1. The Mean-Field Method in the Theory of Superconductivity:
A Wolf in Sheep’s Clothing?
Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign
joint work with Yiruo Lin support: NSF-DMR

2. Spontaneously broken U(1) symmetry: the weakly interacting Bose gas
Spontaneously broken U(1) symmetry: BCS theory of superconductivity
The Bogoliubov-de Gennes equations: s-wave case
The Bogoliubov-de Gennes equations: general case
Application to topological quantum computing in Sr2RuO4
What could be wrong with all this? A couple of (rather trivial) illustrations.
A simple but worrying thought-experiment.
Conclusion

3. Ψ 𝑜 𝑁 = 𝑎 𝑜 + 𝑁 | vac ≡ 𝑎 𝑜 + 𝑎 𝑜 + 𝑁/2 | vac
Spontaneously Broken U(1) Symmetry: The General Idea
1. Dilute Bose gas with weak repulsion (Bogoliubov, 1947)
𝐻 = 𝑘 𝜀 𝑘 𝑎 𝑘 + 𝑎 𝑘 + 𝑉
𝐻 , 𝑁 = 𝐻 , 𝑃 =0
Total particle
number
Total
momentum
ℏ 2 𝑘 2 /2𝑚
To zeroth order in 𝑉 , groundstate of N particles is that of free Bose gas, i.e. (apart from normalization)
Ψ 𝑜 𝑁 = 𝑎 𝑜 + 𝑁 | vac ≡ 𝑎 𝑜 + 𝑎 𝑜 + 𝑁/2 | vac
Most important scattering processes: V O k -k
+ inverse process , i.e. 𝑉 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + 𝑎 𝑜 𝑎 𝑜 +𝐻.𝑐.
𝑉 pair
The simplest ansatz for N-particle GSWF incorporating these processes: (apart from normalization)
Ψ 𝑜 (𝑁) = 𝑎 𝑜 + 𝑎 𝑜 + − 𝑘 𝑐 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + 𝑁/2 | vac
particle- conserving!

4. Bogoliubov: in “reduced” Hamiltonian
Coefficients 𝑐 𝑘 can be found by minimizing 𝐻 including 𝑉 pair terms*. However, calculations rather messy…
Bogoliubov: in “reduced” Hamiltonian
𝐻 𝑟𝑒𝑑 ≡ 𝑘 𝜀 𝑘 𝑎 𝑘 + 𝑎 𝑘 + 𝑘 𝑉 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + 𝑎 𝑜 𝑎 𝑜 +H.c.
treat quantity 𝑎 𝑜 𝑎 𝑜 (or actually 𝑎 𝑜 ) as c-number 𝑎 𝑜 → 𝑁 𝑜 so 𝑎 𝑜 𝑎 𝑜 → 𝑁 𝑜 ;
then
𝐻 𝐵𝑜𝑔 = 𝑘 𝜀 𝑘 𝑎 𝑘 + 𝑎 𝑘 + 𝑁 𝑜 𝑉 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + +H.c.
(but see below…)
*see e.g. AJL, Revs. Mod. Phys. 73, 307 (2001), section VIII.

5. 𝐻 𝐵𝑜𝑔 ′ ≡ 𝐻 𝐵𝑜𝑔 −𝜇 𝑁 (i.e. 𝜀 𝑘 →𝜀 𝑘 −𝜇)
What are the implications of treating 𝑎 𝑜 𝑎 𝑜 as a c-number?
Implies a nonzero expectation value
Ψ 𝑎 𝑜 𝑎 𝑜 Ψ ≅ 𝑁 𝑜 ≡ 𝑎 𝑜 + 𝑎 𝑜
This is not true for the (particle-conserving) wave function
Ψ 𝑜 𝑁 ≡ 𝑎 𝑜 + 𝑎 𝑜 + − 𝑘 𝑐 𝑘 𝑎 𝑘 + 𝑎 −𝑘 𝑁 2 | vac
𝑎 𝑜 𝑎 𝑜 ≡𝑂
but is true e.g. for
Ψ 𝑜 𝐵𝑜𝑔 ≡ exp 𝜆 𝑎 𝑜 + 𝑎 𝑜 + − 𝑘 𝑐 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + ≡ 𝑁=even 𝜆 𝑁 Ψ 𝑜 𝑁 𝑁!
(neglect details about normalization, etc.)
What fixes 𝑁 ~𝜆 ? If we just minimize 𝐻 𝐵𝑜𝑔 as it stands this automatically gives 𝜆~ 𝑁 ~0! Hence must add chemical potential term, i.e. minimize
𝐻 𝐵𝑜𝑔 ′ ≡ 𝐻 𝐵𝑜𝑔 −𝜇 𝑁 (i.e. 𝜀 𝑘 →𝜀 𝑘 −𝜇)
and adjust  so that 𝑁 𝜇 =𝑁  actual number of particles

6. “Spontaneously broken U(1) symmetry”
This procedure is standard in conventional statistical mechanics (macrocanonical  ground canonical), but then resultant state is a mixture of different N. Now by contrast we have a quantum superposition of states of different N.
Let’s define an operator 𝜑 which is canonically conjugate to 𝑁 , so that its eigenfunctions | 𝜑 are related to those of 𝑁 by
| 𝜑 = 𝑁 exp 𝑖𝑁𝜑 | 𝑁 . | 𝑁 = 𝑑𝜑 exp−𝑖𝑁𝜑 | 𝜑
Since original 𝐻 (not 𝐻 𝐵𝑜𝑔 !) satisfies 𝑁 𝐻 𝑁′ ~ 𝛿 𝑁𝑁′ , it is invariant under “rotation” of 𝜑 (“U(1)symmetry”) . However Ψ 𝑜 𝐵𝑜𝑔 picks out a definite value (0) of 𝜑:
“Spontaneously broken U(1) symmetry”

