  Ultracold atoms in optical lattices are one of the most promising
experimental setups to simulate strongly correlated systems. However, efficient
numerical algorithms able to benchmark experiments at low-temperatures in
interesting 3d lattices are lacking. To this aim, here we introduce an
efficient tensor network algorithm to accurately simulate thermal states of
local Hamiltonians in any infinite lattice, and in any dimension. We apply the
method to simulate thermal bosons in optical lattices. In particular, we study
the physics of the (soft-core and hard-core) Bose-Hubbard model on the infinite
pyrochlore and cubic lattices with unprecedented accuracy. Our technique is
therefore an ideal tool to benchmark realistic and interesting optical-lattice
experiments.
