# Quasiparticle-continuum level repulsion in a quantum magnet

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. Such level repulsion, or avoided level crossing, is commonly used to describe the dispersion relation of quasiparticles in solids. However, little is known about the level repulsion when more than two quasiparticles are present; for example, in an open quantum system where a quasiparticle can spontaneously decay into many particle continuum. Here we show that even in this case level repulsion exists between a long-lived quasiparticle state and a continuum. In our fine resolution neutron spectroscopy study of magnetic quasiparticles in a frustrated quantum magnet BiCuPO, we observe a renormalization of quasiparticle dispersion relation due to the presence of the continuum of multi-quasiparticle states. Our results have a broad implication for understanding open quantum systems described by non-hermitian Hamiltonian.

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 long-lived quasiparticles. Within a quasiparticle picture, complex collective excited states in a many body 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 magnetic moments and having valence bond solid ground states are ideal systems in which to study bosonic quasiparticles in an interacting quantum many body system Giamarchi et al. (2008). The elementary magnetic excitations in these materials are triply-degenerate quasiparticles called triplons, and their momentum and energy resolved dynamics can be probed directly though inelastic neutron scattering.

In particular, when the system’s Hamiltonian has an interaction term coupling
single and multi-particle states, the single quasiparticles may decay
into the continuum of multiparticle states Stone et al. (2006); Zhitomirsky and Chernyshev (2013).
Therefore, an ensemble of quasiparticles as may be realized in a quantum magnet
is a good example of an open quantum system, in which particle number is not
conserved. ^{1}^{1}1The term, open quantum system, is often used in a different context, referring to a system that interacts with an environment, and follows stochastic dynamics. [See for example, H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford Univ. Press, Clarendon, 2007).] In this article, however, we are only concerned with a system described by non-hermitian Hamiltonian, in which particles can decay.
In such a system, the energy eigenvalues are in general complex,
and the particles have a finite lifetime, which is given by the imaginary part
of the complex eigenvalue Rotter (2009). It was shown that even in the case of an
open quantum system, the avoided level crossing could occur in the complex
plane OkoÅowicz et al. (2003).

Despite broad interest in interacting open quantum systems, experimentally realizing an ideal condition to study the interaction between a quasiparticle and a multi-particle continuum turns out to be extremely difficult. Here we show that the quantum magnet BiCuPO presents a rare model system that allows one to study quasiparticle level repulsion in the complex plane. This phenomena arises due to the presence of the very large anisotropic exchange interactions in BiCuPO, originating from spin-orbit coupling (SOC). These anisotropic interactions play a dual role in BiCuPO. First, they break the degeneracy of the triplet excitations, making the triplon dispersion relation non-degenerate in phase space. In addition, the anisotropic exchange interaction is responsible for the strong anharmonicity which couples the single and multi-triplon excitations. As a result, the triplon lifetime may be reduced in the region of phase space where (single) triplon dispersion overlaps with multi-triplon continuum. This decay process may in fact be so strong that the quasiparticle description ceases to be valid within the continuum. Furthermore, we show that the dispersion relation of the triplons is strongly renormalized near the boundary of the multi-triplon continuum due to the level repulsion between the single quasiparticle and the continuum.

In the following, we first present inelastic neutron scattering (INS) measurements of the full triplon dispersion in BiCuPO, which reveal a rich excitation spectrum including a multi-triplon continuum of scattering. Analysis of the triplon excitation spectrum using bond-operator theory enables us to determine the magnetic Hamiltonian of BiCuPO 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.

