  Parallel ("nested") regions of a Fermi surface (FS) drive instabilities of
the electron fluid, for example the spin density wave in elemental chromium. In
one-dimensional materials, the FS is trivially fully nested (a single nesting
vector connects two "Fermi dots"), while in higher dimensions only a fraction
of the FS consists of parallel sheets. We demonstrate that the tiny angle
regime of twist bilayer graphene (TBLG) possess a phase, accessible by
interlayer bias, in which the FS consists entirely of nestable "Fermi lines":
the first example of a completely nested FS in a 2d material. This nested phase
is found both in the ideal as well as relaxed structure of the twist bilayer.
We demonstrate excellent agreement with recent STM images of topological states
in this material and elucidate the connection between these and the underlying
Fermiology. We show that the geometry of the "Fermi lines" network is
controllable by the strength of the applied interlayer bias, and thus that TBLG
offers unprecedented access to the physics of FS nesting in 2d materials.
