Abstract
When the energy eigenvalues of two coupled quantum states approach each other in a certain parameter space, their energy levels repel each other and level crossing is avoided^{1}. Such level repulsion, or avoided level crossing, is commonly used to describe the dispersion relation of quasiparticles in solids^{2}. However, little is known about the level repulsion when more than two quasiparticles are present; for example, in a strongly interacting quantum system where a quasiparticle can spontaneously decay into a manyparticle continuum^{3,4,5}. Here we show that even in this case level repulsion exists between a longlived quasiparticle state and a continuum. In our fineresolution neutron spectroscopy study of magnetic quasiparticles in the frustrated quantum magnet BiCu_{2}PO_{6}, we observe a renormalization of the quasiparticle dispersion relation due to the presence of the continuum of multiquasiparticle states.
Main
A fundamental concept in condensed matter physics is the idea that strongly interacting atomic systems can be treated as a collection of weakly interacting and longlived quasiparticles. Within a quasiparticle picture, complex collective excited states in a manybody system are described in terms of effective elementary excitations. The quanta of these excitations carry a definite momentum and energy, and are termed quasiparticles. Magnetic insulators containing localized S = 1/2 magnetic moments and having valencebond solid ground states are ideal systems in which to study bosonic quasiparticles in an interacting quantum manybody system^{6}. The elementary magnetic excitations in these materials are triply degenerate S = 1 quasiparticles called triplons, and their momentum and energyresolved dynamics can be probed directly though inelastic neutron scattering (INS) measurements.
In particular, when the system’s Hamiltonian has an interaction term coupling singleparticle and multiparticle states, the single quasiparticles may decay into the continuum of multiparticle states^{3,4}. In such a system, the Hamiltonian for the single quasiparticles is nonHermitian and the energy eigenvalues are in general complex. The singleparticle decay typically occurs in two ways. Often the singleparticle mode stays as a resonance inside the continuum, but the lifetime becomes short and the mode is highly damped^{3}. Sometimes the single quasiparticle simply ceases to exist, and the dispersion abruptly terminates when it crosses the continuum boundary^{5}. However, there is a third possibility, in which the singlequasiparticle dispersion is significantly renormalized to avoid the multiparticle continuum. This is analogous to the wellknown avoided level crossing behaviour of coupled modes, but in the complex plane of energy eigenvalues^{7}.
Despite broad interest in strongly interacting quantum systems, experimentally realizing an ideal condition to study the interaction between a quasiparticle and a multiparticle continuum turns out to be extremely difficult. One realization occurs in semiconducting quantum dots, where quantum interference effects between a local bound state and a continuum (Fano resonance) produce physics similar to quasiparticles in a continuum^{8,9}. Here we show that the quantum magnet BiCu_{2}PO_{6} presents a rare model system that allows one to study quasiparticle level repulsion in the complex plane. This phenomenon arises owing to the presence of the very large anisotropic exchange interactions in BiCu_{2}PO_{6}, originating from spin–orbit coupling. These anisotropic interactions play a dual role in BiCu_{2}PO_{6}. First, they break the degeneracy of the triplet excitations. Second, the anisotropic exchange interaction is responsible for a strong anharmonic term in the magnetic Hamiltonian which couples the singletriplon and multitriplon excitations. As a result, the triplon lifetime may be reduced in the region of phase space where the (single) triplon dispersion overlaps with multitriplon continuum. This decay process may in fact be so strong that the quasiparticle description ceases to be valid within the continuum. Both types of decay processes are observed in BiCu_{2}PO_{6}. Furthermore, we observe that the dispersion relation of the triplons is strongly renormalized near the boundary of the multitriplon continuum owing to the level repulsion between the single quasiparticle and the continuum.
In the following, we first present INS measurements of the full triplon dispersion in BiCu_{2}PO_{6}, which reveal a rich excitation spectrum including a multitriplon scattering continuum. Analysis of the triplon excitation spectrum using bondoperator theory enables us to determine the magnetic Hamiltonian of BiCu_{2}PO_{6} accurately. We find that strong spin–orbit coupling plays an essential role in this compound through substantial symmetric and antisymmetric anisotropic interactions. We will finally discuss the interaction of the triplon quasiparticles with the continuum, which manifests as a drastic renormalization of the quasiparticle spectra and ultimately spontaneous quasiparticle decay.
