Abstract
The notion of a quasiparticle, such as a phonon, a roton or a magnon, is used in modern condensed matter physics to describe an elementary collective excitation. The intrinsic zerotemperature magnon damping in quantum spin systems can be driven by the interaction of the onemagnon states and multimagnon continuum. However, detailed experimental studies on this quantum manybody effect induced by an applied magnetic field are rare. Here we present a highresolution neutron scattering study in high fields on an S=1/2 antiferromagnet C_{9}H_{18}N_{2}CuBr_{4}. Compared with the noninteracting linear spin–wave theory, our results demonstrate a variety of phenomena including fieldinduced renormalization of onemagnon dispersion, spontaneous magnon decay observed via intrinsic linewidth broadening, unusual nonLorentzian twopeak structure in the excitation spectra and a dramatic shift of spectral weight from onemagnon state to the twomagnon continuum.
Introduction
Quasiparticles, first introduced by Landau in Fermiliquid theory as the excitations for the interacting fermions at low temperatures, have become a fundamental concept in condensed matter physics for an interacting manybody system. Naturally, quasiparticles are assumed to have long, or even infinite intrinsic lifetimes, because of either weak interactions, or the prohibiting energymomentum conservation for scatterings. However, this picture can break down spectacularly in some rare conditions. Quasiparticle decay was first predicted^{1} and then discovered in the excitation spectrum of the superfluid ^{4}He (refs 2, 3, 4), where the phononlike quasiparticle beyond a threshold momentum decays into two rotons. In magnetism, spontaneous (T=0) magnon decays into the twomagnon continuum were observed by inelastic neutron scattering (INS) in zero field in various valencebond type quantum spin systems including piperazinium hexachlorodicuprate (PHCC)^{5}, IPACuCl_{3} (ref. 6) and BiCu_{2}PO_{6} (ref. 7) as well as in some triangularlattice compounds^{8,9}. The mechanism for this magnon instability is the threemagnon scattering process, which is enhanced in the vicinity of the threshold for the decays of onemagnon to twomagnon states^{10,11,12}.
By contrast, the phenomenon of the fieldinduced spontaneous magnon decay in ordered antiferromagnets (AFMs) is much less mature experimentally. Although the key threemagnon coupling term is forbidden for the collinear ground state, it is present in an applied magnetic field when spins are canted along the field direction, that is, the coupling is facilitated via a fieldinduced spin noncollinearity. To date, spontaneous magnon decay in canted AFMs has been thoroughly studied theoretically^{12,13,14,15,16,17,18}, but there have been very few detailed experimental studies due to the lack of materials with suitable energy scales. The only experimental evidence was reported by Masuda et al. in an S=5/2 squarelattice AFM Ba_{2}MnGe_{2}O_{7} (ref. 19), where the INS spectra become broadened in a rather narrow part of the Brillouin zone (BZ). For the quantum spin1/2 systems, the magnon decay effect is expected to be much more pronounced, with the analytical^{13,14} and numerical studies^{15,16} predicting overdamped onemagnon excitations in a large part of the BZ.
Recently, a novel spin1/2 metalorganic compound (dimethylammonium)(3,5dimethylpyridinium)CuBr_{4} (C_{9}H_{18}N_{2}CuBr_{4}) (DLCB) was synthesized^{20}. Figure 1 shows the molecular twoleg ladder structure of DLCB with the chain direction extending along the crystallographic b axis. At zero field, the interladder coupling is sufficiently strong to drive the system into the ordered phase and the material orders magnetically at T_{N}=1.99(2) K coexisting with a spin energy gap due to a small Ising anisotropy^{21}.
In this paper, we report neutron scattering measurements on DLCB in applied magnetic fields up to 10.8 T and at temperature down to 0.1 K. In finite fields, our study shows strong evidence of the fieldinduced spontaneous magnon decay manifesting itself by the excitation linewidth broadening and by a dramatic loss of spectral weight, both of which are associated with the threemagnon interactions that lead to magnon spectrum renormalization and to the kinematically allowed onemagnon decays into the twomagnon continuum.
