  Topological quantum many-body systems, such as Hall insulators, are
characterized by a hidden order encoded in the entanglement between their
constituents. Entanglement entropy, an experimentally accessible single number
that globally quantifies entanglement, has been proposed as a first signature
of topological order. Conversely, the full description of entanglement relies
on the entanglement Hamiltonian, a more complex object originally introduced to
formulate quantum entanglement in curved spacetime. As conjectured by Li and
Haldane, the entanglement Hamiltonian of a many-body system appears to be
directly linked to its boundary properties, making it particularly useful for
characterizing topological systems. While the entanglement spectrum is commonly
used to identify complex phases arising in numerical simulations, its
measurement remains an outstanding challenge. Here, we perform a variational
approach to realize experimentally, as a genuine Hamiltonian, the entanglement
Hamiltonian of a synthetic quantum Hall system. We use a synthetic dimension,
encoded in the electronic spin of dysprosium atoms, to implement spatially
deformed Hall systems, as suggested by the Bisognano-Wichmann prediction. The
spectrum of the optimal variational Hamiltonian exhibits a chiral dispersion
akin to a topological edge mode, revealing the fundamental link between
entanglement and boundary physics. Our variational procedure can be easily
generalized to interacting many-body systems on various platforms, marking an
important step towards the exploration of exotic quantum systems with
long-range correlations, such as fractional Hall states, chiral spin liquids
and critical systems.
