Bosonic spinons in anisotropic triangular antiferromagnets

Anisotropic triangular antiferromagnets can host two primary spin excitations, namely, spinons and triplons. Here, we utilize polarization-resolved Raman spectroscopy to assess the statistics and dynamics of spinons in Ca3ReO5Cl2. We observe a magnetic Raman continuum consisting of one- and two-pair spinon-antispinon excitations as well as triplon excitations. The twofold rotational symmetry of the spinon and triplon excitations are distinct from magnons. The strong thermal evolution of spinon scattering is compatible with the bosonic spinon scenario. The quasilinear spinon hardening with decreasing temperature is envisaged as the ordering of one-dimensional topological defects. This discovery will enable a fundamental understanding of novel phenomena induced by lowering spatial dimensionality in quantum spin systems.

Before proceeding, we analyze the observed phonon modes by the factor-group theory.
Ca3ReO5Cl2 possesses the space group Pnma (No.62) [1]. According to the point group representation of D2h (mmm), the factor-group analysis predicts a total of 66 Raman-active modes: Γ = 20Ag(aa,bb,cc) + 13B1g(ab) + 20B2g(ac) + 13B3g(bc). In the chosen polarizations 20Ag(bb, cc) and 13B3g(bc) modes are symmetry-allowed. We observe a total of 22Ag and 8B3g modes. Some weak peaks cannot be unambiguously assigned. We ascribe the extra modes to polarizer leakage. The missing modes may be due to the weakness of their scattering intensity or the overlap of their frequencies within a spectral resolution.
Shown in Supplementary Fig. 1b is the temperature dependence of the frequency, full width at half maximum (FWHM), and normalized intensity of the representative phonons at 518.5 cm -1 , 403.9 cm -1 , and 24.7 cm -1 . With decreasing temperature, the high-energy phonon modes at 518.5 cm -1 and 403.9 cm -1 exhibit a hardening by ~ 4 cm -1 , being consistent with lattice anharmonicity. In sharp contrast, the low-energy 24.7 cm -1 mode shows a small softening by ~ 1 cm -1 with decreasing temperature. This anomaly points to a strong coupling of the low-energy phonons to magnetic degrees of freedom.
For quantitative analysis of the observed phonon anomalies, we fit the experimental data to an anharmonic phonon interaction model [2]: and Here, 0 and 0 are the phonon frequency and linewidth at T=0 K, respectively, and A and B are constants. In the temperature range of T=80 -300 K, the phonon harmonicity gives a nice description to the data with the fitting parameters A=-4.24 cm -1 and B=7.14 cm -1 for 518.45 cm -1 and A=-2.51 cm -1 and B=6.5 cm -1 for 403.87 cm -1 , and A=0.04 cm -1 and B=0.07 cm -1 for 24.7 cm -1 . The high-energy modes show small deviations from the fitted curves for temperatures below T = 80 K. This discrepancy becomes apparent for the low-energy 24.7 cm -1 mode in its frequency and FWHM. As the 24.7 cm -1 mode lies on top of a magnetic continuum, the phonon anomaly suggests the existence of another relaxation channel in addition to an anharmonic phonon process. As discussed below, the magnetic specific heat and entropy become appreciable for temperatures below T=80 K, indicative of the development of shortrange magnetic correlations. As such, a spin-phonon coupling is responsible for the renormalization of the phonon energy and lifetime. According to the aforementioned factor group analysis, we can observe the Ag and B3g modes in the (bc) scattering geometry. Within the semiclassical theory, the Raman intensity of the phonon scatterings is given as [3]