7. 𝐻 𝑀𝐹 ≡ 𝑘 𝜖 𝑘 −𝜇 𝑎 𝑘 + 𝑎 𝑘 + 𝑁 𝑜 𝑉 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + +H.c.
The Mean-Field (Bogoliubov) Method: Quasiparticles:
“Mean-field” Hamiltonian 𝐻 𝑀𝐹 ≡ 𝐻 𝐵𝑜𝑔 ′ , i.e.
𝐻 𝑀𝐹 ≡ 𝑘 𝜖 𝑘 −𝜇 𝑎 𝑘 + 𝑎 𝑘 + 𝑁 𝑜 𝑉 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + +H.c.
with standard Bose C.R.’s
𝑎 𝑘 , 𝑎 𝑘′ = 𝑎 𝑘 + 𝑎 𝑘′ + =0, 𝑎 𝑘 , 𝑎 𝑘′ + = 𝛿 𝑘𝑘′
This can be diagonalized by Bogoliubov transformation
𝛼 𝑘 + = 𝑢 𝑘 𝑎 𝑘 + + 𝑣 𝑘 𝑎 −𝑘 (etc.), 𝑢 𝑘 2 − 𝑣 𝑘 2 =1
which preserves Bose C.R.’s.:
𝐻 𝑀𝐹 ≡ 𝑘 𝐸 𝑘 𝛼 𝑘 + 𝛼 𝑘 +
const.
where (after  is fixed to give 𝑁 =𝑁 and 𝑉 𝑘 𝑠𝑒𝑡 ≅ 𝑉 𝑂 )
𝐸 𝑘 = 𝜀 𝑘 𝜀 𝑘 + 2𝑁𝑉 𝑂 
Bogoliubov spectrum

8. 𝛼 𝑘 + = 𝑢 𝑘 𝑎 𝑘 + + 𝑣 𝑘 𝑎 𝑜 + 𝑎 𝑜 + / 𝑁 𝑜 𝑎 −𝑘
Note: All these results, including formula for 𝐸 𝑘 , can be obtained by particle-conserving method based on ansatz Ψ 𝑂 𝑁 . However, in that method we have
𝛼 𝑘 + = 𝑢 𝑘 𝑎 𝑘 + + 𝑣 𝑘 𝑎 𝑜 + 𝑎 𝑜 + / 𝑁 𝑜 𝑎 −𝑘
particle- conserving!
𝑁→𝑁+1 +1 +2 -1
By contrast, in Bogoliubov method
𝑢 𝑘 𝑎 𝑘 + adds a particle 𝑁→𝑁+1
𝑣 𝑘 𝑎 −𝑘 subtracts a particle 𝑁→𝑁−1
So qp state is not an eigenstate of N. (but neither is groundstate…)
[Density fluctuation in number-conserving formalism:
𝜌 𝑘 = 𝛼 𝑘 + 𝑎 𝑜 = 𝑢 𝑘 𝑎 𝑘 + 𝑎 𝑜 + 𝑣 𝑘 𝑎 𝑜 + 𝑎 𝑜 + 𝑁 𝑜 𝑎 −𝑘 𝑎 𝑜
= 𝑢 𝑘 𝑎 𝑘 + 𝑎 𝑜 + 𝑣 𝑘 𝑎 −𝑘 𝑎 𝑜 +
[Generalization of Bogoliubov method to inhomogeneous situations]

9. 𝑉 𝑘𝑘′ 𝑎 𝑘′↑ + 𝑎 −𝑘′↓ 𝑎 −𝑘↓ 𝑎 𝑘↑ +H.c.
Spontaneously Broken U(1) Symmetry: The General Idea (cont.)
2. Superconductors (BCS 1957)
𝐻 = 𝑘𝜎 𝜀 𝑘 𝑎 𝑘𝜎 + 𝑎 𝑘𝜎 + 𝑉
𝐻 , 𝑁 = 𝐻 , 𝑃 =𝑂
Most important scattering process: V
𝑘′↑ 𝑘↑
−𝑘↓
−𝑘′↓
(+ inverse process),
i.e.
𝑉 𝑘𝑘′ 𝑎 𝑘′↑ + 𝑎 −𝑘′↓ 𝑎 −𝑘↓ 𝑎 𝑘↑ +H.c.
𝑉 pair
even
Simplest ansatz for N-particle GSWF incorporating these processes (apart from normalization)
Ψ 𝑂 𝑁 = 𝑘 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ 𝑁 2 | vac
particle- conserving!
with coefficients found by minimizing 𝐻 including 𝑉 pair terms*.
*See e.g. AJL, Quantum Liquids, section V.

