Figure 1

Schematic plot of the hydrodynamic regime in quantum spin liquids. The hydrodynamic frequency range is bounded on either side by the rate of fluctuations in the $U{\left(1\right)}_{\mathrm{flux}}$ quantum number ${\tau }_{\mathbf{a}}^{-1}$ and the rate of spinon scattering ${\tau }_{\psi }^{-1}$. At timescales smaller than ${\tau }_{\psi }$, the spinon degrees of freedom are no longer in equilibrium with each other, and the system is in a “spinon plasma” regime. At timescales larger than ${\tau }_{\mathbf{a}}$, compactness of the microscopic gauge field becomes apparent and leads to confinement. A magnetohydrodynamic regime of locally equilibrated spinon degrees of freedom coupled to an approximately noncompact gauge field arises between these two extremes. The origin of the associated wave-number scales is discussed in the main text. The temperature dependence of these cutoffs is not well studied in general and depends on the model at hand. For the $U\left(1\right)$ spin liquid with a Fermi surface, estimates are provided in Eqs. (13) and (14), while for Dirac spin liquids with a large number of fermion flavor $N_f\gg 1$, estimates are provided in Eqs. (31) and (32).