Electron Spin Resonance of the Interacting Spinon Liquid

We report experimental verification of the recently predicted collective modes of spinons, stabilized by backscattering interaction, in a model quantum spin chain material. We exploit the unique geometry of uniform Dzyaloshinskii-Moriya interactions in ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Br}}_2$ to measure the interaction-induced splitting between the two components of the electron spin resonance (ESR) response doublet. From that we directly determine the magnitude of the “marginally irrelevant” backscattering interaction between spinons for the first time.

Figure 1

Electron spin resonance in ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Br}}_2$. (a) Several low-$T$ resonance lines. Magnetic field is applied along the DM axis $b$. Gray points is the measured rate of absorption of the microwave radiation by the sample. It is well fitted by several Lorentzian lines (dark red line). The contributions of modes ${\nu }^{+}$ and ${\nu }^{-}$ is shown in orange. At low frequencies an additional parasitic paramagnetic resonance can be detected (light green). The inset shows the sketch of ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Br}}_2$ crystal structure. (b) Calculated spectrum of small-$q$ transverse spin fluctuations in a magnetized spin chain without the backscattering interaction. (c) The same for the interacting spinons, $\hslash u=2.38Ja$. For ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Br}}_2$ considered here $J=20.5\mathrm{K}$, and the magnetic field is 0.3 T in both panels. The solid lines show $\nu (q)$ for the poles of the transverse spin susceptibility according to Refs. [22, 31], and the color shows their intensity. Black (red) horizontal arrows indicate ESR frequencies of the spin chain without (black) and with (red) the DM interaction.

Figure 2

ESR spectrum of ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Br}}_2$ and its theoretical description in terms of interacting spinons. (a) The frequency-field diagram including the $H\parallel D$ data at two different temperatures (circles and squares). Error bars are obtained from the lineshape fit and for the majority of ESR lines are estimated as 0.02 T and are within the symbol size. For some of frequencies above 50 GHz, the line shape was distorted by the parasitic electrodynamic size effect, here errors in $H$ value are estimated as a whole linewidth and are seen as error bars on the left and middle panels. The dotted line is the $h\nu =g{\mu }_{B}H$ paramagnetic resonance, the dashed lines correspond to the noninteracting approximation (4) with $D=0.26\mathrm{K}$. The solid lines and their color (intensity, as indicated in the inset) correspond to the interacting spinon theory (5). The best-fit value of $\hslash u/a=48.8\mathrm{K}$ is used. (b) The same diagram, but in the “reduced” representation with $\gamma H$ subtracted, $\nu \to \nu -\gamma H$. (c) The mode intensity ratio versus the observation frequency. Points are the experimental data (as in the other panels), error bars stem from the experimental uncertainty in determining the widths and amplitudes of the two overlapping peaks. Solid line is the theoretical prediction (8) for the obtained best-fit value $\delta =0.12$. The dashed line shows noninteracting spinons result.