Entanglement dynamics and eigenstate correlations in strongly disordered quantum many-body systems

The many-body localised phase of quantum systems is an unusual
dynamical phase wherein the system fails to thermalise and yet,
entanglement grows unboundedly albeit very slowly in time. We present a
microscopic theory of this ultraslow growth of entanglement in terms of
dynamical eigenstate correlations of strongly disordered, interacting
quantum systems in the many-body localised regime. These correlations
involve sets of four or more eigenstates and hence, go beyond correlations
involving pairs of eigenstates which are usually studied in the context of
eigenstate thermalisation or lack thereof. We consider the minimal case,
namely the second Rényi entropy of entanglement, of an initial product
state as well as that of the time-evolution operator, wherein the
correlations involve quartets of four eigenstates. We identify that the
dynamics of the entanglement entropy is dominated by the spectral
correlations within certain special quartets of eigenstates. We uncover
the spatial structure of these special quartets and the ensuing statistics
of the spectral correlations amongst the eigenstates therein, which
reveals a hierarchy of timescales or equivalently, energy scales. We show
that the hierarchy of these timescales along with their non-trivial
distributions conspire to produce the logarithmic in time growth of
entanglement, characteristic of the many-body localised regime. The
underlying spatial structures in the set of special quartets also provides
a microscopic understanding of the spacetime picture of the entanglement
growth. The theory therefore provides a much richer perspective on
entanglement growth in strongly disordered systems compared to the
commonly employed phenomenological approach based on the ℓ-bit picture.

Reference: arXiv:2406.09392