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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 BiCu2PO6, we observe a renormalization of quasiparticle dispersion relation due to the presence of the continuum of multi-quasiparticle states. Our results have a broadimplication for understanding open quantum systems described by non-hermitian Hamiltonian.

Through comprehensive inelastic neutron scattering experiments we have measured the spin excitation spectrum of the valence-bond ordered quantum magnet BiCu2PO6. We find that remarkably large anisotropic interactions, which originate from spin orbit coupling, are essential to explain the low energy excitation spectrum and its magnetic field dependence. The low energy excitation spectrum is captured by a quadratic bond-operator theory which includes large symmetric and antisymmetric anisotropy terms D/J1 ∼ 0.6 and Γ/J1 ∼ 0.14. The giant anisotropies place BiCu2PO6 in a unique regime among 3d transition metal oxides and we discuss implications of the large spin anisotropies for the high field phases in BiCu2PO6. Anisotropic magnetic interactions are always present in real materials but are often treated as a complicating and extraneous detail. Recently, it has been discovered that in certain limits large anisotropies induced by spin-orbit coupling (SOC) can play a fundamental role, offering routes to realize unexpected new phases of matter [1]. In 3d transition metal oxides SOC is very weak compared with the energy scales set by electronic correlations and crystal field levels, and is generally considered to be a small perturbation in an effective spin Hamiltonian. For magnetic compounds with non-centrosymmetric crystal structures SOC is manifest through the antisymmetric Dzyaloshinskii-Moriya (DM) and symmetric (Γ) anisotropic exchange terms [2][3][4]. Both exchange terms arise in an effective magnetic Hamiltonian when the Anderson superexchange mechanism [5] is extended to include spin-orbit coupling [3]. The antisymmetric term D · (S i ×S j ) is linear in SOC and so was originally suggested to represent the leading anisotropy [3], while the symmetric term Γ µν ij S µ i S ν j is second order in SOC and is often neglected.
Although typically much smaller than the isotropic Heisenberg interactions, spin-orbit induced anisotropies can qualitatively affect the ground states of ordered magnetic compounds. Dzyaloshinsky-Moriya interactions often favor non-collinear structures, giving rise to weak ferromagnetism in antiferromagnetic compounds [2,3,6] as well as stabilizing helical or chiral order [7][8][9]. For materials which have spiral magnetic ordering, DM interactions can also act to induce a ferroelectric polarization [10][11][12], giving rise to multiferroic behavior. In gapped quantum magnets, such as valence-bond solids, the effect of small spin-orbit induced anisotropies is more subtle. Here the main effect of DM interactions is to break the degeneracy of the triplet excitations, and mix singlet and triplet excited states [13][14][15]. As a result, the ground state can no longer be described as a singlet state and the spin excitation spectrum becomes very complex. The DM physics of a gapped system can be studied experimentally by applying a high magnetic field; the DM term stabilizes exotic high-field phases above a certain critical field, and may alter the universality class of the field-driven phase transition [16][17][18][19][20]. In the valence-bond magnet BiCu 2 PO 6 , high field specific heat and NMR measurements have revealed a rich phase diagram above an anisotropic critical field [20,21]. The intricate phase diagram was speculated to arise from a combination of frustration and the spin-orbit interaction [20]; however, the underlying Hamiltonian of BiCu 2 PO 6 was unknown and a quantitative understanding of the microscopic physics underlying observed high-field properties has not yet been offered.
In this letter we report inelastic neutron scattering (INS) measurements of the complete spin excitation spectrum in BiCu 2 PO 6 which enable an accurate determination of the magnetic Hamiltonian. We find that remarkably strong spin-orbit coupling plays an essential role in this compound through giant symmetric and antisymmetric exchange interactions D 1 /J 1 ∼ 0. 6  0.14. The anisotropic interactions result in a complete loss of spin rotational symmetries in zero field and an anomalous Zeeman splitting of the excitation spectrum in applied magnetic fields. Elucidation of the magnetic Hamiltonian for BiCu 2 PO 6 allows us to clarify the nature of high field phases as described later.