Results
Neutron diffraction results in high magnetic fields
The magnetic structure at zero field is collinear with the staggered spin moments aligned along an easy axis, that is, the c* axis in the reciprocal lattice space^{21}. The size of staggered moment is 0.40μ_{B}, much smaller than the 1 μ_{B} expected from free S=1/2 ions, due to quantum fluctuations^{21}. When a magnetic field is applied perpendicular to the easy axis, the ordered moments would cant gradually toward the field direction and saturate at μ_{0}H_{s}∼16 T. Figure 2 shows the fielddependent neutron scattering peak intensity (inset: the size of the ordered staggered moment m) measured at q=(0.5, 0.5, −0.5) and T=0.25 K. The intensity initially grows due to the gradual suppression of quantum fluctuations with field and reaches a peak value around 6 Tesla. It then decreases with the increase of field because the spin canting angle becomes large. There is no evidence of a phase transition, which confirms this quasiparticle picture. Within the linear spin–wave theory (LSWT), one may expect a similar semiclassical scenario to apply for the spin dynamics.
Inelastic neutron scattering results in high magnetic fields
The spin dynamics, in contrast, undergoes a dramatic change in applied magnetic fields. The corresponding Hamiltonian for a twodimensional (2D) spin interacting model with nearest neighbour interactions can be written as:
where J_{γ} is either the rung, leg or interladder exchange constant—and i and j are the nearestneighbour lattice sites. is the Landé gfactor and μ_{B} is the Bohr magneton. The parameter λ identifies an interaction anisotropy, with λ=0 and 1 being the limiting cases of Ising and Heisenberg interactions, respectively. We assume that λ is the same for all three J′s in order to minimize the number of fitting parameters to be determined from the experimental dispersion data (Note: this assumption would not affect the main conclusion of the study and is made to prevent overparameterization).
Figure 3 shows falsecolour maps of the backgroundsubtracted spinwave spectra along two highsymmetry directions in the reciprocal lattice space measured at μ_{0}H=0, 4, 6 and 10 T and T=0.1 K. Spectral weights of the transverse optical branches are weak in the current experimental configurations. Figure 3a,b show the observed spin gapped magnetic excitation spectra at zero field. We employed LSWT for the description of the energy excitations in the Hamiltonian equation (1). The calculated dispersion curves shown as the red lines in Fig. 3a,b using SPINW^{22} agree well with the experimental data. The best fit yields the spinHamiltonian parameters as J_{leg}=0.60(4) meV, J_{rung}=0.70(5) meV, J_{int}=0.17(2) meV and λ=0.95(2), which are fairly close to the values reported in ref. 21. At H>0, the transverse acoustic mode splits into two branches owing to the broken uniaxial symmetry. The highenergy mode (TM_{high}) corresponds to the spin fluctuations along the field direction. The lowenergy mode (TM_{low}) corresponds to the spin fluctuations perpendicular to both the field direction and the staggered spin moment (≡) and is indeed experimentally evidenced in Fig. 3d,f,h (pointed out by the red arrows). Since the neutron scattering probes the components of spin fluctuations perpendicular to the wavevector transfer, TM_{low} is expected to be weak and is consistent with the LSWT calculations in Fig. 4d,f,h. The dispersion bandwidths of the TM_{high} mode collapse with field, which can also be captured at this LSWT level with the same set of parameters. For instance, the bandwidths along the (H, H, −0.5) direction at μ_{0}H=4 and 6 T are reduced to 0.44 and 0.25 meV, respectively, from 0.80 meV at zero field. Surprisingly, however, we notice that the TM_{high} mode near the BZ centre in Fig. 3c,e visibly bends away from the LSWT calculation in Fig. 4c,e for μ_{0}H=4 and 6 T (pointed out by the white arrows). Such a renormalization of onemagnon dispersion, which is attributed to the strongly repulsive interaction with the twomagnon continuum to avoid the overlap between them, was also recently reported in a different ladder compound, BiCu_{2}PO_{6} (ref. 7). Moreover, the spectral weight of onemagnon modes at μ_{0}H=10 T in Fig. 3g,h is much less than what is expected from the LSWT in Fig. 4g,h, indicative of a shift of the spectral weight from onemagon state to multimagnon continuum.
