The number entropy as an accessible measure to detect transport and topological phases

For a system with particle number conservation, the entanglement entropy of a subsystem can be split into configurational and number entropy. While a measurement of the entanglement entropy requires a full quantum tomography, the number entropy is based on particle distributions alone and accessible, for example, via single site spectroscopy. Here I show that in equilibrium the number entropy can be used as indicator for topological phase transitions in systems with symmetry protected topological order. Furthermore, after a quantum quench, the number entropy shows whether or not particles spread through the system or remain localized. Our results indicate, in particular, that one of the widely studied models for many-body localization (MBL) - the spin-1/2 Heisenberg chain with random magnetic fields - never localizes for finite interactions and disorder strengths which are numerically accessible. This is in contrast to earlier studies which predicted an MBL transition and calls into question whether non-trivial MBL phases exist at all.