10. Δ 𝑘 ≡ 𝑘′ 𝑉 𝑘𝑘′ 𝑎 −𝑘′↓ 𝑎 𝑘′↑ Again, calculations messy…
BCS: in “reduced” Hamiltonian
𝐻 𝑟𝑒𝑑 = 𝑘𝜎 𝜖 𝑘 𝑎 𝑘𝜎 + 𝑎 𝑘𝜎 + 𝑘𝑘′ 𝑉 𝑘𝑘′ 𝑎 𝑘↑ + 𝑎 −𝑘↓ + 𝑎 −𝑘′↓ 𝑎 𝑘′↑ +H.c.
treat the quantity
Δ 𝑘 ≡ 𝑘′ 𝑉 𝑘𝑘′ 𝑎 −𝑘′↓ 𝑎 𝑘′↑
as c-number. Then (adding chemical potential term as in Bose case)
𝐻 𝑀𝐹 = 𝑘𝜎 𝜖 𝑘 −𝜇 𝑎 𝑘𝜎 + 𝑎 𝑘𝜎 + 𝑘 Δ 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + +H.c.
Δ 𝑘 ≡ 𝑘′ 𝑉 𝑘𝑘′ 𝑎 −𝑘′↓ 𝑎 𝑘′↑
Again, this makes sense only if U(1) symmetry spontaneously broken, e.g.
Ψ 𝐵𝐶𝑆 =exp 𝑘 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ + | vac ≡ 𝑘 𝑢 𝑘 |𝑜𝑜 𝑘 + 𝑣 𝑘 |11 𝑘

11. The Mean-Field (BCS) Method: Quasiparticles
As above, mean-field Hamiltonian is
𝐻 𝑀𝐹 = 𝑘𝜎 𝜖 𝑘 −𝜇 𝑎 𝑘𝜎 + 𝑎 𝑘𝜎 + 𝑘 Δ 𝑘 𝑎 𝑘↑ 𝑎 −𝑘↓ + +H.c.
with standard Fermi ACR’s
𝑎 𝑘𝜎 , 𝑎 𝑘′𝜎′ = 𝑎 𝑘𝜎 + , 𝑎 𝑘′𝜎′ + =𝑂, 𝑎 𝑘𝜎 , 𝑎 𝑘′𝜎′ + = 𝛿 𝑘𝑘′ 𝛿 𝜎𝜎′
This can be diagonalized by Bogoliubov (-Valatin) transformation
𝛼 𝑘𝜎 + = 𝑢 𝑘 𝑎 𝑘𝜎 + +𝜎 𝑣 𝑘 𝑎 −𝑘−𝜎 (etc.), 𝑢 𝑘 𝑣 𝑘 2 =1
which preserves Fermi ACR’s:
𝐻 𝑀𝐹 = 𝑘𝜎 𝐸 𝑘 𝛼 𝑘𝜎 + 𝛼 𝑘𝜎
𝐸 𝑘 ≡ 𝜀 𝑘 Δ 𝑘

12. 𝛼 𝑘𝜎 + = 𝑢 𝑘 𝑎 𝑘𝜎 + + 𝜎𝑣 𝑘 𝑎 −𝑘,−𝜎 𝑐 †
Note: All these results can be obtained by particle-conserving method based on ansatz Ψ 𝑜 𝑁 . However, in that method we have
𝛼 𝑘𝜎 + = 𝑢 𝑘 𝑎 𝑘𝜎 + + 𝜎𝑣 𝑘 𝑎 −𝑘,−𝜎 𝑐 †
particle- conserving!
𝑁→𝑁+1 +1 -1 +2
where 𝐶 † is normalized Cooper-pair creation operator:
𝐶 † ≡𝑛∙ 𝑘 𝑐 𝑘 𝑎 𝑘↑ + 𝑎 −𝑘↓ +
normalization
By contrast, in BCS method
𝑢 𝑘 𝑎 𝑘𝜎 + adds an electron 𝑁→𝑁+1
𝜎𝑣 𝑘 𝑎 −𝑘,−𝜎 + subtracts an electron 𝑁→𝑁−1
so qp state is not an eigenstate of N (but neither is GS …)
[In recent literature on TI’s, etc., 𝐻 𝑀𝐹 often described as
“noninteracting-electron” Hamiltonian (!)]

13. The Mean-Field Method for Spatially Inhomogeneous Problems:
The Bogoliubov-De Gennes Equations
If the single-particle potential and/or the pairing interaction varies in space, the construction of a GSWF which generalizes Ψ 𝑂 𝑁 , while possible in principle*, is usually unfeasible in practice. However, the mean-field method is simply generalized (de Gennes 1964):
Consider for illustration the simple Hamiltonian
(obtained by setting 𝑉 𝒓−𝒓′ =𝑔𝛿 𝒓−𝒓′ and using 𝜓 𝜎 𝒓 2 ≡𝑂)
single-particle potential
𝑑𝒓 𝜎
ℏ 2 2𝑚 𝜵 𝜓 𝜎 † 𝒓 ⋅𝜵 𝜓 𝜎 𝑟 + ∪ 𝒓 −𝜇 𝜓 𝜎 † 𝑟 𝜓 𝜎 𝑟 ℏ 2 2𝑚 𝜵 𝜓 𝜎 † 𝒓 ⋅𝜵 𝜓 𝜎 𝑟 + ∪ 𝒓 −𝜇 𝜓 𝜎 † 𝑟 𝜓 𝜎 𝑟
𝐻 =
+𝑔 𝑑𝑟 𝜓 ↑ † 𝑟 𝜓 ↓ † 𝑟 𝜓 ↓ 𝑟 𝜓 ↑ 𝑟
The mean-field approximation: take the quantity
Δ 𝒓 ≡𝒈 𝜓 ↓ 𝒓 𝜓 ↑ 𝒓
to be a c-number. Thus (↑: caution re factors of 2 etc!)
*Stone & Chung, Phys. Rev. B 73, (2006)

