom two-dimensional sheets of triangular-lattice planes a r X i v : . [ c ond - m a t . s t r- e l ] J a n K (r.l.u.)0.6 0.80510152025 E ne r g y ( m e V ) –9.5 –9 –8.5 –10–9–8–7–615105 Energy (meV) ln(I) (arb. units) n =1 n= n= l n ( I ) ( a r b . un i t s ) (a) (b) FIG. 1: Neutron scattering data collected with E i = 30 meVat T = 4 K. The data were integrated throughout the entirezone in both H and L . Panel (a) shows an intensity of thedata on a logarithmic scale. The dashed, white line showsthe direction of the energy cut plotted in panel (b). Panel(b) shows an energy cut through the intensity map centeredat the AF zone center, K = 0 .
5, with the n = 1, n = 2, and n = 3 magnon modes. of Mn moments, yet the Jahn-Teller effect inherent tothe Mn-cations drives a coherent distortion of the tri-angular lattice into coupled isosceles triangles comprisedof two long Mn-Mn distances and one short Mn-Mn ex-change pathway [12]. This breaks the frustration of thetriangular lattice and defines a strong AF nearest neigh-bor exchange energy ( J ) along the short bond lengththat defines the AF chain direction. Coupling betweenthese chains is frustrated by equivalent AF next nearestneighbor couplings ( J ) via the two longer legs of the tri-angular lattice. The result is a highly one-dimensionalspin system [13, 14] with effective J /J = 8 [11] thatalso possesses a weak Ising-like single-ion anisotropy, D ,which we will estimate to be D/J = 0 . α -NaMnO to the droplet problemby uncovering a three-body bound state in α -NaMnO .Both n = 2 and n = 3 magnon bound states are ob-served in inelastic neutron scattering measurements withbinding energies consistent with a minimal model of AFspin chains possessing an easy-axis uniaxial, single-ionanisotropy. Notably, the proper theory requires consid-eration of the “charge” of the magnons that bind, whichbreaks integrability and leads to a reduction of the bind-ing energy. Our modeling suggests α -NaMnO and otherweakly anisotropic, large S chains can be excellent test K =0.4770.4810.4850.4890.4930.4970.5010.5050.5090.5130.5170.5210.525 11.411.311.211.111.010.910.8 0.530.520.510.500.490.480.47543210 191817161514 K =0.47750.48250.48750.49250.49750.50250.50750.51250.51750.52250.5275 15.915.815.715.615.515.4 0.530.520.510.500.490.480.47 I n t e n s i t y ( a r b . un i t s ) I n t e n s i t y ( a r b . un i t s ) Energy (meV)Energy (meV) K (r.l.u.) K (r.l.u.) En e r g y ( m e V ) En e r g y ( m e V ) (a) (b) (c) (d) FIG. 2: Constant K cuts parameterizing the dispersions of the n = 2 and n = 3 modes. The data were integrated throughoutthe entire zone in both H and L and represent cuts throughthe color plot in Fig. 1. (a) Individual K -cuts parameterizingthe dispersion of the n = 2 mode. (b) The fit energies of the n = 2 mode plotted as a function of K . (c) Individual K -cutsparameterizing the dispersion of the n = 3 mode (d) Theenergies of the n = 3 mode as a function of K . The K cutsin panels (a) and (c) are offset from one another for clarity,and the solid black lines are fits parameterizing the modes’dispersions as described in the text. The solid orange lines inpanels (b) and (d) are fits to the 1D dispersion described inthe text. platforms for few-body physics.A single crystal of α -NaMnO was grown via thefloating zone method [15], and neutron scattering mea-surements were performed on the time-of-flight neutronspectrometer SEQUOIA [16] at the Spallation NeutronSource at Oak Ridge National Laboratory. The crys-tal was sealed in a cryostat with 4He exchange gas, andthe (0 , ,
0) and (1 , ,
1) crystallographic directions werealigned in the horizontal scattering plane. Data were col-lected with incident neutron energies of E i = 30 and 60meV with the Fermi chopper in high-resolution mode,and the sample was rotated about the ( − , ,
1) axis in1 ◦ increments over a range of 180 ◦ . Data were reducedand analyzed using the software package Horace [17].Throughout the paper, positions in momentum space, Q , are reported in reciprocal lattice units (r.l.u.), where H , K , and L reflect Q [˚ A − ] = ( πa sin β H, πb K, πc sin β L )with a = 5 .