BiCu 2 PO 6 has an orthorhombic crystal structure containing zig-zag chains of Cu 2+ ions running parallel to the b-axis as shown in Fig. 1. Previously we have identified the dominant Heisenberg exchange pathways [22]. Along the chains neighboring Cu 2+ interact via competing nearest-neighbor (NN) and next-nearest-neighbor (NNN) antiferromagnetic exchange terms J 1 and J 2 . The chains are coupled along the c-axis by J 4 to form a frustrated two-leg ladder, with an additional weak interladder exchange J 3 . The frustrated ladder can alternatively be viewed as a bilayer structure with spacing a = 0.162a. Owing to the presence of two crystallographically inequivalent copper sites (Cu A and Cu B in Fig. 1), inversion symmetry is broken across all magnetic bonds; consequently, DM interactions are permitted by symmetry with limited constraints. The potentially relevant DM vectors are illustrated as arrows occupying the bonds in Fig. 1 Magnetic excitations in BiCu 2 PO 6 were mapped through INS measurements performed on a number of instruments. Time-of-flight measurements were conducted on the SEQUOIA spectrometer [23] at the Spallation Neutron Source in zero field, and in applied magnetic fields on the DCS spectrometer [24] at the NIST center for neutron research. High resolution zero field measurements were also performed on the SPINS triple axis spectrometer at NIST. Details of each instrument and configuration are provided in the Supplemental Material. All measurements used the same 4.5 g single crystal as previous studies [22]. Prior to discussion of the INS data it is helpful to briefly review expectations for the excitation spectrum of the frustrated two-leg ladder. In the absence of any anisotropic interactions, a perturbative expansion around the limit of J 1 , J 2 J 4 is an informative starting point. This expansion produces the excitation spectrum schematically illustrated in the inset of Fig. 2 (a) [26,27]. The dispersion has a distinct W shape with minimum at an incommensurate wavevector determined by the competition between J 1 and J 2 . BiCu 2 PO 6 contains two dimers per unit cell and so there are two separate bands of triply degenerate spin excitations, or 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 mag-netic Hamiltonian may then further split the degeneracy of each primary triplon band.
An overview of the zero field INS measurements is presented in Fig. 2. The gapped, W-shaped, 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 in (a) and (b) respectively. We have not observed any dispersion along the h-direction confirming the weak interbilayer coupling; however, intensities are strongly modulated with momentum transfers along h. This effect is most clearly shown in Fig. 2  (d) where the intensity along the two dimensional rods of scattering, at k = 0.5 ± 0.075 positions, is plotted. The modulation arises from a bilayer structure factor, analogous to what is observed in the bilayer parent cuprate YBa 2 Cu 3 O 6+x [28]. For weak interbilayer coupling the bilayer structure factor can be written simply in the following form B i (Q) = A cos 1 2 Q·a ĥ + φ i , where a is the intrabilayer spacing, shown in Fig. 1 (a),ĥ is a unit vector directed along h, φ i is a phase factor for the mode indexed by i, and A is an arbitrary amplitude factor. Solid lines in Fig. 2 (d) are the bilayer structure factor with the known intrabilayer spacing for BiCu 2 PO 6 of a = 0.162a, and φ i = 0 and π/2 for the modes with minima at k = 0.575 and k = 0.425 respectively.
High resolution measurements around the incommensurate wavevector at Q = (0, 0.575, 1) reveal that strong anisotropies in BiCu 2 PO 6 completely split the degeneracy of each primary branch. Note that the intensity modulation of the primary modes by the bi-layer structure factor enables the independent probing of each mode. A representative scan is shown in Fig. 2 (c), where three resolution limited modes are clearly resolved. Anisotropy splitting is significant, with the minima of each band corresponding to gap values of ∆ 1 = 1.67(2) meV, ∆ 2 = 2.85(5) meV, and ∆ 3 = 3.90(5) meV.
