Download Spontaneously Broken U(1) - University of Illinois Urbana

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 UrbanaChampaign
joint work with Yiruo Lin
support: NSF-DMR-09-06921
UT 2
1. Spontaneously broken U(1) symmetry:
the weakly interacting Bose gas
2. Spontaneously broken U(1) symmetry:
BCS theory of superconductivity
3. The Bogoliubov-de Gennes equations:
s-wave case
4. The Bogoliubov-de Gennes equations:
general case
5. Application to topological quantum
computing in Sr2RuO4
6. What could be wrong with all this?
A couple of (rather trivial) illustrations.
7. A simple but worrying thought-experiment.
8. Conclusion
UT 3
SPONTANEOUSLY BROKEN U(1) SYMMETRY: THE GENERAL IDEA
1. Dilute Bose gas with weak repulsion (Bogoliubov, 1947)
𝜀𝑘 𝑎𝑘+ 𝑎𝑘 + 𝑉
𝐻=
𝑘
𝐻, 𝑁 = 𝐻, 𝑃 = 0
Total particle
number
2 2
ℏ 𝑘 /2𝑚
Total
momentum
To zeroth order in 𝑉, groundstate of N particles is that of
free Bose gas, i.e. (apart from normalization)
𝑁
= 𝑎𝑜+
Ψ𝑜
𝑁 |vac
≡ 𝑎𝑜+ 𝑎𝑜+
𝑁/2 |vac
Most important scattering processes:
O
k + inverse
V
O
-k
process
+
, i.e. 𝑉𝑘 𝑎𝑘+ 𝑎−𝑘
𝑎𝑜 𝑎𝑜 +𝐻. 𝑐.
𝑉pair
The simplest ansatz for N-particle GSWF incorporating these
processes: (apart from normalization)
𝑁/2
(𝑁)
Ψ𝑜
+
𝑐𝑘 𝑎𝑘+ 𝑎−𝑘
= 𝑎𝑜+ 𝑎𝑜+ −
𝑘
|vac
particleconserving!
UT 4
Coefficients 𝑐𝑘 can be found by minimizing 𝐻
including 𝑉 pair terms*. However, calculations
rather messy…
Bogoliubov: in “reduced” Hamiltonian
𝜀𝑘 𝑎𝑘+ 𝑎𝑘 +
𝐻𝑟𝑒𝑑 ≡
𝑘
+
𝑉𝑘 𝑎𝑘+ 𝑎−𝑘
𝑎𝑜 𝑎𝑜 + H. c.
𝑘
treat quantity 𝑎𝑜 𝑎𝑜 (or actually 𝑎𝑜 ) as c-number
𝑎𝑜 → 𝑁𝑜 so 𝑎𝑜 𝑎𝑜 → 𝑁𝑜 ;
then
+
𝜀𝑘 𝑎𝑘+ 𝑎𝑘 + 𝑁𝑜 𝑉𝑘 𝑎𝑘+ 𝑎−𝑘
+ H. c.
𝐻𝐵𝑜𝑔 =
𝑘
(but see below…)
*see e.g. AJL, Revs. Mod. Phys. 73, 307 (2001), section VIII.
UT 5
What are the implications of treating 𝑎𝑜 𝑎𝑜 as a c-number?
Implies a nonzero expectation value
≡ 𝑎𝑜+ 𝑎𝑜
Ψ 𝑎𝑜 𝑎𝑜 Ψ ≅ 𝑁𝑜
This is not true for the (particle-conserving) wave function
𝑁 2
𝑁
Ψ𝑜
+
𝑐𝑘 𝑎𝑘+ 𝑎−𝑘
≡ 𝑎𝑜+ 𝑎𝑜+ −
𝑎𝑜 𝑎𝑜 ≡ 𝑂
|vac
𝑘
but is true e.g. for
𝐵𝑜𝑔
Ψ𝑜
+
𝑐𝑘 𝑎𝑘+ 𝑎−𝑘
≡ exp 𝜆 𝑎𝑜+ 𝑎𝑜+ −
𝑁
𝜆𝑁 Ψ𝑜
≡
𝑘
𝑁=even
(neglect details about normalization, etc.)
