Magnetic excitations in the site-centered stripe phase: Spin-wave theory of coupled three-leg ladders

The success of models of coupled two-leg spin ladders in describing the magnetic excitation spectrum of La${}_{2-x}$Ba${}_x$CuO${}_4$ had been interpreted previously as evidence for bond-centered stripes. In a recent article, however, we determined the magnetic coupling induced by the charge stripes between bond- or site-centered spin stripes modeled by two- or three-leg ladders, respectively. We found that only the site-centered models order, while the coupling in bond-centered models is insufficient to trigger the quantum phase transition into the magnetically ordered state. We further indicated excellent agreement of a fully consistent analysis of coupled three-leg ladders using a spin wave theory of bond with the experimental data. Here, we provide a full and detailed account of this analysis.

Figure 1

(a) Superpositions of cuts along $(k_x,\hspace{0.5em}\pi )$ and $(\pi ,\hspace{0.5em}{k}_y)$ for the lowest mode $\omega (\mathit{k})$ obtained with the bond operator spin wave theory of coupled three-leg ladders presented here (red line) superimposed with the experimental data obtained by inelastic neutron scattering by Tranquada et al.[15] (black). (b) The neutron data as originally presented, with a triplon dispersion of a two-leg ladder superimposed (red line) (reprinted by permission from Macmillian Publishers Ltd., Nature (London) 429, 534 (2005).

Figure 2

Overall momentum dependence of the magnetic response of superconducting YBa${}_2$Cu${}_3$O${}_{6.85}$ as reported by Bourges et al.[6] There is no stripe ordering in this compound (from Bourges et al., Ref. 6; reprinted by permission from AAAS).

Figure 3

Spin model for (a) bond-centered and (b) site-centered stripes in CuO superconductors.

Figure 4

Single rung on sublattice $\mathcal{A}$.

Figure 5

Bosonic operator for sublattice $\mathcal{A}$.

Figure 6

Bosonic operator for sublattice $\mathcal{B}$.

Figure 7

The microscopic model for site centered spin stripes, with intrarung couplings set to unity, inter-rung–intraladder couplings $J$, and interladder couplings $J^{\prime }$. The real space unit cell contains two rungs and is indicated by the shaded area in gray.

Figure 8

The reduced Brillouin zone corresponding to the real space unit cell indicated in Fig. 7 contains only one-eighth of the full Brillouin zone and is indicated by the shaded area.

Figure 9

Modes described by ${\mathrm{Ĥ}}_{b1,c1,c-1}$ plotted as cuts (a) along $(k_x,\pi )$ and (b) along $(\pi ,k_y)$ using $J_{\text{exp}}=140\hspace{0.5em}\text{meV}$.

Figure 10

The dispersion $\omega (k_x,k_y)$ of the lowest eigenmode of ${\mathrm{Ĥ}}_{b1,c1,c-1}$ in half of the reduced Brillouin assuming $J_{\text{exp.}}=140\hspace{0.5em}\mathrm{meV}$.

Figure 11

Constant energy slices of the neutron scattering intensity ${\chi }^{+-}(k,\omega )$ (see Appendix below) for $J_{\text{exp}}=140\hspace{0.5em}\text{meV}$ and $J^{\prime }=0.07\hspace{0.5em}{J}_{\text{exp}}$ in the magnetic Brillouin zone. In the indicated energy range, only the lowest mode shown in Figs. 10 and 1a contributes. We have replaced the $\delta$-functions in frequency by Lorentzians with half-width $\Delta =0.05\hspace{0.5em}{J}_{\text{exp}}$ and averaged over both stripe orientations (i.e., horizontal and vertical).

Figure 12

The saddle point energy $\omega (\pi ,\pi )$ of the lowest magnetic mode $\omega (\mathit{k})$ calculated for various values of $J^{\prime }/J$ (points). Fitting yields $\omega (\pi ,\pi )\approx 1.47\sqrt{J^{\prime }J}$ (solid line).

Figure 13

Auxiliary models of (a) two weakly coupled three-leg ladders and (b) two weakly coupled spin $\frac{1}{2}$ chains used in the discussion to understand the square root dependence of $\omega (\pi ,\pi )$ on $J^{\prime }$ depicted in Fig. 12.

Figure 14

Finite-size geometries with (a) unfrustrated and (b) frustrated periodic boundary conditions for site-centered stripe models. The spin stripes are localized by a staggered magnetic field $B$, as indicated by the signs in the grey shaded areas.

Figure 15

Finite size geometries with (a) unfrustrated and (b) frustrated periodic boundary conditions for bond-centered stripe models. The spin stripes are localized by a staggered magnetic field $B$ as indicated by the signs in the gray shaded areas.

Figure 16

Modes described by ${\mathrm{Ĥ}}_{a0}$ (blue), ${\mathrm{Ĥ}}_{c0}$ (green), ${\mathrm{Ĥ}}_{a1}$ (red), ${\mathrm{Ĥ}}_{c-2}$ (black) plotted as cuts (a) along $(k_x,\pi )$ and (b) along $(\pi ,k_y)$ using $J_{\text{exp}}=140\hspace{0.5em}\text{meV}$.

Figure 17

Numerical evaluation of the matrix elements $|\langle 0|{\mathrm{Ŝ}}_{\mathit{k}}^{+}|{\gamma }_{1,\mathit{k}}\rangle |{}^2$ for the entire Brillouin zone $\mathit{k}\in [0,2\pi ]\times [0,2\pi ]$. Note the strong enhancement aground the antiferromagnetic ordering wave vectors $\mathit{k}=(\pi \pm \pi /4,\pi )$.