Magnetic Excitations in a Bond-Centered Stripe Phase: Spin Waves Far from the Semiclassical Limit

Using a spin-only model, we compute spin excitation spectra in a bond-centered stripe state with long-range magnetic order. We employ a bond operator formalism, which naturally captures both dimerization and broken spin symmetry in a unified framework. At low energies, the spin excitations resemble spin waves, but at higher energies they are very similar to spin-one excitations of isolated spin ladders. Our theory does well describe neutron scattering data [J. M. Tranquada et al., Nature (London) 429, 534 (2004)] on ${\mathrm{L}\mathrm{a}}_{2-x}{\mathrm{B}\mathrm{a}}_x{\mathrm{C}\mathrm{u}\mathrm{O}}_4$, pointing towards bond order in this material.

Figure 1

Sketch of the bond-centered stripe state considered here. We use a 2D spin-only model, where the effect of the hole-rich stripes is to mediate a weak coupling, $J_s<0$, between hole-poor regions, accross the hole-rich ones. The hole-poor regions are spin ladders, characterized by couplings $J,J^{\prime }$. The unit cell consists of eight Cu sites, i.e., four spins in the effective Heisenberg model.

Figure 2

Dispersions $\omega (\mathbf{k}\mathbf{)}$ of the magnetic modes, calculated for vertical bond-centered stripes with charge period 4, see Fig. 1, using the harmonic approximation (see text). Solid: Doubly degenerate transverse mode. Dashed: Longitudinal mode. Parameter values are $J=J^{\prime }=100\mathrm{m}\mathrm{e}\mathrm{V}$, $J_s=-0.06J$, and the $J$ renormalization (see text) has been used [20, 22]. For these parameters the “condensation” amplitude is $\lambda =0.27$ and the staggered magnetization $M_s=0.30$; where $\lambda =1$, $M_s=0.61$ would correspond to the spin-wave result in the semiclassical (Néel) limit of a square-lattice antiferromagnet.

Figure 3

Neutron scattering intensity, ${\chi }^{\prime \prime }(\mathbf{k},\omega )$: The panels show fixed energy cuts as function of momentum in the magnetic Brillouin zone. Parameter values are as in Fig. 2. The $\delta$ peaks have been replaced by Lorentzians with width $\Gamma =0.2J$, and the responses of horizontal and vertical stripes have been added. The background arising from the two-particle continuum is not shown. The figure can be directly compared to Fig. 2 of Ref. 11.