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 $\mathrm{U}\left(1\right)\times \mathrm{SO}\left(3\right)$ symmetry. The experimental method based on the Stern-Gerlach experiment is proposed for studying the Ward-Takahashi identity.

中文翻译：

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多组分玻色-爱因斯坦凝聚物的Hugenholtz-Pines定理

Hugenholtz-Pines（HP）定理是针对具有内部自由度的Bose-Einstein凝聚物（BEC）推导的。系统具有线性和二次对称破断项，从而提供了低能量的Ward-Takahashi身份。此身份可用于组织针对多分量BEC（例如二进制BEC和自旋的BEC）的HP定理。$F$ 磁场破坏时自旋BEC $\mathrm{ü}（\mathrm{1个}）\times \mathrm{所以}（3）$对称。提出了一种基于Stern-Gerlach实验的实验方法来研究Ward-Takahashi身份。