To investigate this quantum effect in more detail, we have measured excitations at the magnetic zone centre q=(0.5, 0.5, −0.5) in fields up to 10.8 T and T=0.1 K using a highflux coldneutron spectrometer. Figure 5a summarizes the backgroundsubtracted field dependence of the magnetic excitations. Besides the TM_{high} and TM_{low} modes, interestingly, there is also evidence of a mode induced by the spin fluctuations along but not anticipated by LSWT^{23}. At zero field, it is called the longitudinal mode (LM)^{24,25}, which is predicted to appear in quantum spin systems with a reduced moment size in a vicinity of the quantum critical point^{26}. A detailed study of this type of excitation at zero field will be reported elsewhere.
Figure 5b shows the representative backgroundsubtracted energy scans at the magnetic zone centre in different fields. To extract the peak position Δ and the intrinsic (instrumental resolution corrected) excitation width Γ, we used the same twoLorentzian damped harmonicoscillator model in equation (2) as cross section as used in our previous highpressure studies^{27,28} and numerically convolved it with the instrumental resolution function as described in the Methods section.
The results are plotted in solid (dashed) lines shown in Fig. 5b. The spectral line shape at zero field is the superposition of two such damped harmonicoscillators and the best fits gives the location of two spin gaps at Δ_{TM}=0.32(3) and Δ_{LM}=0.48(3) meV, respectively. The LM mode increases with field and is traceable up to 3 T. For the TM_{high} mode, the observed peaks are instrumental resolution limited (Γ→0) up to 4 T although the lineshape looks narrow at 4 T due to the shallow dispersion slope. The steeper slope at zero field induces a broad peak due to the finite instrumental wavevector resolution. At μ_{0}H=6 and 7.5 T, the line shapes become broadened and the best fits give fullwidth at halfmaximum (FWHM) 2Γ=0.03(1) and 0.07(1) meV, respectively.
With a further increase of field, the spectral line shape becomes complex. As shown in Fig. 5c, the twopeak structures, distinct from the onemagnon state, appear in the spectrum and are accompanied by a suppression of the magnon intensity. These complex features are consistent with the theoretical prediction for spontaneous magnon decay and spectral weight redistribution from the quasiparticle peak to the twomagnon continuum at high fields^{13,18}.
Discussion
Spontaneous magnon decay was also observed in noncollinear transitionmetal oxide compounds due to a strong phononmagnon coupling^{29,30}. In DLCB, our consideration of the broadening concerns antiferromagnetic magnons in the proximity of the zone boundary of phonon modes; thus a direct crossing with the longwavelength acoustic phonon can be excluded. For the metalorganic materials, the optical phonon and spinwave magnon are usually wellseparated, making a phonon branchcrossing scenario unlikely. An entire branch of magnon excitation in DLCB is affected by broadening, making it very hard to reconcile with the phononinduced scenario where only a select area of the qω space is affected. Moreover, the shift of the magnon energy due to the field would make it exceptionally unlikely for the resonantlike condition with the phonon branch to be sustained for all the fields. Therefore, the mechanism of the spinlattice coupling can be ruled out. The observed magnon instability at finite fields in DLCB originates from a hybridization of the singlemagnon state with the twomagnon continuum.
The process of onemagnon decays into the twomagnon continuum is allowed if the following kinematic conditions are satisfied:
where ɛ_{1} is the onemagnon dispersion relation, and and are the lower and upper boundary of the twomagnon continuum, respectively.
Since S_{z} does not commute with the Hamiltonian of equation (1) under the field direction perpendicular to z, it is not a good quantum spin number in a field and the twomagnon continuum at finite field can be obtained from any combination between the acoustic and optical TM_{high} and TM_{low} modes. We calculated the lower boundary of the twomagnon continuum, as described in the Methods, plotted as grey lines in Figs 3 and 5a. The upper boundary, which is above the experimental energy range, is not shown. At zero field and along the reciprocal lattice (H, H, −0.5) direction, the lower boundary already crosses with the TM_{high} mode at H′≈0.15 and 1H′≈0.85, suggesting that DLCB is prone to magnon decays. Upon an increase of the field, the lower boundary of the twomagnon continuum decreases. At 10 T, the continuum lies underneath the singlemagnon branch for the whole BZ as shown in Fig. 3g,h, so the effect of spontaneous decays is expected to be significant.
