Hugenholtz-Pines theorem for multicomponent Bose-Einstein condensates

TL;DR

This paper extends the Hugenholtz-Pines theorem to multicomponent Bose-Einstein condensates with internal degrees of freedom, providing a theoretical framework and proposing an experimental approach for verification.

Contribution

It derives the HP theorem for multicomponent BECs using the Ward-Takahashi identity, accounting for symmetry breaking and internal degrees of freedom.

Findings

Organizes the HP theorem for multicomponent BECs.

Provides a low-energy Ward-Takahashi identity.

Proposes an experimental method via Stern-Gerlach experiment.

Abstract

The Hugenholtz-Pines (HP) theorem is derived for Bose-Einstein condensates (BECs) with internal degrees of freedom. The low-energy Ward-Takahashi identity is provided in the system with the linear and quadratic symmetry breaking terms. This identity serves to organize the HP theorem for multicomponent BECs, such as the binary BEC as well as the spin-$f$ spinor BEC in the presence of a magnetic field with broken U$(1)$$\times$SO$(3)$ symmetry. The experimental method based on the Stern-Gerlach experiment is proposed for studying the Ward-Takahashi identity.