Yoichuro Nambu (1921-2015): spontaneously broken symmetry in particle physics

Yoichuro Nambu died earlier this month, and there was an obituary in the New York Times yesterday. He shared the Nobel Prize in Physics in 2008, and is best known for this paper

Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I 
 Y. Nambu and G. Jona-Lasinio

I reproduce the abstract below because it really does summarise the work and is a nice example of a beautifully written abstract.

It is suggested that the nucleon mass arises largely as a self-energy of some primary fermion field through the same mechanism as the appearance of energy gap in the theory of superconductivity. The idea can be put into a mathematical formulation utilizing a generalized Hartree-Fock approximation which regards real nucleons as quasi-particle excitations. We consider a simplified model of nonlinear four-fermion interaction which allows a γ5-gauge group. An interesting consequence of the symmetry is that there arise automatically pseudoscalar zero-mass bound states of nucleon-antinucleon pair which may be regarded as an idealized pion. In addition, massive bound states of nucleon number zero and two are predicted in a simple approximation.
 The theory contains two parameters which can be explicitly related to observed nucleon mass and the pion-nucleon coupling constant. Some paradoxical aspects of the theory in connection with the γ5 transformation are discussed in detail.

I offer a few minor contextual comments, in order of decreasing significance.

1. Nambu's work is a very nice example of the cross-fertilisation between solid state physics and elementary particle physics. Before Nambu's paper it went mostly one way: solid state theorists used field theoretical techniques. However, Nambu showed how significant new insights in particle physics could be obtained from solid state analogues.

2. Before Nambu there was a lot of concern about the fact that BCS theory was not gauge invariant. He clarified this to the point that these objections were considered dealt with. However, I still get confused about this because of subtle issues about the Goldstone boson [associated with the broken U(1) gauge symmetry of electromagnetism] being "renormalised" by the Coulomb interaction leading to gapped plasmons. Even today there is still debate about whether there is a spontaneously broken symmetry or whether superconductors are topologically ordered, as advocated here.

3. One elegant and technical aspect of this paper was that he introduced the Nambu matrices for describing superconductivity. These and the associated Lie algebras naturally generalise to more complicated situations such field theories and superfluid 3He where the order parameter has 3 spin and 3 orbital degrees of freedom. I found this approach incredibly useful when I did my Ph.D thesis on order parameter collective modes in superfluid 3He-B. Some of this is described here.

4. Was Nambu at the right place at the right time?
 In a previous post, Born for success in quantum many-body theory, I noted how more than half of the founders of the application of field theory techniques to solid state physics were born between 1923 and 1926. Nambu was born in 1921.