Valencebond solid
The orthorhombic crystal structure of BiCu_{2}PO_{6} is shown in Fig. 1a; the structure contains zigzag chains of Cu^{2+} ions running parallel to the baxis. Magnetic interactions along the chains are frustrated because of a competition between the nearestneighbour (NN) and nextnearestneighbour (NNN) antiferromagnetic exchange terms J_{1} and J_{2}. The frustrated chains are coupled strongly along the caxis by antiferromagnetic coupling J_{4} to form a twoleg ladder. As a result of the strong antiferromagnetic coupling J_{4}, two spins on each rung can form spinsinglets, and the ground state is described as an array of singlets termed a valencebond solid. The elementary excitations in this case are spintriplets that can propagate along the chain direction owing to J_{1} and J_{2}. Within a single (ladder) bilayer, there are two crystallographically inequivalent copper sites (Cu_{A} and Cu_{B} as shown in Fig. 1a). This results in the breaking of inversion symmetry across all magnetic bonds, and consequently anisotropic interactions are permitted in the magnetic Hamiltonian, as will be discussed later.
The dispersion relation, which contains essential information regarding the triplon dynamics such as the effective mass and velocity, is revealed directly by INS measurements. Before discussing our neutron scattering measurements, it is helpful to briefly review the expected excitation spectrum of the frustrated twoleg ladder as realized by BiCu_{2}PO_{6}, first ignoring any anisotropic interactions. In the strong coupling limit of J_{1}, J_{2} ≪ J_{4}, the expected excitation spectrum is schematically illustrated in the inset of Fig. 1b^{10,11}. The dispersion has a distinct W shape with a minimum at an incommensurate wavevector; this incommensurate minimum of the gapped spectrum is the manifestation of the magnetic frustration in BiCu_{2}PO_{6}. As BiCu_{2}PO_{6} contains two singlet dimers per unit cell, there are two separate bands of triplons. The bands become degenerate at the point k = 0.5 and are related by a simple folding of the zone, with the minima of each band appearing at incommensurate wavevectors k = 0.5 ± δ. Anisotropic interactions entering the magnetic Hamiltonian may then further split the degeneracy of each triplon band.
An overview of the zerofield INS measurements is presented in Fig. 1b. The gapped, Wshaped, dispersion of each branch is clearly visible at both h = 0 and 3 with a bandwidth of 25 meV, and incommensurate minima at k = 0.575 and k = 0.425. We have not observed any dispersion along the hdirection confirming the weak interbilayer coupling; however, intensities are strongly modulated with momentum transfers along h. This effect is most clearly shown in Fig. 1d, where the intensity along the twodimensional rods of scattering, at k = 0.5 ± 0.075 positions, is plotted. The modulation arises from the interference between the scattering from the two layers within a bilayer, and enables us to probe each mode independently.
What makes BiCu_{2}PO_{6} unique among valencebond solids is the presence of strong anisotropic interactions that qualitatively alter the nature of the triplons. The evidence for anisotropic interactions in BiCu_{2}PO_{6} is first borne out by highresolution measurements around the incommensurate wavevector at Q = (0,0.575,1) shown in Fig. 1c. These reveal that anisotropies in BiCu_{2}PO_{6} completely split the degeneracy of each primary branch such that three distinct modes are observed. Anisotropy splitting is significant, with the minima of each mode corresponding to gap values of Δ_{1} = 1.67(2) meV, Δ_{2} = 2.85(5) meV and Δ_{3} = 3.90(5) meV.
The quantum states of each mode can be further explored through INS measurements performed with applied magnetic fields. The neutron intensities for applied fields of 4, 8 and 11 T are plotted in Fig. 2. No additional splitting of the modes was observed, indicating that anisotropic interactions in the Hamiltonian have completely lifted the SU(2) spin rotation symmetry. Constant momentum transfer cuts around the incommensurate wavevector, Fig. 2d–f, reveal an anomalous Zeeman behaviour. Rather than splitting into the conventional ordering in energy of S_{z} = {+1,0, −1} (refs 6,12), the lowestenergy mode exhibits negligible field dependence and is assigned a S_{z} = 0 quantum state, whereas the two higherenergy modes have the Zeeman character of S_{z} = +1 and S_{z} = −1, respectively; see the field dependence plotted in Fig. 3c. We note that in the presence of anisotropic interactions the singlet and triplet wavefunctions are mixed and S_{z} is no longer a good quantum number. Here we assign each mode a pseudo S_{z} based on on its Zeeman energy, as this provides a convenient labelling scheme.
Noninteracting triplons
The complete dispersion extracted from INS measurements is plotted in Fig. 3. There are six triplon modes: two primary bands each split into three nondegenerate modes by anisotropic interactions.
