Lectures on flavor superfluidity & superconductivity

Flavor Superconductivity/ 

Superfluidity 

Fifth Aegean Summer School, Adamas, Milos, September 2009 

q 

¯q 

AdS/CFT 

[arXiv: 0810.2316] 

[arXiv: 0903.1864] 

by Matthias Kaminski (IFT-UAM/CSIC Madrid) 

in collaboration with M.Ammon, J.Erdmenger, P.Kerner (MPI Munich)

Outline 

I. Invitation: Superconductivity & Holography 

II. Review: Holographic Concepts 

III. Details: Flavored Plasma (D3/D7) 

IV. Results: Flavor Superconducting Phase (D3/D7) 

V. Discussion 

Remarks on references: 

All references are hyperlinked in this document. 

Publications quoted here are chosen because they review or explain certain 

aspects in a (pedagogical) way which is accessible to the unexperienced reader. 

Matthias Kaminski Flavor Superconductivity/Superfluidity Page 

2

I. Invitation: Conventional Superconductors 

Examples 

Color superconducting phase at high densities 

[Alford, Rajagopal, Wilczek ‘97] 

Higgs mechanism: Superconductivity of vacuum 

Weak coupling concepts 

charged condensate of Cooper-pairs 

(gauge) symmetry: electromagnetic U(1) em 

local symmetry spontaneously broken 

Goldstone bosons eaten 

[see lectures by Horowitz] 

[e.g.Weinberg] 

(photons in SC become massive Meissner effect) 

Theory: BCS (Bardeen-Cooper-Schrieffer) well established 

Superfluidity: global symmetry spontaneously broken, 

Goldstone bosons survive (become hydro modes) 

weakly gauge boundary theory 

Matthias Kaminski Flavor Superconductivity/Superfluidity Page 3

I. Invitation: Unconventional Superconductors 

[see talk by Panagopoulos] 

Typical signatures 

magnetic field expulsion (c) 

energy gap (peak at edge) 

pseudo gap 

underdoped: strong coupl. 

T c 

Figure: Tunneling spectra measured in 

high temperature 

superconductor Bi 2 Sr 2 CaCu 2 O 8+δ . 

[Renner et al., Phys. Rev. Lett. 80, 

149 - 152 (1998)] 

Theory? Pairing mechanism? Meissner effect? 

[see lectures by Sadchdev] 


I. Invitation: Unconventional Superconductors 

[see talk by Panagopoulos] 

Holographic result 

Typical signatures 

Energy gap 

Re σ 

4πw 

sigma4Piw 

magnetic field expulsion (c) 

1.4 

energy gap (peak at edge) 

1.2 

pseudo gap 

1.0 

underdoped: strong coupl. 

0.8 

0.6 

0.4 

Preview 

Figure: Tunneling spectra measured in 

0.2 

high temperature 

superconductor Bi 2 Sr 2 CaCu 2 O 8+δ . 

0.0 

0 5 10 15 20 w 

[Renner et al., Phys. Rev. Lett. 80, 

149 - 152 (1998)] 

w 

T c 

Theory? Pairing mechanism? Meissner effect? 

[see lectures by Sadchdev] 


I. Invitation: Building a Holographic SC 

Field Theory 

Gravity 

[Gubser, Pufu 0805.2960] 

What do we need? 

AdS-boundary 

condensate 

curvature 

electromag. 

charged condensate (vev) 

no source 

condensate of charge carriers 

finite temperature 

horizon 

gravity 

introduce normalizble mode 

no non-normalizable mode 

condensate hovers over horizon 

black hole 

Is this stable? 


I. Invitation: Get some intuition 

Field Theory 

Gravity 

L ∼ D ν φD ν φ ∼ (M q 2 − µ isospin 2 )φ 2 

5 

4 

VΦ 

3 

2 

1 

0 

-2 -1 0 1 2 

Φ 

charged particles condense at 

large enough chemical potential 

µ isopin ∼ M q 

strings (D3-D7) give FT charges 

cannot put infinitely many 

second brane is important 

Why do we need a non-Abelian structure? 


