  We study the Hubbard and Heisenberg models on hyperbolic lattices with open
boundary conditions by means of mean-field approximations, spin-wave theory and
quantum Monte Carlo (QMC) simulations. For the Hubbard model we use the
auxiliary-field approach and for Heisenberg systems the stochastic series
expansion algorithm. We concentrate on bipartite lattices where the QMC
simulations are free of the negative sign problem. The considered lattices are
characterized by a Dirac like density of states, Schl\"afli indices
$\{p,q\}=\{10,3\}$ and $\{8,3\}$, as well as by flat bands, $\{8,8\}$. The
Dirac density of states cuts off the logarithmic divergence of the staggered
spin susceptibility and allows for a finite $U$ semi-metal to insulator
transition. This transition has the same mean-field exponents as for the
Euclidean counterpart. In the presence of flat bands we observe the onset of
magnetic ordering at any finite $U$. The magnetic state at intermediate
coupling can be described as a global antiferromagnet. It breaks the $C_p$
rotational and time reversal symmetries but remains invariant under combined
$C_p \mathcal{T}$ transformations. The state is characterized by macroscopic
ferromagnetic moments, that globally cancel. We observe that fluctuations on
the boundary of the system are greatly enhanced: while spin wave calculations
predict the breakdown of antiferromagnetism on the boundary but not in the
bulk, QMC simulations show a marked reduction of the staggered moment on the
edge of the system.