To understand the spin dynamics in BiCu_{2}PO_{6} we consider the following generic model for interacting spins, S_{i}, on a quasi2D lattice in the b–c plane where relevant Heisenberg exchange terms J_{ij} are shown in Fig. 1a, and H is an external magnetic field. The model Hamiltonian is an extended J_{1}–J_{2}–J_{3}–J_{4} model including symmetryallowed anisotropic spin interactions. The antisymmetric Dzyaloshinskii–Moriya (D_{ij}) and symmetric (Γ_{ij}^{μν}) anisotropic interactions are constrained by the relation Both the anisotropic exchange terms originate from spin–orbit coupling^{13,14,15,16}. Employing a quadratic (noninteracting) bondoperator theory (BOT; refs 17,18,19) for the valencebond ordered ground state with valence bonds on J_{4} links, we have found that the INS data is best described with the following coupling constants: J_{1} = J_{2} = J_{4} = 8 meV, J_{3} = 1.6 meV, D_{1}^{a} = 0.6J_{1}, D_{1}^{b} = 0.45J_{1}, Γ_{1}^{aa} = 0.039J_{1}, Γ_{1}^{bb} = −0.039J_{1} and Γ_{1}^{ab} = Γ_{1}^{ba} = 0.135J_{1}; the calculated triplon dispersion is plotted in Fig. 3a, b. The quadratic BOT captures important details of the lowenergy spectra, such as the slight shift of incommensurate minima between each branch, and the overall bandwidth of the excitations. Importantly, this calculation appropriately describes the anomalous Zeeman splitting plotted in Fig. 3c. Furthermore, extending the BOT to determine the field dependence of each mode for fields applied along the b and c directions correctly predicts the hierarchy of critical fields measured previously: H_{c}^{a} > H_{c}^{b} > H_{c}^{c} (ref. 20).
Although the quadratic BOT describes the measured triplon dispersion and the field dependence very well, it is essentially a meanfield expansion and so overestimates the coupling constants. In addition, this quadratic theory fails to capture some very distinct features of the spectrum, including a bending of the triplon modes around k_{c} ≈ 1 ± 0.26 in Fig. 3a and a excitation continuum resulting from the decay of single triplons in the same region. As we will discuss below, these dynamics ultimately arise as a consequence of the anharmonic magnetic interactions which couple singletriplon quasiparticles with a continuum of multitriplon states.
Multiparticle continuum and level repulsion
In addition to exciting a singletriplon quasiparticle, a neutron can create two or more triplon excitations simultaneously. For example, a neutron with momentum Q can create two triplons with momentum q and Q − q. These twotriplon excitations form a continuum with a lower bound determined by conservation of momentum and energy ω_{2t}^{αβ}(Q) = min_{q}{ω^{α}(q) + ω^{β}(Q − q)}, where ω^{α}(q) is the singletriplon dispersion for a band indexed by α. In many magnetic materials, the twotriplon (or twomagnon) excitations are usually directly observable only at energies much higher than the onetriplon energy, where the twotriplon density of states becomes large. At low energies the presence of the twotriplon continuum can be revealed through the decay behaviour of a single triplon. In the presence of significant anharmonic (cubic order in bond operators) couplings, the triplon lifetime is significantly reduced, even in the absence of any thermal fluctuations. The decay phenomena manifest in a neutron scattering experiment as a strong damping of the quasiparticle peak and a renormalization of the singletriplon dispersion.
In Fig. 4b the quantity ℏωS(Q, ω) is plotted to highlight the effects of multitriplon interactions, including a strongly damped triplon mode at high energies around k = 1, and the bending and extinction of the lowest triplon modes around k = 0.8. We note that the observation of triplon decay is possible owing to the crystal structure of BiCu_{2}PO_{6}, which breaks ladder permutation symmetry explicitly^{21}. Another consequence of this lowsymmetry structure is that the twotriplon continuum scattering intensity modulates in phase with the singletriplon scattering, preventing us from separating the two, unlike in the case of symmetric ladder compounds^{22,23}.
