  In many physical scenarios, close relations between the bulk properties of
quantum systems and theories associated to their boundaries have been observed.
In this work, we provide an exact duality mapping between the bulk of a quantum
spin system and its boundary using Projected Entangled Pair States (PEPS). This
duality associates to every region a Hamiltonian on its boundary, in such a way
that the entanglement spectrum of the bulk corresponds to the excitation
spectrum of the boundary Hamiltonian. We study various specific models, like a
deformed AKLT [1], an Ising-type [2], and Kitaev's toric code [3], both in
finite ladders and infinite square lattices. In the latter case, some of those
models display quantum phase transitions. We find that a gapped bulk phase with
local order corresponds to a boundary Hamiltonian with local interactions,
whereas critical behavior in the bulk is reflected on a diverging interaction
length of the boundary Hamiltonian. Furthermore, topologically ordered states
yield non-local Hamiltonians. As our duality also associates a boundary
operator to any operator in the bulk, it in fact provides a full holographic
framework for the study of quantum many-body systems via their boundary.
