Is a phonon excitation of a superfluid Bose gas a Goldstone boson?

TL;DR

This paper investigates whether phonons in a superfluid Bose gas are true Goldstone bosons, concluding that in finite systems they are not, but in infinite systems the classification becomes paradoxical due to degeneracy issues.

Contribution

The study challenges the conventional view by analyzing finite and infinite Bose gases with multiple methods, showing phonons are not Goldstone bosons in finite systems.

Findings

Phonons in finite superfluid Bose gases are not Goldstone bosons.

In infinite systems, the Goldstone nature of phonons is ambiguous due to degeneracy.

Different theoretical approaches yield consistent results.

Abstract

It is generally accepted that phonons in a superfluid Bose gas are Goldstone bosons. This is justified by spontaneous symmetry breaking (SSB), which is usually defined as follows: the Hamiltonian of the system is invariant under the $U(1)$ transformation $\hat{\Psi}(\mathbf{r},t)\rightarrow e^{i\alpha}% \hat{\Psi}(\mathbf{r},t)$, whereas the order parameter $\Psi(\mathbf{r},t)$ is not. However, the strict definition of SSB is different: the Hamiltonian and the boundary conditions are invariant under a symmetry transformation, while the ground state is not. Based on the latter criterion, we study a finite system of spinless, weakly interacting bosons using three approaches: the standard Bogoliubov method, the particle-number-conserving Bogoliubov method, and the approach based on the exact ground-state wave function. Our results show that the answer to the question in the title is ``no''. Thus, phonons in a real-world (finite) superfluid Bose gas are similar to sound in a classical gas: they are not Goldstone bosons, but quantised collective vibrational modes arising from the interaction between atoms. In the case of an infinite Bose gas, however, the picture becomes paradoxical: the ground state can be regarded as either infinitely degenerate or non-degenerate, making the phonon both similar to a Goldstone boson and different from it.