Post on 31-Aug-2020
transcript
Condensed Matter Physics 2016 Lectures 29/11, 2/1: Superconductivity
1. Attractive electron-electron interaction 2. 103 years of superconductivity 3. BCS theory 4. Ginzburg-Landau theory 5. Mesoscopic superconductivity 6. Josephson effect
References: Ashcroft & Mermin, 34 Taylor & Heinonen, 6.5, 7.1-7.5, 7.7
blackboard
1911: Discovery of superconductivity
Resistivity R = 0 in Hg below Tc = 4.2 K
H. Kamerlingh Onnes (Nobel prize 1913)
”Mercury has passed into a new state, which on account of its extraordinary electrical properties may be called the superconducting state…”
”Superconductivity can also be destroyed by a magnetic field larger than the critical field …” (Kamerlingh Onnes 1914)
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
1933: Meissner-Ochsenfeld effect
W. Meissner
A superconductor is a perfect diamagnet: Expels magnetic fields from the interior.
An applied magnetic field induces a superconducting current on the surface of the superconductor. This creates an induced magnetic field which compensates the applied field.
1933: Meissner-Ochsenfeld effect
W. Meissner
A superconductor is a perfect diamagnet: Expels magnetic fields from the interior.
An applied magnetic field induces a superconducting current on the surface of the superconductor. This creates an induced magnetic field which compensates the applied field.
London penetration depth
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
1933: Meissner-Ochsenfeld effect
W. Meissner
A superconductor is a perfect diamagnet: Expels magnetic fields from the interior.
An applied magnetic field induces a superconducting current on the surface of the superconductor. This creates an induced magnetic field which compensates the applied field.
”type I”
London penetration depth
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
Superconductor = ”perfect conductor” and ”perfect diamagnet”
1935: Discovery of type II superconductors
L. Shubnikov
B
TTc
Bc1
Bc2
Meissner phase
mixed (vortex) phase
normal phase
* The magnetic field penetrates as quantized * flux lines (each carrying flux ). * Supercurrents around the flux lines produce
vortices. Flux pinning (or quantum locking) makes possible stable ”magnetic levitation”.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e**
J. Rjabinin & L. Shubnikov
http://www.ted.com/talks/boaz_almog_levitates_a_superconductor
Early theories of superconductivity
London & London (1935): Model of the Meissner effect using classical electromagnetism
Ginzburg & Landau (1950): Macroscopic order parameter theory (”effective field theory”)
Experimental hint: the isotope effect (1950)E. Maxwell, 1950 1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
BCS: Microscopic theory of superconductivity (1957)
For their jointly developedtheory of superconductivity,called BCS theory…
Basic (”classical”) idea: An electron moving through the lattice interacts with the ions and creates a region of relative positive charge. This attracts another electron. Retarded effect!
The first electron can travel a distance of ~104 Å before the second electron gets attracted by the net local positive charge from the perturbed ions. Thus: negligible effect from the e-e repulsion between the electrons (assuming a screened Coulomb potential).
Basic (”classical”) idea: An electron moving through the lattice interacts with the ions and creates a region of relative positive charge. This attracts another electron. Retarded effect!
The first electron can travel a distance of ~104 Å before the second electron gets attracted by the net local positive charge from the perturbed ions. Thus: negligible effect from the e-e- repulsion between the electrons (assuming a screened Coulomb potential).
Quantization: Attractive electron-electron
interaction mediated by phonons…
… back to the blackboard!
Summary from last lecture:
Fröhlich Hamiltonian
Schrieffer-Wolff transformation, keep only quartic interaction terms
1
2
3
4
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
Fröhlich Hamiltonian
Schrieffer-Wolff transformation, keep only quartic interaction terms
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p 1 3
42
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
Fröhlich Hamiltonian
Schrieffer-Wolff transformation, keep only quartic interaction terms
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
Attractive electron-electron interaction for
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
”Cooper instability” (L. Cooper, 1956)
Formation of Cooper pairs (weakly bound state of of two electrons, making up a boson)
Lowest energy when CM momentum = 0: To experience the weak attractive interaction, the electrons can’t be too far from each other: (antisymmetric spin wave function)
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
1
2
3
4
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0 +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k�q)2 � (~!q)2
c†k0+q,s0
c†k�q,s0ck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.
HBCS =X
k
✏k(c†kck + c†
�kc�k)�X
k,k0Vkk0c†
k0c†�k0c�kck
1
Hc(T ) = Hc(0)(1� (T/Tc)2)
B(x) = B(0) exp(�x/�L)
�0 = h/2e
Tc ⇠ 1/pM Bc ⇠ 1/
pM
H 0= H0e +
1
2
X
k,k0,q;s,s0
|Mq |22~!q
(✏k � ✏k0 )2 � (~!q)2c†k0+q,s0
c†k�q,sck,sck0,s0
+ Hp�p +He�e (1)
H = H0 +He�p
H0e
|✏k � ✏k�q | < ~!q
k0= �k s0 = �s
Vkk0q
k � k k � q ⌘ k0 � k + q ⌘ �k0
1
2
X
k,k0,q;s,s0
Vkk0qc†k0+q,s0
c†k�q,sck,sck0,s0
!X
k,k0Vkk0c†
k0c†�k0c�kck (2)
where c†k ⌘ c†k", c†
�k ⌘ c†�k#
, etc.