*Introduction to Many Body Physics*(free online). It is a nice clear and helpful discussion.

One of the key ideas first emphasized by Phil Anderson in 1963 was that a massless gauge field can aquire a mass in the presence of a coupling to a spontaneously broken field. A concrete realisation of this occurs in superconductors. In the Meissner effect a superconductor thicker than the penetration depth expels magnetic fields. This is like the photon acquires a mass.

In the electro-weak theory of Weinberg-Salam there is a combined U(1) x SU(2) gauge symmetry. Due to coupling to the Higgs field (whose symmetry is spontaneously broken)

one gauge field remains massless (the photon) and the other three become massive. These massive particles are the W+, W-, and Z bosons.

In a type II superconductor, vortices are allowed in the superconducting order parameter field.

**Can such vortices occur in the Higgs field?**They may have been important in the early universe.

On fascinating thing I learnt is that for the Higgs field the crucial ratio [between the London penetration length and the superconducting coherence length] that determines whether type II behaviour is possible is the ratio of Higgs boson mass to W mass. The

**LHC results suggest that type II behaviour is possible!**

In summary, here is an extract from Coleman's book (page 246).

Shortly after the importance of this mechanism for relativistic Yang Mills theories was noted by Higgs and Anderson, Weinberg and Salem independently applied the idea to develop the theory of “electro-weak” interactions. According to this picture, the universe we live is a kind of cosmological Meissner phase, formed in the early universe, which excludes the weak force by making the vector bosons which carry it, become massive. It is a remarkable thought that the very same mechanism that causes superconductors to levitate lies at the heart of the weak nuclear force responsible for nuclear fusion inside stars. In trying to discover the Higg’s particle, physicists are in effect trying to probe the cosmic superconductor above its gap energy scale.Aside: Later Coleman discusses how (in a slave boson formulation) "the Anderson-Higgs effect in the Kondo problem endows the composite f−electron with charge."

The other half of the Higgs story is the way it gives fermions mass by coupling a left-handed fermion to a right-handed fermion. This goes back to Nambu 1960.

ReplyDeleteI think, one key difference in superconductors is that photons play the role of Higgs (correct me, I might be wrong). And a photon becomes massive itself, which implies that the magnetic field becomes heavy to enter the superconducting sample.

ReplyDeleteI agree with Mitchell too. Nambu started with a theory for a BCS superconductor and ended up with a theory for the pions (NLJ theory).

Photons do NOT play the role of the Higgs. The superconducting order parameter (the symmetry breaking field).

ReplyDeleteYou are correct that the photon becomes massive.

Sorry. I probably said without thinking much. Actually Higgs mechanism doesn't require a Higgs field (Just needs SSB+local gauge invariance). Here in case of superconductors, originally the quantum field is a massive complex scalar field. And if we don't impose the local gauge invariance (LGI) principle, SSB makes one component of the scalar field massless, which is known as the Goldstone mode. However, in presence of photons, which have U(1) LGI, the Goldstone modes are "eaten" by the photons (gauge boson) and hence the photons becomes massive. This is, in a phrase, called the Higgs mechanism. If we go to higher gauge symmetry, we get an extra massive scalar field, which is the so-called Higgs field.

ReplyDeleteHope this time I'm correct.