Cooper L.N., Feldman D. (Eds.) BCS: 50 Years

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BCS from Nuclei and Neutron Stars to Quark Matter and Cold Atoms 521

7. Concluding Remarks

To close on a personal note, I would like to dedicate this paper to the memory

of John Bardeen, who was a warm colleague and caring mentor of me as a

young faculty member in Urbana. It was a source of great pleasure to John

to see how BCS expanded our understanding of modern physics beyond

laboratory superconductors — from nuclear physics to astrophysics to high

energy physics, a shaping that continues to this day, from cold atomic physics

to quark matter.

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October 4, 2010 11:13 World Scientiﬁc Review Volume - 9.75in x 6.5in ch21

ENERGY GAP, MASS GAP, AND SPONTANEOUS

SYMMETRY BREAKING

Yoichiro Nambu

Enrico Fermi Institute, University of Chicago,

5640 South Ellis Ave., Chicago, IL 60637, USA

This article is based on a talk given at a Symposium at the University of

Illinois on the occasion to commemorate the 50th anniversary of BCS — I

gave a historical overview of how BCS theory has come to be transplanted

to particle physics and has helped solve its problems.

In the early 1950s while I was at the Institute for Advanced Study, Professor

Pines invited me to give a seminar at the University of Illinois as I was

interested in the plasma theory of Bohm and Pines. That was before BCS,

and before I joined the theorists Wentzel and Goldberger at the University

of Chicago.

Now to provide a short story about my background, I studied physics at

the University of Tokyo, graduating at the MS level according to the system

of those days. I was attracted to particle physics because of three famous

names, Y. Nishina, S. Tomonaga and H. Yukawa, who were the founders of

particle physics in Japan. In fact, since this year and last there have been

consecutive centennial celebrations of Tomonaga and Yukawa. But these

people were at diﬀerent institutions than mine. On the other hand, con-

densed matter physics was pretty good at Tokyo. R. Kubo, well known for

his work in statistical mechanics, was my contemporary. I got into particle

physics only when I came back to Tokyo as a kind of research associate after

the war. Tomonaga had just started to develop the renormalization theory

at his university nearby, and some of my room mates were working with him,

as he also had served as a visiting professor at Tokyo after my graduation.

The remarkable events of the Lamb shift and the pion-muon decay chain

soon occurred, too. In hindsight, though, I must say that my early exposure

to condensed matter physics has been quite beneﬁcial to me.

525

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526 Y. Nambu

I will now brieﬂy review the history of particle physics to the extent that

is necessary for the main theme of my talk. Particle physics is an outgrowth

of nuclear physics which began in the early 1930s with the discovery of the

neutron by J. Chadwick, the invention of the cyclotron by E. Lawrence,

and the “invention” of meson theory by H. Yukawa. The appearance of an

ever increasing array of new particles in the subsequent decades, and the

advances in quantum ﬁeld theory gradually led to our understanding of the

basic laws of nature, culminating in the present Standard Model of parti-

cle physics. We have come to know that the elementary particles, or the

fundamental fermions, consist of two kinds, leptons (so named by L. Rosen-

feld) and quarks (so named by M. Gell-Mann). They are characterized by

the internal quantum numbers, ﬂavor and color. The forces or interactions

among them (except for gravity) are described by the various gauge ﬁelds.

A set of two leptons and six quarks make up a so-called generation, and

there exist three generations, so far, with increasingly heavier masses. The

electromagnetic and weak forces are due to the ﬂavor gauge ﬁelds and are

shared by all the fermions. The strong color forces are unique to the quarks,

and lead to formation of permanent quark composites called hadrons (named

by L. B. Okun), i.e. baryons and mesons.

On the theorists’ side, faced with those new particles, their ﬁrst attempts

were to make sense out of them by ﬁnding some regularities in their prop-

erties. They invoked symmetry principles to classify them. A symmetry

in physics leads to a conservation law. Some conservation laws are ex-

act, like energy and electric charge, but these classiﬁcation attempts were

mostly based on approximate similarities of masses and interactions, but not

exact symmetries.

Nevertheless, seeing similarities is a natural trait of the human mind, and

in fact it has proved very useful. The near equality of proton and neutron

masses and their interactions led to the concept of isospin SU(2) symmetry

of N. Kemmer, Wentzel’s former student. One could argue, or hope, that the

diﬀerences were attributed to the electromagnetic interactions which had a

diﬀerent symmetry property from the strong interaction of the meson theory.

When strange hadrons showed up, the search for an enlarged symmetry led

to the SU(3) symmetry of M. Gell-Mann and Y. Ne’eman that covered all

hadrons known up to the 1960s, although its origin was, and still is, not

clear.

