  The exact nature of the many-body localization transition remains an open
question. An aspect which has been posited in various studies is the emergence
of scale invariance around this point, however the direct observation of this
phenomenon is still absent. Here we achieve this by studying the logarithmic
negativity and mutual information between disjoint blocks of varying size
across the many-body localization transition. The two length scales, block
sizes and the distance between them, provide a clear quantitative probe of
scale invariance across different length scales. We find that at the transition
point, the logarithmic negativity obeys a scale invariant exponential decay
with respect to the ratio of block separation to size, whereas the mutual
information obeys a polynomial decay. The observed scale invariance of the
quantum correlations in a microscopic model opens the direction to probe the
fractal structure in critical eigenstates using tensor network techniques and
provide constraints on the theory of the many-body localization transition.
