Evidence of two-spinon bound states in the magnetic spectrum of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$

Recent inelastic neutron scattering (INS) experiments of the triangular antiferromagnet ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ revealed strong deviations from semiclassical theories. We demonstrate that key features of the INS data are well reproduced by a parton Schwinger boson theory beyond the saddle-point approximation. The measured magnon dispersion is well reproduced by the dispersion of two-spinon bound states (poles of the emergent gauge fields propagator), while the low-energy continuum scattering is reproduced by a quasifree two-spinon continuum, suggesting that a free spinon gas is a good initial framework to study magnetically ordered states near a quantum melting point.

Figure 1

Diagrammatic representation of different contributions to the dynamical spin susceptibility: (a) saddle-point and (b) Gaussian fluctuations around the SP solution [27, 29]. The dashed lines represent the external fields. The full lines represent the single-spinon propagator for the SP solution. The wavy lines represent the propagator of the auxiliary fields [26], whose poles correspond to the true magnons of the SBT.

Figure 2

Comparison between the INS measurements of ${\mathrm{Ba}}_3{\mathrm{CoSb}}_2{\mathrm{O}}_9$ reproduced from Ref. [30] (left column) and the zero-temperature $I(\mathit{q},\omega )$ computed with the SBT [27, 28, 29] (middle and right columns). The middle column shows the mean-field result corresponding to the diagram depicted in Fig. 1. The right column includes contributions from both diagrams shown in Fig. 1. In all figures included in this work, the calculated $I(\mathit{q},\omega )$ was multiplied by a single overall intensity scale factor to compare with the observed experimental intensity data. The brackets on the right-hand side in (c) indicate the energy cuts shown in Fig. 5. The wave-vector path in (a)–(c) is ${\mathrm{\Gamma }}^{\prime }(0,1,1)\to {\mathrm{K}}_1^{\prime }(1/3,1/3,1)\to {\mathrm{M}}_2^{\prime }(0,1/2,1)\to {\mathrm{K}}_2^{\prime }(-1/3,2/3,1)\to {\mathrm{\Gamma }}^{\prime }$, shown in the inset. In (d)–(f), the path is ${\mathrm{K}}_2(-1/3,2/3,0)\to {\mathrm{M}}_2(0,1/2,0)\to \mathrm{\Gamma }(0,1,0)\to {\mathrm{K}}_2\to {\mathrm{K}}_2^{\prime }(-1/3,2/3,1)\to {\mathrm{M}}_2^{\prime }(0,1/2,1)\to {\mathrm{\Gamma }}^{\prime }(0,1,1)$.

Figure 3

Comparison between the two-spinon bound state dispersion computed with SBT (full lines) and the best parametrization (open circles) of the experimentally observed magnon dispersions from Ref. [30].

Figure 4

Comparison between INS intensity averaged over $l$ (including several zones as in Ref. [30]) and the corresponding average of the intensity $I(\mathit{q},\omega )$ obtained from the two diagrams shown in Fig. 1 (solid line). A Lorentzian broadening, which accounts for the estimated experimental energy resolution, has been introduced in the calculated $I(\mathit{q},\omega )$ via the replacement $\omega \to \omega +i\eta$.

Figure 5

Intensity maps of $I(\mathbf{q},\omega )$ [Eq. (12)] as a function of momentum in the $hk$ plane at a series of constant energies. The results have been integrated over an energy range that is indicated at the top of each panel to facilitate the comparison with the INS data reproduced from Ref. [30].