1. Cooper Pairing in “Exotic” Fermi Superfluids: An Alternative Approach
Lecture 1
Cooper Pairing in “Exotic” Fermi Superfluids: An Alternative Approach
Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign
based largely on joint work with Yiruo Lin
supported in part by the National Science Foundation under grand no. DMR

2. The established wisdom*:
BCS theory (including “spontaneously broken U(1) symmetry”) 
“topological superconductors” (e.g. 𝑆𝑟 2 𝑅𝑢𝑂 4 , 3 𝐻𝑒−𝐴 )
Majorana fermions: (nonabelian statistics)
(Ising) topological quantum computation (and other exotica)
The $64K question:
Is the established wisdom correct?
* E.g. Read & Green, Phys. Rev. B 61, (2000)
Ivanov, Phys. Rev. 86, 268 (2001)
Stern, von Oppen & Mariani, Phys. Rev. B 70, (2004)
Chung & Stone, J. Phys. A 40, 4923 (2007)
Nayak, et al. Rev. Mod. Phys. 80, 1083 (2008)
Read, Phys. Rev. B 79, (2009)

3. ≡ 𝑟 𝑖𝑛𝑡 𝑉(𝑟) 𝜒(𝑟) 𝑟 ~ 𝑎 𝑠 Basic description of Cooper pairing 𝑇=0
Illustrative example: 𝑁(=even) spin −1 2 fermions, mass m, in volume , interacting via isotropic attractive potential 𝑉 𝑟 of range 𝑟 𝑜 ≪ Ω 𝑁 but adjustable strength. (example: ultracold Fermi gas, Feshbach resonance). Effect of potential encapsulated in s-wave scattering length 𝑎 𝑠 .
≡ 𝑟 𝑖𝑛𝑡
1. “BEC limit” (strong attraction)
2 fermions form simple s-wave diatomic molecule, with radius ≲ 𝑎 𝑠 >0 ≪ 𝑟 𝑖𝑛𝑡 , binding energy ~ ℏ 2 𝑚𝑎 𝑠 2 , 𝑟𝑙 𝑣𝑐 . w.f. 𝜒 𝑟 Since 𝑎 𝑠 ≪ 𝑟 𝑖𝑛𝑡 , molecules do not overlap ⇒ can be regarded as 𝑁 2 simple bosons, with (com) coords 𝑹 𝑖 , small residual interaction and fixed relative w.f. 𝜒 𝜌 𝑖
𝑉(𝑟) 𝑟
~ 𝑎 𝑠
𝜒(𝑟)
relative coord

4. Zeroth approximation: Ψ 𝑁 𝑹 𝑖 , 𝝆 𝑖 =𝑛 𝑖=1 𝑁 2 𝜑 𝑜 𝑹 𝑖 𝜁 𝝆 𝑖
Ψ 𝑁 𝑹 𝑖 , 𝝆 𝑖 =𝑛 𝑖=1 𝑁 2 𝜑 𝑜 𝑹 𝑖 𝜁 𝝆 𝑖
can usually forget
simple Bose condensation (BEC)
Slightly better approximation:
still treat as structureless bosons, (i.e. still ignore 𝜁 𝝆 𝑖 ) but allow for interactions:
Ψ 𝑁 𝑹 𝑖 ≠ 𝑖 𝜑 𝑜 𝑅 𝑖
Generalized concept of BEC (Yang):
d.f. 𝜌 1 𝑹,𝑹′
≡𝑁 Ψ 𝑁 ∗ 𝑅 1 = 𝑅 1 ,𝑅 2 … 𝑅 𝑁 2 Ψ 𝑁 𝑅 1 = 𝑅 ′ , 𝑅 2 … 𝑅 𝑁/2
𝑑 𝑹 2 …𝑑 𝑹 𝑁/2 ≡ 𝜓 † 𝑹 𝜓 𝑹′ 𝑜
single-particle density matrix
𝑖 𝑛 𝑖 =𝑁
𝜌 1 𝑹,𝑹′ = 𝑖 𝑛 𝑖 𝜉 𝑖 ∗ 𝑹 𝜉 𝑖 𝑹′
If one and only one of 𝑛 𝑖 is 𝑂 𝑁 , not 𝑜(1), define for this value of 𝑖
𝑛 𝑖 ≡ 𝑁 𝑜 ≡ condensate no (<𝑁 in general)
𝜉 𝑖 𝑅 ≡Ψ 𝑹 ≡ condensate w.f. 𝑂.𝑃.≡ 𝑁 𝑜 Ψ 𝑜 𝑅