On the other hand, one could also go in the opposite direction, and elevate

the symmetry to a general dynamical principle. Then symmetry will deter-

mine all physics, a most attractive possibility. Thus the beautiful properties

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Energy Gap, Mass Gap, and Spontaneous Symmetry Breaking 527

of electromagnetism was extended to the SU(2) non-Abelian gauge ﬁeld by

C. N. Yang and R. L. Mills

as an ideal model for the strong interaction.

But this is a huge oversimpliﬁcation. Strong interactions are short range,

and the mesons are all massive. Giving a mass to a gauge ﬁeld destroys

gauge invariance. Nevertheless, this gauge ﬁeld analogy arose in the 1950s

in two diﬀerent contexts.

1. The existence of vector mesons ρ (770) (isospin 1) and ω (782) (isospin

0). Indeed the ρ (770) can give attractive forces to hadrons, and seemed

to have a universal coupling strength (g

/4π

) to isospin current, which

led J. J. Sakurai,

my colleague, to promote a vector dominance theory

of strong interactions.

2. The V −A property of the weak interaction as was found by R. Feynman

and M. Gell-Mann,

and by R. Marshak and G. Sudarshan,

which also

seemed almost universal for those particles known at that time. This

raised the possibility that the weak processes might also be mediated by

a gauge ﬁeld. The V −A structure and apparent massless neutrinos also

directed our attention to the chirality, or the γ

charge, as a possible

symmetry which, however, showed up in subtle ways, in spite of the fact

that it is violated by the masses of the fermions.

So the problem of mass somehow seemed to be a sticking point in our

search for a dynamical symmetry principle for both strong and weak inter-

actions. It is at this point that analogies learned from other branches of

physics have turned out to be very fruitful. I will follow this based on my

own experiences in the early 50s here and in Japan. There are three topics

in this connection.

1. The Debye screening of the Coulomb interaction turns the 1/r potential

into a Yukawa form exp(−µr)/r. In ionized media the screening gives

rise to the plasma oscillations, ﬁrst noted by I. Langmuir and Tonks.

It is a mode where the dispersion relation ω versus k has a mass (in

particle physics language). I was intrigued when D. Bohm, E. P. Gross,

and D. Pines

gave a systematic description of this phenomenon, and

generalized it to transverse modes. I tried hard, though unsuccessfully, to

apply it to the problems of nuclear physics, like the saturation of nuclear

forces and the large spin-orbit eﬀects.

2. Tomonaga wrote a paper

on a description of collective modes in one-

dimentsonal fermionic systems. Stimulated by this, I found that, for

bosonic media in general, one could apply Dirac’s canonical transforma-

tion from the creation and annihilation operators to number and phase

October 4, 2010 11:13 World Scientiﬁc Review Volume - 9.75in x 6.5in ch21

528 Y. Nambu

operators, and convert a Hamiltonian for an interacting medium of the

type

H =

|∇ψ|

ρV ρ

, ψ =

√

ρe

iθ

into an approximate eﬀective form quadratic in ρ and θ

H ∼



|∇θ|

4ρ

|∇ρ|



ρV ρ

One immediately obtains the eigenvalues,

ω(k)

∼



V (k) +

4mρ



For the Coulomb case V = e

, we get the plasma “mass”

= e

/m .

3. Then there is the more important and mysterious London theory of super-

conductivity, based on the ansatz j = ΛA. The magnetic ﬁeld B acquires

the “mass” Λ(= ω

) with an appropriate deﬁnition of ρ

and m. This

was to become the problem to be understood. I learned, among others,

of a work by M. R. Schafroth,

another former student of Wentzel’s, on

the Meissner eﬀect of a Bose gas. All these examples gave a hint that

the mass of gauge ﬁelds in particle physics might also be resolved if one

considered the vacuum as a kind of medium. Indeed Dirac had already

invoked this analogy in his interpretation of the negative energy solution

of his equation.

Now I come to the BCS theory.

I ﬁrst heard about it when our boss

Wentzel invited Mr. Schrieﬀer for a seminar before the work was published.

I understood that he was still a graduate student. Subsequently he would

join our faculty, and I greatly beneﬁted from our interaction. I remember

the excitement I felt at the seminar, but at the same time I became wor-

ried about the fact that their Hartree–Fock state was a superposition of

diﬀerent number of charges and did not appear to respect gauge invariance.

Soon thereafter N. N. Bogoliubov

and J. G. Valatin

independently in-

troduced the concept of quasiparticles as fermionic excitations in the BCS

medium. The quasiparticles did not carry a deﬁnite charge as they were a

superposition of electron and hole, with their proportion depending on the

momentum.

