Local Conservation Laws and the Structure of the Many-Body Localized States

We construct a complete set of local integrals of motion that characterize the many-body localized (MBL) phase. Our approach relies on the assumption that local perturbations act locally on the eigenstates in the MBL phase, which is supported by numerical simulations of the random-field $XXZ$ spin chain. We describe the structure of the eigenstates in the MBL phase and discuss the implications of local conservation laws for its nonequilibrium quantum dynamics. We argue that the many-body localization can be used to protect coherence in the system by suppressing relaxation between eigenstates with different local integrals of motion.

Figure 1

(a) Averaged entanglement entropy is of the order 1 and varies weakly with $L$ for strong disorder, indicating that eigenstates are short-range entangled. Interaction strength is $V=1$. (b) Inverse participation ratios for the product of two eigenstates of the $\mathcal{L}$ and $\mathcal{R}$ subsystems are close to 1 and do not depend on $L$ for strong disorder.

Figure 2

The weight of the perturbed eigenstate $|{\lambda }^{\alpha }⟩$ in the subspace with index $\alpha$. For strong disorder, the action of the projector is contained within the subspace $\alpha$, irrespective of the interaction: the case of no interaction $V=0$ is shown in (a), and $V=1$ in (b). The weight increases with system size. For weak disorder ($W<3$), the presence of interactions causes the weight to decrease with the system size, suggesting the onset of the delocalized phase.