5. 2. Now weaken interaction:
𝛼 𝑠 increases, eventually becomes ~ 𝑟 𝑖𝑛𝑡 .
now can no longer treat as 𝑁 2 structureless bosons! (can no longer ignore 𝜁 𝝆 𝑖 , nor more importantly underlying Fermi statistics) ⟹ must formulate in terms of fermion coordinates 𝒓 𝑖 𝑖=1,2…𝑁 and spins 𝜎 𝑖 , i.e.
Ψ N = Ψ N 𝒓 1 𝒓 2 … 𝒓 𝑁 : 𝜎 1 𝜎 2 … 𝜎 𝑁
But no reason to think “BEC “ goes away!
Generalized definition of (“pseudo -”) BEC (Yang):
define 𝜌 2 𝜂 1 Σ 1 , 𝜂 2 Σ 2 : 𝜂 ′ Σ ′ 1 , 𝜂 ′ Σ ′ 2
≡𝑁 𝑁−1 𝜎 3 … 𝜎 𝑁 𝑑 𝑟 3 …𝑑 𝑟 𝑁 ,
Ψ 𝑁 ∗ 𝑟 1 = 𝜂 1 , 𝜎 1 = Σ 1 , 𝑟 2 = 𝜂 2 , 𝜎 2 = Σ 2 : 𝑟 3 𝜎 3 … 𝑟 𝑁 𝜎 𝑁 ×
Ψ 𝑁 𝑟 1 = 𝜂′ 1 , 𝜎 1 = Σ′ 1 , 𝑟 2 = 𝜂′ 2 , 𝜎 2 = Σ′ 2 : 𝑟 3 𝜎 3 … 𝑟 𝑁 𝜎 𝑁
(~ “best description of behavior of 2 particles averaged over that of N-2 others”)

6. From now on: 𝜂 1 → 𝑟 1 , Σ 1 → 𝜎 1 , etc., so
𝜌 2 ≡ 𝜌 2 𝑟 1 𝜎 1 , 𝑟 2 𝜎 2 , : 𝑟′ 1 𝜎′ 1 , 𝑟1 2 𝜎′ 2
≡ 𝜓 𝜎 1 † ( 𝑟 1 ) 𝜓 𝜎 2 † 𝑟 2 𝜓 𝜎′ 2 𝑟′ 2 𝜓 𝜎′ 1 𝑟′ 1 𝑜
Since 𝜌 2 is Hermitian, can expand:
𝜌 2 𝑟 1 𝜎 1 , 𝑟 2 𝜎 2 : 𝑟 1 ′ 𝜎 1 ′ , 𝑟 2 ′ 𝜎 2 ′ = 𝑖 𝑛 𝑖 𝜒 𝑖 ∗ 𝑟 1 𝜎 1 , 𝑟 2 𝜎 2 𝜒 𝑖 𝑟 1 ′ 𝜎 1 ′ , 𝑟 2 ′ 𝜎 2 ′
𝑖 𝑛 𝑖 =𝑁 𝑁−1
Max. of any single eigenvalue 𝑛 𝑖 is 𝑂 𝑁 (not 𝑂 𝑁 𝑁−1 (Yang)
If one and only one 𝑛 𝑖 is 𝑂 𝑁 , rest 𝑂 1 , define for that value of 𝑖,
𝑛 𝑖 ≡ Ν 𝑜 ≡condensate number
𝜒 𝑖 𝑟 1 𝜎 1 , 𝑟 2 𝜎 2 : ≡ 𝜒 𝑜 𝑟 1 𝜎 1 , 𝑟 2 𝜎 2 ≡condensate wave function
(One possible) definition of OP:
F 𝒓 1 𝜎 1 , 𝒓 2 𝜎 2 ≡ N 𝑜 𝜒 𝒓 1 𝜎 1 , 𝒓 2 𝜎 2
←2 particle quantity!
[Note: at this point, still have 𝑎 𝑠 >0!]