14. 𝜓 𝜎 𝑟 , 𝜓 𝜎′ † 𝑟′ =𝛿 𝒓−𝒓′ 𝛿 𝜎𝜎′ (etc.)
𝐻 𝑀𝐹 = 𝐻 𝑂 +𝑔 𝑑𝑟 ∆ 𝑟 𝜓 ↑ † 𝑟 𝜓 ↓ 𝑟 +H.c.
𝑑𝒓 𝜎
ℏ 2 2𝑚 𝜵 𝜓 𝜎 † 𝑟 ⋅𝜵 𝜓 𝜎 𝑟 + ∪ 𝒓 −𝜇 𝜓 𝜎 † 𝑟 𝜓 𝜎 𝑟
𝐻 𝑜 ≡
with standard Fermi ACR’s
𝜓 𝜎 𝑟 , 𝜓 𝜎′ † 𝑟′ =𝛿 𝒓−𝒓′ 𝛿 𝜎𝜎′ (etc.)
Since 𝐻 𝑀𝐹 is a quadratic form in 𝜓 𝜎 † 𝑟 , 𝜓 𝜎 𝒓 , it can be diagonalized by introducing operators 𝛾 𝑖 + defined by
𝛾 𝑖𝜎 + = 𝑑𝒓 𝑢 𝑖 𝑟 𝜓 𝜎 † 𝑟 +𝜎 𝑣 𝑖 𝑟 𝜓 −𝜎 𝒓
particle- nonconserving!
not H.c’s
which will satisfy Fermi ACR’S provided the 𝑢 𝑖 𝑟 , 𝑣 𝑖 𝑟 satisfy
𝑑𝑟 𝑢 𝑖 𝑟 , 𝑢 𝑗 𝑟 + 𝑣 𝑖 𝑟 , 𝑣 𝑗 𝑟 = 𝛿 𝑖𝑗
a condition in fact guaranteed by the BdG equations (below). Thus 𝑖𝜎
𝐻 𝑀𝐹 =
𝐸 𝑖 𝛾 𝑖𝜎 + 𝛾 𝑖𝜎 + const.

15. 𝐻 𝑜 Δ 𝑟 Δ ∗ 𝑟 − 𝐻 𝑜 ∗ 𝑢 𝑖 𝑟 𝑣 𝑖 𝑟 = 𝐸 𝑖 𝑢 𝑖 𝒓 𝑣 𝑖 𝒓
𝑢 𝑖 𝑣 𝑖
The eigenfunctions and eigenvalues 𝐸 𝑖 of the mean-field Hamiltonian satisfy the Bogoliubov-de Gennes (BDG) equations:
𝐻 𝑜 Δ 𝑟 Δ ∗ 𝑟 − 𝐻 𝑜 ∗ 𝑢 𝑖 𝑟 𝑣 𝑖 𝑟 = 𝐸 𝑖 𝑢 𝑖 𝒓 𝑣 𝑖 𝒓
This guarantees
(a) orthonormality of spinors (c.f. above)
(b) if is a solution with energy 𝐸 𝑖 , then
is also a solution with energy −𝐸 𝑖 : hence now be
able to find solutions with 𝐸 𝑖 =0, provided
𝑢 𝑖 𝑟 = 𝑣 𝑖 ∗ 𝑟 (“pseudo-Majorana”)
𝑢 𝑖 𝑣 𝑖
𝑣 𝑖 ∗ 𝑟 𝑢 𝑖 ∗ 𝑟
𝑢 𝑖 𝑟 𝑣 𝑖 𝑟

16. Mean-field method in more general case:
𝜎𝜎′ 𝑑𝒓
𝛿 𝜎𝜎′ ℏ 2 2𝑚 𝜵 𝜓 𝜎 † 𝑟 ⋅𝜵 𝜓 𝜎′ 𝑟
𝐻 =
+ ∪ 𝝈𝝈′ 𝒓 −𝜇 𝛿 𝜎𝜎′ 𝜓 𝜎 † 𝒓 𝜓 𝜎′ 𝒓
𝛼𝛽𝛾 𝛿 𝑑𝒓𝑑𝒓′ 𝑉 𝛼𝛽𝛾𝛿 𝑟,𝑟′ 𝜓 𝛼 𝑟 𝜓 𝛽 † 𝑟′ 𝜓 𝛾 𝑟 𝜓 𝛿 𝑟
Define
∆ 𝛼𝛽 𝑟,𝑟′ ≡ 𝛾 𝛿 𝑉 𝛼𝛽𝛾𝛿 𝒓,𝒓′ 𝜓 𝛾 𝒓′ 𝜓 𝛾 𝒓
and treat it as a c-number. Then
𝐻 𝑀𝐹 = 𝐻 𝑜 + 𝑑𝒓𝑑𝒓′ Δ 𝛼𝛽 𝑟,𝑟′ 𝜓 𝛼 † 𝑟 𝜓 𝛽 † 𝑟′ +𝐻.𝑐.
Now the BdG equations will in general be 4x4. But note that in the special case of a single spin species (𝛼, 𝛽, 𝛾, 𝛿≡1, e.g. spin-polarized ultracold Fermi alkali gas) we have
𝛾 𝑖 † ≡ 𝑑𝑟 𝑢 𝑟 𝜓 † 𝑟 +𝑣 𝑟 𝜓 𝑟
H.c’s
so in particular 𝐸 𝑖 =0, 𝑢 𝑟 = 𝑣 ∗ 𝑟 modes (if any) may be true Majoranas.

