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

This paper argues that phonons in a finite superfluid Bose gas are not Goldstone bosons, but rather quantized collective vibrational modes, because the strict definition of spontaneous symmetry breaking—which requires a non-invariant ground state under symmetric Hamiltonian and boundary conditions—is not satisfied in finite systems.

The Gist

The Big Question

Imagine you have a jar of tiny, invisible balls (atoms) that are so cold they start acting like a single, giant wave. This is a superfluid. In this state, if you poke the fluid, it ripples. These ripples are called phonons (sound waves).

For decades, physicists have believed these ripples are special "Goldstone bosons."

• The Goldstone Analogy: Imagine a pencil balanced perfectly on its tip. The laws of physics (gravity) treat all directions the same (symmetry). But the pencil must fall in one specific direction. Once it falls, the symmetry is "broken." The ripples that happen as the pencil wobbles while falling are the Goldstone bosons.
• The Old Belief: Physicists thought that in a superfluid, the atoms "fall" into a specific state (condensate), breaking a symmetry called U(1) (which is like a hidden dial that can be turned to any angle). They thought the sound waves (phonons) were the direct result of this broken symmetry, just like the wobbling pencil.

Maksim Tomchenko's paper asks: Is this actually true?

The Twist: Finite vs. Infinite

The author argues that the answer depends entirely on whether you are talking about a real jar of atoms (finite) or a theoretical, infinite universe of atoms.

1. The Real World (Finite Systems)

The Analogy: Imagine a choir of 1,000 singers in a room.

• The Symmetry: The song they are singing doesn't care if they all start singing a tiny bit louder or quieter together; the music sounds the same. This is the "symmetry."
• The Old View: People thought that because the choir is so big, they spontaneously decide to sing in a specific key, breaking the symmetry.
• Tomchenko's Discovery: He ran the math on a choir of a fixed number of people (a real system). He found that if you count the people exactly, the "hidden dial" (the phase) cannot be broken. The choir remains perfectly symmetrical.
• The Result: In a real, finite system, phonons are NOT Goldstone bosons.
  • Why? Because the sound waves aren't caused by a broken symmetry. They are just the result of the atoms bumping into each other, like sound in a hot gas. The atoms are interacting, but they haven't "broken" any fundamental rules. The "Goldstone" label is a mistake caused by using too many approximations.

2. The Theoretical World (Infinite Systems)

The Analogy: Now imagine a choir with an infinite number of singers.

• The Paradox: With an infinite number of singers, the math gets weird. You can add or remove a few singers, and it doesn't change the total count (because $\infty + 1 = \infty$).
• The Result: In this infinite world, the "broken symmetry" can happen. The ground state becomes infinitely degenerate (there are infinite ways to arrange the choir that all have the same energy).
• The Catch: Even here, the author argues this degeneracy isn't because of the "U(1" dial. It's because the number of particles is undefined. It's a mathematical paradox of infinity, not a physical law of nature.

The Three Ways He Checked

To be sure, the author didn't just guess; he used three different mathematical "lenses" to look at the problem:

1. The Standard Lens (Bogoliubov Method): This is the old, popular way of doing the math. It uses a shortcut where it treats the main group of atoms as a fixed number.
   • What it showed: It looked like symmetry was broken. But the author realized this was an illusion created by the shortcut. The math was inconsistent.
2. The Strict Lens (Particle-Conserving Method): This method refuses to use shortcuts. It insists that the number of atoms never changes.
   • What it showed: When you do the math strictly, the symmetry is never broken. The ground state stays perfectly symmetrical.
3. The Exact Lens (Exact Wave Function): This uses the most precise, complex math possible to describe the exact state of every atom.
   • What it showed: Confirmed the second lens. In a real, finite system, the symmetry holds firm. The "broken" state is a myth.

The Takeaway: What Does This Mean?

The paper concludes that for any real-world superfluid (like liquid helium in a lab or a Bose gas in a trap):

• Phonons are not Goldstone bosons. They are just quantized sound waves caused by atoms interacting.
• Superfluidity does not require "Spontaneous Symmetry Breaking." You don't need the "broken dial" to explain why the fluid flows without friction.
• The "Goldstone" idea is a ghost from infinity. It only appears when you pretend the system is infinite. Since real systems are always finite, the Goldstone explanation is technically incorrect for reality.