7. 1. 𝜇=0 (“strong pairing  weak pairing”)
3. Decrease attraction (increase 𝑎 𝑠 ) further: two qualitatively significant points:
1. 𝜇=0 (“strong pairing  weak pairing”)
2. 𝑎 𝑠 −1 = (“unitarity”)
onset of (pseudo-) BEC N S T
BEC
𝜇=0
−𝑎 𝑠 −1
BCS
Strong 
weak
unitarity
In s-wave case, no qualitative change at either 1 or 2.
Finally, let 𝑎 𝑠 →0 𝑎 𝑠 <0 ⟹ Fermi system with very weak attraction. (BCS problem)
In this limit, N-particle GS believed to be special case of generalized pairing ansatz
all same!
Ψ 𝑁 𝑝𝑎𝑖𝑟 =𝔑𝒜 𝜑 𝒓 1 𝜎 1 , 𝒓 2 𝜎 2 𝜑 𝒓 3 𝜎 3 , 𝒓 4 𝜎 4 ….. 𝜑 𝒓 𝑁−1 𝜎 𝑁−1 , 𝒓 𝑁 𝜎 𝑁
normalizer
antisymmetrizer
formally identical to BEC of molecule!
Note: can still define the “condensate wave function” 𝜒 𝒓 1 𝜎 1 , 𝒓 2 𝜎 2 , but (unlike at the BEC end) it is not equal to 𝜑 𝒓 1 𝜎 1 , 𝒓 2 𝜎 (see below)

8. For illustration, specialize to case (BCS problem)
COM at rest ⇒ 𝜑 𝒓 1 𝜎 1 , 𝒓 2 𝜎 2 = 𝜑 𝒓 1 − 𝒓 2 , 𝜎 1 𝜎 2
Spin singlet ⇒ 𝜑 𝒓 −1 − 𝒓 −2 , 𝜎 1 𝜎 2 = 𝜑 𝒓 1 − 𝒓 ↑ 1 ↓ 2 − ↓ 1 ↑ 2
Isotropic ⇒ 𝜑 𝒓 1 − 𝒓 2 =𝜑 𝒓 1 − 𝒓 2
Then straightforward to show* that in 2nd – quantized notation
Ψ 𝑁 =𝔑 𝒌 𝑐 𝒌 𝑎 𝒌↑ + 𝑎 −𝒌↓ 𝑁 2 |𝑣𝑎𝑐> ,
𝑐 𝑘 =F.T.of 𝜑 𝒓 1 − 𝒓 2 =𝑓 𝒌
𝔑= 1 𝑁! 𝑘 𝑐 𝑘 −1 2
Normalization:
with constraint
𝑘 𝑐 𝑘 𝑐 𝑘 = 𝑁 2
*see e.g. AJL, Quantum Liquids section 5.4

