  In real paramagnets, there is always a subtle many-body contribution to the
system's energy, which can be regarded as a small effective local magnetic
field $B_{loc}$. Usually, it is neglected, since it is very small when compared
with thermal fluctuations and/or external magnetic fields $B$. Nevertheless, as
both the temperature $T \rightarrow$ 0K and $B \rightarrow$ 0T, such many-body
contributions become ubiquitous. Here, employing the magnetic Gr\"uneisen
parameter $\Gamma_{mag}$ and entropy arguments, we report on the pivotal role
played by the mutual interactions in the regime of ultra-low-$T$ and vanishing
$B$. Our key results are: $i$) absence of a genuine zero-field quantum phase
transition due to the presence of $B_{loc}$; $ii$) connection between the
canonical definition of temperature and $\Gamma_{mag}$; and $iii$) possibility
of performing adiabatic magnetization by only manipulating the mutual
interactions. Our findings unveil unprecedented aspects emerging from the
mutual interactions.