Summary Metaphor

Think of a marching band.

• Old Theory: The band spontaneously decides to march in a specific direction, breaking the symmetry of "marching anywhere." The sound they make is a "Goldstone sound."
• Tomchenko's Theory: In a real band (finite number of people), they are just marching in step because they are listening to the drum major and bumping into each other. They haven't "broken" any universal rule of marching. The sound is just the result of their coordination, not a broken symmetry. The idea that they "broke" the symmetry only exists if you imagine a band with an infinite number of people, which doesn't exist in our universe.

In short: The paper tells us to stop calling sound waves in superfluids "Goldstone bosons." They are just sound waves, and the "broken symmetry" story is a mathematical artifact that doesn't apply to the real, finite world.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in condensed matter physics: Are phonons in a superfluid Bose gas (at $T < T_\lambda$) Goldstone bosons?

• Conventional View: It is widely accepted that phonons in superfluids are Goldstone bosons arising from the spontaneous symmetry breaking (SSB) of the continuous $U(1)$ gauge symmetry. This view relies on the premise that the Hamiltonian is invariant under $U(1)$ transformations ($\hat{\Psi} \to e^{i\alpha}\hat{\Psi}$), but the order parameter (condensate wavefunction) is not.
• The Conflict: The author argues that the standard definition of SSB used in this context is often imprecise. A strict definition requires that both the Hamiltonian and the boundary conditions (BCs) are invariant, while the ground state is not.
• The Core Question: Does a real-world, finite system of interacting bosons exhibit true SSB of the $U(1)$ symmetry? If not, then phonons are not Goldstone bosons but rather quantized collective vibrational modes arising from interatomic interactions, similar to sound in a classical gas.

2. Methodology

The author employs a rigorous quantum-mechanical approach, analyzing a system of spinless, weakly interacting bosons using three distinct theoretical frameworks to determine the symmetry properties of the ground state:

1. Standard Bogoliubov Approach:

   • Uses the conventional approximation where the zero-momentum operator $\hat{a}_0$ is replaced by a c-number ($a_0 = \sqrt{N_0}e^{i\theta}$).
   • Analyzes the invariance of the approximate Hamiltonian ($\hat{H}_{app}$) and the resulting ground state wavefunction ($|\theta\rangle$) under $U(1)$ rotations.

2. Particle-Number-Conserving (PNC) Bogoliubov Approach:

   • Utilizes modified operators ($\hat{\varsigma}_k$) that preserve the total particle number operator $\hat{N}$.
   • Avoids the introduction of c-numbers, ensuring the Hamiltonian commutes with $\hat{N}$ ($[\hat{H}, \hat{N}] = 0$).
   • Derives the ground state wavefunction for a finite system with a fixed number of particles $N$.

3. Exact Ground-State Wavefunction Approach:

   • Uses the exact many-body wavefunction expressed in terms of collective variables (density operators $\hat{\rho}_k$) and correlation functions.
   • This approach is non-perturbative and valid for any interaction strength (weak to strong).
   • Explicitly constructs the ground state $|0\rangle$ and excited states $|p\rangle$ to test their transformation properties under $U(1)$ rotations.

Key Analytical Tool: The author applies the unitary transformation $\hat{U}_\phi = e^{i\phi \hat{N}}$ to the ground state. If SSB occurs, $\hat{U}_\phi |0\rangle$ should not be proportional to $|0\rangle$ (i.e., the state is not an eigenstate of $\hat{N}$ or the symmetry is broken).

3. Key Contributions and Results

A. Finite Systems: No Spontaneous Symmetry Breaking

The most significant finding is that in any finite system, the $U(1)$ symmetry is not spontaneously broken.

