Unifying Magnons and Triplons in Stripe-Ordered Cuprate Superconductors

Based on a two-dimensional model of coupled two-leg spin ladders, we derive a unified picture of recent neutron scattering data of stripe-ordered ${\mathrm{L}\mathrm{a}}_{15/8}{\mathrm{B}\mathrm{a}}_{1/8}{\mathrm{C}\mathrm{u}\mathrm{O}}_4$, namely, of the low-energy magnons around the superstructure satellites and of the triplon excitations at higher energies. The resonance peak at the antiferromagnetic wave vector ${\mathbf{Q}}_{\mathrm{A}\mathrm{F}}$ in the stripe-ordered phase corresponds to a saddle point in the dispersion of the magnetic excitations. Quantitative agreement with the neutron data is obtained for $J=130–160\mathrm{m}\mathrm{e}\mathrm{V}$ and $J_{\mathrm{c}\mathrm{y}\mathrm{c}}/J=0.2–0.25$.

Figure 1

(a) 2D square lattice split into two-leg $S=1/2$ ladders (full dots) coupled by bond-centered charge stripes (open dots). (b) Momentum space, coordinates in reciprocal lattice units $2\pi /a$ (rlu). Dashed line: magnetic Brillouin zone around the antiferromagnetic wave vector ${\mathbf{Q}}_{\mathrm{A}\mathrm{F}}=(1/2$, 1/2). Open (full) dots: superstructure peaks from vertical (horizontal) ladders. (c) Momentum space rotated by 45° with $h^{\prime }=h+k$ and $k^{\prime }=k-h$, thus ${\mathbf{Q}}_{\mathrm{A}\mathrm{F}}^{\prime }=(1,0)$ in magnetic reciprocal lattice units $2\pi /\sqrt{2}a$ (mrlu). Dotted line: see caption of Fig. 4.

Figure 2

(a), (b): dispersion ${\omega }_k^0$ and spectral weight $a^2(k)$ of one triplon in a two-leg $S=1/2$ ladder with $J_{\parallel }=J_{\perp }=J$ and $x_{\mathrm{c}\mathrm{y}\mathrm{c}}=J_{\mathrm{c}\mathrm{y}\mathrm{c}}/J_{\perp }$. (c) One-triplon dispersion ${\omega }_{h,k}$ for coupled ladders at the quantum critical point $J^{\prime }/J=-0.072$ for $x_{\mathrm{c}\mathrm{y}\mathrm{c}}=0.2$.

Figure 3

Constant-energy slices of ${\tilde{S}}_{h^{\prime },k^{\prime }}(\omega )$ for $J=127\mathrm{m}\mathrm{e}\mathrm{V}$, $J_{\mathrm{c}\mathrm{y}\mathrm{c}}=0.2J$, and $J^{\prime }/J=-0.072$ with $\omega \pm \Delta \omega$ given in each panel for direct comparison with Figure 2 in Ref. 10. The experimental $\Delta \omega$ is accounted for by $\omega \to \omega +i\Delta \omega$ in (6).

Figure 4

(a) Momentum-integrated structure factor $S(\omega )$. (b) $S(\omega )$ integrated only over the magnetic Brillouin zone: Symbols: INS data from Ref. 10. Lines: For each value of $x_{\mathrm{c}\mathrm{y}\mathrm{c}}=0$, 0.1, 0.2, and 0.25 with $J^{\prime }/J$ chosen to meet the quantum critical point (see text), we fix $J$ such that the lower singularity peaks at 44 meV ($J=62$, 85, 127, and 162 meV). The intensity is not fitted but fixed by the value of $J$ (using a $g$ factor of $g=2$). The broadening $\Delta \omega$ is set to 2.5 meV. Thin solid lines ($x_{\mathrm{c}\mathrm{y}\mathrm{c}}=0.2$) mimic experimental details: (i) at high energies, Lorentzian broadening $\Delta \omega =20\mathrm{m}\mathrm{e}\mathrm{V}$; (ii) at low energies, integration only in the dotted region of Fig. 1c [30].