  We develop the theory of an Andreev junction, which provides a method to
probe the intrinsic topology of the Fermi sea of a two-dimensional electron gas
(2DEG). An Andreev junction is a Josephson $\pi$ junction proximitizing a
ballistic 2DEG, and exhibits low-energy Andreev bound states that propagate
$\textit{along}$ the junction. It has been shown that measuring the nonlocal
Landauer conductance due to these Andreev modes in a narrow linear junction
leads to a topological Andreev rectification (TAR) effect characterized by a
quantized conductance that is sensitive to the Euler characteristic $\chi_F$ of
the 2DEG Fermi sea. Here we expand on that analysis and consider more realistic
device geometries that go beyond the narrow linear junction and fully adiabatic
limits considered earlier. Wider junctions exhibit additional Andreev modes
that contribute to the transport and degrade the quantization of the
conductance. Nonetheless, we show that an appropriately defined
$\textit{rectified conductance}$ remains robustly quantized provided large
momentum scattering is suppressed. We verify and demonstrate these predictions
by performing extensive numerical simulations of realistic device geometries.
We introduce a simple model system that demonstrates the robustness of the
rectified conductance for wide linear junctions as well as point contacts, even
when the nonlocal conductance is not quantized. Motivated by recent
experimental advances, we model devices in specific materials, including InAs
quantum wells, as well as monolayer and bilayer graphene. These studies
indicate that for sufficiently ballistic samples observation of the TAR effect
should be within experimental reach.
