energy tDOS of a CFL varies as $A(\omega)\propto e^{-\omega_{0}/\omega}$ as a
function of bias $\omega$ [119]. The tDOS can be measured through the
tunneling current in two setups. Using scanning tunneling microscope (STM),
electrons can be directly injected into the sample. Alternatively, the
tunneling current between two CFLs separated by a tunneling barrier can be
measured. The contrast in zero bias features have been shown directly in Ref.
[5], in which the second setup is employed in the Landau level setting.

Transport measurements give distinct signatures for the CFL. If we denote
$\tilde{\rho}$ the (dimensionless) resistivity tensor for the CFL itself, in
the half filled Landau level setting, the physical conductivity is given by:

$\sigma=\frac{e^{2}}{2h}\left({\bf\epsilon}+\frac{\tilde{\rho}}{2}\right),$
(62)

where $\epsilon$ is the antisymmetric $2\times 2$ tensor. Now, in the absence
of disorder and of scattering of the composite fermions,
$\tilde{\rho}\rightarrow 0$, whch implies the longitudinal conductivity
$\sigma_{xx}$ vanishes. In contrast, in a clean Fermi liquid in a Chern band,
the longitudinal conductivity is expected to grow as disorder is reduced
[120]. One may further seek to measure thermoelectric transport signatures,
encapsulated in the tensor ${\bf E}=\hat{S}\bf\nabla T$, where for the CFL in
the half filled LL, $\hat{S}=\sigma^{-1}\epsilon\tilde{\rho}\tilde{\alpha}/2$,
where $\tilde{\alpha}$ is the thermoelectric conductivity of the composite
Fermi surface Ref. [121, 122]. Finally, thermal conductivity measurements,
although challenging, will serve as a smoking gun signature of the nFL, since
the CFL will strongly violate the Wiedemann-Franz law [117].

Away from the CFL filling, the composite fermions experience a effective
magnetic field tied to the filling deviation: $\Delta\nu=\nu-1/2$, leading to
Landau levels of the composite Fermi surface. FCIs correspond to fully
occupying these Landau levels[1, 3], corresponding to fillings
$\nu=-(p+1)/(2p+1)$, and the many-body gap of the FCI is identified with the
cyclotron gap of the Landau levels. The observation of FCIs at filling
$\nu=-2/3,\,-3/5$ in moiré TMDs [26, 27], corresponding to $p=1,\,2$ is
consistent with this expectation. On the other hand, an FL would not exhibit
such quantum oscillations when the filling is changed. Analogously, imposing a
one-dimensional periodic grating is expected to lead to geometric resonances
on varying the filling $\Delta\nu$ in the moiré TMD setting, as have been
observed for CFLs in Landau levels [3, 6, 7].

A notable prediction of the CFL picture [3], verified in experiments [8] in
the Landau level context, relates to the finite wave-vector conductivity
$\sigma_{xx}(\boldsymbol{q})$ which can be probed by surface acoustic waves
(SAWs). The CFL conductivity is predicted to be linear in the wave-vector
$|{\bf q}|$. This follows directly from the form of the CFL conductivity
tensor 62 combined with the fact that the intrinsic resistivity of the CFL in
the clean limit is set by the finite wave-vector of the SAW probe
$\tilde{\rho}\propto|{\boldsymbol{q}}|$.

Implementing the proposals above will require overcoming significant
experimental challenges - from performing transport measurements on TMD moiré
materials to working with the smaller size of moiré samples compared to
GaAs/GaAlAs samples. Nevertheless, given the experimental opportunity to study
a zero field non-Fermi liquid which might give rise to of a non-Abelian
topological phase, we anticipate there will be strong push to tackle these
challenges.