A translation or spatial rotation symmetry breaking leads to phonons. What is the analogue symmetry for plasmons? Should it be translation invariance also?
I wonder if Golstein bosons should read here gondstone boson. pl. clarify.
Sorry for my spelling mistake. It should read as "Goldstone Bosons" and not "Golstein Bosons".
In particle physics these kind of bosons are common: W-, Z-bosons, when gauge or vacuum symmentry is spontaneousely broken.
Similar things do occur in solid systems with high translation symmetry. When the symmetry is broken phonons appear as a result. These phonons are kind of Goldstone Bosons!
Now my question, what symmetry breaking mechanism is responsible for emergence of plasmons?
Should it be some kind of translation?
Plasmons are not associated with any (continuous) symmetry breaking. They correspond to collective excitations of a charged liquid (say, an assembly of electrons) in an oppositely-charged background. The energies {e_i | i} of small-amplitude plasmon excitations are solutions of the equation ε(q,e) = 0, where ε is the dielectric function and q the wave vector. Since ε(q,e) = 1 - v(q) P(q,e), where v(q) is the Fourier transform of the two-body potential (say, the Coulomb potential in realistic models) and P(q,e) the polarization function, the plasmon energy dispersion follows from the equation v(q) P(q,e) = 1 (the 1 both here and in the definition of ε is the contribution of the vacuum). Now since for small values of ||q|| (the long-wave-length, or the hydrodynamic, region) P(q,e) is proportional to some positive power of ||q||, unless v(q) diverges like the inverse of the same power of ||q||, v(q) P(q,e) = 1 cannot be satisfied (no plasmon excitations). In d dimensions, with d >1, where P(q,e) vanishes like ||q||^(d-1) and the Fourier transform of the Coulomb potential v(q) diverges like 1/||q||^(d-1) for ||q|| approaching zero, one expects plasmon excitations to exist in this region, that is that the equation v(q) P(q,e) = 1 to have a solution for small values of ||q||. That this is indeed the case follows from the fact that for small values of ||q|| the function P(q,e) has a wide range of variation along the e axis so that for small values of ||q|| there must be some e > 0 at which v(q) P(q,e) = 1 is indeed satisfied (since P(q,e) is an even function of e, e = e0 satisfying the equation v(q) P(q,e) = 1, this equation is also satisfied at e = -e0).
Incidentally, the polarization function P(q,e) consists of the contributions arising from both the charged particles and the oppositely-charged background (to leading order, these contributions are additive). Often the background contribution to P(q,e), on account of the background being less polarizable than the charged liquid, is neglected. In conventional superconductors where superconductivity arises from the electron-phonon interaction, this background contribution plays a vital role, in changing the sign of the effective electron-electron interaction for q in the neighbourhood of the underlying Fermi surface and thus bringing about the Cooper pairing of the electrons.
Thank you very much for the enlightenment. It looks like this explanation for plasmonics is within Random Phase Approximation (RPA) or other classical method, which utilizes polarization of the fluctuating charged system. This is, in fact, very real.
However, my point is, I want see if there can be another way of explaining the same physical phenomenon which may be due to a broken symmetry within Gauge Theory. As I have indicated in the first post, I want to believe that this collective charge fluctuations/oscillations can be related to certain symmetry breaking mechanism, especially in crystalline systems. I am of the conviction the kind of symmetry is SU(2)-Rotation symmetry which could be broken as a consequence! Though I am not quite sure! How?
Thank you once more for your clearance.
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