9. 𝑐 𝑞 ∗ 𝑐 𝑞′ 1+ 𝑐 𝑞 2 1+ 𝑐 𝑞′ 2 ≡ 𝐹 𝑞 ∗ 𝐹 𝑞′
Two vital quantities:
Recall
𝑁! −1 𝑘 𝑐 𝑘 − 𝑘 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ 𝑁 2 |𝑣𝑎𝑐
Ψ 𝑁 𝑐 𝑘 ≡
Pick out a particular pair of states 𝒒↑, −𝒒↓ and define
𝑘 ′≡ 𝑘≠𝑞
𝑘 ′≡ 𝑘≠𝑞 , ,
Ψ′ 𝑁 ≡ 𝑁! −1 𝑘 ′ 𝑐 𝑘 − 𝑘 ′ 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ 𝑁 2 |𝑣𝑎𝑐
Ψ 𝑁 = 𝑐 Ψ′ 𝑁 + 𝑐 𝑞 𝑎 𝑞↑ + 𝑎 −𝑞↓ + Ψ′ 𝑁−1
Then we have
in words:
amplitude for pair in 𝑞↑,−𝑞↓ no pair in 𝑞↑,−𝑞↓ = 𝑐 𝑞 𝑐 𝑞 𝑐 𝑞 2
Hence
(a)
(b)
𝑛 𝑞↑ = 𝑛 −𝑞↓ = 𝑐 𝑞 𝑐 𝑞 2
𝑎 𝑞↑ + 𝑎 −𝑞↓ + 𝑎 −𝑞′↓ 𝑎 𝑞′↑ =
𝑐 𝑞 ∗ 𝑐 𝑞′ 𝑐 𝑞 𝑐 𝑞′ 2 ≡ 𝐹 𝑞 ∗ 𝐹 𝑞′ ,
𝐹 𝑞 ≡ 𝑐 𝑞 𝑐 𝑞 2
where
≡ “anomalous amplitude”
= Ψ 𝑁−1 𝑎 −𝑞↓ 𝑎 𝑞↑ Ψ 𝑁 ≤ 1 2
All the above is general, for any choice of 𝑐 𝑞 ’s.

10. 𝑐 𝑘 =Θ 𝑘 𝐹 −𝑘 𝑘< 𝑘 𝐹 𝑎 𝑘↑ + 𝑎 −𝑘↓ + 𝑁/2 ≡ 𝑘< 𝑘 𝐹 𝑎 𝑘↑ + 𝑎 −𝑘↓ +
Notes:
1. Normal GS is special choice, with
𝑐 𝑘 =Θ 𝑘 𝐹 −𝑘 𝑘< 𝑘 𝐹 𝑎 𝑘↑ + 𝑎 −𝑘↓ + 𝑁/2 ≡ 𝑘< 𝑘 𝐹 𝑎 𝑘↑ + 𝑎 −𝑘↓ +
2. Multiplication of all 𝑐 𝑘 ’s by the same phase factor 𝑒 𝑖𝜑 is equivalent to multiplying complete MBWF by exp 𝑖𝑁 𝜑/ 2 ⇒ no physical significance
3. The substitution 𝑐 𝑞 → −𝑐 𝑞 ∗−1 produces a paired state orthogonal to Ψ 𝑁 𝑐 𝑘 , with
𝑛 𝑞 →1− 𝑛 𝑞 , 𝐹 𝑞 →− 𝐹 𝑞
4. An alternative representation of Ψ 𝑁 𝑐 𝑘 : start from
|FS ≡ 𝑘< 𝑘 𝐹 𝑎 𝑘↑ + 𝑎 𝑘↓ + |𝑣𝑎𝑐
Ψ 𝑁 =
𝔑 𝑜 2𝑛 𝑑𝜑 exp 𝑒 𝑖𝜑 𝑘> 𝑘 𝐹 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓Λ + + 𝑒 𝑖𝜑| 𝑘> 𝑘 𝐹 𝑑 𝑘 𝑎 −𝑘↓ 𝑎 𝑘↑ |𝐹𝑆
For s-wave case (only!) this is just an alternative (equivalent) way of writing Ψ 𝑁 . But…

