  Entanglement membrane theory is an effective coarse-grained description of
entanglement dynamics and operator growth in chaotic quantum many-body systems.
The fundamental quantity characterizing the membrane is the entanglement line
tension. However, determining the entanglement line tension for microscopic
models is in general exponentially difficult. We compute the entanglement line
tension in a recently introduced class of exactly solvable yet chaotic unitary
circuits, so-called generalized dual-unitary circuits, obtaining a non-trivial
form that gives rise to a hierarchy of velocity scales with $v_E<v_B$. For the
lowest level of the hierarchy, $\bar{\mathcal{L}}_{2}$ circuits, the
entanglement line tension can be computed entirely, while for the higher levels
the solvability is reduced to certain regions in spacetime. This partial
solvability enables us to place bounds on the entanglement velocity. We find
that $\bar{\mathcal{L}}_{2}$ circuits saturate certain bounds on entanglement
growth that are also saturated in holographic models. Furthermore, we relate
the entanglement line tension to temporal entanglement and correlation
functions. We also develop new methods of constructing generalized dual-unitary
gates, including constructions based on complex Hadamard matrices that exhibit
additional solvability properties and constructions that display behavior
unique to local dimension greater than or equal to three. Our results shed
light on entanglement membrane theory in microscopic Floquet lattice models and
enable us to perform non-trivial checks on the validity of its predictions by
comparison to exact and numerical calculations. Moreover, they demonstrate that
generalized dual-unitary circuits display a more generic form of information
dynamics than dual-unitary circuits.