## Valence Bond Solid

The orthorhombic crystal structure of BiCuPO is shown in Fig. 1 (a); the structure contains zig-zag chains of ions running parallel to the b-axis. Magnetic interactions along the chains are frustrated because of a competition between the nearest-neighbor (NN) and next-nearest-neighbor (NNN) antiferromagnetic exchange terms and . The frustrated chains are coupled strongly along the c-axis by antiferromagnetic coupling to form a two-leg ladder. As a result of the strong antiferromagnetic coupling , two spins on each rung can form singlets, and the ground state is described as an array of singlets termed a valence bond solid. The elementary excitations in this case are triplets that can propagate along the chain direction due to and . Within a single (ladder) bilayer, there are two crystallographically inequivalent copper sites (Cu and Cu as shown in Fig. 1 (a)). 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 inelastic neutron scattering. Before discussing our neutron scattering measurements, it is helpful to briefly review the expected excitation spectrum of the frustrated two-leg ladder as realized by BiCuPO, first ignoring any anisotropic interactions. In the strong coupling limit of , the expected excitation spectrum is schematically illustrated in the inset of Fig. 1 (b)Tsirlin et al. (2010); Lavarélo et al. (2011). The dispersion has a distinct W shape with minimum at an incommensurate wavevector. The incommensurate minimum of the gapped spectrum is the manifestation of the magnetic frustration in BiCuPO; since BiCuPO contains two singlet dimers per unit cell, there are two separate bands of triplons. The bands become degenerate at the point and are related by a simple folding of the zone with the minima of each band appearing at incommensurate wavevectors . Anisotropic interactions entering the magnetic Hamiltonian may then further split the degeneracy of each triplon band.

An overview of the zero field INS measurements is presented in Fig. 1 (b). The gapped, W-shaped, dispersion of each branch is clearly visible at both and with a bandwidth of 25 meV, and incommensurate minima at and in (a) and (b) respectively. We have not observed any dispersion along the -direction confirming the weak inter-bilayer coupling; however, intensities are strongly modulated with momentum transfers along . This effect is most clearly shown in Fig. 1 (d) where the intensity along the two dimensional rods of scattering, at positions, is plotted. The modulation arises from the bilayer structure factor: interference between the scattering from the two layers within a bilayer. For weak inter-bilayer coupling the bilayer structure factor can be written simply in the following form , where is the intrabilayer spacing, shown in Fig. 1 (a), is a unit vector directed along , is a phase factor for the mode indexed by , and is an arbitrary amplitude factor. Solid lines in Fig. 1 (d) are the bilayer structure factor with the known intrabilayer spacing for BiCuPO of , and and for the modes with minima at and respectively.

What makes BiCuPO unique among valence bond solids is the presence of strong anisotropic interactions that qualitatively alter the nature of the triplons. The evidence for anisotropic interactions in BiCuPO is first borne out by high resolution measurements around the incommensurate wavevector at shown in Fig. 1 (c). These reveal that anisotropies in BiCuPO completely split the degeneracy of each primary branch such that three distinct modes are observed. Note that the intensity modulation of the primary modes by the bi-layer structure factor enables the independent probing of each mode. Anisotropy splitting is significant, with the minima of each mode corresponding to gap values of meV, meV, and meV.

The quantum states of each mode can be further explored via 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
further splitting of the modes was observed indicating that anisotropic
interactions in the Hamiltonian have completely lifted the SU(2) spin rotation
symmetry ^{2}^{2}2In this work, measurements in applied fields are all for the
very narrow region r.l.u. because the magnet aperture
limited the detection of neutrons with momentum transfer out of the
horizontal scattering plane. Constant momentum-transfer cuts around the
incommensurate wavevector, Fig. 2 (d)-(f), reveal an anomalous
Zeeman behaviour. Rather than splitting into the conventional ordering in
energy of Matsumoto et al. (2002); Giamarchi et al. (2008), the
lowest energy mode exhibits negligible field dependence and is assigned a quantum state, while the two higher energy modes have the Zeeman character
of and respectively. (See the field dependence
plotted in Fig. 3c.) ^{3}^{3}3 Of course, in the presence of anisotropic
interactions the singlet and triplet wave functions are mixed and is
no longer a good quantum number. Here we assign each mode a pseudo
based on on its Zeeman energy as this provides a convenient labelling
scheme.

## Non-interacting triplons

The complete dispersion extracted from INS measurements, combining the data for and , is plotted in Fig. 3. There are six triplon modes, two bands arise from two inequivalent dimers per unit cell which are each further split into three non-degenerate modes by anisotropic interactions.