11. Relation to Yang’s ideas:
At first sight tempting to identify the “single macroscopic eigenvalue” of 2-particle d.m. 𝜌 2 as 𝑁 2 and the corresponding eigenfunction as 𝜑 𝑜 𝒓 1 − 𝒓 This is right in the BEC limit, but gets progressively worse as we cross over to the BCS limit because of effects of Pauli principle (need to antisymmetrize Ψ 𝑁 ). Rather, consider
𝜌 2 𝒓 1 𝜎 1 , 𝒓 2 𝜎 2 : 𝒓′ 1 𝜎′ 1 , 𝒓′ 2 𝜎′ 2 ≡
𝜓 𝜎 1 † 𝑟 1 𝜓 𝜎 2 † 𝑟 2 𝜓 𝜎′ 𝑟′ 2 𝜓 𝜎′ 𝑟 1
= 𝑖 𝑛 𝑖 𝜒 𝑖 ∗ 𝑟 1 𝜎 1 , 𝑟 2 𝜎 2 𝜒 𝑖 𝑟 1 ′ 𝜎 1 ′ : 𝑟 2 ′ 𝜎 2 ′
Most intuitive to take F.T.’s and rearrange:
F.T. = 𝑎 𝑘 + 𝑎 𝑙 + 𝑎 𝑚 𝑎 𝑛 𝑁,0 = 𝑎 𝑚 𝑎 𝑛 𝑎 𝑘 + 𝑎 𝑙 + 𝑁,0 +𝑜 𝑁 −1
Quite generally,
𝑎 𝑚 𝑎 𝑛 𝑎 𝑘 + 𝑎 𝑙 + 𝑁,0 =
𝑖 𝑁,0 𝑎 𝑚 𝑎 𝑛 𝑁+2,𝑖 | 𝑁+2,𝑖 𝑎 𝑘 + 𝑎 𝑙 + 𝑁,0
(𝑖 any complete orthonormal set of 𝑁+2−particle wave functions).

12. Can we find any 𝑁+2−particle state 𝑖 and any combination
So question is:
Can we find any 𝑁+2−particle state 𝑖 and any combination
Ω † ≡ 𝑘𝑙 𝑐 𝑘𝑙 𝑎 𝑘 + 𝑎 𝑙 + s.t. 𝑁+2,𝑖 Ω 𝑁,0 =0 𝑁 ?
For “normal” state |𝑁,0 (e.g. Fermi sea) this is not possible. But for |𝑁,0 a Cooper-paired state we can choose
𝑐 𝑘𝑙 = 𝛿 𝑘,−ℓ 𝑖.𝑒 Ω † = 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + ,
𝑖=0 (i.e. 𝑁+2−particle GS)
and then since
𝑁+2, 0 𝑎 𝑘 + 𝑎 −𝑘 + 𝑁,0 ≡ 𝐹 𝑘
𝑁+2, 0 Ω † 𝑁,0 ≡ 𝑘 𝐹 𝑘
Thus the “condensate wave function” 𝜒 𝑜 𝒓 1 𝜎 1 𝒓 2 𝜎 2 is just the F.T. 𝑤.𝑟.𝑡. 𝒓 1 − 𝒓 2 of 𝐹 𝑘 , and the corresponding eigenvalue
𝑁 𝑜 is 𝑘 𝐹 𝑘 In the BCS limit when 𝐹 𝑘 = Δ 𝑘 2 𝐸 𝑘 , this
quantity is 𝑂 Δ∙𝑁 𝑂 ~𝑁 ∆ 𝐸 𝐹 , i.e.
in BCS limit, condensate fraction ~∆ 𝐸 𝐹
For the purposes of evaluating pairing contribution to any 2–particle quantity (e.g. V), F.T. of 𝐹 𝑘 , 𝐹 𝑟 plays exactly role of 2–particle wave function
e.g.
𝑉 = 𝑉 𝑟 𝜓 𝑟 2 𝑑𝑟⇒ 𝑉 = 𝑉 𝑟 𝐹 𝑟 2 𝑑𝑟
2 – particle wave function
pair wave function

