Probing multi-spinon excitations outside of the two-spinon continuum in the antiferromagnetic spin chain cuprate Sr2CuO3

One-dimensional (1D) magnetic insulators have attracted significant interest as a platform for studying quasiparticle fractionalization, quantum criticality, and emergent phenomena. The spin-1/2 Heisenberg chain with antiferromagnetic nearest neighbour interactions is an important reference system; its elementary magnetic excitations are spin-1/2 quasiparticles called spinons that are created in even numbers. However, while the excitation continuum associated with two-spinon states is routinely observed, the study of four-spinon and higher multi-spinon states is an open area of research. Here we show that four-spinon excitations can be accessed directly in Sr2CuO3 using resonant inelastic x-ray scattering (RIXS) in a region of phase space clearly separated from the two-spinon continuum. Our finding is made possible by the fundamental differences in the correlation function probed by RIXS in comparison to other probes. This advance holds promise as a tool in the search for novel quantum states and quantum spin liquids.

A sketch of the scattering geometry is given in Figure 1 along the chain di RIXS scattering intensity, related to involve plaquette Supplementary geometry. the experiment to ch scattering angle function of solid line) a A sketch of the scattering geometry is given in 1b shows the corresponding mapping between the incident angle and momentum transfer along the chain direction scattering intensity, related to involve plaquette-connecting oxygen sites function of (magenta solid line) related to intra

Supplementary Note 1:
A sketch of the scattering geometry is given in he corresponding mapping between the incident angle and momentum transfer rection (x-axis) scattering intensity, related to connecting oxygen sites (magenta solid line) intra-LHB RIXS excitations

Supplementary Note 1:
A sketch of the scattering geometry is given in he corresponding mapping between the incident angle and momentum transfer axis) in O Kscattering intensity, related to excitations within the Lower Hubbard Ba connecting oxygen sites A schematic of the experimental 1D chain lie in the scattering plane. ange the projection of the momentum transfer along is kept constant. Panel ) -as well as LHB RIXS excitations 2

Supplementary Note 1: Experimental
A sketch of the scattering geometry is given in he corresponding mapping between the incident angle and momentum transfer -edge RIXS excitations within the Lower Hubbard Ba connecting oxygen sites. shows the scattering is varied during whereas the momentum transferred along the chains, q x , as of the scattering intensity (blue

Supplementary Note 2: RIXS spectra in the full Brillouin zone
Due to the relative low energy of x-ray photons used for the excitations at the oxygen Kedge, the technique can only access about half of the first Brillouin zone (BZ). One can, however, measure out to the Brillouin zone boundary at other elemental edges such as the Cu L 2,3 -edge. Similarly, we can also compute the RIXS spectra throughout the entire BZ. By examining the magnetic excitations throughout the 1 st BZ, we can gain further insight into their identities. Supplementary Figure 2

Supplementary Note 3: Low-energy excitations and background in O K-and Cu L 3 -edge line spectra at the Γ-point
Supplementary Figure 3 shows zoom into O K-and Cu L 3 -edge line spectra measured at 0. The data are normalized to acquisition time. The low-energy excitations below 1 eV can be clearly seen in the O K-edge spectrum. They are well separated from dd-excitations and there is little background in the region 1.2 -1.6 eV. In contrast, the Cu L 3 -edge spectrum has a high background in the low-energy region, which extends from the very strong dd-excitations (see Fig. 1 and Supplementary Reference 1) all the way down to the elastic peak. This makes it much more difficult to study excitations of weak spectral weight at the Cu L 3 -edge, such as four-spinon excitations. The statistics of the presented O K-edge data is better, due to the four times longer acquisition time. The combined energy resolution of the O K-edge data is factor 1.8 better (see Methods section).

Supplementary Note 4: Boundaries of the two-and four-spinon continua
The dispersion of a single spinon is given by |sin | and the boundaries of the two-spinon continuum is given by a simple convolution of with itself. Similarly, the convolution of four single spinon dispersions will produce the four-spinon continuum. The upper and lower boundaries of the two-spinon continuum are given by

Supplementary Note 5: Incidence energy dependence
Supplementary Figure 5 shows how the various magnetic excitations depend on the incident photon energy in our model. The multi-spinon excitations that are prominent at 0 are very sensitive to the incident energies compared to the dispersing two-spinon excitations, and the multi-spinon excitations have the largest spectral weight for +, in the range of 0.3 eV to 0.6 eV. This sensitivity to the incident photon energy indicates that the multi-spinon excitations found outside of the two-spinon continuum are more effectively reached via particular intermediate states.