17. The “Topological Insulator – Topological Superconductor” Analogy
Consider a “superconductor” with only one spin species, and assume for the moment spatially homogeneous conditions. Then we can take spatial Fourier transforms and get
≡ 𝜀 𝑘 −𝜇
𝐻 𝑀𝐹 = 𝑘 𝜉 𝑘 𝑎 𝑘 + 𝑎 𝑘 + Δ 𝑘 𝑎 𝑘 + 𝑎 −𝑘 + +H.c. Δ −𝑘 ≡ −Δ 𝑘
The eigenfunctions of 𝐻 𝑀𝐹 are of form
𝜓 𝑘 𝑢 𝑘 𝑣 𝑘 ≡ 𝑢 𝑘 𝑎 𝑘 + + 𝑣 𝑘 𝑎 −𝑘 |GS
Creates particle in state k
Creates hole in
state -k
Thus, if we define a 2D Hilbert space corresponding to “pseudo spin”
| ↑ ≡| particle in state 𝒌
| ↓ ≡| hole in state −𝒌
then 𝐻 𝑀𝐹 can be written (after subtraction of a constant term from the KE) in the “Nambu” form
𝐻 𝑀𝐹 𝑘 𝜉 𝑘 𝜎 𝑧𝑘 + Δ 𝑥𝑘 𝜎 𝑧𝑘 + Δ 𝑦𝑘 𝜎 𝑦𝑘 ≡ 𝑘 𝜉 𝑘 𝜎 𝑧𝑘 + 𝜟 𝑘 ∙ 𝝈 ⊥𝑘
≡𝑅𝑒 ∆ 𝑘
≡𝐼𝑚 ∆ 𝑘

18. Δ 𝑥𝑘 Δ 𝑦𝑘 = 𝑅𝑒 𝐼𝑚 part of interband matrix element
Now compare this with a simple 2-band solid with nonzero interband coupling. For this case define (for say real spin ↑) pseudospin states
| ↑ ≡| particle in state 𝒌 in band 1
| ↓ ≡| particle in state 𝒌 in band 2
𝜉 𝑘 ≡ 𝜀 𝑘 1 − 𝜀 𝑘 2
Δ 𝑥𝑘 Δ 𝑦𝑘 = 𝑅𝑒 𝐼𝑚 part of interband matrix element
Then 𝐻 2-band is formally identical to 𝐻 𝑀𝐹 !
(: physical interpretation is completely different!)

19. Topological superconductors, cont.
A particularly interesting case is when the system is 2D and
Δ 𝑘 =const. 𝑘 𝑥 + 𝑖𝑘 𝑦 , 𝜉 𝑘 changes sign at 𝒌 = 𝑘 𝑜 .
(e.g. due to spin-orbit coupling
In the 2-band case this corresponds to a topological insulator.* If we now allow 𝑘 𝑜 to be a 𝑓 𝒓 and to cross zero on some contour (e.g. the physical boundary of the system) it is easy to show (Volkov & Pankratov 1985) that a localized mode appears on this contour and disperses across the bulk band gap, with zero energy for 𝒌⊥ contour. For a (real geometrical) “hole” in the T⊥, 𝒌 is always ⊥ contour so E=0. This mode is a quantum superposition of | ↑ and | ↓ (i.e. | 1 and | 2 ) with equal weights (and, in the usual representation, a 𝜋 2 phase shift between them).
Now consider the superconducting analog – a p-wave superconductor with “p + ip” pairing Δ 𝑘 =const. 𝑘 𝑥 + 𝑖𝑘 𝑦 Δ 𝑘 =const. 𝑘 𝑥 + 𝑖𝑘 𝑦 . By analogy we expect boundary modes, and more interestingly, at vortices (“holes”) a single E=0 mode with equal weight for the | ↑ and | ↓ components. But | ↑ now represents a particle and | ↓ a hole, so
the E=0 state has exactly equal weight for particle & hole components.
*cf. Bernevig et al. 2006

20. So: What could be wrong with all this?
By an appropriate choice of global phases we can set 𝑢 𝑟 = 𝑣 ∗ 𝑟 , so particle is its own antiparticle (“Majorana fermion”).
Actually, M.F.’s always come in pairs (2 MF’s = 1 Bogoliubov qp), so any pair of vortices may carry either no MF’s "| 0 " or 2 "| 1 " .
Ivanov (2001): if 2 vortices exchanged, | 1 picks up Berry phase of 𝜋 2 relative to | 0 . Argument relies heavily on use of BdG equations.
This result is fundamental to the idea of topological quantum computing using p + ip Fermi superfluids (e.g UC alkali gases, 3He-A, Sr2RuO ).
So: What could be wrong with all this?

