  We explore the Fock-space structure of eigenstates across the many-body
localisation (MBL) transition in a disordered, interacting quantum spin-1/2
chain. Eigenstate expectation values of spatially local observables, which
distinguish an MBL phase from an ergodic one, can be represented in terms of
eigenstate amplitudes on the Fock space. Motivated by this, we introduce and
study spatial correlations on the Fock space. From these, a correlation length
emerges, which is found to vary discontinuously across the MBL transition; and
is intimately connected to the discontinuous jump in the multifractal exponents
characterising the Fock-space wavefunctions. Exploiting the direct connection
between the local observables and Fock-space correlations, we show that the
discontinuity in the lengthscale also implies discontinuous behaviour of the
local observables across the transition. A scaling theory based on these
Fock-space correlations is constructed, which is closely connected to that for
the inverse participation ratio. It yields a volume-scale in the ergodic phase
and a length-scale in the MBL phase, whose critical properties suggest a
Kosterlitz-Thouless-like scenario for the MBL transition, as is predicted by
recent phenomenological theories. Finally, we also show how correlation
functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes
on the Fock space in the MBL phase.

Disordered Systems and Neural Networks;Quantum Gases;Statistical Mechanics;Strongly Correlated Electrons;Quantum Physics