Many-body localization: a quantum frontier

A closed quantum system is ergodic and satisfies equilibrium statistical physics when it completely loses local information of its initial condition under time evolution, by 'hiding' the information in non-local properties like entanglement. In the last decade, a flurry of theoretical work has shown that ergodicity can be broken in an isolated, quantum many-body system even at high energies in the presence of disorder, a phenomena known as many-body localization (MBL). In this novel phase of matter, highly excited states of an interacting system can serve as quantum memory and even protect exotic forms of quantum order. Recent claims of experimental observation of MBL in two dimensions using ultra-cold atoms has further raised a plethora of intriguing questions.

In one dimension, the strongly localized regime is described in terms of quasi-local integrals of motion, also known as l-bits. Based on this picture I will describe an efficient tensor network method to efficiently represent the entire spectrum of fully many-body localized systems. This ansatz is also successful at capturing features of the MBL to thermal transition. For higher dimensions, I will develop a refined phenomenology of MBL in terms of l*-bits which are only 'approximately' conserved, based on the stability of the localized phase to perturbations on the boundary. I will conclude with a bird's-eye view of some of the open problems in this rapidly growing field.