# Conformal symmetry of trapped Bose-Einstein condensates and massive Nambu-Goldstone modes

## TL;DR

This paper explores the conformal symmetry properties of trapped Bose-Einstein condensates and identifies massive Nambu-Goldstone modes arising from symmetry breaking, providing universal equations of motion and an effective action.

## Contribution

It reveals the presence of massive Nambu-Goldstone modes in trapped BECs and derives universal equations of motion based on modified Schr"odinger symmetry.

## Key findings

- Massive Nambu-Goldstone modes correspond to variance and center of trapped gases.
- Universal equations of motion describe harmonic, cyclotron, and breathing oscillations.
- Exact effective action for NG modes is constructed.

## Abstract

The Gross-Pitaevskii (GP) or nonlinear Schr\"odinger equation relevant to ultracold atomic gaseous Bose-Einstein condensates possess a modified Schr\"odinger symmetry in two spatial dimensions, in the presence of a harmonic trapping potential, an (artificial) constant magnetic field (or rotation) and an electric field of a quadratic electrostatic potential. We find that a variance and a center of a trapped gas with or without a vorticity can be regarded as massive Nambu-Goldstone (NG) modes associated with spontaneous breaking of the modified Schr\"odinger symmetry. We show that the Noether theorem for the modified Schr\"odinger symmetry gives universal equations of motion which describe exact time-evolutions of the trapped gases such as a harmonic oscillation, a cyclotron motion and a breathing oscillation with frequencies determined by the symmetry independently of the details of the system. We further construct an exact effective action for all the NG modes.