• Standard Bogoliubov Failure: The standard approach yields a ground state $|\theta\rangle$ that appears to break symmetry. However, the author demonstrates that this is an artifact of the approximation. The approximate Hamiltonian $\hat{H}_{app}$ itself is not invariant under $U(1)$ because it does not conserve particle number. Therefore, the non-invariance of the state is a consequence of the non-invariance of the approximate Hamiltonian, not true SSB.
• PNC and Exact Results: When using particle-number-conserving methods or exact wavefunctions, the ground state $|0\rangle$ for a system with $N$ particles transforms as:
  $$\hat{U}_\phi |0\rangle = e^{iN\phi} |0\rangle$$
  Since the state remains an eigenstate of the symmetry operation (up to a phase factor), the symmetry is preserved.
• Order Parameter: Consequently, the expectation value of the field operator vanishes:
  $$\langle 0 | \hat{\Psi}(r, t) | 0 \rangle = 0$$
  This contradicts the postulate of a non-zero order parameter in finite systems.
• Conclusion for Finite Systems: Phonons in finite superfluids are not Goldstone bosons. They are collective excitations (sound waves) resulting from atom-atom interactions, existing regardless of whether the system is in a superfluid or normal phase.

B. Infinite Systems: The Paradox of Infinity

The paper explores why the infinite system (thermodynamic limit) appears different.

• Indeterminacy of Particle Number: In the thermodynamic limit ($N, V \to \infty$), the ground state can be described by states with indefinite particle numbers (e.g., coherent states).
• Degeneracy Source: The author argues that the infinite degeneracy of the ground state in infinite systems arises from the indeterminacy of the particle number $N$, not strictly from the $U(1)$ symmetry of the Hamiltonian.
  • If $N$ is fixed (even at infinity), the state is non-degenerate.
  • If $N$ is indefinite, the system allows for a continuous family of ground states with different phases, leading to apparent SSB.
• Paradox: The infinite system can be viewed as both degenerate (allowing SSB) and non-degenerate (if $N$ is treated as a fixed macroscopic variable). This duality is a mathematical artifact of infinity.

C. Re-evaluation of the Goldstone Theorem and 1/q² Theorem

• Goldstone Theorem: The author notes that the Goldstone theorem strictly applies to Quantum Field Theory (where particle number is not conserved). In Quantum Mechanics with fixed particle number, the theorem does not apply directly.
• 1/q² Theorem: Bogoliubov's 1/q² theorem links gapless dispersion to SSB. The author shows that while the gapless dispersion law exists in finite systems, SSB is absent. This implies the gapless nature of phonons is not a consequence of SSB, but a result of the interaction dynamics. The similarity between the Goldstone theorem and the 1/q² theorem is merely formal in this context.

4. Significance and Implications

1. Nature of Superfluidity: The paper decouples the phenomenon of superfluidity and the existence of a condensate from the concept of Spontaneous Symmetry Breaking in finite, real-world systems. Superfluidity is a property of the interaction and the collective mode structure, not a broken symmetry phase.
2. Critique of Thermodynamic Limit: The study highlights that taking the thermodynamic limit ($N \to \infty$) can lead to physically misleading conclusions when applied to real, finite systems. Properties like SSB and infinite degeneracy are artifacts of the limit, not features of actual physical systems.
3. Phonon Identity: Phonons in superfluids are re-characterized as quantized collective vibrational modes (similar to sound in a classical gas) rather than Goldstone bosons. This unifies the description of sound modes above and below the transition temperature $T_\lambda$, as the underlying mechanism (interatomic interaction) remains the same.
4. Experimental Consistency: The findings align with experimental observations (e.g., in liquid $^4$He) where the structure factor and dispersion laws change continuously across $T_\lambda$, suggesting no fundamental symmetry-breaking phase transition in the strict quantum-mechanical sense for finite systems.

Conclusion

Maksim Tomchenko concludes that the answer to the title question is "No." In real-world (finite) superfluid Bose gases, phonons are not Goldstone bosons because the $U(1)$ symmetry is not spontaneously broken; the ground state remains invariant (up to a phase) under $U(1)$ transformations. The perception of SSB and Goldstone bosons is an artifact of approximations (c-number substitution) and the thermodynamic limit, which introduces particle-number indeterminacy.