Elastic gauge fields and zero-field 3D quantum Hall effect in hyperhoneycomb lattices*

We derive the elastic gauge fields that emerge from lattice deformations in the hyperhoneycomb lattice, a three-dimensional structure with trigonally connected sites. In its semimetallic form, this lattice is a nodal-line semimetal with a closed loop of Dirac nodes. Using strain engineering, we find the strain fields that create nearly uniform 3D Landau level quantization. In the 3D quantum anomalous Hall phase, we show that the elastic Hall viscosity is η^H=β²√3/(16πa³), where a is the lattice constant.