Supplementary
where in and out are the polarization vectors of the incident and scattered light, respectively, and R is the Raman tensor. In order to describe the angular dependence of Raman intensities, we take a light absorption effect into account [3,4]. In an absorptive material, thus, the Raman tensors of the Ag and B3g modes should be in a complex form, given as and Given that the incident and scattered light lies in the bc plane, the polarization unit vectors are expressed as in = (0, 1, 0) and out = (0, cos , sin ) for the (bθ) configuration, where the angle θ is measured relative to the b-axis. Then, we obtain the angular dependence of the Raman intensity for the Ag and B3g modes: and In the (bθ) polarization, the angular-dependent intensity of both the Ag and B3g modes reveals 180º periodicity. The Ag mode vanishes when the incident and scattered polarization are perpendicular to each other. On the contrary, the intensity of the B3g mode reaches a maximum in this perpendicular polarization configuration.
For the parallel polarizations, we have in = out = (0, cos , sin ) . The Raman intensity for each mode is given by ( g ) = (| |cos 2 cos + | |sin 2 ) 2 + | | 2 cos 4 sin 2 , and where = − is the phase difference between the b and c components of the Raman tensor. From Equations (8) and (9), it is apparent that the angular dependence of the Ag and B3g modes has commonly a 90º period.
We next compare the calculated intensities to the experimental angular-dependent data. In Supplementary Fig. 4a,d, we present the angular dependence of phonon-subtracted Raman spectra in (bθ) and (θθ) configurations at T=300 K. The Raman scattering intensity is maximum at θ=0° (in the intrachain polarization). The intensity decreases continuously with increasing the angle θ=0° to 90° in two configurations. We further visualize an angular dependence of the Raman response in a color plot in Supplementary Fig. 4b,e. To analyze the symmetry of the Raman response at T=300K, we depict the integrated intensity in polar plots as shown in Supplementary Fig. 4c In Supplementary Fig. 7a-c, we present Raman susceptibility χ˝(ω) versus temperature in c Magnetic specific heat Cm evaluated from the relation ( ) ∝ ′′ ( )/( ). The solid line is the thermodynamic magnetic specific heat taken from Ref. [1].
In a quasi-1D spin system, quasielastic light scattering arises from diffusive magnetic fluctuations [5][6][7] or fluctuations of the spin energy density [8,9]. We find that our quasielastic response is described by a Lorentzian spectral function, implying that the fluctuations of the magnetic energy density provide a dominant contribution. In this case, we can calculate the scattering intensity using the Fourier components of a correlation function of the magnetic energy density In the high-temperature and hydrodynamical conditions [8,10], Equation (10) is simplified to where Cm is the magnetic specific heat, DT is the thermal diffusion coefficient DT= K/Cm, and K is the magnetic thermal conductivity. Equation (11) can be rewritten in terms of a Raman susceptibility ′′( ) In this light, the magnetic specific heat can be extracted from the Raman conductivity ′′ ( )/ . In Supplementary Fig. 7 and Fig. 8a, we plot the temperature dependence of Supplementary Fig. 9 shows the temperature dependence of the dc magnetic susceptibility χ(T) of Ca3ReO5Cl2 for μ0H//b and μ0H//c. We measured χ(T) in an external magnetic field of μ0H=7 T. For both field directions, χ(T) exhibits a broad maximum at Tmax=27 K, which is typical for the short-range ordering in low-dimensional spin systems. We observe no discernible anomaly pertinent to long-range magnetic order down to 2 K. Despite the apparent anisotropic magnetic sublattice, the in-plane magnetic anisotropy is not bigger than 3 %. This suggests that the Re 6+ spins are predominantly exchange-coupled by Heisenberg-type interactions.
The negative Curie-Weiss temperature indicates predominant antiferromagnetic exchange interactions between the Re 6+ spins. The effective magnetic moments are evaluated to be μeff=1.567 μB for μ0H//b and μeff=1.515 μB for μ0H//c. These values are rather smaller than the spin-only value of 1.73 μB for S=1/2, indicating the presence of substantial spin-orbit coupling.
All the obtained parameters are in perfect agreement with the previous result [11].
To examine the one-dimensionalization induced by geometrical frustration, we attempted to fit χ(T) using two different models: a S=1/2 Heisenberg anisotropic triangular lattice (ATL) model for high T versus a Bonner-Fisher model for low T [12,13]. In the temperature range of