21. 𝛾 𝑖 † = 𝑢 𝑘 𝑎 𝑘+ 𝑚𝑣 ℏ ,↑ † + 𝑣 𝑘 𝑎 −𝑘+ 𝑚𝑣 ℏ ,↓ ← nb. not 𝑎 − 𝑘+ 𝑚𝑣 ℏ !
Failure of (a naïve interpretation of ) the BdG equations: a trivial example:
Consider an even-number-parity BCS superfluid at rest. (so 𝑷 𝑜 =0)
Now create a single fermionic quasiparticle in state k:
For this, the appropriate BdG qp creation operator 𝛾 𝑖 † turns out to be
𝛾 𝑖 † = 𝑢 𝑘 𝑎 𝑘↑ + + 𝑣 𝑘 𝑎 −𝑘↓ , 𝑢 𝑘 ≡ ∈ 𝑘 𝐸 𝑘 , 𝑣 𝑘 ≡ − ∈ 𝑘 𝐸 𝑘
𝐸 𝑘 ≡ ∈ 𝑘 Δ 𝑘 /2 (so 𝑢 𝑘 𝑣 𝑘 2 =1)
The total momentum 𝑷f of the (odd-number-parity) many-body state created by 𝛾 𝑖 is
𝑷 𝑓 = 𝑢 𝑘 2 ℏ𝒌 − 𝑣 𝑘 2 −ℏ𝒌 = 𝑢 𝑘 𝑣 𝑘 2 ℏ𝒌≡ℏ𝒌
so Δ 𝑷 𝑓 ≡ 𝑷 𝑓 - 𝑷 𝑜 =ℏ𝒌
Now consider the system as viewed from a frame of reference moving with velocity −𝒗, so that the condensate COM velocity is 𝒗. Since the pairing is now between states with wave vector 𝒌+𝑚𝒗/ℏ and −𝒌+𝑚𝒗/ℏ , intuition suggests (and explicit solution of the B&G equations confirms) that the form of 𝛾 𝑖 † is now
𝛾 𝑖 † = 𝑢 𝑘 𝑎 𝑘+ 𝑚𝑣 ℏ ,↑ † + 𝑣 𝑘 𝑎 −𝑘+ 𝑚𝑣 ℏ ,↓ ← nb. not 𝑎 − 𝑘+ 𝑚𝑣 ℏ !
Thus the added momentum is
Δ 𝑷′ 𝑓 = 𝑢 𝑘 2 ℏ𝒌+𝑚𝒗 − 𝑣 𝑘 2 −ℏ𝒌+𝑚𝒗
=ℏ𝒌+( 𝑢 𝑘 2 − 𝑣 𝑘 2 )𝑚𝒗
total
momentum
standard textbook result

22. Recap: BdG ® Δ 𝑃′ 𝑓 =ℏ𝒌+( 𝑢 𝑘 2 − 𝑣 𝑘 2 )𝑚𝒗
However, by Galilean invariance, for any given condensate number N,
𝑷′ 𝑜 = 𝑷 𝑜 +𝑁𝑚𝒗, 𝑷′ 𝑓 = 𝑷 𝑓 + 𝑁+1 𝑚𝒗.
⇒Δ 𝑃′ 𝑓 ≡ 𝑃 ′ 𝑓 − 𝑃 ′ 𝑜 = Δ 𝑷′ 𝑓 +𝑚𝒗=ℏ𝒌+𝑚𝒗.
and this result is independent of N (so involving “spontaneous breaking of U(1) symmetry” in GS doesn’t help!)
So: BdG ⇒Δ 𝑃′ 𝑓 =ℏ𝒌+( 𝑢 𝑘 2 − 𝑣 𝑘 2 )𝑚𝒗
GI → Δ 𝑃′ 𝑓 =ℏ𝒌+𝑚𝒗
Galilean invariance
What has gone wrong?
Solution: Conserve particle no.!
When condensate is at rest, correct expression for fermionic correlation operator 𝛾 𝑖 † is
𝛾 𝑖 † = 𝑢 𝑘 𝑎 𝑘↑ + + 𝑣 𝑘 𝑎 −𝑘↓ 𝐶 (0) † ← creates extra Cooper pair (with COM velocity 0)
Because condensate at rest has no spin/momentum/spin current …, the addition of 𝐶 has no effect. However:
when condensate is moving
𝛾 𝑖 † = 𝑢 𝑘 𝑎 𝒌+ 𝑚𝒗 ℏ ,↑ + + 𝑣 𝑘 𝑎 −𝑘+ 𝑚𝒗 ℏ ,↓ 𝐶 𝑣 †
⇒Δ 𝑃′ 𝑓 = Δ 𝑃 ′ 𝑓 𝐵𝑑𝐺 +2𝑚 𝑣 𝑘 2 =ℏ𝒌+𝑚𝒗
in accordance with GI argument
← creates extra Cooper pair(with COM velocity v)
Moral: addition of fermionic particle 𝑢 𝑘 𝑎 𝒌 produces total extra momentum ℏ𝒌, regardless of whether condensate is moving.

23. Failure of BdG/MF: a slightly less trivial example
For a thin slab of (appropriately oriented) 3He-B, calculations based on BdG equations predict a Majorana fermion state on the surface. A calculation* starting from the fermionic MF Hamiltonian then predicts that in a longitudinal NMR experiment, appreciable spectral weight should appear above the Larmor frequency 𝜔 𝐿 ≡𝛾 Β 𝑜 : B
𝐼𝑚 𝜒 𝑧𝑧 𝜔
Brf wL w→
The problem: no dependence on strength of dipole coupling 𝑔 𝐷 , (except for initial orientation)! But for 𝑔 𝐷 =0, 𝑆 𝑧 commutes with 𝐻 ⇒
𝜔 𝐼𝑚 𝜒 𝑧𝑧 𝜔 𝑑𝜔 ~ 𝑆 𝑧 , 𝑆 𝑧 , 𝐻 =𝑂
⇒𝐼𝑚 𝜒 𝑧𝑧 𝜔 ~ 𝛿 𝜔 !
For nonzero gD,
𝜔 𝑙𝑚 𝜒 𝑧𝑧 𝜔 𝑑𝜔 ~ 𝑔 𝐷 ⇒ result cannot be right for 𝑔 𝐷 →𝑂
Yet – follows from analysis of MF Hamiltonian!
MORAL: WE CANNOT IGNORE THE COOPER PAIRS!
*M.A. Silaev, Phys. Rev. B 84, (2011)

