  We study a three-dimensional (3D) classical Ising model that is exactly
solvable when some coupling constants take certain imaginary values. The
solution combines and generalizes the Onsager-Kaufman solution of the 2D Ising
model and the solution of Kitaev's honeycomb model, leading to a
three-parameter phase diagram with a third order phase transition between two
distinct phases. Interestingly, the phases of this model are distinguished by
topological features: the expectation value of a certain family of loop
observables depend only on the topology of the loop (whether the loop is
contractible), and are quantized at rational values that differ in the two
phases. We show that a related exactly solvable 3D classical statistical model
with real coupling constants also shows the topological features of one of
these phases. Furthermore, even in the model with complex parameters, the
partition function has some physical relevance, as it can be interpreted as the
transition amplitude of a quantum dynamical process and may shed light on
dynamical quantum phase transitions.
