High-resolution inelastic neutron scattering on the frustrated spin-ladder magnet BiCu2_2PO6_6 revealed a striking branch-dependent coupling of triplon quasiparticles to the two-triplon continuum: the two lowest branches bend away from the non-interacting dispersion and terminate abruptly near k≈0.75k\approx0.75 r.l.u. (reciprocal lattice units), while the highest branch merges smoothly into the continuum with strong damping; moreover, fits with non-interacting (quadratic) bond-operator theory require anomalously large Dzyaloshinskii–Moriya and symmetric anisotropic exchange. The open problem is a controlled microscopic theory of interacting triplons — treating the trilinear (one-triplon to two-triplon) decay terms and the quadrilinear (two-triplon binding) terms on equal footing — that quantitatively reproduces this phenomenology, derives the momentum- and branch-(pseudo-spin-)dependent selection rules governing decay, fixes the renormalized anisotropy parameters, and clarifies the scope of a non-Hermitian effective description of quasiparticle–continuum coupling in this realistic lattice model. Partial progress by Müller and Uhrig (2023) attributed the dominant ~7 meV downturn to a two-triplon bound state rather than continuum level repulsion, but at a numerical resolution explicitly insufficient to settle the remaining features, leaving the full interacting-triplon account open.

Background

In gapped quantum antiferromagnets whose ground state is a valence-bond solid — a crystalline arrangement of singlet (S=0S=0) spin pairs — the elementary excitations are triplons: localized triplet (S=1S=1) states of a valence bond that propagate through the lattice and behave as bosonic quasiparticles carrying definite energy and momentum. A neutron scattering experiment can create one triplon, or two or more simultaneously; at fixed total momentum the set of all two-triplon energies forms a continuous band, the two-triplon continuum. A one-triplon branch lying inside this continuum can decay into it whenever the Hamiltonian contains terms that do not conserve the triplon number; the triplon is then effectively an open quantum system with a finite lifetime, naturally described by a non-Hermitian effective Hamiltonian whose complex eigenvalues encode the renormalized energy (real part) and inverse lifetime (imaginary part). Level repulsion, or avoided level crossing — the mutual pushing-apart of the energies of two coupled quantum states — is familiar for two discrete levels; the source study asked how it generalizes when one of the coupled objects is a many-particle continuum. The general theory of spontaneous decay of magnetic quasiparticles was reviewed by Zhitomirsky and Chernyshev (2013).

BiCu2_2PO6_6 (Tsirlin et al. 2010) realizes weakly coupled frustrated spin ladders of S=1/2S=1/2 Cu2+^{2+} moments with a spin gap. Because inversion symmetry is broken on the Cu–Cu bonds, sizable Dzyaloshinskii–Moriya (DM) exchange Dij⋅(Si×Sj)\mathbf{D}_{ij}\cdot(\mathbf{S}_i\times\mathbf{S}_j) and symmetric anisotropic exchange ΓijαβSiαSjβ\Gamma^{\alpha\beta}_{ij}S_i^\alpha S_j^\beta, both rooted in spin-orbit coupling, are symmetry-allowed. Plumb et al. (Nat. Phys. 12, 224 (2016)) mapped the full triplon dispersion by inelastic neutron scattering and analyzed it with bond-operator theory (BOT) — an exact rewriting of a dimerized spin Hamiltonian in singlet/triplet creation and annihilation operators, kept at quadratic (non-interacting) mean-field order. The quadratic BOT captures the overall bandwidth, the incommensurate minima, and the field dependence, but requires anomalously large anisotropies (D≈0.3D\approx0.3–0.6 J10.6\,J_1, where J1J_1 is the dominant antiferromagnetic exchange of the ladder) and fails where the one-triplon branches meet the two-triplon continuum near kc≈0.75k_c\approx0.75 r.l.u. (reciprocal lattice units): the two lowest branches bend away from the quadratic dispersion (an avoided crossing) and then terminate abruptly, while the highest branch merges smoothly into the continuum with strong damping. The authors interpreted the bending as level repulsion between the quasiparticle and the continuum mediated by the anharmonic (cubic-in-bond-operator) DM terms, and speculated that branch-dependent selection rules — tied to the pseudo-spin label, the pseudo-SzS^z assigned to each anisotropy-split branch from its Zeeman character (a convenient labeling scheme, since anisotropy mixes the triplet components so that SzS^z itself is no longer a good quantum number) — decide which branches decay. The companion theory of Hwang and Kim (Phys. Rev. B 93, 235130 (2016)), which treated the decay at one-loop self-energy level, found the level repulsion at that order too small and the fitted DM values anomalously large, and explicitly deferred a treatment of the single- and two-triplon sectors on equal footing to future work. What remained missing was a controlled, quantitative, non-perturbative account of interacting triplons for this material.

Problem Statement

Develop a controlled microscopic theory of interacting triplons for the established spin Hamiltonian of BiCu2_2PO6_6 — frustrated spin ladders with DM and symmetric anisotropic exchange and weak interladder coupling — that treats the trilinear (one-triplon ↔ two-triplon) terms and the quadrilinear (two-triplon interaction and binding) terms on equal footing, and use it to: (a) quantitatively reproduce — i.e. recover the dispersion measured by Plumb et al. (2016) to within that dataset's experimental error bars over the momentum range of the reported data — the measured inelastic-neutron-scattering phenomenology, namely the strong dispersion downturn near 7 meV at k≈0.75k\approx0.75 r.l.u. (reciprocal lattice units), the weaker downturn near 9.5 meV, the abrupt termination of the two lowest branches beyond kc≈0.75k_c\approx0.75 r.l.u., and the damped smooth merger of the highest branch into the two-triplon continuum — deciding, for each feature, whether its origin is level repulsion from the continuum, a two-triplon bound state, or their interplay; (b) derive the momentum- and pseudo-spin-dependent selection rules — pseudo-spin being the pseudo-SzS^z label assigned to each anisotropy-split triplon branch from its Zeeman character — that determine which triplon branches hybridize with, and decay into, which continuum channels; (c) determine the renormalized Dzyaloshinskii–Moriya and symmetric anisotropy parameters including interaction corrections beyond the quadratic/one-loop level, and check their consistency with magnitudes expected from spin-orbit coupling; and (d) establish to what extent a controlled non-Hermitian effective description (complex eigenvalues encoding renormalized dispersions and lifetimes) of the quasiparticle–continuum coupling in this realistic lattice model reproduces the observed avoided-crossing phenomenology. These four deliverables (a)–(d) are all mandatory and jointly serve one objective: a quantitative, experimentally accountable account of when the triplon in this magnet survives as a particle and when it dissolves into, or binds within, the multi-triplon continuum; no proper subset of (a)–(d) constitutes a resolution on its own.

References

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