  Tensor network algorithms have proven to be very powerful tools for studying
one- and two-dimensional quantum many-body systems. However, their application
to three-dimensional (3D) quantum systems has so far been limited, mostly
because the efficient contraction of a 3D tensor network is very challenging.
In this paper we develop and benchmark two contraction approaches for infinite
projected entangled-pair states (iPEPS) in 3D. The first approach is based on a
contraction of a finite cluster of tensors including an effective environment
to approximate the full 3D network. The second approach performs a full
contraction of the network by first iteratively contracting layers of the
network with a boundary iPEPS, followed by a contraction of the resulting
quasi-2D network using the corner transfer matrix renormalization group.
Benchmark data for the Heisenberg and Bose-Hubbard models on the cubic lattice
show that the algorithms provide competitive results compared to other
approaches, making iPEPS a promising tool to study challenging open problems in
3D.