What fixes 𝑁 ~𝜆 ? If we just minimize 𝐻𝐵𝑜𝑔 as it stands
this automatically gives 𝜆~ 𝑁 ~0! Hence must add
chemical potential term, i.e. minimize
′
𝐻𝐵𝑜𝑔
≡ 𝐻𝐵𝑜𝑔 − 𝜇𝑁
(i.e. 𝜀𝑘 → 𝜀𝑘 −𝜇)
and adjust  so that 𝑁 𝜇 = 𝑁
 actual number of
particles
𝑁!
UT 6
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 =
exp 𝑖𝑁𝜑 |𝑁 . |𝑁 =
𝑑𝜑 exp − 𝑖𝑁𝜑 |𝜑
𝑁
Since original 𝐻 (not 𝐻𝐵𝑜𝑔 !) satisfies 𝑁 𝐻 𝑁′ ~𝛿𝑁𝑁′ , it is
invariant under “rotation” of 𝜑 (“U(1)symmetry”) .
𝐵𝑜𝑔
However Ψ𝑜 picks out a definite value (0) of 𝜑:
“Spontaneously broken U(1) symmetry”
UT 7
THE MEAN-FIELD (BOGOLIUBOV) METHOD: QUASIPARTICLES:
′
“Mean-field” Hamiltonian 𝐻𝑀𝐹 ≡ 𝐻𝐵𝑜𝑔
, i.e.
+
𝜖𝑘 − 𝜇 𝑎𝑘+ 𝑎𝑘 + 𝑁𝑜 𝑉𝑘 𝑎𝑘+ 𝑎−𝑘
+ H. c.
𝐻𝑀𝐹 ≡
𝑘
with standard Bose C.R.’s
+
𝑎𝑘 , 𝑎𝑘′ = 𝑎𝑘+ 𝑎𝑘′
= 0,
+
𝑎𝑘 , 𝑎𝑘′
= 𝛿𝑘𝑘′
This can be diagonalized by Bogoliubov transformation
𝛼𝑘+ = 𝑢𝑘 𝑎𝑘+ + 𝑣𝑘 𝑎−𝑘 (etc.), 𝑢𝑘
2
− 𝑣𝑘
2
=1
which preserves Bose C.R.’s.:
𝐸𝑘 𝛼𝑘+ 𝛼𝑘 +const.
𝐻𝑀𝐹 ≡
𝑘
where (after  is fixed to give 𝑁 = 𝑁 and 𝑉𝑘 𝑠𝑒𝑡 ≅ 𝑉𝑂 )
𝐸𝑘 = 𝜀𝑘 𝜀𝑘 + 2𝑁𝑉𝑂
1 2
 Bogoliubov spectrum
UT 8
Note: All these results, including formula for 𝐸𝑘 , can be
obtained by particle-conserving method based on ansatz
𝑁
Ψ𝑂 . However, in that method we have
𝛼𝑘+ = 𝑢𝑘 𝑎𝑘+ + 𝑣𝑘 𝑎𝑜+ 𝑎𝑜+ /𝑁𝑜 𝑎−𝑘
+1
+2
-1
 particleconserving!
𝑁 →𝑁+1
By contrast, in Bogoliubov method
𝑢𝑘 𝑎𝑘+ adds a particle 𝑁 → 𝑁 + 1
𝑣𝑘 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:
𝜌𝑘 = 𝛼𝑘+ 𝑎𝑜 = 𝑢𝑘 𝑎𝑘+ 𝑎𝑜 + 𝑣𝑘 𝑎𝑜+ 𝑎𝑜+ 𝑁𝑜 𝑎−𝑘 𝑎𝑜
= 𝑢𝑘 𝑎𝑘+ 𝑎𝑜 + 𝑣𝑘 𝑎−𝑘 𝑎𝑜+
[Generalization of Bogoliubov method to inhomogeneous
situations]
UT 9
SPONTANEOUSLY BROKEN U(1) SYMMETRY: THE GENERAL IDEA (CONT.)
2. Superconductors (BCS 1957)
+
𝜀𝑘 𝑎𝑘𝜎
𝑎𝑘𝜎 + 𝑉
𝐻=
𝐻, 𝑁 = 𝐻, 𝑃 = 𝑂
𝑘𝜎
Most important scattering process:
𝑘′ ↑
𝑘↑
V
−𝑘 ↓
(+ inverse process),
i.e.