63 ˚A, b = 2 .
86 ˚A, c = 5 .
77 ˚A, and α = γ =90 ◦ ; β = 113 ◦ .Fig. 1 shows inelastic neutron scattering data collectedabout the 1D AF zone center, (0, 0.5, 0), in the AF or-dered state ( T = 4 K). As the spin dynamics in thissystem are quasi-1D, data are integrated across the en-tire zone in H (the interchain) and L (the interplane)directions. The resulting color map of intensities plottedin Fig. 1 (a) shows three branches of excitations dispers-ing along K (the intrachain direction), each of which arecentered at the AF zone center K = 0 . E = 6 .
15 meV and is comprised of 4 nearlydegenerate modes superimposed from the two crystallo-graphic and two magnetic twins inherent to this material.A second mode, which is purely 1D, is seen centered at E = 10 . n = 2) bound state. Remark-ably, an additional 1D mode appears at higher energy at E =15.5 meV—this third mode is newly resolved andsuggests the formation of a higher order three-magnon( n = 3) bound state.The dispersions of the n = 2 and n = 3 bound statesare illustrated about the AF zone center via constant K cuts (offset from one another for clarity) and plotted inFigs. 2 (a) and (c). Fits parameterizing the energiesof each mode as a function of K were performed usingGaussian peaks on a sloping background with the result-ing dispersion relations shown in Figs. 2 (b) and (d).These relations were then empirically quantified via a fitto the form E ( K ) = (cid:112) ∆ + c sin πK with ∆ reflec-tive of the gap energy and c an empirical spin stiffnessparameter.For the n = 2 branch of excitations, the results ofthe fit are shown in Fig. 2 (b), where ∆ and c werefound to be 10 . ± .
006 meV and 16 . ± . n = 3branch shown in Fig. 2 (d) yields ∆ = 15 . ± . c = 16 ± n = 3 branch iscoupled to the K bin size chosen. This is due to theweak nature of n = 3 mode (which is nearly two ordersof magnitude weaker than the n = 1 peak). Using theparameterization for n = 3 mode shown in Fig. 2 (d),the effective masses of the modes m i ∝ ∆ /c have ratiosof m : m : m =1 : 3 . . T = 4 K(in the AF ordered state), 30 K (in the incommensurateshort-range ordered state), and 50 K (in the high tem-perature regime of quasi-one dimensional correlations)[18]. True long-range order along the chains (divergent K -axis correlation lengths) occurs only below T = 22 K[18], and both n = 2 and n = 3 modes vanish above thistemperature. The n = 1 single magnon peak persists to I n t en s i t y ( a r b . un i t s ) K = 0.5 r.l.u. I n t en s i t y ( a r b . un i t s ) FIG. 3: E i = 30 meV data showing the temperature de-pendence of spin excitations determined via an energy cut atthe quasi-1D zone center, Q = (0 , . , n = 3 mode. Data are integrated across awidth of 0.01 r.l.u. in K . high temperature as it becomes increasingly damped andbroadens into the single-ion anisotropy gap with increas-ing temperature.We now proceed to interpret the above results theoret-ically. Prior work on this compound developed a semi-classical theory to leading order in 1 /S in the quasi-1Dlimit [11]. In that theory, 1 /S corrections induce an at-tractive delta-function-like interaction between magnons,which creates the n = 2 two magnon bound state. Withinthe same theory higher bound states are expected, as de-scribed in the supplementary materials [19]. Here to ob-tain a more quantitative comparison with the experimen-tal data that does not rely on the 1 /S expansion, we car-ried out numerically exact matrix product state (MPS)-based [20] calculations in the one dimensional limit usingthe ITensor library [21].Given the quasi-1D nature of α -NaMnO , in our nu-merical simulation, we consider a S = 2 antiferromag-netic Heisenberg chain with single-ion anisotropy H = J (cid:88) j (cid:126)S j · (cid:126)S j +1 − D (cid:88) j ( S zj ) (1)where the sum over j indicates a sum over spins on a 1Dlattice. We calculate the spectral function, which at zerotemperature is given by S ( k, ω ) = (cid:90) ∞−∞ dte iωt ∞ (cid:88) j = −∞ e − ikj (cid:68) (cid:126)S j ( t ) · (cid:126)S (0) (cid:69) , (2) ln(I) (arb. units) En e r g y ( m e V ) K (r.l.u.) FIG. 4: Spectral function obtained numerically using the 1Dmodel, for J = 5 .