To understand the spin dynamics in BiCuPO we consider a generic spin Hamiltonian with Heisenberg , as well as antisymmetric and symmetric anisotropic interactions

(1) | ||||

in which the symmetric anisotropy term is constrained by the relation

(2) |

Both the antisymmetric Dzyaloshinskii-Moriya (DM) and symmetric ()
anisotropic exchange terms originate from spin-orbit
coupling Dzyaloshinsky (1958); Moriya (1960); Shekhtman et al. (1992); Yildirim et al. (1995). While
the symmetric anisotropy term is the smaller of the two, and is often
neglected, it can have pronounced effects on the magnetic ground state and
excitation spectrum Zheludev et al. (1998). Employing a quadratic (non-interacting)
bond operator theory (BOT) Sachdev and Bhatt (1990); Gopalan et al. (1994); Matsumoto et al. (2004), for
the valence-bond ordered ground state with valence bonds on links, we
have found that the INS data is best described with the following coupling
constants: meV, meV,
, , ,
, and
^{4}^{4}4See supplementary information
for details of the bond operator theory.; the calculated triplon
dispersion is plotted in Fig. 3 (a) and (b). At quadratic
order, the BOT captures important details of the low energy spectra including
the slight shift of incommensurate minima between each branch,
as well as the overall bandwidth of the excitations. Importantly, this calculation appropriately describes the anomalous Zeeman splitting
plotted in Fig. 3 (c). Furthermore, extending the BOT ^{5}^{5}5See supplementary information
for the calculated Zeeman behaviors of
excitations for . to
determine the field dependence of each mode for fields applied along the
and directions correctly predicts the hierarchy of
critical fields measured previously
.Kohama et al. (2012) The
quadratic BOT is essentially a mean-field expansion and so
is likely to overestimate the coupling constants. However, even within the
mean field estimation, it is remarkable that such large anisotropic
interactions are required to describe magnetic excitations in BiCuPO.

While the BOT describes the overall features of the measured triplon dispersion, this quadratic theory fails to capture some very distinct features of the spectrum, including a bending of the triplon modes around k in Fig. 3 (a). Here the single triplon dispersions are strongly renormalized, bending away from the quadratic dispersion, in an avoided crossing with a multi-triplon continuum. In addition, the dispersion abruptly stops beyond the critical wavevector of . As we will discuss below, these dynamics ultimately arise as a consequence of the anharmonic magnetic interactions which couple single triplon quasiparticles with multi-triplon continuum states.

## Multiparticle continuum and level repulsion

The presence of large anisotropic exchange interactions has dramatic implications on the behaviour of triplons in this system. In contrast to isotropic quantum magnets where the triplon dynamics are typically well described in a harmonic expansion, the anisotropic exchange interactions in BiCuPO produce significant anharmonic (cubic order in bond operators) couplings appearing as non-particle conserving terms in the bond-operator Hamiltonian. This anharmonicity modifies the triplon dispersion relation in BiCuPO qualitatively. In figure 4 (b) the quantity is represented in the false color map. highlights the effects of multi-triplon interactions, including an avoided crossing and extinction of the lowest bands around and a continuum of scattering at high energies around .

In addition to exciting a single-triplon quasiparticle, a neutron can create two or more triplon excitations simultaneously. For example, a neutron with momentum can create two triplons with momentum and . These two-triplon excitations form a continuum with a lower bound determined by conservation of momentum and energy , where is the single-triplon dispersion. This lower boundary for two-triplon scattering, as determined from the quadratic dispersion, approximately delineates the region of continuum scattering shown in Fig. 4 (b). It is within this kinematic bound that the qualitative effects of anharmonic interactions in BiCuPO are most apparent. Here the non-particle conserving terms offer decay channels for single triplon excitations. A physical consequence of this is that the triplon lifetime is significantly reduced, even in the absence of any thermal fluctuations. The effect manifests in a neutron scattering experiment as a strong damping of the quasiparticle peak. In Figs. 4 (c) and (d) the momentum dependent intensity and linewidth of each mode around are shown, highlighting the different behaviour of each mode in this region. While we observe a continuous increase in the damping of the highest energy mode as it smoothly merges into the continuum, a much more dramatic effect is apparent in the two low energy triplon branches. The single triplon dispersions for these branches are strongly renormalized by interactions with the continuum, bending away from the quadratic dispersion, in an avoided crossing as shown in Fig. 3 (a). In addition, these branches remain resolution limited in energy over all wave vectors, but terminate abruptly upon entering 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 Zhitomirsky (2006); Zhitomirsky and Chernyshev (2013).