24. ↑ ↑ (Conjectured) further failure of MF/BdG approach
Imagine a many-body Hamiltonian (e.g. of a 𝑝+𝑖𝑝 superconductor) which admits single Majorana fermion solutions at two widely separated points. These M.F.’s are “halves” of a single 𝐸=0 Dirac-Bogoliubov fermion state. Intuitively, by injecting an electron at 1 one projects on to this single state and hence gets an instantaneous response at 2. Semenoff and Sodano (2008) discuss this problem in detail and conclude that unless we allow for violation of fermion number parity, we must conclude that this situation allows instantaneous teleportation.
Does it? Each Majorana is correctly described, not as usually assumed by
𝛾 𝑀 † = 𝑢 𝑟 𝜓 † 𝑟 + 𝑢 ∗ 𝑟 𝜓 𝑟 𝑑𝒓 ≡ 𝛾 𝑀
but rather by
𝛾 𝑀 † = 𝑢 𝑟 𝜓 † 𝑟 + 𝑢 ∗ (𝑟) 𝜓 (𝑟) 𝐶 † 𝑑𝒓 1 2 ↑
M.F. ↑
M.F. L
Cooper pair creation operator
But it is impossible to apply the “global” operator 𝐶 † instantaneously! What the injection will actually do (inter alia) is to create, at 1, a quasihole–plus–extra–Cooper –pair at 1, and the time needed for a response to be felt at 2 is bounded below by L/c when C is the C. pair propagation velocity (usually ~vf).

25. ⇒ replace by problem of neutral liquid in torus
The $64K question:
What is the Berry phase ( 𝜑 𝐵 ) accumulated by a Bogoliubov
quasiparticle circling the core of an Abrikosov vortex?
(Note: (a) this is in principle an experimentally answerable
question:
(b) at large distances (𝑟≫ 𝜆 𝐿 ),
simple AB effect ⇒ 𝜑 𝐵 =𝜋
However, we are interested in 𝜉≪𝑟≪ 𝜆 𝐿
⇒ replace by problem of neutral liquid in torus qp
Setup of problem:
A. Neutral superfluid (2N fermions) with s-wave pairing circulating with minimum nonzero velocity: 𝑣 𝜃 𝑅
𝑣= ℏ 2𝑚𝑅
𝑘= ℎ 2𝑚
note: Δ/ ∈ 𝐹 may be ~1
B. Create Zeeman trap: 𝜃𝑅 ↑ 𝐿
𝑉 0 𝐵
𝑉 𝑟 =− 𝜎 𝑧 𝐵(𝜃𝑅)
with 𝜉≪𝐿≪2𝜋𝑅, V 0 ≪Δ
𝑉 0 𝐿 2 ≥1 (so single ↑-spin fermion in free space would be bound)

26. WHAT BERRY PHASE IS ACCUMULATED?
C. Add single extra fermion with spin ↑.
What happens? 𝐿
↑𝑉(𝑟) 𝑝↑ ℎ↓
Δ𝜌(𝑟)
𝑠(𝑟)
For superfluid at rest,
Bog. qp presumably sits in
Zeeman trap: since for 𝐿≫𝜉
qp is equal admixture
of p ↑ and h ↓, (+ extra C. pairs) no extra mass density associated with trap. (Also, by TR symmetry, no mass current anywhere in system.)
If superfluid moving, presumably above is still true to zeroth order in 𝑣
D. Now imagine moving Zeeman trap adiabatically around torus ( 𝑣 trap ≪𝑣,𝑐…). Then 𝜏 trap ≫ℏ/Δ𝐸 (⇐ any excitation energy of system) ⇒ Berry phase well-defined.
($64K) question:
WHAT BERRY PHASE IS ACCUMULATED?
note: if superfluid at rest, 𝜑 𝐵 is fairly obliviously 0

27. 1st METHOD OF SOLUTION (BdG):
← particle component
← hole component
with
Δ 𝑟 = Δ 0 exp 𝑖𝜃
Compare this with standard textbook problem of spin ½ in “rotating” magnetic field:
𝐻 MF = 𝐵 𝑧 𝐵(𝑟) 𝐵 ∗ (𝑟) − 𝐵 𝑧
← spin-↑ component
← spin-↓ component 𝜑 𝜒 𝑩
𝐵 𝑟 =| 𝐵 0 | exp 𝑖𝜑
In textbook problem, known result is
𝜑 𝐵 =2𝜋 cos 2 𝜒 /2
and in particular for𝜒=𝜋/2 (equal weight of ↑ and ↓ spin components), 𝜑 𝐵 =𝜋 (“4𝜋 symmetry of fermions”, verified in neutron interferometry experiments)
Hence, by analogy (or direct calculation!) in BdG problem, since “particle” and “hole” components have equal weight, conclude
𝜑 𝐵 =𝜋
BdG calculation