34 meV and D = 0 .
46 meV. The intensityis normalized to a log scale. Three modes are clearly seenin the vicinity of K = 0 .
5. Blue dashed lines plotted ontop correspond to dispersions obtained from experiment. The n = 1 data is obtained from a ( H, H,
0) cut, with integrationin the L direction from − . < L ( r.l.u ) < . − H, H,
0) direction from − . < H ( r.l.u ) < . n = 2 and n = 3 data were obtained via cuts about theAF zone center from Fig. 1(b) and Fig. 2(c) respectively. where the expectation value is taken in the ground stateof the system. We consider a finite chain with N =500 sites and start by obtaining the ground state ofthe system using density matrix renormalization group(DMRG) [22]. We then perform time evolution up totimes t max = 20 J − using time evolving block decima-tion (TEBD) [19, 23, 24].Using this model, we obtain an independent estimatefor the couplings J and D from a least squares fit of thedispersion of the n = 1 mode, obtained from the time-of-flight neutron experiment. The single-magnon mode mo-mentum slice along the ( H, H,
0) direction (correspond-ing to a high-symmetry direction in the 3D reciprocalspace) was modeled via the spectral function calculatednumerically. This fit yields J = 5 . ± . D = 0 . ± . J and D is shown in Fig. 4, with the ex-perimentally measured dispersions for the three modesplotted on top of it as blue dashed lines. The n = 1, n = 2, and n = 3 magnon modes are clearly visible in the spectral function in the vicinity of K = 0 .
5. Thecorresponding gaps are given by E = 6 . ± . E = 10 . ± . E = 14 . ± . n = 2 and n = 3 modes without further fitting pa-rameters. The success of this purely 1D theory is alsoconsistent with the observation that these higher ordermodes are dispersionless along both the interchain andinterplane directions.A physical interpretation of the bound states is as fol-lows. The elementary excitations of the system (whichcomprise the E mode) are magnons, which behave asmassive relativistic particles. These come in two “fla-vors”, with spin S z = ± E mode arises as a bound state of two unlike particles, andhence carries S z = 0, which is why it is longitudinal.The third E mode corresponds to a bound state of twolike and one unlike particles, e.g. two S z = +1 andone S z = − S z = ± /S expansion discussed in theSM confirms the presence of these bound states, thoughthe DMRG calculations are more quantitative. Multiplemagnon binding in NaMnO is therefore a remarkableand unexpected manifestation of new physics accessiblein higher spin one dimensional chains.In summary, we have shown that the quasi-1D spindynamics endemic to the anisotropic triangular lattice ofNaMnO manifest n = 2 and n = 3 high order magnonbound states. This result is captured by a semiclassicaltheory of interacting magnons within a 1D chain weaklybound by uniaxial single-ion anisotropy, which effectivelymaps to few body droplet models of interacting bosonsbound via weak delta function potentials. The result isa striking manifestation of strongly 1D quantum effectswithin the dynamics of a classical ( S = 2), planar anti-ferromagnet. Therefore NaMnO , and we propose likelyother anisotropic triangular antiferromagnets with weak,Ising-like single ion anisotropy, are appealing platformsfor exploring few body interactions in a condensed mattersetting.This work was supported by DOE, Office of Sci-ence, Basic Energy Sciences under Award de-sc0017752(S.D.W., R.D. and M.B.). Work by L.B. was supportedby the DOE, Office of Science, Basic Energy Sciences un-der Award No. DE-FG02-08ER46524. This research isfunded in part by the Gordon and Betty Moore Foun-dation