# Collective excitations: localized or delocalized?

When condensed matter physicists talk about collective excitations in a crystal such as phonons, magnons etc, do they picture these excitations as necessarily delocalized in the crystal?

If they do not i.e., if they can be thought of as local excitations, will the name "collective excitations" be justified and make sense?

EDIT: If they are delocalised, why don't the sound waves travel instantaneously?

• Since they emerge from the periodicity of the crystal, they are 'delocalized' almost by definition. – Jon Custer Nov 21 '16 at 15:58
• The title of the question is misleading. Please change it. Collective excitations, as defined in Wikipedia : en.wikipedia.org/wiki/Quasiparticle do not need to be delocalised. It is true that a simplifying hypothesis is to expand these modes in plane waves (which are intrinsically delocalised) in infinite and periodic systems, but there is nothing which forbids to calculate the space dependency of the density of quasi-particles in a solid. Examples are edge states, vortex, ... or any kind of topological excitation, which are intrinsically quasi-particles. – FraSchelle Nov 22 '16 at 10:31
• As I said, there is nothing which forbids to calculate the space dependency of the density of quasi-particles in a solid. Phonon is one example of collective excitation which is delocalised (in the sense you gave, or even simply because it is well described by plane waves in a particle-field description). In fact, most of the bosons quasi-particles are delocalised (though some are not). Some fermions quasi-particles are delocalised, most are localised (at least in principle, though the spatial dependency is sometimes complicated to extract from the model). – FraSchelle Nov 22 '16 at 11:50
• Explicitly, my previous answer was exactly : once you constructed a quasi-particle scheme with creation operator $a^{\dagger}$, what forbids you to calculate something like $\left\langle x\right|a^{\dagger}a\left|x\right\rangle$ and call it a density of quasi-particle in the real space ? Then your question is : is this object localised (e.g. $\left\langle x_{1}\right|a^{\dagger}a\left|x_{2}\right\rangle \sim e^{-\left|x_{1}-x_{2}\right|}$) or not ? The answer is : it depends on the $a$'s... Nothing forbids localisation, nothing forces delocalisation. – FraSchelle Nov 22 '16 at 11:51
• On a disordered lattice (like glass), there are many Anderson localized and collective phonon modes. On a translation invariant lattice (crystal), phonons arise as Goldstone modes, which can not be localized by definition. The velocity of a delocalized mode is determined by $v=\partial\epsilon_k/\partial k$, which can not be infinity. – Everett You Nov 22 '16 at 14:26

Collective excitations can be either localized or delocalized. For example, in a conductor the low-energy excitations are particles or holes near a Fermi surface, which are delocalized. But in an insulator, the excitations are often localized.

Several commentators have claimed that Bloch's theorem requires all collective excitations to be delocalized in a translationally invariant system. But this is incorrect; Bloch's theorem only applies to noninteracting systems. The Coulomb repulsion can result in an interaction-driven Mott insulator, whose low-energy excitations are usually very localized.

Localized excitations can still be legitimately thought of as "collective," because they're usually not perfectly localized to a single site in the microscopic lattice - they typically have an exponentially-decaying tail in space. E.g. this can arise from screening effects. These localized quasiparticles are not identical to the microscopic particles: they get "dressed" by their interactions with nearby particles - but not with those far away. These interactions can renormalize their mass, Lande g-factor, etc., as in Fermi liquid theory. Thus the presence of nearby microscopic particles is still important, even if the excitations are localized.

Regarding the finite speed of sound: for a lattice system whose interactions decay exponentially (e.g. because of screening), there are Lieb-Robinson bounds that mathematically limit the speed at which information and entanglement can spread. This is true regardless of whether the excitations are localized or delocalized.

Take an electron or hole in a semiconductor crystal. If an electron is in an "energy eigenstate", it is delocalized across the whole crystal. (See Bloch wave). But you can also have an electron in a very localized state, like a Wannier function state. These are related: A Wannier function state is a quantum superposition of different Bloch wave states, and a Bloch wave state is a quantum superposition of different Wannier function states.

Phonons are the same. In a perfect crystal, the "standing wave" or energy eigenstates are delocalized across the whole crystal, but you can take a superposition of these states to get a localized distortion that travels at the speed of sound.

In reality, phonons, electrons, and other particles / quasiparticles / collective excitation may be anywhere from very localized to very delocalized. It depends on the processes that create it and influence it.