Non-Abelian statistics in light-scattering processes across interacting Haldane chains

The $S=1$ Haldane state is constructed from a product of local singlet dimers in the bulk and topological states at the edges of a chain. It is a fundamental representative of topological quantum matter. Its well-known archetype, the quasi-one-dimensional ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ shows both conventional as well as unconventional magnetic Raman scattering. The former is observed as one- and two-triplet excitations with small linewidths and energies corresponding to the Haldane gap ${\mathrm{\Delta }}_H$ and the exchange coupling $J_c$ along the chain, respectively. Well-defined magnetic quasiparticles are assumed to be stabilized by interchain interactions and uniaxial single-ion anisotropy. Unconventional scattering exists as broad continua of scattering with an intensity $I\left(T\right)$ that shows fermionic statistics. Such statistics has also been observed in Kitaev spin liquids and could point to a non-Abelian symmetry. As the ground state in the bulk of ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ is topologically trivial, we suggest its fractionalization to be due to light-induced interchain exchange processes. These processes are supposed to be enhanced due to a proximity to an Ising ordered state with a quantum critical point. A comparison with ${\mathrm{SrCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$, the $S=1/2$ analog to our title compound, supports these statements.

Figure 1

(a) Four twisted Ni chains along the $c$ axis in two different projections. The scattering geometries are indicated by the green arrows for inter-chain and intra-chain configurations. Red arrows denote the applied magnetic field direction along the chains and perpendicular to the chains. (b) Temperature dependent Raman spectra of ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ obtained in ($ac$) polarization. The blue and red arrows mark the one- and two-triplet modes, respectively. For comparison, the 7-K spectrum in ($cc$) polarization is plotted in black. (c) Bose-corrected integrated intensity ($I\left(T\right)$) of the magnetic scattering contributions as a function of temperature. The blue line corresponds to a fit using a Bose factor. (d) Effect of a transverse magnetic field on the assigned one- and two-triplet modes. Zero field positions are marked by dashed lines. Transverse magnetic field effect on inelastic neutron scattering (full line), electron spin resonance (ESR) (dashed line) [17, 18, 19, 20] and our one-triplet Raman mode (diamonds).

Figure 2

(a) Bose-corrected Raman spectra of ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ taken perpendicular to the Haldane chain direction in ($aa$) scattering geometry at 7 and 390 K together with low-temperature spectra in ($LL$), ($LR$), and ($ab$) polarization. The yellow-shaded backgrounds represent the unconventional scattering continuum with two maxima, $C1$ and $C2$. (b) Bose-corrected integrated intensities of the $C1, C2$, quasielastic scattering (QES, $E\approx 0$), and their sum. The lines are guides to the eye. (c) Fermionic statistics (solid red line) fitted to the summed intensities from (b) (squares)

Figure 3

(a) Color-contour plot of as-measured Raman data in ($aa$) polarization across a temperature range 5–390 K. (b) An asymmetric Fano line shape observed for the 145-${\mathrm{cm}}^{-1}$ phonon. (c) Temperature dependence of its Fano asymmetry parameter $|1/q|$.

Figure 4

(a) Raman spectrum obtained at $T=5$ K (black curve) together with a fit of the phonons (red line). (b) Zoom-in at small intensities. (c) Comparison between $T=390$ K data and phonon fit.

Figure 5

(a) Raman data obtained at $T=6$ K and in ($aa$) polarization using various laser energies. (b) Magnetic field ($B$) dependence of the scattering continuum in ($ab$) polarization. The inset plots its integrated intensity over $B$. (c) Polarization-resolved Raman data of the isostructural $S=1/2$ system ${\mathrm{SrCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$.