I. Invitation: Get some intuition 

Field Theory 

Gravity 

L ∼ D ν φD ν φ ∼ (M q 2 − µ isospin 2 )φ 2 

1 

VΦ 

0.5 

0 

-0.5 

-2 -1 0 1 2 

Φ 

charged particles condense at 

large enough chemical potential 

µ isopin ∼ M q 

strings (D3-D7) give FT charges 

cannot put infinitely many 

second brane is important 

Why do we need a non-Abelian structure? 


I. Invitation: Why so complicated? 

Bottom-up 

Top-down 

String Theory 

? 

Phenomenological 

Gravity Dual of SC 

Phenomenological 

requirements, 

SC Field Theory 


[Hartnoll, Herzog, Horowitz 0803.3295] 

study effects in clean setup 

seperate effects 

[see lectures by Horowitz] 

String Theory 

Geometric construction 

(e.g. Dp/Dq-branes) 

Holographic Dual of SC 

string theory derived 

identification of FT degrees o.f. 

‘dirty’ 

many effects at once 

Pairing mechanism! 

[0810.2316] 

[0903.1864] 


Navigator 

✓ Invitation: Superconductivity & Holography 

II. Review: Holographic Concepts 

also reviewed in [M.K. 0808.1114] 





8

II. Review: Boundary Asymptotics 

non-normalizable 

(source) 

normalizable 

(vev) 

A = A (0) + A(2) 

ρ 2 + . . . 

(ρ → ∞) 

ρ 

AdS 

QFT 

ρ bdy 

[see lectures by Argyres] 

Dictionary 

QFT FEATURE 

(energy scale) 

GEOMETRY 

(radial coord. ) 

ρ 

operator J µ field A µ (gauge) 

vev A (2) (charge) 

source A (0) (chem. pot.) 


9

II. Review: Correlators & Spectral Functions 

˜J µ 

J µ A µ 

δ 2 

w 

Gauge Theory 

Problem (strong): 

Find retarded twopoint 

function of 

flavor current 

in YM-plasma. 

Gravity problem 

(weak): 

Find solution for 

equation of motion 

of vector field 

in SUGRA. 

ret 

δA bdy δA bdy 

S Sugra 

Recipe: 

S Sugra ∼ 

∫ 

∂ ρ A ∂ ρ A 

S on-shell ∼ 

∫ 

A ∂ ρ A 

G ret ∼ lim 

ρ→∞ 

A 

A 

∂ ρ A 

A 

Thermal spectral function: 

[Son,Starinets hep-th/0205051] 

R(ω, q) = −2 ImG ret (ω, q) 


10

II. Review: Quasinormal modes 

ω n ∈ C 

Special frequencies: 

; 

(quasinormal) 

e −iωr = e −iRe{ω}r e Im{ω}r 

Example: 

G ret = N f N c T 2 

lim |Ã(ω n)| 2 = 0 

ρ→ρ bdy 

8 

[Berti et al. 0905.2975] 

( ) 

lim ρ 3 ∂ρÃ(ρ) 

ρ→ρ bdy Ã(ρ) 

R = −2 Im G ret 

R 

ω 2 

ω 1 

ω 0 

Gravity: 

quasinormal 

frequencies 

Gauge theory: 

poles of correlator 

(energy, damping, stability 

of mesonic excitations) 


II. Review: D3/D7-Brane Setup (1) 

Flavor Probe Branes (D7) 

0123 

4567 

89 

N D3 

conventional 

open/closed string duality 

SYM 

3!3 

AdS 5 

distinct approaches: 

[lectures by Kiritsis] 

[talk by Cotrone] 

N probe D7 

f 

quarks 

3!7 

flavour open/open 

string duality 

Dirac-Born-Infeld (DBI) action 

S DBI = −T 7 

∫ 

d 8 σ 

R 4 

7!7 

AdS 5 

brane 

[Karch, Katz hep-th/0205236] 

Review: [Erdmenger et al. 0711.4467] 

(√− det(P [G + B] µν + (2πα ′ ) 2 )F µν 

) 

