Spinon excitations and spin correlations in the one-dimensional quantum magnet $β$-VOSO$_4$ probed by Raman spectroscopy

TL;DR

This study investigates fractionalized spinon excitations and spin correlations in the quasi-one-dimensional $S=1/2$ magnet beta-VOSO4 using Raman spectroscopy. The Raman data reveal a low-energy, gapless spinon continuum along the chain direction, together with a temperature-dependent quasi-elastic signal, indicating four-spin correlations and spinon dynamics. Finite-temperature DMRG modeling of a Heisenberg chain with small interchain coupling reproduces the observed spectral evolution and enables extraction of the quantum Fisher information, showing a rapid growth of spin correlations as temperature decreases. Together, the results demonstrate that Raman spectroscopy can serve as a practical entanglement witness for quantum magnets and that quantum Fisher information provides a quantitative link between Raman weight, spin correlations, and entanglement in low-dimensional spin systems.

Abstract

Fractionalized excitations such as spinons and anyons have emerged as a central theme in condensed matter physics with broad implications for superconductivity, quantum statistics, and quantum computation. The nearly ideal one-dimensional $S=1/2$ system $β$-VOSO$_4$ without long-range order down to 85 mK provides a promising platform to experimentally explore such fractionalized excitations. Here, we employ Raman spectroscopy to probe magnetic excitations and the evolution of spin correlations in $β$-VOSO$_4$. Spinon signatures are found along the chain direction, evidenced by a broad, gapless scattering continuum at low temperatures. The temperature dependence of the spinon spectral weight aligns considerably with numerical density matrix renormalization group calculations. By comparing the experimental spinon spectral weight with calculated results and evaluating the associated quantum Fisher information (QFI) therefrom, we observe a steep increase in QFI upon cooling, indicating rapidly growing correlation lengths. Our study showcases QFI as a probe of spin correlations in quantum magnets.

Figures (5)

• Figure 1: Structure and symmetry properties of $\beta$-VOSO$_4$.a) Quasi-1D chains of VO$_6$ octahedra (red) interconnected by SO$_4$ tetrahedra (green). b) Comparison of phonon spectra at $T = 300$ K and 2 K measured in parallel polarization. c) and d): Color-contour plots of the as-measured Raman scattering intensity recorded with $\textbf{e}_{\mathrm{in}} \parallel \textbf{e}_{\mathrm{out}}$ as a function of in-plane light polarization taken at $T = 300$ K and at $T = 2$ K, respectively. e) Comparison of phonon spectra at $T = 300$ K and 2 K measured in crossed polarization. f) and g): Color-contour plots of the as-measured Raman scattering intensity recorded with $\textbf{e}_{\mathrm{in}} \perp \textbf{e}_{\mathrm{out}}$ as a function of in-plane light polarization taken at $T = 300$ K and at $T = 2$ K, respectively. h) Polar plot of quasi-elastic scattering intensity measured at $T = 300$ K in parallel polarization. i) Polar plot of integrated intensity of the continuum measured at $T = 2$ K in crossed polarization. The solid lines represent fits to the data. The red arrows in panels c) and g) mark the energy range over which the scattering intensities shown in the polar plot panels were integrated.
• Figure 2: Thermal evolution of spin correlations in $\beta$-VOSO$_4$.a) and b) Color-contour plots of the as-measured Raman scattering intensity as a function of temperature, focusing on the low-energy regime up to 25 meV. Spectra measured in parallel polarization and in crossed polarization, respectively. The dashed red box guides to the temperature-energy range in which spinon excitations are observed. c) As-measured Raman spectra recorded in crossed polarization with a focus on low energies and low temperatures. The spinon continuum, extending up to $\approx$ 10 meV, is strongly diminished with increasing temperatures. d) Scattering intensity of the spinon continuum, integrated from 0.6 meV to 10 meV, as a function of temperature. The symbols correspond to data obtained with a $\lambda = 561$ nm laser (yellow circles) and a $\lambda = 660$ nm laser (red squares). The solid blue line is a fit to the data based on a fermionic model ([$1-f(\omega)^2$]; see the main text for details).
• Figure 3: Finite-temperature dynamical structure factors for $\beta$-VOSO$_4$ as a function of energy $\hbar \omega$ and momentum $q$. The structure factors $S(\textbf{k},\hbar \omega)$ are calculated for a)$T$ = 20 K, b)$T$ = 15 K, c)$T$ = 11 K, d)$T$ = 9 K, e)$T$ = 7 K, f)$T$ = 5 K, g)$T$ = 3 K, and h)$T$ = 2 K, using DMRG for a chain with $L=50$ sites using open boundary conditions and keeping up to a total of $m_{\mathrm{max}} = 1000$ states. The colour scale represents the weight of the dynamical structure factor at a given momentum and energy.
• Figure 4: Normalized quantum Fisher information (nQFI). The plotted datapoints are obtained from the calculated spectra shown in Fig. 3 for a system of $L = 50$ sites at momentum $\textbf{q}=\pi$ and an exchange interaction $J = 3.83$ meV. Multipartite entanglement is witnessed for $2<\mathrm{nQFI}$. In the light shaded area with $1<\mathrm{nQFI}<2$, at least bipartite entanglement is witnessed. For $\mathrm{nQFI}<1$ (dark shaded area), QFI does not witness (but also does not rule out) entanglement.
• Figure 5: Comparison of Raman-measured $\chi"$ to DMRG results.a) Normalized quantum Fisher information integrand related to Bose-corrected Raman data $\chi"$ measured at four selected temperatures $T=2$ K, 7 K, 11 K, and 20 K (black lines). Red lines highlight the spectral weight of the spinon continuum. b) Normalized quantum Fisher information (nQFI) as entanglement witness. Blue triangles: Raman-measured spectral weight integrated from 0 meV to 15 meV; red squares: Raman-measured spectral weight integrated from 2.5 meV to 15 meV; black circles: DMRG results shown in Fig. 4. The successively darkening shading along the vertical direction marks the increasing multipartite entanglement. All experimental values have been normalized to the DMRG value at 2 K. The error bars denote the standard deviation obtained from the fitting procedures.