through Grant GBMF8690 to UCSB to supportthe work of A.K. A.J.R.H. thanks the Nanyang Techno-logical University for financial support through the CNYang Scholars Programme. M.B. also received partialsupport from the National Science Foundation GraduateResearch Fellowship Program under Grant No. 1650114.Use was made of the computational facilities adminis-tered by the Center for Scientific Computing at the CNSIand MRL (an NSF MRSEC; DMR-1720256) and pur-chased through NSF CNS-1725797. ∗ Contributed equally to this work † [email protected] ‡ [email protected][1] F. Haldane and M. Zirnbauer, Physical review letters ,4055 (1993).[2] S. Eggert, I. Affleck, and M. Takahashi, Phys. Rev. Lett. , 332 (1994), URL https://link.aps.org/doi/10.1103/PhysRevLett.73.332 .[3] F. D. M. Haldane, Phys. Rev. Lett. , 1153 (1983), URL https://link.aps.org/doi/10.1103/PhysRevLett.50.1153 .[4] I. Affleck, Journal of Physics: Condensed Matter , 3047(1989).[5] F. D. M. Haldane, Physics Letters A , 464 (1983).[6] D. C. Mattis, Rev. Mod. Phys. , 361 (1986), URL https://link.aps.org/doi/10.1103/RevModPhys.58.361 .[7] Y. Sekino and Y. Nishida, Phys. Rev. A ,011602 (2018), URL https://link.aps.org/doi/10.1103/PhysRevA.97.011602 .[8] M. Zaccanti, B. Deissler, C. DErrico, M. Fattori,M. Jona-Lasinio, S. M¨uller, G. Roati, M. Inguscio, andG. Modugno, Nature Physics , 586 (2009).[9] J. Sudan, A. L¨uscher, and A. M. L¨auchli, Phys. Rev.B , 140402 (2009), URL https://link.aps.org/doi/10.1103/PhysRevB.80.140402 .[10] L. Kecke, T. Momoi, and A. Furusaki, Phys. Rev. B , 060407 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.76.060407 .[11] R. L. Dally, Y. Zhao, Z. Xu, R. Chisnell, M. B. Stone,J. W. Lynn, L. Balents, and S. D. Wilson, Nat. Com-mun. , 2188 (2018), URL https://doi.org/10.1038/s41467-018-04601-1 .[12] M. Giot, L. C. Chapon, J. Androulakis, M. A. Green,P. G. Radaelli, and A. Lappas, Phys. Rev. Lett. ,247211 (2007), URL https://link.aps.org/doi/10. 1103/PhysRevLett.99.247211 .[13] A. Zorko, S. El Shawish, D. Arˇcon, Z. Jagliˇci´c, A. Lap-pas, H. van Tol, and L. C. Brunel, Phys. Rev. B , 024412 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.77.024412 .[14] C. Stock, L. Chapon, O. Adamopoulos, A. Lappas,M. Giot, J. Taylor, M. Green, C. Brown, and P. Radaelli,Physical review letters , 077202 (2009).[15] R. Dally, R. J. Clment, R. Chisnell, S. Taylor,M. Butala, V. Doan-Nguyen, M. Balasubramanian,J. W. Lynn, C. P. Grey, and S. D. Wilson, Jour-nal of Crystal Growth , 203 (2017), ISSN 0022-0248, URL .[16] G. E. Granroth, A. I. Kolesnikov, T. E. Sherline, J. P.Clancy, K. A. Ross, J. P. C. Ruff, B. D. Gaulin, andS. E. Nagler, Journal of Physics: Conference Series , 012058 (2010), URL https://doi.org/10.1088%2F1742-6596%2F251%2F1%2F012058 .[17] R. Ewings, A. Buts, M. Le, J. van Duijn, I. Bustinduy,and T. Perring, Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers,Detectors and Associated Equipment , 132 (2016),ISSN 0168-9002, URL .[18] R. L. Dally, R. Chisnell, L. Harriger, Y. Liu,J. W. Lynn, and S. D. Wilson, Phys. Rev. B ,144444 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.144444 .[19] See supplemental information, which includes references[25].[20] U. Schollw¨ock, Annals of Physics , 96 (2011).[21] ITensor Library, http://itensor.org/.[22] S. R. White, Phys. Rev. Lett. , 2863 (1992).[23] G. Vidal, Phys. Rev. Lett. , 147902 (2003).[24] G. Vidal, Phys. Rev. Lett. , 040502 (2004), URL https://link.aps.org/doi/10.1103/PhysRevLett.93.040502 .[25] S. R. White and I. Affleck, Phys. Rev. B ,134437 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.77.134437https://link.aps.org/doi/10.1103/PhysRevB.77.134437