Emergent spin-1 Haldane gap and ferroelectricity in a frustrated spin-$\frac{1}{2}$ ladder

We report experimental and theoretical evidence that ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ has a nonmagnetic tetramer ground state of a two-leg ladder comprising antiferromagnetically coupled frustrated spin-$\frac{1}{2}$ chains and exhibits a Haldane spin gap of emergent spin-1 pairs. Three spin excitations split from the spin-1 triplet by a Dzyaloshinskii-Moriya interaction are identified in inelastic neutron-scattering and electron spin resonance spectra. A tiny magnetic field generates ferroelectricity without closing the spin gap, indicating a unique class of ferroelectricity induced by a vector spin chirality order.

Figure 1

Structures of spin chains. (a) Antiferromagnetic spin chain. (b) Frustrated spin-$\frac{1}{2}$ chain with emergent spin-1 pairs (blue clouds). Solid (broken) lines represent antiferromagnetic (ferromagnetic) interactions. (c) Short-range resonating valence bond state involving tetramers (yellow plaquettes) connecting emergent spin-1 pairs. (d) Crystal structure of a pair of spin-$\frac{1}{2}$ chains comprising edge-sharing distorted ${\mathrm{CuO}}_6$ octahedra in ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$. (e) An ideal centrosymmetric chain of edge-sharing regular ${\mathrm{CuO}}_6$ octahedra (black points), compared with the noncentrosymmetric one in ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$. Electric dipole moments due to ionic displacements are shown on the first-neighbor Cu spin pairs by dark green arrows. (f) A unit cell of ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$. Two-leg ladders are located in translucent orange tubes.

Figure 2

Temperature dependence of thermodynamic properties of polycrystalline ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$. (a) Dielectric constant $\varepsilon /{\varepsilon }_0$. (b) Electric polarization $P$ at magnetic fields. Note that a powder average of the magnetic field direction broadens the transition. (c) Magnetic susceptibility $\chi$ (open circles). The impurity contribution ${\chi }_{\mathrm{imp}}$, responsible for the upturn of $\chi$ below 0.5 K, was fitted by the Curie-Weiss law with the spin vacancy concentration of $0.5\%$ and the Weiss temperature $-0.5\mathrm{K}$ (blue dashed curve). Red points represent the data ${\chi }_{\mathrm{spin}}$ subtracted by ${\chi }_{\mathrm{imp}}$. Also shown is $d\chi /dT$ (green points). The inset shows a high-temperature fitting of $\chi$ (black curve) with a powder average of the exact diagonalization results (red curve). (d) Specific heat $C$ at $B=0$. The solid curves in (c) and (d) are the fitting curves proportional to $exp(-E_g/T)$ with the energy gap $E_g=1.7\mathrm{K}$.

Figure 3

Magnetic field dependence of thermodynamic properties of polycrystalline ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$. (a) Magnetization $M$ per Cu atom. (b) Derivatives of $M$ with respect to $B$. (c) Ferroelectric polarization $P$ for both $\mathit{P}\parallel \mathit{B}$ and $\mathit{P}\perp \mathit{B}$.

Figure 4

Neutron-scattering results on polycrystalline ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ at $B=0$. (a) Experimental and (b) theoretical low-energy powder-averaged spectra, measured at 1.6 K and calculated at $T=0$, respectively. (c) Theoretical low-energy spectra along the $b$ axis without the powder average. The results obtained from a 28-site cluster by taking the parameter set for the thermodynamic limit [27] have been interpolated. The incommensurate wavevector $Q=0.3{\AA }^{-1}$ is marked by black arrows in (a), (b), and (c). (d) Experimental and (e) theoretical powder-averaged spectra, measured at 6.5 K and calculated at $T=0$, respectively, in a wider energy range. Note that the incommensurate wavevector is shifted downwards from the maximum position of the powder-averaged spectra to the onset in the panels (a) and (b). (f) Neutron powder diffraction patterns of ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$ measured at 9.89 $\mathrm{K}>T_{\mathrm{FE}}$ (black) and 1.58 $\mathrm{K}<T_{\mathrm{FE}}$ (red) in $B=0\mathrm{T}$. The four peaks with $*$ symbols are derived from a nonmagnetic impurity phase ${\mathrm{Rb}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{10}$. A cold neutron wavelength $\lambda =4.5\text{Å}$ was chosen.

Figure 5

Electron spin resonance spectra of polycrystalline ${\mathrm{Rb}}_2{\mathrm{Cu}}_2{\mathrm{Mo}}_3{\mathrm{O}}_{12}$. (a) Temperature dependence of the experimental transmission spectra at 81 GHz. Arrows represent the resonance fields. (b) Experimental transmission spectra at 1.6 K for designated frequencies. Green and blue arrows denote two sequences of resonance fields. (c) Theoretical optical absorption power at 1.6 K. Experimentally observed resonance fields indicated by blue and green arrows in (a) are plotted by $▵$ and $▿$, respectively, for comparison. (d) Energy levels at $Q_b=0$ (left) and $Q_b=1/4$ r.l.u. (right) computed under $\mathit{B}$ applied along the $x$, $y$, and $z$ directions, where ${\mathit{D}}_u\parallel z$ and ${\mathit{D}}_s\parallel x$. Transitions denoted by the arrows in the left (right) panel produce resonance spectra shown by dashed (solid) curves in (c).