13. Which choice of 𝑐 𝑘 makes Ψ 𝑁 𝑐 𝑘 the groundstate of the N—particle system?
Must minimize 𝐻 ≡ 𝑇 + 𝑉 (note since N fixed, no −𝜇𝑁)
kinetic en.
potential en.
𝑇 ≡ 𝑘𝜎 𝜉 𝑘 𝑛 𝑘𝜎 =2 𝑘 𝑐 𝑘 2 / 1+ 𝑐 𝑘 2
ℏ 2 𝑘 2 2𝑚
𝑘𝑞𝜎 𝑉 𝑞 𝑎 𝑘+𝑞/2,𝜎 + 𝑎 𝑘−𝑞/2𝜎 𝑎 𝑘 ′ −𝑞/2,𝜎′ + 𝑎 𝑘 ′ +𝑞/2,𝜎′
What about 𝑉 ≡ ?
In any completely paired state Ψ 𝑁 𝑐 𝑘 , only 3 types of nonzero term:
(a) Hartree 𝑞=0 : 𝑉 𝐻 = 𝑁 2 𝑉 𝑜 ≠𝑓 𝑐 𝑘 ⇒ can neglect in minimization.
(b) Fock: 𝑘 ′ =𝑘−𝑞 𝑉 𝐹 =− 𝑘𝑞 𝑉 𝑞 𝑛 𝑘+𝑞/2,𝜎 𝑛 𝑘−𝑞/2,𝜎 in principle affects BCS gap equation, but under most conditions changes little in 𝑁→𝑆 transaction, so usually neglected.
(c) Pairing (BCS) 𝑘=− 𝑘 ′ :
𝑉 𝐵𝐶𝑆 = 𝑘𝑘′ 𝑉 𝑘−𝑘′ 𝑎 𝑘↑ + 𝑎 −𝑘↓ + 𝑎 −𝑘′↓ 𝑎 𝑘 ′ ↑ = 𝑘𝑘′ 𝑉 𝑘−𝑘′ 𝐹 𝑘 ∗ 𝐹 𝑘′

14. 𝐻 ′ = 𝑇 + 𝑉 𝐵𝐶𝑆 =2 𝑘 𝜉 𝑘 𝑛 𝑘 + 𝑘𝑘′ 𝑉 𝑘−𝑘′ 𝐹 𝑘 ∗ 𝐹 𝑘′
Thus, minimize
𝐻 ′ = 𝑇 + 𝑉 𝐵𝐶𝑆 =2 𝑘 𝜉 𝑘 𝑛 𝑘 + 𝑘𝑘′ 𝑉 𝑘−𝑘′ 𝐹 𝑘 ∗ 𝐹 𝑘′
with
𝑛 𝑘 = 𝑐 𝑘 𝑐 𝑘 2
𝐹 𝑘 = 𝑐 𝑘 𝑐 𝑘 2 ( ,
Convenient to note:
1−4 𝐹 𝑘 −1 2 =2 𝑛 𝑘 − 𝑛 𝑘 𝑁 2
𝑛 𝑘 𝑁 =𝜃 𝑘− 𝑘 𝐹
and to subtract a constant term − 𝐸 𝐹 𝑁=−𝜇𝑁 from 𝐻 ′, so, 𝑇 𝑒𝑓𝑓 =2 𝑘 𝜖 𝑘 𝑛 𝑘 , 𝜖 𝑘 ≡ 𝜉 𝑘 −𝜇. Then minimization yields* a Schrödinger-like equation for 𝐹 𝑘
2 𝐸 𝑘 𝐹 𝑘 1−4 𝐹 𝑘 𝑘′ 𝑉 𝑘𝑘′ 𝐹 𝑘′ =0
This is just BCS gap equation in disguise! (put 𝐸 𝑘 ≡ 𝐸 𝑘 / 1−4 𝐹 𝑘 , 𝐹 𝑘 ≡ Δ 𝑘 /2 𝐸 𝑘 )
Note: NO USE OF “SPONTANEOUSLY BROKEN U(1) SYMMETRY”!
* See e.g. Q𝐿 section 5.4