28. 2nd Method Of Solution (Particle-Conserving Calculation):
Two general theorems:
(provided (2N+1) GS is any superposition of single fermion excited states, i.e., condensate not affected by Zeeman trap to order v):
𝜑 𝐵 =2𝜋⋅Δ J 0
difference in (angular momentum/ ℏ ) in GS of 2N+1 and 2N system (in lab. frame)
Galilean invariance
⇒ Δ 𝐽 𝑣 −Δ 𝐽 0 = 1 2
Δ𝐽 in rest frame of superfluid
Δ𝐽 in lab. frame ↑ 𝑣
(sys. at rest)
As a result, can state with confidence
𝜑 𝐵 =𝜋(1−2Δ J 𝑣 )
So: if with superfluid at rest and trap rotating, angular momentum 𝐽 𝑣 is nonzero*, BdG must be wrong
QN: WHAT IS 𝐽 𝑣 ?
(QUESTION UNEXPECTEDLY FULL OF TRAPS! DO WE NEED TO RE-INVENT THE WHEEL?)
* Since unarguably 𝐽 2𝑁,𝑣 =0

29. Pairs at rest, walls (traps) rotating↑
Pairs at rest, walls (traps) rotating
pairs at rest
WHAT IS 𝑱?
[note: for free particle, certainly ½ !]
Qp state must generically be of form
Ψ 𝑞𝑝 = 𝑝 𝑓 𝑝 ( 𝑢 𝑝 𝑎 𝑝↑ + + 𝑢 𝑝 𝑎 −𝑝↓ 𝐶 † )|2𝑁〉
≡ 𝑢 𝑟 𝜓 † 𝑟 +𝑣 𝑟 𝜓 𝑟 𝐶 † ) |2𝑁〉
ARGTS. FOR 𝑱=0 ( 𝜑 𝐵 =𝜋):
∫ 𝑢 𝑟 2 𝑑𝑟=∫ 𝑣 𝑟 2 𝑑𝑟= 1 2
1. For “low-energy” Bog. qp.,
⇒ no mass density deviation
⇒ no mass transport
⇒ 𝑱=0
2. In lab. Frame, no time-dependence
⇒ div𝑱=0=div Δ𝐽
⇒Δ𝐽(r) far from trap = Δ𝐽/𝐿
but far from trap 𝑢=𝑣=0, so only contribution to Δ𝑱 is from C. pairs: this is = 𝑚𝒗 =𝑚𝒗
⇒Δ 𝐽 0 =1/2⇒Δ 𝐽 v ≡𝑱=0

30. 𝑄 ≡− 𝑖 𝜎 𝑧𝑖 𝜕𝑉 𝑟 𝜕𝒓 𝑟= 𝑟 𝑖 Argument for 𝑱≠0 𝜑 𝐵 ≠𝜋 :
Apply adiabatic perturbation theory, then (quite generally, for any state |0 ) notation of trap generates perturbation
𝑉 𝑛𝑜 𝑒𝑓𝑓 =𝑖ℏ 𝜕 𝐻 𝜕𝑡 𝑛𝑜 1 𝐸 𝑛 − 𝐸 𝑜 =𝑖ℏ𝒗 𝑛 𝑸 0 𝐸 𝑛 − 𝐸 𝑜
𝑄 ≡− 𝑖 𝜎 𝑧𝑖 𝜕𝑉 𝑟 𝜕𝒓 𝑟= 𝑟 𝑖
where
Hence provided 𝑱 𝑜 =0, value of 𝑱 induced by rotation of trap is
𝐴≡𝑖ℏ 𝑛 𝑱 𝑛 𝑛 𝑸 𝐸 𝑛 − 𝐸 0 2
𝑱=𝐴𝒗 ,
Now if |0 is the 2N-particle GS, both |0 and |𝑛 can be chosen to be eigenstates of 𝑇 ⇒𝐴=0 by symmetry.
“pseudo” time reversal 𝜎→−𝜎,𝐵→𝐵

31. " 𝐽 𝑙𝑜𝑛𝑔 = 𝐽 𝑡𝑟𝑎𝑛𝑠𝑣 "? If so then from f-sum rule
However, if |0 is the GS of the 2𝑁+1 − particle system (with S=1), this is not true!  no symmetry principle requires A to be nonzero.
Indeed, since 𝑱 , 𝐻 𝑏𝑢𝑙𝑘 =0 ,
𝑑 𝑱 𝑑𝑡 =𝑖ℏ 𝐽 , 𝐻 𝑡𝑟𝑎𝑝 =−iℏ 𝑖 𝜎 𝑧𝑖 𝜕𝑉 𝜕𝑟 𝑟𝑧𝑟𝑖 =−𝑖ℏ 𝑸
𝑛 𝑄 0 = 1 𝑖ℏ 𝐸 𝑛 − 𝐸 0 𝑛 𝑱 0
so and so
𝐴= 𝑛 𝑱 𝑛 𝐸 𝑛 − 𝐸 𝑜 ≡ 𝜒 𝐽𝐽 𝑞→0, 𝜔→0
↑: Since 𝐴=0 for 2𝑁− particle GS, must take 𝑱 to be
𝑞→0 𝑙𝑖𝑚 𝑱 𝒒 , 𝒒⊥𝑱
(Meissner effect!). But for extra quasiparticle,
" 𝐽 𝑙𝑜𝑛𝑔 = 𝐽 𝑡𝑟𝑎𝑛𝑠𝑣 "? If so then from f-sum rule
⇒𝑱=𝑚𝒗⇒ 𝜑 𝐵 =0
Wait for the next thrilling installment . . .

32. Conclusions What are implications for TQC program?
𝑣= 5 2 QHE: no implications
(p + ip) Fermi superfluids (e.g. Sr2RuO4) Literature may be right.
But urgent to (dis)-confirm using physical (particle-conserving) many-body wave functions
More generally:
CMP / QI !