Suppose we have a generic system with hamiltonian

\begin{equation} H= JH_1 +\mu H_2 \end{equation} where $J$ and $\mu$ are couplings and $H_1$ and $H_2$ are just two dimensionless parts of the hamiltonian. How can we estimate which term is going to dominate over the other? My first approach was to say "if $\mu$ is much bigger than $J$ then we can ignore $H_1$. But then I thought that, actually, the important parameters are not $J$ and $\mu$ but $\beta J$ and $\beta \mu$ so at $T\rightarrow 0$ both terms would be equally important, even if $\mu>>J$.

What is the right approach to think about these things? does it make sense to think about it in this generic example or further information is relevant to make a statement?


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