The association of broken symmetries with phase transitions is ubiquitous in condensed matter physics: crystals break translational symmetry, magnets break rotational symmetry, and superconductors break gauge symmetry. However, despite the frequency with which it is made, this last statement is a paradox. A gauge symmetry, in this case the U(1) gauge symmetry of electromagnetism, is a redundancy in our description of nature, so the notion of breaking such a “symmetry” is unphysical. Here, we will discuss how gauge symmetry breaks, and doesn't, inside a superconductor, and explore the fundamental relationship between gauge invariance and the striking phenomena observed in superconductors.

1.
S.
Weinberg
,
The Quantum Theory of Fields
, 1st ed. (
Cambridge U.P.
,
Cambridge, UK
,
1995
), Vol. II, pp.
295
358
.
2.
Steven
Weinberg
, “
Superconductivity for particular theorists
,”
Prog. Theor. Phys. Suppl.
86
(1),
43
53
(
1986
).
3.
John
David Jackson
,
Classical Electrodynamics
, 2nd ed. (
John Wiley & Sons
,
New York, NY
,
1975
), pp.
547
556
.
4.

This is because the Lagrangian has a term qA·ẋ coupling the particle to the field. The canonical momentum p=L/ẋ then has an additional term ∼ qA which must be subtracted out to get the “physical” momentum mẋ.

5.
L. D.
Landau
and
E. M.
Lifshitz
,
Quantum Mechanics: Non-Relativistic Theory
(
Pergamon Press
,
London, UK
,
1958
), pp.
471
474
.
6.

Since α(x, t) is dependent on space and time, derivatives acting on the gauge transformed wave-functions give (eiqαψ)=eiqαψ+iq(α)eiqαψ, and similarly for time derivatives.

7.
Lewis H.
Ryder
,
Quantum Field Theory
, 2nd ed. (
Cambridge U.P.
,
Cambridge, UK
,
1996
), pp.
282
293
.
8.
Mehran
Kardar
,
Statistical Physics of Fields
, 1st ed. (
Cambridge U.P.
,
Cambridge, UK
,
2007
), pp.
19
31
.
9.
Alexander
Altland
and
Ben
Simons
,
Condensed Matter Field Theory
, 2nd ed. (
Cambridge U.P.
,
Cambridge, UK
,
2010
), pp.
43
50
.
10.

Since for any system of definite particle number the energy eigenstates will be simultaneous eigenstates of the number operator, there is no way the system can time evolve into a state with different particle number. Put another way, since the number operator commutes with the Hamiltonian, particle number must be conserved.

11.
James F.
Annett
,
Superconductivity, Superfluids and Condensates
, 1st ed. (
Oxford U.P.
,
New York, NY
,
2004
), pp.
106
108
.
12.

Even though the pairs in the condensate are comprised of electrons, we treat them as different objects. To be more precise, the electrons we speak of are quasiparticles, not bare electrons.

13.
J.
Bardeen
,
L. N.
Cooper
, and
J. R.
Schrieffer
, “
Theory of superconductivity
,”
Phys. Rev.
108
,
1175
1204
(
1957
).
14.
Piers
Coleman
,
Introduction to Many-Body Physics
, 1st ed. (
Cambridge U.P.
,
Cambridge, UK
,
2015
), pp.
372
374
.
15.
Reference 14, pp.
505
560
.
16.
Klaus
Dietrich
,
Hans J.
Mang
, and
Jean H.
Pradal
, “
Conservation of particle number in the nuclear pairing model
,”
Phys. Rev.
135
,
B22
B34
(
1964
).
17.
Yukihisa
Nogami
, “
Improved superconductivity approximation for the pairing interaction in nuclei
,”
Phys. Rev.
134
,
B313
B321
(
1964
).
18.
L. D.
Landau
, “
On the theory of superconductivity
,” in
Collected Papers of L.D. Landau
, edited by
D.
Ter Haar
(
Gordon and Breach
,
New York, NY
,
1965
), pp.
546
568
.
19.

Expanding the free energy, which is known to be discontinuous at a phase transition, as an analytic series near the discontinuity can only be justified if we understand the free energy as the result of a saddle point approximation of the appropriate partition function, see Ref. 8.

20.

Details of the Ginzburg-Landau theory, which we have glossed over, ensure that in the superconducting state r < 0, so the gap magnitude Δ†Δ = –r/u is a positive quantity, see Ref. 14.

21.
Reference 9, pp.
281
316
.
22.
R.
Shankar
,
Quantum Field Theory and Condensed Matter
, 1st ed. (
Cambridge U.P.
,
Cambridge, UK
,
2017
), pp.
52
104
.
23.
Henrik
Bruus
and
Karsten
Flensberg
,
Introduction to Many-Body Quantum Theory in Condensed Matter Physics
, 1st ed. (
Oxford U.P.
,
Oxford, UK
,
2004
), pp.
65
85
.
24.
Yoichiro
Nambu
, “
Quasi-particles and gauge invariance in the theory of superconductivity
,”
Phys. Rev.
117
,
648
663
(
1960
).
25.
Matthew D.
Schwartz
,
Quantum Field Theory and the Standard Model
(
Cambridge U.P.
,
Cambridge, UK
,
2014
), pp.
49
65
.
26.
Valery
Rubakov
,
Classical Theory of Gauge Fields
, 1st ed. (
Princeton U.P.
,
Princeton, NJ
,
2002
), pp.
3
8
.
27.

Once could conceivably ask why the quadratic function must be centered about zero, i.e., why could we not have something like FK[(μϕ2eAμ)2j2]2? The effective action would then be minimized when jμ=μϕ2eAμ, which we know from Eq. (55) represents current flow.

28.
S.
Elitzur
, “
Impossibility of spontaneously breaking local symmetries
,”
Phys. Rev. D
12
,
3978
3982
(
1975
).
29.
P. W.
Anderson
, “
Plasmons, gauge invariance, and mass
,”
Phys. Rev.
130
,
439
442
(
1963
).
30.

The Anderson-Higgs mechanism also generates the mass of quarks and electrons due to a Yukawa coupling that has no analogue in superconductors, and thus is not discussed here.

31.
M. A.
Measson
 et al, “
Amplitude Higgs mode in the 2H-NbSe2 superconductor
,”
Phys. Rev. B
89
,
060503
(
2014
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.