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

## TL;DR

This paper introduces the concept of upper branch magnetism in quantum magnets, emphasizing the importance of excited crystal field states in understanding magnetic phase transitions and emergent selection rules.

## Contribution

It develops a microscopic theory accounting for the role of excited crystal field states, revealing new magnetic phases and transition mechanisms in rare-earth magnets.

## Key findings

- Excited crystal field states can significantly influence magnetic properties.
- Phase transitions driven by excited states can lead to new magnetic orders.
- Emergent selection rules help detect underlying excitations.

## Abstract

In many quantum magnets especially the rare-earth ones, the low-lying crystal field states are not well separated from the excited ones and thus are insufficient to describe the low-temperature magnetic properties. Inspired by this simple observation, we develop a microscopic theory to describe the magnetic physics due to the collapses of the weak crystal field states. We find two cases where the excited crystal field states should be seriously included into the theory. One case is when the bandwidth of the excited crystal field states is comparable to the crystal field gap. The other case is when the exchange energy gain between the low-lying and excited crystal field states overcomes the crystal field gap. Both cases could drive a phase transition and result in magnetic orders by involving the excited crystal field states. We dub the above physics as upper branch magnetism and phase transition. We discuss the multitude of magnetic phases and the emergent selection rules for the detection of the underlying excitations. We expect our results to help improve the understanding of many rare-earth magnets with weak crystal field gaps such as Tb$_2$Ti$_2$O$_7$ and Tb$_2$Sn$_2$O$_7$, and also provide a complementary perspective to the prevailing local "$J$" physics in $4d$/$5d$ magnets.