In the consideration of the magnetic zone centre, the lower boundary of the twomagnon continuum crosses over with the TM_{high} mode at 4 T as shown in Fig. 5a. Figure 6a shows the derived intrinsic FWHM, characteristic of magnon damping, as a function of field up to 7.5 T for the TM_{high} mode. The excitation spectra become even more broadened at higher fields. However, the spectral line shapes become nonLorenzian so FWHM can not be reliably determined for higher fields. Additional evidence of observation of spontaneous magnon decays is indicated from FWHM versus Δ_{2} in Fig. 6b, where Δ_{2} is the energy difference between the onemagnon state and the lower boundary of the twomagnon continuum, the quantity of which can be seen as a proxy of the phase space volume for the decay process.
Figure 6c summarizes the field dependence of the energyintegrated intensity of the experimental data plotted with the calculations by LSWT. For comparison purposes, the calculated intensities were scaled by the intensity ratio of TM_{high} at zero field between data and the calculation. Clearly for the TM_{low} mode, data agree well with the calculation. For the TM_{high} mode, data are consistent with the calculation up to μ_{0}H=4 T, above which the intensity drops much faster than the calculation, suggesting the scenario of magnon breakdown. This is consistent with the result shown in Fig. 5a where the TM_{high} mode lies between the lower and upper boundary of the twomagnon continuum above 4 T so the spontaneous magnon decay becomes possible. Due to the maximum field limit accessible for the experimental study, we cannot trace down the critical field where the TM_{high} mode disappears, but its trend points to a much smaller value of such a field than the saturation field H_{s}∼16 T, predicted by LSWT. We also notice the similar scenario of dramatic intensity change for the LM mode beyond the crossover with the lower boundary of the twomagnon continuum at 1.5 T (see Fig. 6c), where the crossover with the lower boundary of the twomagnon continuum takes place as shown in Fig. 5a.
With the aid of calculations by the LSWT, our neutron scattering measurements on DLCB show the indication of fieldinduced magnon decays in the excitation spectra. Direct evidence for the strong repulsion between the onemagon state and the twomagnon continuum is renormalization of the onemagnon dispersion. Our results establish DLCB as the first experimental realization of an ordered S=1/2 AFM that undergoes fieldinduced spontaneous magnon decays and our study provides muchneeded experimental insights to the understanding of these quantum manybody effects in lowdimensional AFMs.
Methods
Single crystal growth
Deuterated single crystals were grown using a solution method^{20}. An aqueous solution containing a 1:1:1 ratio of deuterated (DMA)Br, (35DMP)Br, where DMA^{+} is the dimethylammonium cation and 35DMP^{+} is the 3,5dimethylpyridinium cation, and the corresponding copper(II) halide salt was allowed to evaporate for several weeks; a few drops of DBr were added to the solution to avoid hydrolysis of the Cu(II) ion.
Neutron scattering measurements
The highfield neutrondiffraction measurements were made on a 0.3 g single crystal with a 0.4° mosaic spread, on a cold tripleaxis spectrometer (CTAX) at the HFIR. Highfield inelastic neutronscattering measurements were performed on a disk chopper timeofflight spectrometer (DCS)^{31} (data not shown) and a multiaxis crystal spectrometer (MACS)^{32} at the NIST Center for Neutron Research, on two coaligned single crystals with a total mass of 2.5 g and a 1.0° mosaic spread. At CTAX, the final neutron energy was fixed at 5.0 meV and an 11T cryomagnet with helium3 insert was used. At DCS, disk choppers were used to select a 167 Hz pulsed neutron beam with 3.24 meV and a 10T cryogenfree magnet with dilution fridge insert was used. At MACS, the final neutron energy was fixed at 2.5 meV and an 11T magnet with dilution refrigerator was used. The background was determined at T=15 K at zero field under the same instrumental configuration and has been subtracted. In all experiments, the sample was oriented in the (H,H,L) reciprocallattice plane to access the magnetic zone centre. The magnetic field direction is vertically applied along the [1 0] direction in real space and is thus perpendicular to the staggered moment direction. Reduction and analysis of the data from DCS and MACS were performed by using the software DAVE^{33}. Neutron diffraction at the antiferromagnetic wavevector measures only the staggered moment component of the total spin moment and the size of the staggered moment m is proportional to the square root of the backgroundsubtracted neutron scattering intensity. The fielddependent m(H) at q=(0.5, 0.5, −0.5) in the inset of Fig. 2 was derived as m(H)=m(H=0) × .
