Disorder, Cluster Spin Glass, and Hourglass Spectra in Striped Magnetic Insulators

Hourglass-shaped magnetic excitation spectra have been detected in a variety of doped transition-metal oxides with stripelike charge order. Compared to the predictions of spin-wave theory for perfect stripes, these spectra display a different intensity distribution and anomalous broadening. Here we show, based on a comprehensive modeling for ${\mathrm{La}}_{5/3}{\mathrm{Sr}}_{1/3}{\mathrm{CoO}}_4$, how quenched disorder in the charge sector causes frustration, and consequently cluster-glass behavior at low temperatures, in the spin sector. This spin-glass physics, which is insensitive to the detailed nature of the charge disorder, but sensitive to the relative strength of the magnetic interstripe coupling, ultimately determines the distribution of magnetic spectral weight: The excitation spectrum, calculated using spin waves in finite disordered systems, is found to match in detail the observed hour-glass spectrum.

Figure 1

(a) Perfect-stripe order in the $\mathrm{Co}{\mathrm{O}}_2$ planes of ${\mathrm{La}}_{5/3}{\mathrm{Sr}}_{1/3}{\mathrm{CoO}}_4$. Circles (arrows) represent nonmagnetic ${\mathrm{Co}}^{3+}$ (magnetic ${\mathrm{Co}}^{2+}$) ions. Both magnetic couplings $J$ and $J^{\prime }$ are AF. (b) Frustrated bonds caused by disorder in the charge sector. (c) Disordered charge configuration with ${\xi }_{\mathrm{C}}^{\parallel }\approx 5$ and ${\xi }_{\mathrm{C}}^{\perp }\approx 2$, obtained from the Ising model [24] for $L=48$. Black (gray) squares correspond to ${\mathrm{Co}}^{3+}$ (${\mathrm{Co}}^{2+}$) ions. (d) Charge structure factor $S_{\mathrm{C}}(\overrightarrow{q})$ for the same parameters as in (c) averaged over ${10}^3$ charge configurations.

Figure 2

Magnetic excitations, ${\chi }^{\prime \prime }(\overrightarrow{q},\omega )$, of disordered stripes. From (a) to (e) the disorder in the charge sector increases, as characterized by the correlation lengths ${\xi }_{\mathrm{C}}^{\parallel }$, ${\xi }_{\mathrm{C}}^{\perp }$. Each panel is divided at $\omega ={\omega }_{\mathrm{S}}$, with the lower [upper] portion showing data along $\overrightarrow{q}=(q,q)$ [($q$, $\pi$)] in order to represent the high-intensity features. Parameters are $J^{\prime }=0.05J$, $L=36$, and (a) perfect stripes, ${\omega }_{\mathrm{S}}=0.88JS$; (b) ${\xi }_{\mathrm{C}}^{\parallel }\simeq 19$, ${\xi }_{\mathrm{C}}^{\perp }\simeq 8$, ${\omega }_{\mathrm{S}}=0.82JS$; (c) ${\xi }_{\mathrm{C}}^{\parallel }\simeq 10$, ${\xi }_{\mathrm{C}}^{\perp }\simeq 4$, ${\omega }_{\mathrm{S}}=0.70JS$; (d) ${\xi }_{\mathrm{C}}^{\parallel }\simeq 5$, ${\xi }_{\mathrm{C}}^{\perp }\simeq 2$, ${\omega }_{\mathrm{S}}=0.50JS$; (e) ${\xi }_{\mathrm{C}}^{\parallel }\simeq 3$, ${\xi }_{\mathrm{C}}^{\perp }\lesssim 1$, ${\omega }_{\mathrm{S}}=0.45JS$ [24]. The $\delta$ peaks in Eq. (2) were replaced by Lorentzians with width $\Gamma =0.1JS$.

Figure 3

Constant-energy cuts of ${\chi }^{\prime \prime }(\overrightarrow{q},\omega )$ with model parameters as in Fig. 2d: $J^{\prime }=0.05J$, ${\xi }_{\mathrm{C}}^{\parallel }\simeq 5$, ${\xi }_{\mathrm{C}}^{\perp }\simeq 2$, $L=36$.

Figure 4

Comparison of ${\chi }^{\prime \prime }(\overrightarrow{q},\omega )$ for parameters as in Fig. 2d and $JS=25\mathrm{meV}$ (solid) to the ${\mathrm{La}}_{5/3}{\mathrm{Sr}}_{1/3}{\mathrm{CoO}}_4$ data (dots) taken from Fig. 3 of Ref. 9. Panels (a–c): $\overrightarrow{q}=(\pi +\xi ,\pi -\xi )$; (d–f): $\overrightarrow{q}=(\pi ,\pi +\xi )$. Also shown are ad hoc broadened perfect-stripe spectra (dashed) following Refs. 9, 24, with $JS=18\mathrm{meV}$ adjusted to match the energy ${\omega }_{\mathrm{S}}$ of the saddle point. For both theories, the energy broadening was $\Gamma =1\mathrm{meV}$, and an overall intensity factor and a constant background were adjusted by fitting the 14 meV data and then were used for all $\omega$ [32].