+
𝑉𝑘𝑘′ 𝑎𝑘′↑
𝑎−𝑘′↓ 𝑎−𝑘↓ 𝑎𝑘↑
−𝑘′ ↓
+H. c.
𝑉pair
even
Simplest ansatz for N-particle GSWF incorporating these
processes (apart from normalization)
𝑁 2
𝑁
Ψ𝑂
+ +
𝑐𝑘 𝑎𝑘↑
𝑎−𝑘↓
=
|vac  particle-
𝑘
conserving!
with coefficients found by minimizing 𝐻 including 𝑉pair
terms*.
*See e.g. AJL, Quantum Liquids, section V.
UT 10
Again, calculations messy…
BCS: in “reduced” Hamiltonian
+ +
𝑉𝑘𝑘′ 𝑎𝑘↑
𝑎−𝑘↓ 𝑎−𝑘′↓ 𝑎𝑘′↑ + H. c.
+
𝜖𝑘 𝑎𝑘𝜎
𝑎𝑘𝜎 +
𝐻𝑟𝑒𝑑 =
𝑘𝜎
𝑘𝑘′
treat the quantity
Δ𝑘 ≡
𝑉𝑘𝑘′ 𝑎−𝑘′↓ 𝑎𝑘′↑
𝑘′
as c-number. Then (adding chemical potential term as in
Bose case)
+ +
Δ𝑘 𝑎𝑘↑
𝑎−𝑘↓ + H. c.
+
𝜖𝑘 − 𝜇 𝑎𝑘𝜎
𝑎𝑘𝜎 +
𝐻𝑀𝐹 =
𝑘𝜎
𝑘
Δ𝑘 ≡
𝑉𝑘𝑘′ 𝑎−𝑘′↓ 𝑎𝑘′↑
𝑘′
Again, this makes sense only if U(1) symmetry spontaneously
broken, e.g.
+ +
𝑐𝑘 𝑎𝑘↑
𝑎−𝑘↓ |vac
Ψ𝐵𝐶𝑆 = exp
𝑘
≡
𝑢𝑘 |𝑜𝑜
𝑘
𝑘
+ 𝑣𝑘 |11
𝑘
UT 11
The Mean-Field (BCS) Method: Quasiparticles
As above, mean-field Hamiltonian is
+
Δ𝑘 𝑎𝑘↑ 𝑎−𝑘↓
+ H. c.
+
𝜖𝑘 − 𝜇 𝑎𝑘𝜎
𝑎𝑘𝜎 +
𝐻𝑀𝐹 =
𝑘𝜎
𝑘
with standard Fermi ACR’s
+
+
+
𝑎𝑘𝜎 , 𝑎𝑘′𝜎′ = 𝑎𝑘𝜎
, 𝑎𝑘′𝜎′
= 𝑂, 𝑎𝑘𝜎 , 𝑎𝑘′𝜎′
= 𝛿𝑘𝑘′ 𝛿𝜎𝜎′
This can be diagonalized by Bogoliubov (-Valatin)
transformation
+
+
𝛼𝑘𝜎
= 𝑢𝑘 𝑎𝑘𝜎
+ 𝜎𝑣𝑘 𝑎−𝑘−𝜎 (etc.), 𝑢𝑘
2
+ 𝑣𝑘
2
=1
which preserves Fermi ACR’s:
+
𝐸𝑘 𝛼𝑘𝜎
𝛼𝑘𝜎
𝐻𝑀𝐹 =
𝑘𝜎
𝐸𝑘 ≡
𝜀𝑘2
+ Δ𝑘
2
1
2
UT 12
Note: All these results can be obtained by particle𝑁
conserving method based on ansatz Ψ𝑜
that method we have
+
+
𝛼𝑘𝜎
= 𝑢𝑘 𝑎𝑘𝜎
+ 𝜎𝑣𝑘 𝑎−𝑘,−𝜎 𝑐 †
+1
-1
+2
. However, in
 particleconserving!
𝑁 →𝑁+1
where 𝐶 † is normalized Cooper-pair creation operator:
+ +
𝑐𝑘 𝑎𝑘↑
𝑎−𝑘↓
𝐶† ≡ 𝑛 ∙
𝑘
normalization
By contrast, in BCS method
+
𝑢𝑘 𝑎𝑘𝜎
adds an electron 𝑁 → 𝑁 + 1
+
𝜎𝑣𝑘 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., 𝐻𝑀𝐹 often described as
“noninteracting-electron” Hamiltonian (!)]