B ≡ 0 



N c 

D3-branes 



N f 

D7-branes 

N c 

D3-branes 



N f 

D7-branes 

N c 

D3-branes 

(black) 



N f 

D7-branes 

N c 

D3-branes 

(black) 



Â ν 

⊂ U(N f ) 

N f 

D7-branes 

N c 

D3-branes 

(black) 



Â ν 

⊂ U(N f ) 

ρ 

N f 

D7-branes 

L(ρ) 

0 1 2 3 4 5 6 7 8 9 

D3 x x x x 

D7 x x x x x x x x 

8,9 

N c 

D3-branes 

(black) 

4,5,6,7 



Â ν 

⊂ U(N f ) 

ρ 

N f 

D7-branes 

L(ρ) 

0 1 2 3 4 5 6 7 8 9 

D3 x x x x 


8,9 

N c 

D3-branes 

(black) 

4,5,6,7 

Chemical potential: Â µ = δ µ0 A 0 + Ãµ 

[Kobayashi et al. hep-th/0611099] 

[Mateos et al. 0709.1225] 

(cf. therm. FT) 



Â ν 

⊂ U(N f ) 

ρ 

N f 

D7-branes 

L(ρ) 

0 1 2 3 4 5 6 7 8 9 

D3 x x x x 


8,9 

N c 

D3-branes 

(black) 

4,5,6,7 



[Mateos et al. 0709.1225] 




Mesons 

Â ν 

⊂ U(N f ) 

ρ 

N f 

D7-branes 

L(ρ) 

0 1 2 3 4 5 6 7 8 9 

D3 x x x x 


8,9 

N c 

D3-branes 

(black) 

4,5,6,7 



[Mateos et al. 0709.1225] 



Navigator 


✓ Review: Holographic Concepts 





14

III. Abelian Chemical Potential: Background 

AdS black hole metric 

pullback 

Induced metric on D7-brane 

[Myers et al. hep-th/0611099] 

f(ϱ) = 1 − ϱ H 4 

DBI action 

4 

ϱ H ˜f(ϱ) 

ϱ = 1 + ϱ 2 = r 2 + √ r 4 − r 

4 

4 , χ = cos(θ) 

H 

ϱ 4 , , 

0.0002 

p0 

0.00015 

0.0001 

0.00005 

0 

0 10 20 30 40 50 

ρ 

Can be rewritten as 

constant of motion . ˜d 


III. Abelian Chemical Potential: Fluctuations 

∫ ∣∣∣ 

DBI action: S D7 = d x√ 8 det{[g + F ] + ˜F } ∣ F µν = ∂ [µ A ν] 

Equation of motion: 

[Myers,Starinets,Thomson 0710.0334] 

} {{ } 

G 

0 = Ã′′ + ∂ ρ[ √ | det G|G 22 G 44 ] 

√ 

| det G|G 

22 

G 44 

, 

Ã ′ − G00 

G 44 ϱ H 2 ω 2 Ã 



∫ ∣∣∣ 

DBI action: S D7 = d x√ 8 det{[g + F ] + ˜F } ∣ F µν = ∂ [µ A ν] 


} {{ } 

G 

0 = Ã′′ + ∂ ρ[ √ `Curved’ Maxwell | det G|G 22 G 44 ] 

√ equations: Ã ′ − G00 

| det G|G 

22 

G 44 G 44 ϱ H 2 ω 2 Ã 

∂ µ F µν = 0 


∂ µ 

( √−GG µν G ρσ F νσ 

) 

= 0 

∂ µ 

( √−GG µν G ρσ ∂ [ν Ã σ] 

) 

= 0 

, 



DBI action: 


S D7 = 

∫ 


∣∣∣ 

d x√ 8 det{[g + F ] + ˜F } ∣ 

} {{ } 

G 

0 = Ã′′ + ∂ ρ[ √ | det G|G 22 G 44 ] 

√ 

| det G|G 

22 

G 44 

, 

F µν = ∂ [µ A ν] 

Ã ′ − G00 

G 44 ϱ H 2 ω 2 Ã 

Boundary conditions: 

shooting from 


Translation to gauge theory by duality: 

Gauge correlator: 

