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In many quantum magnets especially the rare-earth ones, the low-lying crystalfield states are not well separated from the excited ones and thus areinsufficient to describe the low-temperature magnetic properties. Inspired bythis simple observation, we develop a microscopic theory to describe themagnetic physics due to the collapses of the weak crystal field states. We findtwo cases where the excited crystal field states should be seriously includedinto the theory. One case is when the bandwidth of the excited crystal fieldstates is comparable to the crystal field gap. The other case is when theexchange energy gain between the low-lying and excited crystal field statesovercomes the crystal field gap. Both cases could drive a phase transition andresult in magnetic orders by involving the excited crystal field states. We dubthe above physics as upper branch magnetism and phase transition. We discussthe multitude of magnetic phases and the emergent selection rules for thedetection of the underlying excitations. We expect our results to help improvethe understanding of many rare-earth magnets with weak crystal field gaps suchas Tb$_2$Ti$_2$O$_7$ and Tb$_2$Sn$_2$O$_7$, and also provide a complementaryperspective to the prevailing local "$J$" physics in $4d$/$5d$ magnets.

Upper branch magnetism in quantum magnets: Collapses of excited levels and emergent selection rules