Chiral Order and Electromagnetic Dynamics in One-Dimensional Multiferroic Cuprates

We show by unbiased numerical calculations that the ferromagnetic nearest-neighbor exchange interaction stabilizes a vector spin chiral order against the quantum fluctuation in a frustrated spin-$\frac{1}{2}$ chain relevant to multiferroic cuprates, ${\mathrm{LiCu}}_2{\mathrm{O}}_2$ and ${\mathrm{LiCuVO}}_4$. Our realistic semiclassical analyses for ${\mathrm{LiCu}}_2{\mathrm{O}}_2$ resolve controversies on the helical magnetic structure and unveil the pseudo-Nambu-Goldstone modes as the origin of experimentally observed electromagnons.

Figure 1

(a) A 1D frustrated spin model with $J_1$ and $J_2(>0)$ being the nearest-neighbor and second-neighbor exchange couplings. (b) A spin cycloid in the propagation direction of the $J_1-J_2$ spin chain. Spins rotate within the common plane represented by the disks. (c),(d) Three Nambu-Goldstone modes in the SU(2)-symmetric model: a phason [(c)] and two rotation modes of the spiral plane [(d)]. (e) ${\mathrm{Cu}}^{2+}$, ${\mathrm{Cu}}^{1+}$ and ${\mathrm{O}}^{2-}$ ions in ${\mathrm{LiCu}}_2{\mathrm{O}}_2$, and interchain Heisenberg exchange ($J_{a,\perp }$) and interlayer Dzyaloshinskii-Moriya interactions. On the arrows with four different colors, four distinct DM vectors (${\mathit{D}}^{\prime \prime }$) are assigned. (f) Hypothetical magnetic structure (red arrows) and lowest-energy spin-excitation mode (rotations around blue arrows) in the THz range for ${\mathrm{LiCu}}_2{\mathrm{O}}_2$.

Figure 2

Ground-state phase diagram of the spin-chain model $H_{1D}$. (a) The classical phase diagram and relevant Q1D spin-$\frac{1}{2}$ materials. Estimates of $J_1/J_2$ for materials are taken from Refs. 17, 26, 27. The materials shown in green exhibit an antiferromagnetic LRO, but their detailed magnetic structures have not been established yet. In ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$, a magnetic LRO has not been detected down to 2 K [26]. Usually, $1-\Delta \lesssim 0.05$ in cuprates. (b) The quantum phase diagram and the map of the chiral order parameter ${\kappa }^z$ for $S=\frac{1}{2}$. The horizontal axis $J_1/J_2$ is the same as in the panel (a). The boundary between the dimer phase and the chiral phase in the case of $J_1>0$ agrees with the previous numerical study [22]. For $|J_1|/J_2\gtrsim 4$, Tomonaga-Luttinger liquid (TLL) phases appear. Detailed methods of identifying each phase and the phase boundaries are explained in the supplementary material [29]. It was difficult to determine the phase boundary for small $|J_1|/J_2$ (dashed line).

Figure 3

Semiclassical analyses for ${\mathrm{LiCu}}_2{\mathrm{O}}_2$. (a) Spin-wave dispersions without taking account of the bilinear coupling among $\mathit{q}=0$, $\pm \mathit{Q}$ modes. This coupling only modifies the modes around $\mathit{q}=0$ and $\pm \mathit{Q}$, marked with solid green and dashed blue circles. The dispersion along the $c$ axis is quite small, since $J_{\perp }$, $D_a^{\prime \prime }<J_a\ll |J_1|$, $J_2$. (b),(c) Anisotropy in the electromagnetic absorption spectra due to the magnetic and electric components of light, in units of $(g{\mu }_B{)}^2{\mathrm{meV}}^{-1}$ and ${\mathrm{cm}}^{-1}$, respectively. The spectrum consists of 11 gapped modes after the reconstruction of the $\mathit{q}=0$, $\pm \mathit{Q}$ modes due to their bilinear coupling: $\omega =0.812$, 1.431, 1.525, 1.717 (doubly degenerate), 2.469, 2.634, 2.977, 2.983, 2.991 (doubly degenerate) meV. Inclusion of even higher harmonics such as $\mathit{q}=\pm 2\mathit{Q}$ modes increases the number of the modes and eventually broadens the peaks. The series of $\delta$-function peaks are replaced with the sum of Gaussians broadened with the width 0.05 meV. Points with dashed lines denote the integrated intensity experimentally observed in the THz spectroscopy [12] in the unit of ${\mathrm{cm}}^{-1}$. We note that the tilting of the spiral planes crucially changes the signals of electromagnons.