The lower boundaries for twotriplon scattering from each singletriplon branch, ω_{2t}^{αβ}, have been determined from the quadratic dispersion (Fig. 3), and are overlaid in Fig. 4a, b. The multiparticle continuum is expected to exist only above this kinematic bound, which also represents the phase space where a quasiparticle decay can occur. To aid a more careful examination of the decay behaviour of three lowenergy modes considered in Fig. 2, the momentumdependent intensity and linewidth of each mode around k = 0.8 are shown in Fig. 4c, d. The increasing intensity of the highestenergy mode reflects the inclusion of both single and multitriplon contributions as the mode smoothly merges with the continuum. Although we observe a continuous increase in the damping of the highestenergy mode, much more marked effects are apparent in the two lowenergy triplon branches. The singletriplon dispersions for these branches are strongly renormalized by interactions with the continuum, bending away from the quadratic dispersion. Furthermore, these branches remain resolutionlimited in energy at all wavevectors, but terminate abruptly inside the continuum. This is a spectacular example of a spontaneous quasiparticle breakdown, where the decay channels are so effective that an appropriate description of the system in terms of quasiparticles does not exist^{4,24}.
To understand this behaviour quantitatively, we have extended the BOT to include anharmonic (both cubic and quartic) interaction terms. In BiCu_{2}PO_{6}, the Heisenberg exchange terms, J_{1}, J_{2} and J_{4} cancel at cubic order in an interacting bondoperator theory and do not contribute to the spontaneous decay. It is the Dzyaloshinskii–Moriya interactions, D_{1}^{a, b}, which appear as the strongest anharmonic terms and, thus, are responsible for the spontaneous decay of singleparticle states into multiparticle states. Singletriplon dynamics and decay processes are then identified through the spectral weight function of the triplon Green’s function, A(Q, ω), shown in Fig. 4a. A region of substantial triplon decay, indicated by a broadening of the spectral function, occurs in an identical region of phase space where triplon decay was observed in the neutron scattering experiments (Fig. 4b). We note that the renormalized coupling constants, J_{1} = J_{2} = J_{4} = 10 meV, J_{3} = 0.2J_{1}, D_{1}^{a} = D_{1}^{b} = 0.3J_{1} and Γ_{1}^{ab} = Γ_{1}^{ba} = 0.045J_{1}, were used to produce the spectral function shown in Fig. 4a. With the inclusion of anharmonic terms, the anisotropic interactions have been reduced from their noninteracting values, illustrating the significance of triplon interaction for describing spin dynamics of BiCu_{2}PO_{6}. We expect this set of coupling constants to reflect magnetic interactions in BiCu_{2}PO_{6} more accurately. Although the basic phenomenology of the triplon decay is well captured by our calculation, the bending of the lowenergy branch is still not accounted for. The bending occurs near the boundary of the twotriplon continuum, possibly due to avoided level crossing between the single quasiparticle and the continuum.
Although detailed observations of triplon dynamics have been made in the past^{22,25,26}, clear examples of the spontaneous breakdown of a triplon spectrum as observed here are rare. In earlier studies of organometallic materials, termination of a welldefined triplon peak in the excitation spectrum beyond a critical wavevector was observed^{3,5}. The decay process we observe in BiCu_{2}PO_{6} is unique, as each triplon branch exhibits different decay behaviour. The highestenergy mode does not bend, but merges smoothly with and decays into the continuum, in contrast to the behaviour of the two lowerenergy branches. As each triplon branch carries a different spin quantum number, the different decay behaviours of each branch might be a direct consequence of spindependent selection rules for the quasiparticle decay. Obviously, further theoretical studies are necessary to understand different decay behaviour of triplon branches, and also the strong renormalization of the singletriplon branch observed here. In addition, unlike organometallic materials investigated in earlier studies, BiCu_{2}PO_{6} is an inorganic compound in which magnetic coupling parameters can be tuned more easily. Further investigation of doped BiCu_{2}PO_{6} could also be useful for understanding the triplon decay in this material.
In summary, we have mapped the quasiparticle excitation spectra in the quantum magnet BiCu_{2}PO_{6} through comprehensive INS measurements. The overall triplon dispersion is captured by a quadratic bondoperator theory for the valencebond solid and we find that large anisotropic interactions are necessary to describe the excitation spectrum. These anisotropic couplings appear as anharmonic, nonparticle conserving, terms in the bondoperator Hamiltonian and facilitates strong interaction between the triplon quasiparticles. Strong hybridization between the lowesttriplon branches and multitriplon continuum scattering in the neighbourhood of the critical wavevector results in a renormalization of singletriplon dispersion indicative of avoided level crossing. Perhaps the most important feature of the excitation spectrum in BiCu_{2}PO_{6} is this selective hybridization, renormalization, and termination of the two lowest branches, distinct from the smooth merging of the highestenergy branch into a continuum. Further theoretical investigation of interacting triplons could shed light on the origin of the observed unusual decay behaviour.
