# Symmetry characterization of the collective modes of the phase diagram of the $\nu=0$ quantum Hall state in graphene: Mean-field and spontaneously broken symmetries

## TL;DR

This paper analyzes the phase diagram and collective modes of the $
u=0$ quantum Hall state in graphene, revealing symmetry-breaking mechanisms, collective mode structures, and the robustness of certain symmetries under various conditions.

## Contribution

It provides a comprehensive classification of collective modes and demonstrates the persistence of $SO(5)$ symmetry in the mean-field phase diagram of graphene's quantum Hall states.

## Key findings

- Phase transitions involve singlet orbital pseudospin modes independent of Coulomb strength.
- Goldstone modes reveal valley-spin symmetry breaking mechanisms.
- $SO(5)$ symmetry persists at phase boundaries under certain conditions.

## Abstract

We devote this work to the study of the mean-field phase diagram of the $\nu=0$ quantum Hall state in bilayer graphene and the computation of the corresponding neutral collective modes, extending the results of recent works in the literature. Specifically, we provide a detailed classification of the complete orbital-valley-spin structure of the collective modes and show that phase transitions are characterized by singlet modes in orbital pseudospin, which are independent of the Coulomb strength and suffer strong many-body corrections from short-range interactions at low momentum. We describe the symmetry breaking mechanism for phase transitions in terms of the valley-spin structure of the Goldstone modes. For the remaining phase boundaries, we prove that the associated exact $SO(5)$ symmetry existing at zero Zeeman energy and interlayer voltage survives as a weaker mean-field symmetry of the Hartree-Fock equations. We extend the previous results for bilayer graphene to the monolayer scenario. Finally, we show that taking into account Landau level mixing through screening does not modify the physical picture explained above.