[Son,Starinets 

hep-th/0205051] 

Ã = (ϱ − ϱ H ) −iw [1 + iw 2 (ϱ − ϱ H) + . . . ] 

G ret = N f N c T 2 

8 

AdS/CFT 

A µ ↔ J µ 

(source) 

( 

lim ρ 3 ∂ρÃ(ρ) 

ρ→ρ bdy Ã(ρ) 

Matthias Kaminski Flavor Superconductivity/Superfluidity Page 16 

)

III. Abelian Chemical Potential: Spectral F’s 

Finite baryon density: 

[Erdmenger, M.K., Rust 0710.0334] 

( , 0) − 0 

4 

2 

0 

2 

4 

˜d = 0.25 

χ 0 = 0.1 

χ 0 = 0.5 

χ 0 = 0.7 

χ 0 = 0.8 

6 

0.0 0.5 1.0 1.5 2.0 2.5 

[Karch, O’Bannon ‘07] 

µq/mq 

Phase diagram: 

0.8 

0.6 

0.4 

0.2 

1 

˜d = 0 

0.2 

˜d = 4 

0.4 

T/ ¯M 

0.6 

˜d = 0.25 

˜d = 0.00315 

0.8 

1 

L(ϱ) = ϱ χ(ϱ) 

χ 0 = χ(ρ) ∣ ∣ 

ρ→ρH 

∼ m quark 

T 

χ = χ( ˜d , ρ) 





Lower temperature 


µq/mq 


0.8 

0.6 

0.4 

0.2 

1 

˜d = 0 

0.2 

˜d = 4 

0.4 

T/ ¯M 

0.6 

˜d = 0.25 

˜d = 0.00315 

0.8 

1 

L(ϱ) = ϱ χ(ϱ) 

χ 0 = χ(ρ) ∣ ∣ 

ρ→ρH 

∼ m quark 

T 

χ = χ( ˜d , ρ) 


( , 0) − 0 



200 

150 

100 

50 

0 

˜d = 0.25 

χ 0 = 0.8 

χ 0 = 0.94 

χ 0 = 0.962 

50 

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 

( , 0) 

140 000 

120 000 

100 000 

80 000 

60 000 

40 000 

20 000 


˜d = 0.25 

χ 0 = 0.999 

n = 0 n = 1 n = 2 n = 3 

0 

0 5 10 15 20 25 30 35 


µq/mq 


0.8 

0.6 

0.4 

0.2 

1 

˜d = 0 

0.2 

˜d = 4 

0.4 

T/ ¯M 

0.6 

˜d = 0.25 

˜d = 0.00315 

0.8 

1 

L(ϱ) = ϱ χ(ϱ) 

χ 0 = χ(ρ) ∣ ∣ 

ρ→ρH 

∼ m quark 

T 

χ = χ( ˜d , ρ) 


III. Isospin Chemical Potential: Spectral F’s 


Finite isospin density 

What has changed? 

− 0 

4000 

3000 

2000 

1000 

0 

XY 

E 3 E 3 Y X 

n = 0 

n = 0 n = 0 

XY E 3 E 3 Y X 

n = 1 n = 1 n = 1 

2 4 6 8 10 

Background action contains 

tr( √ g + F a T a ) ∼ N f × Abelian 

Diagonalize action in flavor 

space (X, Y, E) 

triplet splitting 

analog to Rho-vector meson 

analytical results & interpretation: 

[M.K. 0808.1114] 

Fluctuations in three flavor 

directions, so two new: 

X, Y (orthogonal to isospin) 

0 = X ′′ + . . . X ′ + . . . (ω − µ)X 

0 = Y ′′ + . . . Y ′ + . . . (ω + µ)Y 


III. High isospin densities: Instability! 

[Erdmenger, M.K., Kerner, Rust 0807.2663] 

New (lowest) mesonic excitation: 

New phase: 

Stability: 


First summary: QGP Phenomenology 

✓ stable mesons survive deconfinement 

✓ meson mass changes with temperature 

✓ vec-mesons are isospin triplets (QCD’s Rho-meson) 

✓ new excitation/phase: instability 

Lattice QCD: [Umeda et al.’02, ...] 

