Two dimensionality of magnetic excitations on the trellis lattice: ${(\mathrm{La},\mathrm{Sr},\mathrm{Ca})}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ and $\mathrm{Sr}{\mathrm{Cu}}_2{\mathrm{O}}_3$

We explore the properties of magnetic excitations on the trellis lattice which is relevant for the so-called telephone-number compounds $A_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ with $A=(\mathrm{La},\mathrm{Sr},\mathrm{Ca})$ and the system $\mathrm{Sr}{\mathrm{Cu}}_2{\mathrm{O}}_3$. The trellis lattice consists of two-leg ladders which are coupled in a strongly frustrated fashion. We use the effective model obtained for a single two-leg spin ladder to calculate the two-dimensional one-triplon dispersion and the corresponding one-triplon contribution to the dynamical structure factor. Special attention is laid on signatures of the frustrating interladder magnetic exchange. A detailed suggestion is made for an experimental detection of this exchange in inelastic neutron-scattering experiments.

Figure 1

(Color online) Heisenberg exchange couplings $(J_{\perp },J_{\Vert },J_{\mathrm{int}})$ and the four-spin exchange coupling $J_{\mathrm{cyc}}$ (circles) considered in the Hamiltonian [Eq. (1)]. The rung coupling (horizontal solid lines) is $J_{\perp }$, the one along the legs of the ladders (vertical solid lines) is denoted by $J_{\Vert }$, and the coupling between adjacent ladders (dashed lines) is denoted by $J_{\mathrm{int}}$.

Figure 2

(Color online) Unit cell of a trellis lattice spanned by the unit vectors $e_1$ and $e_2$. The distance between two adjacent rungs on the same ladder is $c$. The distance between the legs of a ladder is $a$, the distance perpendicular to the ladders between the centers (dots) of two rungs on adjacent ladders is $a_2$.

Figure 3

(Color online) Dispersion ${\omega }_k^0$ of a single triplon for an isolated ladder. Upper panel: one-triplon dispersion for $x_{\mathrm{cyc}}=0$ and $x=1$ (black curve) and $x=1.5$ (cyan/gray curve). Lower panel: one-triplon dispersion for $x_{\mathrm{cyc}}=0.25$ and $x=1$ (black curve) and $x=1.5$ (cyan/gray curve).

Figure 4

(Color online) Weight $a_k^2$ of a single triplon for an isolated ladder. Upper panel: one-triplon spectral weight for $x_{\mathrm{cyc}}=0$ and $x=1$ (black curve) and $x=1.5$ (cyan/gray curve). Lower panel: one-triplon spectral weight for $x_{\mathrm{cyc}}=0.25$ and $x=1$ (black curve) and $x=1.5$ (cyan/gray curve).

Figure 5

(Color online) The influence of the interladder coupling $x_{\mathrm{int}}$ on the dispersion ${\omega }_{k,q}$ of a single triplon on the trellis lattice, as shown in Fig. 2 for $x=1$ and $x_{\mathrm{cyc}}=0$. Upper panel: data sets for $q=1$ and $x_{\mathrm{int}}=\{-0.4;-0.2;0.0;0.2;0.4\}$. Lower panel: data sets for $q=2$ and $x_{\mathrm{int}}=\{-0.4;-0.2;0.0;0.2;0.4\}$.

Figure 6

(Color online) The influence of the interladder coupling $x_{\mathrm{int}}$ on the spectral weight $a_{k,q}^2$ of a single triplon on the trellis lattice, as shown in Fig. 2 for $x=1$ and $x_{\mathrm{cyc}}=0$. Upper panel: data sets for $q=1$ and $x_{\mathrm{int}}=\{-0.4;-0.2;0.0;0.2;0.4\}$. Lower panel: data sets for $q=2$ and $x_{\mathrm{int}}=\{-0.4;-0.2;0.0;0.2;0.4\}$.

Figure 7

(Color online) Constant $\omega$ scan of the dynamical structure factor at $\omega =0.25J_{\perp }$ with $x=1.5$ and $x_{\mathrm{cyc}}=0.25$ as a function of the momentum $k$ in ladder direction. The black curves correspond to $x_{\mathrm{int}}=0.4$ and the cyan/gray curves correspond to $x_{\mathrm{int}}=0.2$. The upper panel shows $q=1$ and the lower panel shows $q=2$. A Lorentzian broadening of $\eta =0.05$ is used in all panels.