  In this work we extend the notion of universal quantum Hamiltonians to the
setting of translationally-invariant systems. We present a construction that
allows a two-dimensional spin lattice with nearest-neighbour interactions, open
boundaries, and translational symmetry to simulate any local target
Hamiltonian---i.e. to reproduce the whole of the target system within its
low-energy subspace to arbitrarily-high precision. Since this implies the
capability to simulate non-translationally-invariant many-body systems with
translationally-invariant couplings, any effect such as characteristics
commonly associated to systems with external disorder, e.g. many-body
localization, can also occur within the low-energy Hilbert space sector of
translationally-invariant systems. Then we sketch a variant of the universal
lattice construction optimized for simulating translationally-invariant target
Hamiltonians. Finally we prove that qubit Hamiltonians consisting of Heisenberg
or XY interactions of varying interaction strengths restricted to the edges of
a connected translationally-invariant graph embedded in $\mathbb{R}^D$ are
universal, and can efficiently simulate any geometrically local Hamiltonian in
$\mathbb{R}^D$.