In order to explore the quantum states of each excitation branch, INS measurements were performed with applied magnetic fields. The neutron intensities for applied fields of 4, 8, and 11 T are plotted in Fig. 3. No further splitting of the modes was observed indicating that anisotropic interactions in the Hamiltonian have completely lifted the SU(2) spin rotation symmetry [29]. Constant momentum transfer cuts around the incommensurate wavevector, Fig. 3 (d)-(f), reveal an anomalous Zeeman behaviour. Rather than splitting into the conventional ordering in energy of S z = {+1, 0, −1} [30,31], the lowest energy mode exhibits negligible field dependence and is assigned a S z = 0 quantum state, while the two higher energy modes have the Zeeman character of S z = +1 and S z = −1 respectively [32].
The complete dispersion extracted from INS measurements, combining the data for h = 0 and h = 3, is plotted in Fig. 4. 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 BiCu 2 PO 6 we consider a generic spin Hamiltonian with Heisenberg J ij , as well as antisymmetric D ij and symmetric anisotropic Γ µν ij interactions and the symmetric anisotropy term is constrained by the relation Both the antisymmetric and symmetric anisotropy terms originate from spin-orbit coupling [3,4,6]. 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 [33]. Employing a quadratic bond operator theory (BOT) [34][35][36], for the valence-bond 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: [37]; the calculated triplon dispersion is plotted in Fig. 4 (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, avoided crossing of the modes at k = 0.5 [ Fig. 4 (b)], as well as the overall bandwidth of the excitations. Importantly, this set of coupling constants appropriately describes the anomalous Zeeman splitting plotted in Fig. 4 (c). Furthermore, extending the BOT to determine the Zeeman dependence of each mode for field applied along the b and c directions correctly predicts the hierarchy of critical fields measured previously H a c > H b c > H c c [20]. The magnetic field dependent energy of each mode is strongly coupled to the applied field direction and each mode moves in a different direction in energy for fields applied along the a, b, and c axis [38]. Thus, the high field phase for H a is the result of the condensation of a distinct component of the magnetizations from those for H b, c. The quadratic bondoperator theory 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 BiCu 2 PO 6 . Deviations from the quadratic theory arise around the points k = 1 and 0 r.l.u. which are a result of multi-triplon scattering processes appearing at cubic and quartic order in the BOT. Indeed, we observe an abrupt termination of the lowest energy triplon branches and broadening of the highest energy branch around a critical wavevector of k c ∼ 0.8, a likely consequence of strong mixing between single and multi-triplon excitations.
Understanding the microscopic origin of the unusually strong anisotropic interactions will require ab initio calculations, which is beyond the scope of this paper. However, we can speculate that an explanation is offered by the proximity of Bi ions to all dominant superexchange pathways [ Fig. 1(a)]. Atomic Bismuth has an extremely large spin-orbit coupling constant (1.25 eV [39]) such that the anisotropies may arise through a Cu-O-Bi-O-Cu superexchange interaction. Evidence for this odd superexchange pathway exists in another Bi containing cuprate: Bi 2 CuO 4 , where nuclear magnetic resonance measurements of 63,65 Cu and 209 Bi show a large hyperfine component to the internal field on the Bi site. This indicates that the copper electron must spend some time in the neighborhood of the Bi ions plausibly via Cu-O-Bi-O-Cu hopping process [40]. A similar mechanism may give rise to the large anisotropies in BiCu 2 PO 6 .
In summary, we have mapped the complete dynamic structure factor in BiCu 2 PO 6 through comprehensive INS measurements. The low energy excitations could be explained within the framework of a quadratic bond operator theory and we find that extremely large anisotropies originating from spin-orbit coupling are necessary to describe the excitation spectrum. The relative magnitude of spin-orbit driven interactions compared with the Heisenberg terms in BiCu 2 PO 6 is unprecedented for a 3d transition metal oxide. BiCu 2 PO 6 incorporates both frustration and anisotropic interactions, this combination raises the intriguing possibility of a ferroelectric polarization developing along with spiral magnetic order above the critical field [11,12]. Certainly the exceptionally large spin-orbit coupling places BiCu 2 PO 6 in a unique limit among 3d based quantum magnets.
This work benefited from many useful discussions with Bruce Normand and Götz Uhrig. We would also like to thank 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.