15. ≡Ψ BCS 𝑢 𝑘′ v 𝑘 v 𝑘 → v 𝑘 exp 𝑖𝜑 Ψ 𝑁 = 1 2𝜋 0 2𝜋 𝑑𝜑 Ψ BCS 𝜑 exp 𝑖𝑁𝜑/2
Relation of “particle-conserving” (PC) approach to BCS one:
PC:
BCS:
Ψ 𝑁 =𝒩 𝑘 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + 𝑁/2 | 𝑣𝑎𝑐
Ψ 𝑁 →𝒩exp 𝑘 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | 𝑣𝑎𝑐 ≡𝒩 𝑘 exp 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | 𝑣𝑎𝑐
⇒ (Pauli principle)
𝑛 𝑘 1+ 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | 𝑣𝑎𝑐 ⇒ 𝑘 𝒩 𝑘 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | 𝑣𝑎𝑐
with 𝒩 𝑘 = 𝑐 𝑘 −1/2
If we write 𝑐 𝑘 ≡ v 𝑘 / 𝑢 𝑘 with 𝑢 𝑘 v 𝑘 2 =1, this becomes
Ψ BCS = 𝑘 𝑢 𝑘 + v 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | 𝑣𝑎𝑐 ≡ 𝑘 𝑢 𝑘 𝑘 + v 𝑘 𝑘
BCS form.
≡Ψ BCS 𝑢 𝑘′ v 𝑘
From Ψ 𝐵𝐶𝑆 𝑢 𝑘′ 𝑣 𝑘 we can recover Ψ 𝑁 by “Anderson trick”:
v 𝑘 → v 𝑘 exp 𝑖𝜑
Ψ 𝑁 = 1 2𝜋 0 2𝜋 𝑑𝜑 Ψ BCS 𝜑 exp 𝑖𝑁𝜑/2

16. BCS maneuver is equivalent to
Ψ Ν → 𝑁 𝑐 𝑁 Ψ 𝑁
, i.e. “spontaneous breaking of U(1)
gauge symmetry”
Is this ever valid as a description of physical state? NO!!
Superselection rule for particle number prohibits it!
Digression: What if system has leads?
Then indeed 𝑁 𝑠 is not conserved, but 𝑁 𝑠 + 𝑁 𝑖 is, = 𝑁 𝑡𝑜𝑡 (say), so 𝑆 𝐿
Ψ=Ψ 𝑁 𝑠 , 𝑁 𝐿 = 𝑁 𝑆 𝐶 𝑁 𝑆 Ψ 𝑆 𝑁 𝑠 Ψ 𝐿 𝑁 𝑡𝑜𝑡 − 𝑁 𝑠
so reduced density matrix of 𝑆 (obtained by tracing over 𝑁 𝐿 ) still diagonal in 𝑁 𝑠 −representation:
𝜌 𝑁 𝑠 , 𝑁 𝑠 ′ ~𝑓 𝑁 𝑠 𝛿 𝑁 𝑠 , 𝑁 𝑠 ′
⇒ “spontaneous breaking of U(1) symmetry” IS NOT PHYSICAL!

17. Final note: BCS ansatz for CS is inconsistent!
Take a neutral Fermi system and consider the quantity
𝑆 𝑞 ≡ 𝜌 𝑞 𝜌 −𝑞
𝑞≠0
𝜌 𝑞 ≡ 𝑘𝜎 𝑎 𝑘+𝑞/2,𝜎 + 𝑎 𝑘−𝑞/2,𝜎
Assuming compressibility of system 𝑁/𝑚 𝑐 2 is not infinite, 𝑓−sum rule + compressibility sum rule (KK) + Cauchy-Schwarz ⇒
𝑆 𝑞 ≤𝑁𝑞/𝑚𝑐 𝑞→𝑜
For a free Fermi gas, 𝑆 𝑞 has only the “Fock” contribution
𝑆 𝐹 𝑞 = 𝑘𝑟 𝑛 𝑘+𝑞/2,𝜎 1− 𝑛 𝑘−𝑞/ 𝑒 2 ′𝜎 = 3𝑁 2𝑚 𝑣 𝐹 𝑞  FS
𝒌−𝒒/2
v 𝐹 𝑞cos𝜃
𝒌+𝒒/2
and since 𝑐= v 𝐹 / 3 , inequality is satisfied.

18. But for BCS groundstate there is also a pairing term:
So for 𝑞≲𝑚𝑐 Δ/ 𝐸 𝐹 ~ 𝜉 −1 , inequality is violated!
Solution: must build into GSWF zero-point density fluctuations! (i.e. zero-point AB modes)
Anderson-Bogoliubov
Intuitively: BCS GS⇒𝜑=constant. But this then implies huge fluctuations in condensate no. density ⇒ huge repulsion energies. ZP AB modes “smooth out” density!
[For a charged system (metal), problem is “hidden” because it already occurs in the N phase and is taken into account by involving ZP plasmons.]