UT 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
𝑁
generalizes Ψ𝑂 , 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 𝜓𝜎 𝒓
2
≡ 𝑂)
single-particle potential
𝐻=
𝑑𝒓
𝜎
ℏ2
2𝑚
𝜵𝜓𝜎† 𝒓 ⋅ 𝜵𝜓𝜎 𝑟
+𝑔
+ (∪ 𝒓 −
𝑑𝑟 𝜓↑† 𝑟 𝜓↓† 𝑟 𝜓↓ 𝑟 𝜓↑ 𝑟
The mean-field approximation: take the quantity
Δ 𝒓 ≡ 𝒈𝜓 ↓ 𝒓 𝜓 ↑ 𝒓
to be a c-number. Thus (↑: caution re factors of 2 etc!)
*Stone & Chung, Phys. Rev. B 73, 014505 (2006)
UT 14
𝐻𝑀𝐹 = 𝐻𝑂 + 𝑔
𝐻𝑜 ≡
𝑑𝒓
𝜎
ℏ2
2𝑚
𝑑𝑟 ∆ 𝑟 𝜓↑† 𝑟 𝜓↓ 𝑟 + H. c.
𝜵𝜓𝜎† 𝑟 ⋅ 𝜵𝜓𝜎 𝑟
+ ∪ 𝒓 − 𝜇 𝜓𝜎† 𝑟 𝜓𝜎 𝑟
with standard Fermi ACR’s
†
𝜓𝜎 𝑟 , 𝜓𝜎′
𝑟′ = 𝛿 𝒓 − 𝒓′ 𝛿𝜎𝜎′ (etc.)
Since 𝐻𝑀𝐹 is a quadratic form in 𝜓𝜎† 𝑟 , 𝜓𝜎 𝒓 , it can be
diagonalized by introducing operators 𝛾𝑖+ defined by
+
𝛾𝑖𝜎
= 𝑑𝒓 𝑢𝑖 𝑟 𝜓𝜎† 𝑟 + 𝜎𝑣𝑖 𝑟 𝜓−𝜎 𝒓
not H.c’s
 particlenonconserving!
which will satisfy Fermi ACR’S provided the 𝑢𝑖 𝑟 , 𝑣𝑖 𝑟 satisfy
𝑑𝑟
𝑢𝑖 𝑟 , 𝑢𝑗 𝑟
+ 𝑣𝑖 𝑟 , 𝑣𝑗 𝑟
= 𝛿𝑖𝑗
a condition in fact guaranteed by the BdG equations
(below). Thus
+
𝐸𝑖 𝛾𝑖𝜎
𝛾𝑖𝜎 + const.
𝐻𝑀𝐹 =
𝑖𝜎
UT 15
𝑢𝑖
The eigenfunctions
𝑣𝑖 and eigenvalues 𝐸𝑖 of the
mean-field Hamiltonian satisfy the Bogoliubov-de Gennes
(BDG) equations:
𝐻𝑜
Δ∗ 𝑟
Δ 𝑟
−𝐻𝑜∗
𝑢𝑖 𝑟
𝑣𝑖 𝑟
This guarantees
(a) orthonormality of spinors
𝑢𝑖 𝑟
(b) if 𝑣𝑖 𝑟
𝑢𝑖 𝒓
= 𝐸𝑖
𝑣𝑖 𝒓
𝑢𝑖
𝑣𝑖 (c.f. above)
𝑣𝑖∗ 𝑟
is a solution with energy 𝐸𝑖 , then 𝑢𝑖∗ 𝑟
is also a solution with energy −𝐸𝑖 : hence now be
able to find solutions with 𝐸𝑖 = 0, provided
𝑢𝑖 𝑟 = 𝑣𝑖∗ 𝑟 (“pseudo-Majorana”)
UT 16
Mean-field method in more general case:
ℏ2
𝑑𝒓 𝛿𝜎𝜎′
𝜵𝜓𝜎† 𝑟 ⋅ 𝜵𝜓𝜎′ 𝑟
2𝑚
𝐻=
𝜎𝜎′
+ ∪𝝈𝝈′ 𝒓 − 𝜇𝛿𝜎𝜎′ 𝜓𝜎† 𝒓 𝜓𝜎′ 𝒓
+
1
2
𝑑𝒓𝑑𝒓′ 𝑉𝛼𝛽𝛾𝛿 𝑟, 𝑟′ 𝜓𝛼 𝑟 𝜓𝛽† 𝑟′ 𝜓𝛾 𝑟 𝜓𝛿 𝑟
𝛼𝛽𝛾𝛿
Define
∆𝛼𝛽 𝑟, 𝑟′ ≡
𝑉𝛼𝛽𝛾𝛿 𝒓, 𝒓′ 𝜓𝛾 𝒓′ 𝜓𝛾 𝒓
