  Electron tunneling into a system with strong interactions is known to exhibit
an anomaly, in which the tunneling conductance vanishes continuously at low
energy due to many-body interactions. Recent measurements have probed this
anomaly in a quantum Hall bilayer of the half-filled Landau level, and shown
that the anomaly apparently gets stronger as the half-filled Landau level is
increasingly spin polarized. Motivated by this result, we construct a
semiclassical hydrodynamic theory of the tunneling anomaly in terms of the
charge-spreading action associated with tunneling between two copies of the
Halperin-Lee-Read state with partial spin polarization. This theory is
complementary to our recent work (arXiv:1709.06091) where the electron spectral
function was computed directly using an instanton-based approach. Our results
show that the experimental observation cannot be understood within conventional
theories of the tunneling anomaly, in which the spreading of the injected
charge is driven by the mean-field Coulomb energy. However, we identify a
qualitatively new regime, in which the mean-field Coulomb energy is effectively
quenched and the tunneling anomaly is dominated by the finite compressibility
of the composite Fermion liquid.
