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-09-06921

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, 10267 (2000) Ivanov, Phys. Rev. 86, 268 (2001) Stern, von Oppen & Mariani, Phys. Rev. B 70, 205338 (2004) Chung & Stone, J. Phys. A 40, 4923 (2007) Nayak, et al. Rev. Mod. Phys. 80, 1083 (2008) Read, Phys. Rev. B 79, 045308 (2009)

≡ 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 1 3 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

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

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 ′ 1 Σ ′ 1 , 𝜂 ′ 2 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”)

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!]

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 =0 (“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 2 . (see below)

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 2 1 2 ↑ 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 1+ 𝑐 𝑘 2 −1 2 Normalization: with constraint 𝑘 𝑐 𝑘 2 1+ 𝑐 𝑘 2 = 𝑁 2 *see e.g. AJL, Quantum Liquids section 5.4

𝑐 𝑞 ∗ 𝑐 𝑞′ 1+ 𝑐 𝑞 2 1+ 𝑐 𝑞′ 2 ≡ 𝐹 𝑞 ∗ 𝐹 𝑞′ Two vital quantities: Recall 𝑁! −1 𝑘 1+ 𝑐 𝑘 2 − 1 2 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 ′ 1+ 𝑐 𝑘 2 − 1 2 𝑘 ′ 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + 𝑁 2 |𝑣𝑎𝑐 Ψ 𝑁 = 1 1+ 𝑐 2 2 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 1+ 𝑐 𝑞 2 1 1+ 𝑐 𝑞 2 Hence (a) (b) 𝑛 𝑞↑ = 𝑛 −𝑞↓ = 𝑐 𝑞 2 1+ 𝑐 𝑞 2 𝑎 𝑞↑ + 𝑎 −𝑞↓ + 𝑎 −𝑞′↓ 𝑎 𝑞′↑ = 𝑐 𝑞 ∗ 𝑐 𝑞′ 1+ 𝑐 𝑞 2 1+ 𝑐 𝑞′ 2 ≡ 𝐹 𝑞 ∗ 𝐹 𝑞′ , 𝐹 𝑞 ≡ 𝑐 𝑞 1+ 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.

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…

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 2 . 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′ 2 𝑟′ 2 𝜓 𝜎′ 1 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).

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 1 2 ? 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 2 . 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

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′

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 2 1+ 𝑐 𝑘 2 𝐹 𝑘 = 𝑐 𝑘 1+ 𝑐 𝑘 2 ( , Convenient to note: 1−4 𝐹 𝑘 2 −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 2 + 𝑘′ 𝑉 𝑘𝑘′ 𝐹 𝑘′ =0 This is just BCS gap equation in disguise! (put 𝐸 𝑘 ≡ 𝐸 𝑘 / 1−4 𝐹 𝑘 2 , 𝐹 𝑘 ≡ Δ 𝑘 /2 𝐸 𝑘 ) Note: NO USE OF “SPONTANEOUSLY BROKEN U(1) SYMMETRY”! * See e.g. Q𝐿 section 5.4

≡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 1+ 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | 𝑣𝑎𝑐 with 𝒩 𝑘 = 1+ 𝑐 𝑘 2 −1/2 If we write 𝑐 𝑘 ≡ v 𝑘 / 𝑢 𝑘 with 𝑢 𝑘 2 + v 𝑘 2 =1, this becomes Ψ BCS = 𝑘 𝑢 𝑘 + v 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | 𝑣𝑎𝑐 ≡ 𝑘 𝑢 𝑘 00 𝑘 + v 𝑘 11 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

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!

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.

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.]