𝛾𝛿
and treat it as a c-number. Then
𝐻𝑀𝐹 = 𝐻𝑜 +
𝑑𝒓𝑑𝒓′ Δ𝛼𝛽 𝑟, 𝑟′ 𝜓𝛼† 𝑟 𝜓𝛽† 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. spinpolarized ultracold Fermi alkali gas) we have
𝛾𝑖† ≡
𝑑𝑟 𝑢 𝑟 𝜓 † 𝑟 + 𝑣 𝑟 𝜓 𝑟
H.c’s
so in particular 𝐸𝑖 = 0, 𝑢 𝑟 = 𝑣 ∗ 𝑟 modes (if any) may be
true Majoranas.
UT 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
≡ 𝜀𝑘 − 𝜇
+
𝜉𝑘 𝑎𝑘+ 𝑎𝑘 + Δ𝑘 𝑎𝑘+ 𝑎−𝑘
+ H. c.
𝐻𝑀𝐹 =
Δ−𝑘 ≡ −Δ𝑘
𝑘
The eigenfunctions of 𝐻𝑀𝐹 are of form
𝜓𝑘
𝑢𝑘
+
≡
𝑢
𝑎
+ 𝑣𝑘 𝑎−𝑘 |GS
𝑘
𝑘
𝑣𝑘
Creates hole in
state -k
Creates particle
in state k
Thus, if we define a 2D Hilbert space corresponding to “pseudo
spin”
|↑ ≡ |particle in state 𝒌
|↓ ≡ |hole in state − 𝒌
then 𝐻𝑀𝐹 can be written (after subtraction of a constant
term from the KE) in the “Nambu” form
𝐻𝑀𝐹
𝜉𝑘 𝜎𝑧𝑘 + Δ𝑥𝑘 𝜎𝑧𝑘 + Δ𝑦𝑘 𝜎𝑦𝑘 ≡
𝑘
𝜉𝑘 𝜎𝑧𝑘 + 𝜟𝑘 ∙ 𝝈⊥𝑘
𝑘
≡ 𝑅𝑒∆𝑘
≡ 𝐼𝑚∆𝑘
UT 18
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
𝜉𝑘 ≡ 𝜀𝑘 1 − 𝜀𝑘 2
Δ𝑥𝑘
𝑅𝑒
=
part of interband matrix element
Δ𝑦𝑘
𝐼𝑚
Then 𝐻 2-band is formally identical to 𝐻𝑀𝐹 !
(: physical interpretation is
completely different!)
UT 19
TOPOLOGICAL
SUPERCONDUCTORS, cont.
A particularly interesting case is when the system is 2D and
Δ𝑘 = const. 𝑘𝑥 + 𝑖𝑘𝑦 , 𝜉𝑘 changes sign at 𝒌 = 𝑘𝑜 .
(e.g. due to spin-orbit coupling
In the 2-band case this corresponds to a topological
insulator.* If we now allow 𝑘𝑜 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 Δ𝑘 =
the E=0 state has exactly equal weight for particle & hole
components.
*cf. Bernevig et al. 2006
UT 20
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, Sr2RuO4 . . .).
SO: WHAT COULD BE WRONG WITH ALL THIS?
