1. “fixed” object (phonon)
REMINDERS ABOUT BCS MODEL:
free electrons, attraction mediated by exchange of virtual phonons: condensation energy from scattering of pairs
k k
–k
–k 
Result:
BCS wave function
N-particle projection:
Alternative (equivalent) description (Yang): df. 2-particle density matrix: at T=0,
“fixed” object (phonon)
“2-particle state averaged over behavior of all the other particles”
(at nonzero T, take thermal average of over many-body states n.)

2. ____________________________________
CHARACTERISTICS OF BCS STATE:
consider 2-particle DM
In N phase, all ni are 0(1). (General) BCS-paired phase defined by: one (and only one) of the ni is O(N). (nb: also if ni and i time-dependent)
: how do we know? — flux quantization, Josephson effect ….
In BCS-paired phase, if No~O(N), all other ni ~1.
____________________________________
In original BCS theory of superconductivity:
eigenvalue eigenfunction
COM relative spin orbital singlet s-wave
For r fixed (e.g. r = 0),
“macroscopic wave function”
For (e.g.) “internal wave function of Cooper pair”

3. STRUCTURE OF COOPER-PAIR WAVE FUNCTION (in original BCS theory of superconductivity)
Energy gap
“Number of Cooper pairs” (No) = normn of F(r) aB ~
“Number of Cooper pairs” (No) = normn of F(r)
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In original BCS theory of superconductivity,
spin singlet orbital s-wave
PAIRS HAVE NO “ORIENTATIONAL” DEGREES OF FREEDOM
(Stability of supercurrents, etc.)

4. ______________________________
THE FIRST ANISOTROPIC COOPER-PAIRED SYSTEM: SUPERFLUID 3HE
solid
2-PARTICLE DENSITY MATRIX 2 still has one and only one ( !) macroscopic eigenvalue  can still define “pair wave function” F(R,r:12 ) However, even when , A  A B N
T mK
HAS ORIENTATIONAL DEGREES OF FREEDOM!
(i.e. depends nontrivially on )
All three superfluid phases have
______________________________
A phase (“ABM”)
Spin triplet
char. “spin axis”
char. “orbital axis”
Properties anisotropic in orbital and spin space separately,
e.g.
: WHAT IS TOTAL ANG. MOMENTUM?

5.  finite-frequency resonance!
B phase (“BW”)
For any particular direction (in real or k-space) can always choose spin axis s.t.
i.e Alternative description:
BW phase is 3Po state “spin-orbit rotated” by 104o L=S=J=O because of dipole force cos-1(-1/4)=o
Note: rotation (around axis ) breaks P but not T
Orbital and spin behavior individually isotropic, but: properties involving spin-orbit correlations anisotropic!
Example: NMR dipole energy
 of rotation about rf
Field direction
In transverse resonance, rotation around equiv. rotation of with o unchanged
No dipole torque.
In longitudinal resonance, rotation changes o
 finite-frequency resonance!

6. “EXOTIC” PROPERTIES OF SUPERFLUID 3HE (G. E. Volovik)
Orientation const. in space, varying in time: — spin dynamics (NMR) — orbital dynamics (“normal locking”) (A phase) — effect of macroscopic ang. momentum? (A phase) ____________________
Orientation const. in time, varying in space — spin textures (3He-A) ( ) in equation               (carries spin current) — orbital textures — topological singularities (boojums, “half-quantum” vortices ) — instability of supercurrents in 3He-A __________________
Orientation varying in both space and time — spin waves — orbital waves —”flapping” and “clapping” modes
Amplification of ultra-weak effects

9. OTHER BCS-LIKE SYSTEMS WITH
heavy-fermion superconductors (UPt3, UBc13, UNiAl3….) probably anisotropic (but not T-violating)
Sr2 Ru O4
— strongly 2-dimensional (like cuprates) — pairing state probably ABM-like (i.e. px + ipy) — if so, violates T (Kidwiringa et al.)
3He in aerogel (“dirty” system, ) — 2 superfluid phases, “A-like” and “B-like” — but, exptl. characteristics very puzzling (esp. NMR)
ultracold fermions close to p-wave Feshbach resonance — can study “BEC-BCS crossover” — is there a phase transition e.g. at =O? — $64K question: how does total ang. momentum vary in crossover?

10. CUPRATE SUPERCONDUCTORS
Cu O
typical structure:  
6–15Å Ca
“charge reservoir”
 top view of CuO2 phase
side view
____________________
How do we know the cuprates are BCS-paired? Flux quantization (in units of h/2e) ( : only ab-phase!)
Josephson effect
___________________
What do we know about the Cooper pairs? — live in CuO2 planes — spin singlet (Knight shift, T1)  orbital parity even — size prob. ~ Å (Hc, Tc) (?)  much further away from “BCS end” of BCS-BEC crossover then classic suprs
$64K question (early 90’s):
What is orbital symmetry of pair wave function F(r)?

11. ORBITAL SYMMETRY OF PAIR W.F. OF CUPRATES
Symmetry of CuO2 planes (approx) that of square  relevant symmetry group is C4v. Spin singlet  even parity
No second phase transition  only one irreducible representation (irrep)
Even-parity irreps of C4v:
Name +1 -1 S + + + + + + – – – – + +
“Orientation” tied to xtal axis! So, how do we tell? (flux quantization in “twisted” ring)
Actual Josephson expts:
YBCO
“corner SQUID” 
“tricrystal” Nb

12. Anyway: BCS PAIRING RULES OK!
Obvious question about superconductivity in cuprates: is mechanism of pair formation BCS-like, i.e. via exchange of virtual phonons?
Arguments against phonon mechanism: — v. anomolous N-state properties (esp. (T)) — folk theorems on Tc (McMillan, Dynes …) — absence of isotope effect in higher-Tc cuprates
And yet … — substantial phonon effects in ARPES (Lanzara et al.) — possible way around isotope-effect argument? (Newns, Tsuei)
More general question: is mechanism of generalized BCS type (virtual-boson-exchange) at all?
k
-k
k33
k44 k
-k
k11
k22
Interaction individually of pairing interaction modified by pairing (cf. 3He)
Anyway: BCS PAIRING RULES OK!