Disorder free many-body localization transition in two quasiperiodically coupled Heisenberg spin chains

Disorder free many-body localization (MBL) can occur in interacting systems that can dynamically generate their own disorder. We address the thermal-MBL phase transition of two isotropic Heisenberg spin chains that are quasi-periodically coupled to each other. The spin chains are incommensurate and are coupled through a short range exchange interaction of the $XXZ$ type that decays exponentially with the distance. Using exact diagonalization, matrix product states and density matrix renormalization group, we calculate the time evolution of the entanglement entropy at long times and extract the inverse participation ratio in the thermodynamic limit. We show that this system has a robust MBL phase. We establish the phase diagram with the onset of MBL as a function of the interchain exchange coupling and of the incommensuration between the spin chains. The Ising limit of the interchain interaction optimizes the stability of the MBL phase over a broad range of incommensurations above a given critical exchange coupling. Incorporation of interchain spin flips significantly enhances entanglement between the spin chains and produces delocalization, favoring a pre-thermal phase whose entanglement entropy grows logarithmically with time.