UT 21
Failure of (a naïve interpretation of ) the BdG equations: a trivial
example:
Consider an even-number-parity BCS superfluid at rest. (so 𝑷𝑜 = 0)
total
momentum
Now create a single fermionic quasiparticle in state k:
For this, the appropriate BdG qp creation operator 𝛾𝑖† turns out to be
1
+
𝛾𝑖† = 𝑢𝑘 𝑎𝑘↑
+ 𝑣𝑘 𝑎−𝑘↓
𝐸𝑘 ≡ ∈2𝑘 + Δ𝑘
∈
1
∈
, 𝑢𝑘 ≡ 2 1 + 𝐸𝑘 , 𝑣𝑘 ≡ 2 1 − 𝐸𝑘
2 1/2
𝑘
(so 𝑢𝑘
2
+ 𝑣𝑘
𝑘
2
= 1)
The total momentum 𝑷f of the (odd-number-parity) many-body state
created by 𝛾𝑖 is
standard textbook result
𝑷𝑓 = 𝑢𝑘
2
ℏ𝒌 − 𝑣𝑘
2
−ℏ𝒌 =
𝑢𝑘 2 + 𝑣𝑘
2
ℏ𝒌 ≡ ℏ𝒌
so Δ𝑷𝑓 ≡ 𝑷𝑓 - 𝑷𝑜 = 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 𝛾𝑖† is now
†
𝛾𝑖† = 𝑢𝑘 𝑎𝑘+
𝑚𝑣 + 𝑣𝑘 𝑎−𝑘+𝑚𝑣,↓ ← nb. not 𝑎
−
,↑
ℏ
ℏ
𝑚𝑣
𝑘+ ℏ !
Thus the added momentum is
Δ𝑷′𝑓 = 𝑢𝑘 2 ℏ𝒌 + 𝑚𝒗 − 𝑣𝑘 2 −ℏ𝒌 + 𝑚𝒗
= ℏ𝒌 + ( 𝑢𝑘 2 − 𝑣𝑘 2 )𝑚𝒗
Recap: BdG Δ𝑃′𝑓 = ℏ𝒌 + ( 𝑢𝑘
2
− 𝑣𝑘 2 )𝑚𝒗
UT 22
However, by Galilean invariance, for any given condensate
number N,
𝑷′𝑜 = 𝑷𝑜 + 𝑁𝑚𝒗, 𝑷′𝑓 = 𝑷𝑓 + 𝑁 + 1 𝑚𝒗.
⇒ Δ𝑃′𝑓 ≡ 𝑃′𝑓 − 𝑃′ 𝑜 = Δ𝑷′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 ⇒ Δ𝑃′𝑓 = ℏ𝒌 + ( 𝑢𝑘
2
− 𝑣𝑘 2 )𝑚𝒗
GI → Δ𝑃′𝑓 = ℏ𝒌 + 𝑚𝒗
Galilean invariance
What has gone wrong?
Solution: Conserve particle no.!
When condensate is at rest, correct expression for fermionic
correlation operator 𝛾𝑖† is
†
+
𝛾𝑖† = 𝑢𝑘 𝑎𝑘↑
+ 𝑣𝑘 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
† ← creates extra Cooper
+
𝑚𝒗 𝐶
𝛾𝑖† = 𝑢𝑘 𝑎𝒌+
𝑚𝒗 + 𝑣𝑘 𝑎
−𝑘+ ,↓ 𝑣
,↑
ℏ
ℏ
pair(with COM velocity v)
⇒ Δ𝑃′𝑓 = Δ𝑃′𝑓
𝐵𝑑𝐺
+ 2𝑚 𝑣𝑘
2
= ℏ𝒌 + 𝑚𝒗
in accordance with GI argument
Moral: addition of fermionic particle 𝑢𝑘 𝑎𝒌+ +. . . produces
total extra momentum ℏ𝒌, regardless of whether condensate is
moving.
UT 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 𝜔𝐿 ≡ 𝛾Β𝑜 :
B
Brf
𝐼𝑚𝜒𝑧𝑧 𝜔
w→
wL
The problem: no dependence on strength of dipole coupling 𝑔𝐷 ,
(except for initial orientation)! But for 𝑔𝐷 =0, 𝑆𝑧 commutes with 𝐻 ⇒
𝜔 𝐼𝑚𝜒𝑧𝑧 𝜔 𝑑𝜔 ~ 𝑆𝑧 , 𝑆𝑧 , 𝐻
=𝑂
⇒ 𝐼𝑚𝜒𝑧𝑧 𝜔 ~ 𝛿 𝜔 !
For nonzero gD,
𝜔 𝑙𝑚𝜒𝑧𝑧 𝜔 𝑑𝜔 ~𝑔𝐷 ⇒ result cannot be right for 𝑔𝐷 → 𝑂
Yet – follows from analysis of MF Hamiltonian!
