supersymmetric bcs superconductivity -...

Supersymmetric BCS superconductivity

Alejandro Barranco Lopez

arXiv:1204.4157 - Alejandro Barranco, Jorge Russo

October 18, 2012

Alejandro Barranco Lopez Supersymmetric BCS superconductivity 1/13

Outline

1 Review of superconductivity and motivation

2 Review of Bardeen-Cooper-Schrieffer theory of superconductivity

3 The SUSY case


Review of superconductivity

I London:

~J ∝ ~A ⇒E ∝ ∂tJ

∇2B ∝ B

I Landau-Ginzburg: U(1) spontaneous symmetry breaking when T < Tc

FLG = α(T −Tc)|∆|2+β

2|∆|4 + . . .

I Bardeen-Cooper-Schrieffer: Cooper pairs


Motivation

I High Tc superconductors:

The pairing mechanism isnot well understood

it involves strong coupling.

AdS/CFT is an new tool to study strongly coupled field theories.

Holographic Superconductor

Gravity Superconductor

Black Hole T

At µ“Hair” ∆

U(1) gauge U(1) global

This is typically supersymmetric.

Hartnoll, Herzog, Horowitz, Holographic superconductors

I Possible applications to real condensed matter systems with fermionand scalar quasiparticle excitations.

I SUSY softens divergences.




3 The SUSY case


Review of relativistic BCS

L = iψ†σµ∂µψ −mψ†ψ + g 2(ψψ)(ψ†ψ†)

I Add chemical potential for some conserved charge:“gauge field” A0 = µ ⇒ Fermi Surface

Polchinski, Effective Field Theory and the Fermi Surface

I Adding temperature: Euclideum with φ(β, x) = φ(0, x)ψ(β, x) = −ψ(0, x)

I Perform a Hubbard-Stratonovich transformation: ∆ = (ψ†ψ†)

L = ψ†σ0(∂τ+µ)ψ+iψ†σi∂ iψ+mψ†ψ−g 2∆(ψψ)−g 2∆∗(ψ†ψ†)+g 2|∆|2

Classical potential Vcl


Review of relativistic BCS: V1−loop

Energy eigenvalues: ω± =√

(ω0 ± µ)2 + |∆|2 ω0 ≡√

p2 +m2

F = g2∆2 −→ Classical potential

+∫ ΛD d3p

(2π)3(2ω0(p) − ω−(p)− ω+(p)) −→ Coleman-Weinberg

− 2β

∫ d3p

(2π)3

(

log(1 + e−βω−(p)) + log(1 + e−βω+(p))

)

−→ Thermal

100 200 300 400 500D0

-200 000

200 000

400 000

600 000

800 000

1´106

F




3 The SUSY case


SUSY BCS: µ for U(1)B

K (Φ,Φ†) = Φ†xΦx +Φ†

yΦy + g 2(Φ†xΦx)

2 + g 2(Φ†yΦy )

2 W = mΦxΦy

LS = (1 + 4g2|φx |2)∂µφ

∗x ∂

µφx −m2|φy |2

1 + 4g2|φx |2+ (x ↔ y)

LF = i(1 + 4g2|φx |2)(ψ†

x σµ∂µψx ) + 4ig2(ψ†

x σµψx )φ

∗x ∂µφx +

g2(ψxψx )(ψ†xψ

†x )

1 + 4g2|φx |2

+

(

2mg2φyφ∗x

1 + 4g2|φx |2(ψxψx )−

1

2mψxψy + h.c.

)

+ (x ↔ y) .

U(1)B baryonic symmetry: Φx and Φy have opposite charges.

∆x = −∆y ≡ ∆ωF =

√

(√

p2 +m2 ± µ)2 + 4g 4∆2

ωS =√

4g 4∆2 +m2 + p2 ± µ


SUSY BCS: µ for U(1)B

ωF =

√

(√

p2 +m2 ± µ)2 + 4g 4∆2 ωS =√

4g 4∆2 +m2 + p2 ± µ

Fermi surface defined by the minimum of ωF−:√

p2F +m2 = µ ⇒ µ > m

ωS < 0 Vthermal =1

β

∑

ω

∫

d3p

(2π)3log(1− e−βωS ) is ill defined

The occupation number of scalars with zero momentum goes to ∞ as µ→ m

Bose-Einstein Condensation, which spoils BCS mechanism.

Try putting the theory on S1 × S3:

I Scalars couple to curvature ⇒ Extra mass term:(

m2 + R−2)

(φ∗xφx)

I The scalar would be negligible if 1/R > Λ.

I The integral over momentum is replaced by a discrete sumoriginating from the Kaluza-Klein modes of S3.

Scalars: p2 −→ l(l + 2)R−2

Fermions: p2 −→ (l + 1/2)2R−2 l = 0, 1, 2 . . .


R-symmetry

[Qα,R ] = Qα

[Q†α,R ] = −Q

†α

Rψ = Rφ − 1

I Rφ = 0 and Rψ = −1 allows us to introduce µ for only fermions⇒ We avoid BEC but RW 6= 2 ⇒ m = 0.

I R-symmetry must be non-anomalous to have a well defined µ.

∑

fermions

R3 = 0

I Add more chiral fields e.g. ΦZ1,ΦZ2, with RφZ= 2 and canonical

Kahler potential.

I The new scalars couple to µ⇒ BECZ sector is completely decoupled from X ,Y sector.


SUSY BCS: µ for U(1)R

Energy eigenvalues:ω2S x,y = p2 +

4g4∆2x,y

1+4g2v2x,y

ω2F x,y = (p ± µ)2 +

4g4∆2x,y

(1+4g2v2x,y)

2

Classical Potential: Vcl = g 2(1 + 4g 2v2x )|∆x |

2 + (x ↔ y)

10 20 30 40 50 60D

-500

500

1000

1500

2000

2500

F

0 1 2 3 4 5 6

0

5

10

15

20

25

30

T

∆


Conclusions

I We have been able to implement BCS theory in a SUSY field theorywith U(1)R symmetry,

I Scalars make the phase transition first order rather than secondorder.

I SUSY softens divergences VCW ∼ Λ2D → log ΛD .


Conclusions

I We have been able to implement BCS theory in a SUSY field theorywith U(1)R symmetry,

I Scalars make the phase transition first order rather than secondorder.

I SUSY softens divergences VCW ∼ Λ2D → log ΛD .

Thank you for your attention


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