Supersymmetric BCS superconductivity Alejandro Barranco L´ opez arXiv:1204.4157 - Alejandro Barranco, Jorge Russo October 18, 2012 Alejandro Barranco L´opez Supersymmetric BCS superconductivity 1 / 13
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
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 2/13
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
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 3/13
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.
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 4/13
1 Review of superconductivity and motivation
2 Review of Bardeen-Cooper-Schrieffer theory of superconductivity
3 The SUSY case
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 5/13
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
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 6/13
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
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 7/13
1 Review of superconductivity and motivation
2 Review of Bardeen-Cooper-Schrieffer theory of superconductivity
3 The SUSY case
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 8/13
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
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 9/13
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 . . .
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 10/13
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.
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 11/13
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
∆
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 12/13
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 .
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 13/13
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
Alejandro Barranco Lopez Supersymmetric BCS superconductivity 13/13