While detailed observations of triplon dynamics have been made in the past Xu et al. (2000); Stone et al. (2001); Notbohm et al. (2007), clear examples of the spontaneous breakdown of a triplon spectrum as observed here are rare. In the two-dimensional correlated singlet material piperazinium hexachlorodicuprate (PHCC), a well defined triplon peak in the excitation spectrum was observed to abruptly merge with a continuum and vanish beyond a threshold momentum Stone et al. (2006). Somewhat different phenomena were observed in the organometallic two-leg ladder compound IPA-CuCl, in which the single triplon dispersion abruptly terminated beyond a critical wavevector, without damping nor two triplon continuum Masuda et al. (2006). The decay process we observe in BiCuPO is unique, as each triplon branch exhibits different decay behavior, as illustrated schematically in Fig. 4a. The highest energy mode does not bend, but merges smoothly with and decays into the continuum, in contrast to the behavior of the two lower energy branches. Since each single particle mode is associated with a different effective spin quantum number, this behaviour may be associated with branch dependent selection rules in the decay processes. Such an explicit manifestation of spin dependent quasiparticle interactions has not been observed before.

We would like to emphasize that the underlying magnetic interactions in BiCuPO
satisfies two crucial conditions for realizing stong coupling between the
single and multiparticle states. First, the large bandwidth of triplon
excitations relative to the gap energy at the incommensurate minima results in
a large overlap between the single-triplon and continuum states. This overlap
ensures that the kinematic conditions for the decay of a single-triplon are
satisfied over a large region of phase space. Second, strong anharmonic
interactions provide channels for coupling single and multi triplon states
Zhitomirsky (2006); Zhitomirsky and Chernyshev (2013). The existence of magnetic
interactions which couple single and multi-particle states are not always
guaranteed. In BiCuPO, the Heisenberg exchange terms, , , and
cancel at cubic order in an interacting bond operator theory and do not
contribute to the spontaneous decay. It is the DM interactions which appear as
the strongest anharmonic terms and, thus, are responsible for the spontaneous
decay of single particle states into multiparticle ones.^{6}^{6}6A full
theoretical treatment of triplon interactions is beyond the scope of this
article, but such interactions will renormalize the DM parameters to
somewhat smaller values. Nevertheless substantial and terms
are essential for fitting the magnetic excitation spectra.

In summary, we have mapped the quasiparticle excitation spectra in the quantum magnet BiCuPO through comprehensive INS measurements. The low energy triplon excitations are captured by a quadratic bond operator theory for the valence bond solid and we find that very large anisotropic interactions are necessary to describe the excitation spectrum. These anisotropic couplings appear as anharmonic, non particle conserving, terms in the bond operator Hamiltonian and manifest in a complete termination of the quasiparticle spectra beyond a critical wavevector. Strong hybridization between the lowest triplon branches and higher order continuum scattering in the neighbourhood of the critical wavevector results in a renormalization and avoided crossing in this open quantum system. Perhaps the most important feature of the excitation spectrum in BiCuPO is this selective hybridization, renormalization, and termination of the two lowest branches, distinct from the smooth merging of the highest energy branch into a continuum. Further theoretical investigation of interacting triplons could shed light on the origin of the observed unusual decay behaviour.

Acknowledgments: We would also like to thank Götz Uhrig, Oleg Tchernyshyov and Se Kwon 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. DMR-0944772.

Methods: All measurements used the same 4.5 g single crystal as previous studies Plumb et al. (2013). Magnetic excitations in BiCuPO were mapped through INS measurements performed on a number of instruments. High energy time of flight neutron scattering measurements where carried out on the SEQUOIA spectrometer at the Spallation Neutron source covering the full dynamic range of excitations in BiCuPO with high energy resolution meV at the elastic line. Measurements on SEQUOIA were performed with a fixed incident neutron energy of E= 40 meV and the fine resolution Fermi-chopper (FC)Granroth et al. (2010) 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 . In order to map the complete dynamic structure factor 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 high resolution measurements where conducted on the SPINS cold triple axis spectrometer at the NIST center for neutron research. Here the sample was mounted in the (0,k,l) scattering plane and all measurements used a fixed final energy of E = 3.7 meV employing a vertically focusing PG monochrometer, a flat PG analyser, and a BeO filter between the sample and analyser. The spectrometer collimation sequence was Guide-80’-80’-Open resulting in an energy resolution of 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 He dilution refrigerator. Measurements in an applied magnetic field were carried out on the DCS time-of-flight spectrometer at NIST Copley and Cook (2003). All measurements on DCS were performed using a fixed incident neutron wavelength of Å. The energy resolution on DCS was 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 refrigerators insert. A magnetic field between 4 and 11.5 T was applied along the a-axis and the sample was held at T=100 mK for the duration of the measurements. Because of the very narrow magnet aperture, measurements on DCS were confined to the scattering plane with momentum transfers in the vertical direction limited to r.l.u..