MORAL: WE CANNOT IGNORE THE COOPER PAIRS!
*M.A. Silaev, Phys. Rev. B 84, 144508 (2011)
UT 24
(Conjectured) further failure of MF/BdG approach
1
2
↑
↑
M.F.
M.F.
L
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
𝛾𝑀† =
𝑢 𝑟 𝜓 † 𝑟 + 𝑢∗ 𝑟 𝜓 𝑟 𝑑𝒓
≡ 𝛾𝑀
but rather by
𝛾𝑀† =
𝑢 𝑟 𝜓 † 𝑟 + 𝑢∗ (𝑟)𝜓(𝑟)𝐶 † 𝑑𝒓 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).
UT 25
The $64K question:
What is the Berry phase (𝜑𝐵 ) 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 ≫ 𝜆𝐿 ),
simple AB effect ⇒ 𝜑𝐵 = 𝜋
However, we are interested in 𝜉 ≪ 𝑟 ≪ 𝜆𝐿
qp
⇒ replace by problem of neutral liquid in torus
Setup of problem:
𝜃
𝑣
𝑅
A. Neutral superfluid (2N
fermions) with s-wave pairing
circulating with minimum
nonzero velocity:
𝑣=
ℏ
2𝑚𝑅
𝑘=
ℎ
2𝑚
note: Δ/∈𝐹 may be ~1
B. Create Zeeman trap:
𝑉 𝑟 = −𝜎𝑧 𝐵(𝜃𝑅)
𝜃𝑅
𝐵
↑
𝑉0
𝐿
with 𝜉 ≪ 𝐿 ≪ 2𝜋𝑅, V0 ≪ Δ
𝑉0 𝐿2 ≥ 1 (so single ↑-spin
fermion in free space would be
bound)
UT 26
C. Add single extra fermion with spin ↑.
What happens?
𝐿
For superfluid at rest,
↑ 𝑉(𝑟)
Bog. qp presumably sits in
Zeeman trap: since for 𝐿 ≫ 𝜉
𝑝↑
qp is equal admixture
ℎ↓
of p ↑ and h ↓, (+ extra C. pairs)
no extra mass density
Δ𝜌(𝑟)
0
associated with trap. (Also,
0
𝑠(𝑟)
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 (𝑣trap ≪ 𝑣, 𝑐 …). Then 𝜏trap ≫ ℏ/Δ𝐸 (⇐ any
excitation energy of system) ⇒ Berry phase well-defined.
($64K) question:
WHAT BERRY PHASE IS ACCUMULATED?
note: if superfluid at rest, 𝜑𝐵 is fairly obliviously 0
UT 27
1st METHOD OF SOLUTION (BdG):
𝐻BdG =
with
𝐻𝑜↑
Δ(𝑟)
∗
Δ∗ (𝑟) −𝐻𝑜↓
𝑢 (𝑟) ← particle component
𝑣 (𝑟) ← hole component
Δ 𝑟 = Δ0 exp 𝑖𝜃
Compare this with standard textbook problem of spin ½
in “rotating” magnetic field:
𝐻MF =
𝐵𝑧
𝐵(𝑟)
𝐵∗ (𝑟) −𝐵𝑧
← spin-↑ component
← spin-↓ component
𝜑
𝐵 𝑟 = |𝐵0 | exp 𝑖𝜑
𝑩
In textbook problem, known result is
𝜑𝐵 = 2𝜋 cos2 𝜒 /2
𝜒
and in particular for𝜒 = 𝜋/2 (equal weight of ↑ and ↓
spin components), 𝜑𝐵 = 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
𝜑𝐵 = 𝜋
BdG calculation
UT 28
2ND METHOD OF SOLUTION
(PARTICLE-CONSERVING CALCULATION):
Two general theorems:
1. (provided (2N+1) GS is any superposition of single
fermion excited states, i.e., condensate not affected
by Zeeman trap to order v):
𝜑𝐵 = 2𝜋 ⋅ ΔJ0
difference in (angular momentum/ ℏ ) in GS of 2N+1
and 2N system (in lab. frame)
2. Galilean invariance ⇒ Δ𝐽𝑣 − Δ𝐽0 =
1
2
Δ𝐽 in rest frame of Δ𝐽 in lab. frame
superfluid
↑
As a result, can state with confidence
𝑣
𝜑𝐵 = 𝜋(1 − 2ΔJ𝑣 )
So: if with superfluid at rest and trap
rotating, angular momentum 𝐽𝑣 is
nonzero*, BdG must be wrong
(sys. at rest)
QN: WHAT IS 𝐽𝑣 ?
