  It has been suggested previously that an ultra-soft fermionic excitation
develops, albeit with a small spectral weight, in a system of massless fermions
and scalar bosons with Yukawa interaction at high temperature ($T$). In this
paper we study how this excitation is modified at finite chemical potential
($\mu$). We relate the existence of the ultra-soft mode to symmetries, in
particular charge conjugation, and a supersymmetry of the free system which is
spontaneously broken by finite temperature and finite density effects, as
argued earlier by Lebedev and Smilga. A non vanishing chemical potential breaks
both symmetries explicitly, and maximally at zero temperature where the mode
ceases to exist. A detailed calculation indicates that the ultra-soft
excitation persists as long as $T\gtrsim \mu$.
