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.