Ergodic quantum many-body systems satisfy the eigenstate
thermalization hypothesis (ETH). However, strong disorder can destroy
ergodicity through many-body localization (MBL) – at least in one
dimensional systems – leading to a clear signal of the MBL transition in
the probability distributions of energy eigenstate expectation values of
local operators. For a paradigmatic model of MBL, namely the
random-field Heisenberg spin chain, we consider the full probability
distribution of eigenstate correlation functions across
the entire phase diagram. We find gaussian distributions at weak
disorder, as predicted by pure ETH. At intermediate disorder – in the
thermal phase – we find further evidence for anomalous
thermalization in the form of heavy tails of the distributions.
In the MBL phase, we observe peculiar features of the correlator
distributions: a strong asymmetry in S_i^z S_{i+r}^zSizSi+rz
correlators skewed towards negative values; and a multimodal
distribution for spin-flip correlators. A quantitative
quasi-degenerate perturbation theory calculation of these
correlators yields a surprising agreement of the full
distribution with the exact results, revealing, in particular,
the origin of the multiple peaks in the spin-flip correlator
distribution as arising from the resonant and off-resonant admixture of
spin configurations. The distribution of the
S_i^zS_{i+r}^zSizSi+rz
correlator exhibits striking differences between the MBL and Anderson
insulator cases.