How can one then trust the BCS theory for discussing the electro-

magnetic properties like the Meissner eﬀect? As this doubt gradually grew

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Energy Gap, Mass Gap, and Spontaneous Symmetry Breaking 529

more serious in my mind, I decided to attack it, but it actually took two

years before I was able to do so to my satisfaction and publish the result.

the mean time, the problem was addressed by many people,

but I wanted

to understand it in my own way. Essentially it is the presence of a mass-

less collective mode, now known by the generic name of Nambu–Goldstone

(NG) boson, that saves charge conservation or gauge invariance. To describe

the Bogoliubov–Valatin (BV) quasiparticle, I introduced a two-component

notation, which is similar to the Gor’kov formalism.

In this space, the

Hamiltonian for the quasiparticle is given as

H = ψ

†

(τ

ε(k) + τ

∆)ψ

(ε: kinetic energy relative to the Fermi energy; ∆: gap) The charge is di-

agonal (∼ τ

) in this space, but the BV state is not. The collective mode

is an excitation of the BV pairs ∼ ψ

†

ψ. It is not the eigenstate of charge

either, but its interaction with the quasiparticle, i.e. its emission and absorp-

tion, insures that charge is conserved overall when a quasiparticle undergoes

acceleration. In other words, the charge-current operator j

is made up of

contributions from quasiparticle as well as the collective cloud around it; j

is not diagonal in the energy basis, but divergence-free. In addition to the

massless NG mode, I found another collective mode in the direction of the

gap ∼ τ

, with mass 2∆. It is a quasiparticle composite with zero binding

energy, which may be thought of being similar to the so-called sigma meson

and Higgs boson in its relation to the NG mode. I also found that, when the

Coulomb interaction between quasiparticles is included, it mixes strongly

with the massless collective mode because of their common massless nature,

and they turn into the massive plasmon mode. The correct expression for

the charge and current is,

tot

= ρ

∂φ

∂t

, v = v

Fermi

√

tot

= j

− ∇φ ,

where φ is the NG mode, so that

∂ρ

tot

/∂t + ∇ · J

tot

∂

− ∇

φ = 0

The eﬀect of the Coulomb interaction is to screen this longitudinal part of

charge-current, but leaves the London term for the transverse part.

Thinking of general characterizations of the BCS mechanism, the idea

of spontaneous symmetry breaking (SSB), with the associated massless col-

lective modes arose in my mind. (The word SSB was later introduced by

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530 Y. Nambu

M. Baker and S. Glashow.

) The phenomenon is due to the fact that the

lowest energy state of the system is inﬁnitely degenerate, and a Hilbert space

of the quasi-particle excitations is built on just one of the vacua. Hilbert

spaces built from diﬀerent vacua are equivalent but are orthogonal to each

other in the sense that matrix elements of local observables between them are

zero because of the inﬁnite degrees of freedom of the system. They are like

diﬀerent phases of a medium at transition temperature. The NG collective

modes correspond to local ﬂuctuations into other vacua, so the excitation

energy goes to zero in the long wave length limit.

The broken symmetry under question in the BCS case was that of electric

charge conservation, but it immediately reminded me of analogous cases like

ferromagnetism and crystal formation. They correspond to a global break-

ing of symmetry, in these cases that of spin rotation and space of translation

and rotation, respectively, and the corresponding collective modes are the

magnon and the phonons (longitudinal and tranverse). I thought of col-

lecting further examples systematically before writing a general paper on

this problem. I had used the name zeron for the NG boson. Incidentally

I also realized that this SSB mechanism was what Heisenberg had invoked

in his ill-fated nonlinear uniﬁed ﬁeld theory

to generate more degrees of

freedom than are in the original Lagrangian, although this would amount to

simultaneous existence of diﬀerent phases, or worlds.

The emergence of the “zeron” and its subsequent transﬁguration into a

massive gauge ﬁeld was elucidated by the works of a number of people,

starting with J. Goldstone. They made use of a model theory with a scalar

ﬁeld with nonlinear terms, which eventually led to the electroweak uniﬁcation

of Glashow, Salam, and Weinberg and the current Standard Model of particle

physics. In hindsight I regret that I should have explored in more detail the

general mechanism of mass generation for the gauge ﬁeld. But I thought

the plasma and the Meissner eﬀect had already established it. I also should

have paid more attention to the Ginzburg–Landau theory

which was a

forerunner of the present Higgs description. I had pushed it out of my

mind because I had not understood the meaning of their order parameter

ﬁeld. (In writing this, I recall what Wentzel once remarked to me about the

Schroedinger equation: “What a bold step to put a Coulomb potential into

the de Broglie wave!”)

I was actually more intrigued by the similarity of the BV 2-component

fermion formalism and the Dirac equation. Only a spinor of left-handed or

right-handed chirality is suﬃcient to describe a massless fermion. The two

components of opposite charges in a BV fermion get mixed if a energy gap