On performing a sequence of renormalisation group (RG) transformations on a
system of two-dimensional non-interacting Dirac fermions placed on a torus, we
demonstrate the emergence of an additional spatial dimension arising out of the
scaling of multipartite entanglement. The renormalisation of entanglement under
this flow exhibits a hierarchy across scales as well as number of parties.
Geometric measures defined in this emergent space, such as distances and
curvature, can be related to the RG beta function of the coupling g
responsible for the spectral gap. This establishes a holographic connection
between the spatial geometry of the emergent space in the bulk and the
entanglement properties of the quantum theory lying on its boundary. Depending
on the anomalous dimension of the coupling g, three classes of spaces
(bounded, unbounded and flat) are generated from the RG. We show that changing
from one class to another involves a topological transition. By minimising the
central charge of the conformal field theory describing the noninteracting
electrons under the RG flow, the RG transformations are shown to satisfy the
cβtheorem of Zamolodchikov. This is shown to possess a dual within the
emergent geometric space, in the form of a convergence parameter that is
minimised at large distances. In the presence of an Aharonov-Bohm flux, the
entanglement gains a geometry-independent piece which is shown to be
topological, sensitive to changes in boundary conditions, and can be related to
the Luttinger volume of the system of electrons. In the presence of a strong
transverse magnetic field, the system becomes insulating and Luttinger's
theorem does not hold. We show instead that the entanglement contains a term
that can be related to the Chern numbers of the quantum Hall states. This
yields a relation between the topological invariants of the metallic and the
quantum Hall systems.Comment: 41 pages, 16 figure