Lattice QCD: [Umeda et al.’02, ...] 

2-flavor QCD: [Splittorff et al.’03] ; Sakai-Sugimoto: [Aharony et al.’07] 

stabilize the new phase, new ground state 


Navigator 


✓ Review: Holographic Concepts 

✓ Details: Flavored Plasma (D3/D7) 




21

General idea 

IV. Flavor Superconducting Phase 

[Erdmenger, M.K., Kerner, Rust 0807.2663] 

A 3 0 = µ + d ρ 2 + . . . 

superconducting 

??? 

[Ammon, Erdmenger, M.K., Kerner 0810.2316] 

A 3 0 = µ + d3 0 

ρ 2 + . . . 

A 1 3 = 

d 1 3 

ρ 2 + . . . 

spont. breaks U(1) 



IV. Field Theory Picture 

Gravity field A 3 0 = µ + d3 0 

ρ 2 + . . . 

dual to current 

explicitly breaks 

U(2) ∼ U(1) B × SU(2) I → U(1) B × U(1) 3 

New field 

dual to current 

A 1 3 = 

d 1 3 

ρ 2 + . . . 

spontaneously breaks 

U(1) 3 ↔ U(1) em 


C 

Exercise 1: Derive the background EOMs. 

S DBI = −T D7 

∫ 

d 8 ξStr 

( √det √ 

) 

Q det (P ab [E µν + E µi (Q −1 − δ) ij E jν ] + 2πα ′ F ab ) 

Q i j = δ i j + i2πα ′ [Φ i , Φ k ]E kj , E µν = g µν + B µν 

µ, ν = 0, . . . , 9; a, b = 0, . . . , 7; i, j = 8, 9 

8,9 rotation: Φ 9 ≡ 0 ⇒ Q i j = δj 

i 

B ≡ 0 

A = A 3 0 dt τ 3 + A 1 3 dz τ 1 and 

Choose: Φ 8 || τ 0 

( ) 

⇔ 

FT: charge eigenstates are also mass eigenstates 

⇒ scalars Φ 8 , Φ 9 decouple from vectors A 

(set to zero from now on, i.e. massless quarks)



∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

= −T D7 N f 

∫ 

d 8 ξ √ √ 

−g Str 1 + g tt g ρρ (Fρ0 3 )2 (σ 3 ) 2 + g 33 g 44 (Fρ3 1 )2 (σ 1 ) 2 − c 2 g tt g 33 (F03 2 )2 (σ 2 ) 2 

F = dA + [A, A] 

(e.g. [Myers et al. hep-th/0611099])



∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

= −T D7 N f 

∫ 

d 8 ξ √ √ 


F = dA + [A, A] 

= ∂ ρ A 3 0 τ 3 dρ ∧ dt + ∂ ρ A 1 3 τ 1 dρ ∧ dz + iɛ 231 A 3 0A 1 3 dt ∧ dz 




∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

= −T D7 N f 

∫ 

d 8 ξ √ √ 


Problem 1: How to evaluate the symmetrized trace of the square 

root exactly? 

F = dA + [A, A] 


Problem 2: Non-Abelian DBI-action only known to fourth order 

in . 

α ′ 




∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

= −T D7 N f 

∫ 

d 8 ξ √ √ 


Problem 1: How to evaluate the symmetrized trace of the square 

root exactly? 

Solution: Set commutators zero, set 

symmetrized trace. 

F = dA + [A, A] 

(σ i ) 2 = 1 

inside 


Problem 2: Non-Abelian DBI-action only known to fourth order 

in α ′ . 

Solution: Expand square root to fourth order in α . 

