Symmetric and asymmetric excitations of a strong-leg quantum spin ladder

The zero-field excitation spectrum of the strong-leg spin ladder (C${}_7$H${}_{10}$N)${}_2$CuBr${}_4$ is studied with a neutron time-of-flight technique. The spectrum is decomposed into its symmetric and asymmetric parts with respect to the rung momentum and compared with theoretical results obtained by the density matrix renormalization group method. Additionally, the calculated dynamical correlations are shown for a wide range of rung and leg coupling ratios in order to point out the evolution of arising excitations, as, e.g., of the two-magnon bound state from the strong to the weak coupling limit.

Figure 1

Illustration of the crystallographic structure of (C${}_7$H${}_{10}$N)${}_2$CuBr${}_4$ (DIMPY), projected onto the $(a,b)$ plane. Cu${}^{2+}$ and Br${}^{-}$ ions are shown as red and black spheres, respectively. The two ladder systems as well as the relevant interactions are shown, together with the corresponding rung vectors ${\mathbf{d}}_{1,2}$ (blue arrows).

Figure 2

Raw data from neutron time-of-flight experiments in (C${}_7$D${}_{10}$N)${}_2$CuBr${}_4$, measured at (a) $T=1.5$ K and (b) $T=50$ K. Data was integrated along the nondispersive ${\mathbf{b}}^{☆}$ and ${\mathbf{c}}^{☆}$ direction. Intensity is shown as a function of energy transfer $\hslash \omega$ and momentum transfer along the leg $h$, in reciprocal lattice units.

Figure 3

(a) Background subtracted data from neutron time-of-flight experiments in (C${}_7$D${}_{10}$N)${}_2$CuBr${}_4$, integrated along the ${\mathbf{b}}^{☆}$ and ${\mathbf{c}}^{☆}$ direction. Intensity is shown as a function of energy transfer $\hslash \omega$ and momentum transfer along the leg. Black and white points correspond to triple-axis measurements of the magnon and two-magnon bound state excitations (Refs. 24 and 25). (b) Numerical DMRG calculation of the dynamical structure factor ${\mathcal{S}}_{+}(h,\omega )+{\mathcal{S}}_{-}(h,\omega )$, convoluted with experimental resolution. The measured and calculated spectra show remarkable agreement.

Figure 4

Background subtracted data ${\mathcal{I}}_{h,\omega }^{\mathrm{sub}}(k,l)$ as a function of momentum transfer along the perpendicular directions ${\mathbf{b}}^{*}$ and ${\mathbf{c}}^{*}$, in order to visualize (a) the asymmetric and (c) the symmetric structure factors $s_h^{\pm }(k,l)$. Data was integrated along $h=[0.45,0.55]$ rlu and in the energy range (a) $\hslash \omega =[0.2,0.6]$ meV and (c) $[1.75,2.05]$ meV. The corresponding calculation of the asymmetric and symmetric structure factor $s_h^{\pm }(k,l)$ weighted by the magnetic form factor are shown in (b) and (d).

Figure 5

Separation of the asymmetric and symmetric channel in (C${}_7$D${}_{10}$N)${}_2$CuBr${}_4$. (a) Asymmetric and (c) symmetric contribution extracted from raw data as explained in the text. A spurious remainder of the single-magnon excitation is still visible in the separated symmetric contribution (c) (black arrow). DMRG calculation of (b) ${\mathcal{S}}_{-}(h,\omega )$ and (d) ${\mathcal{S}}_{+}(h,\omega )$, convoluted with a similar resolution as in (a).

Figure 6

DMRG calculation of the dynamic structure factor in the asymmetric (left, $q_{\perp }=\pi$) and symmetric (right, $q_{\perp }=0$), as a function of $x=J_{\mathrm{leg}}/J_{\mathrm{rung}}$. White lines correspond to predictions from Refs. 36 and 37. Black lines indicate the boundaries of multiparticle continua as described in the text.