Convolution with the instrumental resolution function
In comparison to the observed magnetic intensity, the calculated dynamic spin correlation function (q, ω) of the spin fluctuation component perpendicular to the wavevector transfer q by LSWT was numerically convolved with the instrumental resolutions function as follows:
where F(q) is the isotropic magnetic form factor for Cu^{2+} (ref. 34) and is a unity normalized resolution function that is peaked on the scale of the FWHM resolution for q≈q′ and ℏω≈ℏω′ and approximated as a Guassian distribution.
Convolution of the excitation spectra in Fig. 4 was considered as Gaussian broadening of the instrumental energy resolution only. Convolution at the magnetic zone centre in Fig. 5b at zero field was obtained as Gaussian broadening of both instrumental energy and wavevector resolutions, which were calculated using the Reslib software^{35}. For all other fields in Fig. 5b, convolution was considered only as Gaussian broadening of the instrumental energy resolution because the dispersion curve becomes flat near the magnetic zone centre and the spectral linewidth broadening due to the instrumental wavevector resolution can be safely neglected.
Determination of the lower boundary of the twomagnon continuum
There are one acoustic and one optical transverse modes in DLCB at zero magnetic field. When a field is applied perpendicular to the easy axis, either the acoustic or the optical mode splits into two branches (TM_{high} and TM_{low}) and it becomes four modes in total. Since S_{z} is not a good quantum spin number in this case, the twomagnon continuum at finite field was then obtained from any combination between the acoustic and optical TM_{high} and TM_{low} modes.
For the interested q=(H,H,L), which is equivalent to (1+H,1+H,L), we find the lower boundary of the twomagnon continuum ɛ_{2}(q)^{min} as follows:
where ɛ_{1} is the onemagnon dispersion relation, q_{1}=(1, 1, 0), and q_{2}=(H, H, L).
The field dependence of ɛ_{1}(q_{1})^{min} obtained by SPINW in limit of linear spin wave approximation^{22} was plotted as the yellow line in Fig. 5a in the main text. The minimum of ɛ_{1}(q_{2}) at μ_{0}H=0, 4, 6 and 10 T can be easily deduced from Fig. 4 in the main text. The determined lower boundary of the twomagnon continuum at μ_{0}H=0, 4, 6 and 10 T was then plotted as the grey lines in Figs 3 and 5a.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Additional information
How to cite this article: Hong, T. et al. Field induced spontaneous quasiparticle decay and renormalization of quasiparticle dispersion in a quantum antiferromagnet. Nat. Commun. 8, 15148 doi: 10.1038/ncomms15148 (2017).
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Acknowledgements
T.H. thanks J. Leao for help with cryogenics. A portion of this research used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. The work at NIST utilized facilities supported by the NSF under Agreement No. DMR1508249. The work of A.L.C. was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award # DEFG0204ER46174.
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T.H. conceived the project. F.F.A. and M.M.T. prepared the samples. T.H., H.A. and Y.Q. performed the neutronscattering measurements. T.H., A.L.C., M.M., D.A.T., K.C. and K.P.S. analysed the data. All authors contributed to the writing of the manuscript.
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Hong, T., Qiu, Y., Matsumoto, M. et al. Field induced spontaneous quasiparticle decay and renormalization of quasiparticle dispersion in a quantum antiferromagnet. Nat Commun 8, 15148 (2017). https://doi.org/10.1038/ncomms15148
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DOI: https://doi.org/10.1038/ncomms15148
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