I'm confused by a discussion in Ashcroft and Mermin's textbook on pg. 512-513. They say that if we have a bunch of ions in a solid and neglect the effect of the conduction electrons, then waves will propagate at the plasma frequency, which contradicts the idea that acoustic phonon energies should vanish as $\mathbf{k} \rightarrow \mathbf{0}$. They then say that the conduction electrons are essential to explain why the dispersion relation in metals is linear.

On the other hand, I've heard that the linearity of the dispersion relation is due to Goldstone's theorem, which relies on the translational invariance of the Hamiltonian and seems more general than Ashcroft and Mermin's argument. Is Ashcroft and Mermin's argument correct?

edit: Here's a quote of relevant passages from Ashcroft and Mermin, on pg. 512-513 in ch. 26:

[...] a set of charged point ions should undergo long-wavelength vibrations at an ionic plasma frequency $\Omega_{p}$[...]

This contradicts the conclusion in Chapter 22 that the long-wavelength normal-mode frequencies of a monatomic Bravais lattice should vanish linearly with $k$. That result is inapplicable because the approximation (22.64) leading to the linear form for $\omega (\mathbf{k})$ at small $k$ is only valid if the forces between ions separated by $R$ are negligibly small for $R$ of order $1/k$. But the inverse square force falls off so slowly with distance that no matter how small $k$ is, interactions of ions separated by $R \geq 1/k$ can contribute substantially to the dynamical matrix (22.59). [They mention that experimental evidence, however, supports a linear dispersion relation.]

To understand why the phonon dispersion is linear at small $k$ it is essential, when considering ionic motion, to take the conduction electrons into account.

[They then discuss how screening results in an effective short-range interaction.] [...] yielding an effective ionic field that is short-ranged, and therefore capable of leading to a phonon dispersion relation that is linear in $k$ at long-wavelengths.


First of all, it is perfectly possible for goldstone modes to have non-linear dispersion relations (e.g. a quadratic $k^2$ dispersion), see more discussion here.

Additionally, acoustic phonons are not always linear in $k$, in fact graphene has an acoustic phonon that disperses quadratically as $k^2$ rather than linear in $k$. This is apparently a generic feature of the out-of-plane polarized acoustic phonons in quasi-2D layered materials.

It is important to point out that the other acoustic phonons have linear dispersions.

In the case of graphene, the quadratic dispersion of the ZA mode seems to come from the rotational and reflection symmetry of the system, more discussion can be followed in this paper.

From "Ab Initio Study of the Electronic and Vibrational Properties of 1-nm-Diameter Single-Walled Nanotubes"


I think what they are trying to say is that in a lattice of ions, the waves propagate at the plasma frequency only if we can assume that the interaction between ions is sufficiently small (this assumption is contained in the part "is only valid if the forces between ions separated by R are negligibly small for R of order 1/k").

But in a bare lattice of ions, without electrons, this is not the case : ions interact with a Coulomb potential which decreases as 1/r, which is a long-range potential. However, if we add the conduction electrons, the Coulomb potential gets screened into a Yukawa potential of the form exp(-r)/r, which is short-ranged. Thus a metal has short-range interaction potential : we get a linear dispersion relation.


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