Magnons and excitation continuum in XXZ triangular antiferromagnetic model: Application to ${\text{Ba}}_3{\text{CoSb}}_2{\text{O}}_9$

We investigate the excitation spectrum of the triangular-lattice antiferromagnetic XXZ model using series expansion and mean field Schwinger boson approaches. The single-magnon spectrum computed with series expansion exhibits rotonic minima at the middle points of the edges of the Brillouin zone, for all values of the anisotropy parameter in the range $0\le J^z/J\le 1$. Based on the good agreement with series expansion for the single-magnon spectrum, we compute the full dynamical magnetic structure factor within the mean field Schwinger boson approach to investigate the relevance of the XXZ model for the description of the unusual spectrum found recently in ${\text{Ba}}_3{\text{CoSb}}_2{\text{O}}_9$. In particular, we obtain an extended continuum above the spin wave excitations, which is further enhanced and brought closer to those observed in ${\text{Ba}}_3{\text{CoSb}}_2{\text{O}}_9$ with the addition of a second neighbor exchange interaction approximately $15\%$ of the nearest-neighbor value. Our results support the idea that excitation continuum with substantial spectral-weight are generically present in two-dimensional frustrated spin systems and fractionalization in terms of bosonic spinons presents an efficient way to describe them.

Figure 1

Relation dispersion predicted by series expansion (magenta dots) and dynamical structure factor computed within the reconstructed SBMF theory (intensity curves), along the path of the Brillouin zone shown in the inset, for several anisotropy values. The LSWT results (thin green line) are shown for comparison. Inset: path of the Brillouin zone, $\text{O}=(0,0)$, $\text{P}=(\frac{2\pi }{3},0)$, $\text{A}=(\pi ,0)$, $\text{B}=(\pi ,\frac{\pi }{\sqrt{3}})$, $\text{C}=(\frac{2\pi }{3},\frac{\pi }{\sqrt{3}})$, $\text{Q}=(\frac{4\pi }{3},0)$, and $\text{E}=(0,\frac{\pi }{\sqrt{3}})$. The values of $J_z$ are in units of $J$.

Figure 2

Dynamical structure factor predicted by the reconstructed SBMF theory (intensity curves) along the same path of the Brillouin zone of Fig. 1 for several second neighbors exchange values. The same anisotropy interaction $J^z/J=0.8$, between first and second neighbors, has been selected. Inset: schematic representation of the exchange interactions on the triangular lattice. The values of $J_2$ are in units of $J$.

Figure 3

Spinon spinon interaction $V_{\mathrm{int}}$ predicted by the SBMF within the first order perturbation theory [Eq. (22)] for different $J_2$ values and $J^z/J=0.8$. The values of $J_2$ are in units of $J$.

Figure 4

Bottom panel: dynamical structure factor predicted by the reconstructed SBMF theory (intensity curves) along the experimental path [see Fig. 4(d) of Ref. [12]$]$ for $J^z/J=0.9$. In the horizontal axis the cut along [H H 0] direction is depicted, in units of $4\pi$, as in Ref. [12]. Top panel: relative weight of the two spinon continuum $\int S_{\mathrm{cont}}^{zz}(\mathbf{k},\omega )d\omega /S^{zz}\left(\mathbf{k}\right)$. The value of $J_2$ is in units of $J$.