Measuring Spin-Charge Separation by an Off-diagonal Dissipative Response

Fractionalization of symmetry - exemplified by spin-charge separation in the 1D Hubbard model and fractional charges in the fractional quantum Hall effect - is a typical strongly correlated phenomena in quantum many-body systems. Despite the success in measuring velocity differences, however, it is still quite challenging in probing emergent excitations' anomalous dimensions experimentally. We propose a off-diagonal dissipative response protocol, leveraging dissipative response theory (DRT), to directly detect spin-charge separation. By selectively dissipating spin-$\downarrow$ particles and measuring the spin-$\uparrow$ response, we uncover a universal temporal signature: the off-diagonal response exhibits a crossover from cubic-in-time ($t^3$) growth at short times to linear-in-time ($t$) decay at long times. Crucially, the coefficients $\varkappa^s$ (short-time) and $\varkappa^l$ (long-time) encode the distinct anomalous dimensions and velocities of spinons and holons, providing unambiguous evidence of fractionalization. This signal vanishes trivially without spin-charge separation. Our predictions, verified numerically via tDMRG, with microscopic parameters linking with Luttinger parameters by Bethe ansatz, establish off-diagonal dissipative response as a probe of quantum fractionalization in synthetic quantum matter.