′ 




∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

= −T D7 N f 

∫ 

d 8 ξ √ √ 


F = dA + [A, A] 





∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

= −T D7 N f 

∫ 

d 8 ξ √ √ 


= −T D7 N f 

∫ 

d 8 ξ √ √ 

−g 1 + g tt g ρρ (∂ ρ A 3 0 )2 + g 33 g 44 (∂ ρ A 1 3 )2 − c 2 g tt g 33 (A 3 0 A1 3 )2 

F = dA + [A, A] 





∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

= −T D7 N f 

∫ 

d 8 ξ √ √ 


= −T D7 N f 

∫ 

d 8 ξ √ √ 


⇒ equations of motion 

F = dA + [A, A] 





∫ 

= −T D7 N f 

∫ 

d 8 ξ Str{ √ det[g + 2πα ′ F ]} 

d 8 ξ √ √ 


= −T D7 N f 

∫ 

d 8 ξ √ √ 


⇒ equations of motion 

F = dA + [A, A] 


Legendre transformed: 

∫ 

˜S DBI = −N f T D7 d 8 ξ √ −g 

[(1 − 2c2 (A 3 0A 1 3) 2 

π 2 ρ 4 f 2 )(1 + 8(p3 0) 2 

ρ 6 f 3 − 8(p1 3) 2 ] 1 

2 

ρ 6 ˜ff ) 2 

= factor * grand potential 


IV. Thermodynamics 

Grand potential: 

Omega 

0.6 0.8 1.0 1.2 1.4 

200 

400 

T 

T 

T c 

Order parameter: 

mdc 

50 

40 

30 

20 

Critical exponent 

is 1/2. 

600 

800 

Specific heat: 

C 

400 

10 

0.70 0 

0.75 0.80 0.85 0.90 0.95 1.00 T 

SC density: 

1.0 

ds 

T 

T c 

300 

0.8 

200 

∆C 

DC 

0.6 

0.4 

Vanishes 

linearly. 

100 

0.8 1.0 1.2 1.4 

T 

T 

T c 

0.2 

0.15 0.20 0.5 0.25 0.30 0.351.00 

T 

T 

T c 


26

IV. Background field configuration 

Gravity fields: 

Conjugate momenta: 

A 3 0 

→ µ 3 0 

p 3 0 

→ d 3 0 

ρ 

ρ 

A 1 3 

p 1 3 

→ d 1 3 

ρ 

ρ 


Exercise 2: Derive the fluctuation EOMs. 

A = A 3 0 dt τ 3 + A 1 3 dz τ 1 +Ãa mdx m τ a 

Linearized fluctuation equation of motion: 

Ã 3 2 

Ã 3 2 

Ã 3 2

IV. Stability 

Poles of 

X, Y , Ã 3 2 

in complex plane: 


IV. Conductivity 

Conductivity: 

with flavorelectric current J m A m ∈ U(1) 

Re σ 

4πw 

sigma4Piw 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

σ = J E = 

Energy gap 

A(2) iA(2) 

∼ 

(0) 

∂ t A 

ωA (0) = − i ω 

[compare: spikes in 

Horowitz’ model] 

ρ 3 A ′ 

0.0 

0 5 10 15 20 w 

2 A = i ω Gret (ω, q = 0) 

w 

Only the first of these 

peaks was seen to 

second order in F. 

[Basu et al. 0810.3970] 

Our expansion to 

fourth order shows all 

peaks at zero quark 

mass! 

Peaks are higher 

order effect in F . 


IV. Higgs mechanism & Meissner effect 

Peaks at finite mass: 

Finite magnetic field: 

b0t⩵1.0 and chi0⩵0.92 

R − R0 

ImG 

600 

400 

200 

0 

200 

0 5 10 15 

w 

20 25 30 

w 

Peaks in conductivity/ 

spectral function approach 

SUSY vector meson spectrum 

at large quark mass. 

Bulk Higgs Mechanism 

generates meson mass 

H 3 3 

d 3 0 

B 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

superconducting 

(SC) 

normal phase 

(NC) 

0.15 0.2 0.25 0.3 

T 

T 

d 3 1/3 

0 

Add background field 

component: 

H 3 3 = F 3 12 = ∂ 1 A 3 2 

Induced currents in 

SC phase with H 


IV. String Theory Picture 

boundary 

ρ 

curvature 

electromag. 