(QUESTION UNEXPECTEDLY FULL OF TRAPS!
DO WE NEED TO RE-INVENT THE WHEEL?)
* Since unarguably 𝐽2𝑁,𝑣 = 0
UT 29
↑
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
+
𝑓𝑝 (𝑢𝑝 𝑎𝑝↑
+ 𝑢𝑝 𝑎−𝑝↓ 𝐶 † )|2𝑁〉
Ψ𝑞𝑝 =
𝑝
≡
𝑢 𝑟 𝜓 † 𝑟 + 𝑣 𝑟 𝜓 𝑟 𝐶 † ) |2𝑁〉
ARGTS. FOR 𝑱 = 0 (𝜑𝐵 = 𝜋):
1. For “low-energy” Bog. qp.,
2 𝑑𝑟
𝑢 𝑟
⇒ no mass density deviation
⇒ no mass transport
⇒𝑱=0
=
𝑣 𝑟
2 𝑑𝑟
1
=
2
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
1
Δ𝑱 is from C. pairs: this is = 2𝑚𝒗 = 𝑚𝒗
2
⇒ Δ𝐽0 = 1/2 ⇒ Δ𝐽v ≡ 𝑱 = 0
UT 30
ARGUMENT FOR 𝑱 ≠ 0
𝜑𝐵 ≠ 𝜋 :
Apply adiabatic perturbation theory, then (quite generally,
for any state |0 ) notation of trap generates perturbation
𝑒𝑓𝑓
𝑉𝑛𝑜
where
𝜕𝐻
= 𝑖ℏ
𝜕𝑡
𝑄≡−
𝑖
Hence provided 𝑱
trap is
𝑱 = 𝐴𝒗
,
𝑜
𝑛𝑸0
1
= 𝑖ℏ𝒗
𝐸
−
𝐸
𝐸𝑛 − 𝐸𝑜
𝑜
𝑛𝑜 𝑛
𝜕𝑉 𝑟
𝜎𝑧𝑖
𝜕𝒓
𝑟=𝑟𝑖
= 0, value of 𝑱 induced by rotation of
𝐴 ≡ 𝑖ℏ
𝑛
0𝑱𝑛 𝑛𝑸0
𝐸𝑛 − 𝐸0 2
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
UT 31
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, 𝐻𝑏𝑢𝑙𝑘 = 0 ,
𝑑𝑱
= 𝑖ℏ 𝐽, 𝐻𝑡𝑟𝑎𝑝 = −iℏ
𝑑𝑡
𝑖
𝜕𝑉
𝜎𝑧𝑖
𝜕𝑟
= −𝑖ℏ𝑸
𝑟𝑧𝑟𝑖
1
𝐸 − 𝐸0 𝑛 𝑱 0 and so
so 𝑛 𝑄 0 =
𝑖ℏ 𝑛
𝐴=
𝑛
0𝑱𝑛 2
≡ 𝜒𝐽𝐽 𝑞 → 0, 𝜔 → 0
𝐸𝑛 − 𝐸𝑜
↑: Since 𝐴 = 0 for 2𝑁 − particle GS, must take 𝑱 to be
𝑙𝑖𝑚
𝑞→0 𝑱𝒒 , 𝒒 ⊥ 𝑱 (Meissner effect!). But for extra quasiparticle,
"𝐽𝑙𝑜𝑛𝑔 = 𝐽𝑡𝑟𝑎𝑛𝑠𝑣 "? If so then from f-sum rule
⇒ 𝑱 = 𝑚𝒗 ⇒ 𝜑𝐵 = 0
Wait for the next thrilling installment . . .
UT 32
CONCLUSIONS
What are implications for TQC program?
1. 𝑣 = 5 2 QHE: no implications
2. (p + ip) Fermi superfluids (e.g. Sr2RuO4)
Literature may be right.
3. But urgent to (dis)-confirm using physical
(particle-conserving) many-body wave
functions
4. More generally:
CMP / QI !