Methods
All measurements used the same 4.5 g single crystal as previous studies^{28}. Magnetic excitations in BiCu_{2}PO_{6} were mapped through inelastic neutron scattering (INS) measurements performed on a number of instruments. Highenergy timeofflight neutron scattering measurements where carried out on the SEQUOIA spectrometer at the Spallation Neutron Source (SNS), covering the full dynamic range of excitations in BiCu_{2}PO_{6} with a fine energy resolution of ΔE ∼ 0.8 meV at the elastic line. Measurements on SEQUOIA were performed with a fixed incident neutron energy of E_{i} = 40 meV and the fineresolution Fermichopper (FC_{2}; ref. 29) rotating at 360 Hz. The sample was mounted with the (h, k, 0) plane lying in the horizontal scattering plane of the instrument and (h, 0,0) initially aligned along the incident neutron wavevector k_{i}. To map the complete dynamic structure factor S(Q, ω) the sample was rotated through 180° in 0.5° steps. All measurements on SEQUOIA were performed with the sample held at a temperature of 4 K. Another set of highresolution measurements where conducted on the SPINS cold tripleaxis spectrometer at the NIST Center for Neutron Research (NCNR). Here the sample was mounted in the (0, k, l) scattering plane and all measurements used a fixed final energy of E_{f} = 3.7 meV employing a vertically focusing PG monochromator, a flat PG analyser, and a BeO filter between the sample and analyser. The spectrometer collimation sequence was Guide80′80′Open, resulting in an energy resolution of ΔE ∼ 0.1 meV at the elastic line. For the duration of the experiment, the sample was mounted on a Cu mount and temperature was controlled in a ^{3}He dilution refrigerator. Measurements in an applied magnetic field were carried out on the DCS timeofflight spectrometer at NIST (ref. 30). All measurements on DCS were performed using a fixed incident neutron wavelength of λ_{i} = 2.9 Å. The energy resolution on DCS was ΔE ∼ 0.3 meV at the elastic line. The sample was mounted in the (0, k, l) scattering plane with (0,0, l) initially at 50° from the incident neutron beam and then rotated through 120° in 0.5° steps throughout the measurement. The sample was fixed on a Cu mount in a 11.5 T vertical field cryomagnet with a dilution refrigerator insert. A magnetic field between 4 and 11.5 T was applied along the aaxis and the sample was held at T = 100 mK for the duration of the measurements. Because of the very narrow magnet aperture, measurements with applied field on DCS were confined to the (0, k, l) scattering plane, with momentum transfers in the vertical direction limited to h = 0 ± 0.2 r.l.u.
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Acknowledgements
We would also like to thank G. Uhrig, O. Tchernyshyov and S. K. Kim for helpful discussions. This research was supported by NSERC of Canada, Canada Foundation for innovation, Canada Research Chairs Program, and Centre for Quantum Materials at the University of Toronto. Work at ORNL was sponsored by the Division of Scientific User Facilities, Office of Basic Energy Science, US Department of Energy (DOE). Work at NIST utilized facilities supported in part by the National Science Foundation under Agreement No. DMR0944772.
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Affiliations
Department of Physics and Center for Quantum Materials, University of Toronto, Toronto, Ontario M5S 1A7, Canada
 K. W. Plumb
 , Kyusung Hwang
 , Yong Baek Kim
 & YoungJune Kim
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
 Y. Qiu
 & Leland W. Harriger
Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA
 Y. Qiu
Neutron Data Analysis and Visualization Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
 G. E. Granroth
Chemical and Engineering Materials Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
 Alexander I. Kolesnikov
Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan
 G. J. Shu
 & F. C. Chou
Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, CH–5232 Villigen, Switzerland
 Ch. Rüegg
Department of Quantum Matter Physics, University of Geneva, CH–1211 Geneva, Switzerland
 Ch. Rüegg
Canadian Institute for Advanced Research/Quantum Materials Program, Toronto, Ontario MSG 1Z8, Canada
 Yong Baek Kim
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Contributions
K.W.P. and Y.J.K. conceived the experiments. K.W.P., Y.Q., L.W.H. G.E.G. and A.I.K. performed the experiments and K.W.P. analysed the data. C.R. provided additional data. K.H. and Y.B.K. developed the theoretical model and performed calculations. G.J.S. and F.C.C. provided the sample. K.W.P. and Y.J.K. wrote the paper with contributions from all coauthors.
Competing interests
The authors declare no competing financial interests.
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Correspondence to YoungJune Kim.
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