Phenomenology of spectral functions in finite disordered spin chains

Studies of disordered spin chains have recently experienced a renewed interest, inspired by the question to which extent the exact numerical calculations comply with the existence of a many-body localization phase transition. For the paradigmatic random field Heisenberg spin chains, many intriguing features were observed when the disorder is considerable compared to the spin interaction strength. Here we introduce a phenomenological theory that may explain some of those features. We develop a theory based on the proximity to the noninteracting limit, in which the system is an Anderson insulator. In particular, we argue that the proximity to the local integrals of motion of the Anderson insulator substantially impacts the dynamical properties in finite interacting systems. Taking the spin imbalance as an exemplary observable, we show that our theory quantitatively describes its integrated spectral function for a wide range of disorders.