Author 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 co-authors.

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Supplementay Information: Quasiparticle-continuum level repulsion in a quantum magnet

## Appendix A Bond Operator Theory

Considering the valence bond crystal order with valence bonds at the links in BiCuPO, we have calculated triplon excitation spectrum within the framework of bond operator theory as described below. To describe spin dynamics in BiCuPO, we employ a general spin Hamiltonian with Heisenberg , Dzyaloshinskii-Moriya , and anisotropic symmetric exchange interactions

(3) |

where summations over are assumed. The symmetric anisotropy term is constrained by the relation

(4) |

Hence, for a given link, and are regarded as free parameters with determined by the former. The coupling constants in the Hamiltonian are constrained with the space group symmetry into , , and as introduced in main part of the paper. In our theory, the anisotropic interactions , and at links are ignored since is already significantly small compared to the other Heisenberg interactions.

Following standard procedures in bond operator theory Sachdev and Bhatt (1990); Gopalan et al. (1994); Matsumoto et al. (2004), we rewrite the Hamiltonian by using the bond operator representations for the spins in each dimer at links

(5) |

where and is the totally antisymmetric tensor. Here, the bond operators and create the spin-singlet and spin-triplet states at the dimer respectively. We require the bosonic statistics for the bond operators. The bond operator representation enlarges the Hilbert space so that we restrict it to the physical Hilbert space by allowing only one bond particle at a dimer via the Lagrange multiplier : . On top of that, the -bosons are condensed to describe the valence bond crystal phase: .

After all above procedures, the bond operator Hamiltonian has the form , where consists of quadratic terms of -boson operators, and so forth. We consider the quadratic part for the description of single-triplon excitations in BiCuPO. The quadratic bond operator Hamiltonian is diagonalized through Bogoliubov transformation:

(6) |

where is the Bogoliubov quasiparticle, here triplon, operator with the excitation energy . Here, is momentum and is band index of the triplon excitation spectrum. The ground state is determined by the equations .

As already discussed in main body of the paper, neutron scattering measurement data is well reproduced in the bond operator theory with the following set of coupling constants: meV, meV, , , , , and . With this set of coupling constants, the ground state has , . Comparing with the case of an isolated dimer (, ), we notice that the valence bonds in BiCuPOare quite soft with significant fluctuations, which is attributed to strong inter-dimer interactions (, , ) comparable to intra-dimer interaction (). An important comment is followed. In the quadratic Hamiltonian , the DM terms with the coupling constant cancel and do not explicitly enter the Hamiltonian, rather it is the symmetric anisotropic interaction which is essential for describing magnetic anisotropy in BiCuPO. Of course the existence of a large symmetric anisotropy term necessarily implies a large antisymmetric DM term through Eq. 4.

The bond operator theory approach was able to successfully capture the magnetic field dependence of triplon excitations for as measured by neutron scattering. However, as an additional test of the theory we may explore predictions for the Zeeman behaviour of the triplon bands with magnetic fields applied along the other axes of the crystal. This can provide information about the anisotropic critical fields which have been measured by high field magnetization experiments Kohama et al. (2012).

The Zeeman energy of the triplon bands at the dispersion minimum calculated within the bond operator theory are plotted in Fig. S1. Each band has a characteristic field direction, where the Zeeman dependence is flat, with negligible field dependence. The critical magnetic field at which the spin gap is closed can be obtained by extrapolating the data points for each applied magnetic field direction, giving T, T, and T roughly in agreement with measured critical fields Kohama et al. (2012). Besides the hierarchy in the critical fields, a rather curious Zeeman behavior is revealed in Fig. S1 which shows the magnetic field dependent energy of each band is strongly coupled to the applied field direction with the modes having the relative Zeeman character of , , and for fields applied along the , , and axis respectively. Here, means the spin component along the field direction.

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- Matsumoto et al. (2004) M. Matsumoto, B. Normand, T. M. Rice, and M. Sigrist, Phys. Rev. B 69, 054423 (2004).
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