  Torsion gravity is a natural extension to Einstein gravity in the presence of
the fermion matter sources. In this paper we adopt Wald's covariant method of
Noether charge to construct the quasi-local energy of the
Einstein-Cartan-fermion system, and find that its explicit expression is
formally independent of the coupling constant between torsion and axial
current. This seemingly topological nature is unexpected and is reminiscent of
similar nature of quantum Hall effect and topological insulator. However, the
coupling dependence does enter when evaluating it on-shell, and thus the
topological nature is pseudo. Based on the expression of the quasi-local
energy, we evaluate it for a particular solution on the entanglement wedge and
find the agreement with the holographic relative entropy obtained before. This
shows that the equivalence of these two quantities in the
Einstein-Cartan-fermion system. Moreover, the quasi-local energy in this case
is not always positive definite so that it provides an example of swampland in
torsion gravity. Based on the covariant Noether charge, we also derive the
nonzero fermion effect on Komar angular momentum. The implication of our
results to the tests of torsion gravity in the future gravitational wave
astronomy is also discussed.