E 3 ρ = F 3 ρ0 = ∂ ρ A 3 0 

B 1 ρ3 = F 1 ρ3 = ∂ ρ A 1 3 


condensate 

gravity 

ρ 

E 2 3 = F 2 30 = A 1 3A 3 0 

bulk charge 

(7-7 strings) 

E 3 ρ 

E 2 3 

A 1 3 

7-7 strings generate 

i.e. they break the U(1) 

and are thus dual to 

Cooper pairs. 

x 3 

charged horizon (3-7 strings generate ) 

Matthias Kaminski Flavor Superconductivity/Superfluidity Page 32 

A 3 0

IV. Discussion 

✓ Rich strong coupling phenomenology (QGP) 

• Resonances are vector mesons (analogous to rho-meson) 

• Vector mesons survive deconfinement 

✓ Top-down approach: direct identification of d.o.f. 

✓ Energy gap, Meissner effect, Higgs mechanism 

✓ Stringy picture of pairing mechanism 

Critical exponents 

Speed of second, fourth sound (backreact) 

Drag on D7-D7 strings 

Fermi surface? 


APPENDIX: Embeddings 

5 

2 

4 

1.5 

L 

3 

2 

L 

1 

1 

0.5 

0 0 

2.5 

1 

2 

r 

3 

˜d = 0.25 

4 

5 

0 

1 

2 3 4 5 

r 

˜d = 

10 −4 

4 

2 

2.5 

Ã0 

1.5 

1 

0.5 

1 

2 3 4 5 

ρ 

Ã0/10 −4 

2 

1.5 

1 

0.5 

1 

2 3 4 5 

ρ 


APPENDIX: Parameters 

The mass parameter m depending on the parameter χ 0 . 

m 

2.5 

2.0 

1.5 

1.0 

˜d = 0 

˜d = 0.002 

˜d = 0.1 

˜d = 0.25 

χ 0 = χ(ρ) ∣ ∣ 

ρ→ρH 

m = 

lim ρ χ(ρ) = 2m quark 

√ 

ρ→ρ bdy λT 

0.5 

0.0 

0.0 0.2 0.4 0.6 0.8 1.0 

Other relations: 

L(ϱ) = ϱ χ(ϱ) 

, 

χ 0 

ρ = 

ϱ 

ϱ H 

Near-boundary expansions: 

χ(ρ) = m ρ + c 

ρ 3 + . . . 

A 0 = µ − 1 ˜d 

ρ 2 2πα ′ + . . . 


APPENDIX: Fluctuations 

Equation of motion written out: 

0 = Ã′′ +∂ ρ ln 

( 

√ 

1 

8 ˜f 2 fρ 3 (1−χ 2 +ρ 2 χ ′ 2 ) 3/2 × 1 − 2 ˜f(1 

) 

− χ 2 )(∂ ρ A 0 ) 2 

Ã ′ +8w ˜f 2 1 − χ 2 + ρ 2 χ ′ 2 

f 2 (1 − χ 2 + ρ 2 χ ′2 ) f 2 ρ 4 (1 − χ 2 ) 

Ã 

ρ = 

ϱ 

˜f(ϱ) = 1 + ϱ H 4 

ϱ 

, , , L(ϱ) = ϱ χ(ϱ) w = ω 

H 

, 

ϱ 4 

ϱ 4 f(ϱ) = 1 − ϱ H 4 

2πT 

The D7-brane embedding 

and gauge field component 

are given numerically. 

χ(ρ) 

A 0 (ρ) 

A 0 ≡ A t 


APPENDIX: Extension of the correspondence 

Original AdS/CFT 

correspondence 

Gauge QCD N = 4 SuperYangMills thermal Yang-Mills 

Gravity ? Type II Sugra in AdS 

Gauge 

theory 

symmetry 

Relations 

Universality 

nonconf. 

non- 

SUSY 

✓c 

✓c 

๏c 

๏c 

g Y M 2 = g s 

R 4 

AdS Schwarzschild 

black hole (D3/D7) 

TypeII Sugra in AdS 

Schwarzschild b.h. 

T 

µ B , µ I 

(α ′ ) 2 = 4πN cg s ≡ λ 

✓c 

✓c 


A 0 (ρ) 


37

Lectures on flavor superfluidity & superconductivity

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