In real paramagnets, there is always a subtle many-body contribution to the systems 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 Gruneisen 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.
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