Construction of many-body critical states from local mirror symmetry points

It is known that a single-particle Hamiltonian has three different kinds of spectra — extended, localized, and critical — with the interacting many-body extensions of the first two being thermalized and many-body localized. Whether there exists an interacting many-body extension of the single-particle critical states at large system size remains inconclusive from numerical results. However, a possible theoretical argument may be developed, inspired by the rigorous constructions of single-particle critical states that have been extensively studied in the mathematical literature. In this work, we theoretically construct many-body critical states for Hamiltonians possessing a series of (approximate) local mirror-symmetry points under sufficiently strong disorder. We demonstrate several key differences between the interacting and non-interacting models. Such many-body critical states can be viewed as a particular type of rare resonance, which naturally appear in certain quasi-periodic spin chains. Finally, we discuss the possibility of thermalization induced by these many-body critical states.