Eigenstate thermalization in spin-$\frac{1}{2}$ systems with SU(2) symmetry

We study the diagonal and off-diagonal matrix elements of observables in the eigenstates of the extended spin-$\frac{1}{2}$ Heisenberg chain, which exhibits the non-Abelian SU(2) symmetry. We explore integrable and nonintegrable regimes, and consider observables that preserve the SU(2) symmetry of the Hamiltonian as well as observables that break it. We study in detail the low-frequency behavior of the off-diagonal matrix elements at and away from integrability. In the nonintegrable regime, we test the non-Abelian eigenstate thermalization hypothesis, paying special attention to the effect of the spin, which is the distinctive conserved quantity introduced by the SU(2) symmetry.