Holographic MetalsValentina Giangreco Marotta Puletti

Chalmers Institute of Technology

XIII Marcel Grossmann MeetingStockholm, July 5th, 2012

in collaboration with S. Nowling, L. Thorlacius, and T. Zingg

based on 1110.4601 [hep-th] + work in progress (and also on 1011.6261 [hep-th])

lördag 7 juli 12

Overview

► Holography + metals:

apply AdS/CFT to Condensed Matter physics (AdS/CMT)

► What do I mean by AdS/CFT here?

- gravity in (asymptotical) (d+1)-AdS (BULK) ⟺ QFT in d 𝐑 (BOUNDARY) - weakly/strongly coupled duality - radial coordinate z ⟺ energy scale

► Why?

- strongly coupled QFT and perturbative methods fail here: in the large N limit and when the QFT is strongly coupled:

where L = AdS curvature radius and Newton’s constant is

- AdS/CFT can give a geometrical translation of these systems

- new geometries

L2

2⇠ N# >> 1

2 = 8⇡GN

lördag 7 juli 12

Outline

► Introductions & Motivations (Fermi liquids vs Non-Fermi liquids)

► Introduction to Friedel oscillations

► Introduction to electron star geometry

► Friedel oscillations in the electron star geometry

► Electron star vs AdS hard wall geometry

► Summary plus Future

lördag 7 juli 12

Intro and motivations: The “broad” picture

► Goal?

- Holographic description of (2+1) strongly correlated charged fermions at finite density (µ) and at very low temperature (or zero) T (T<< µ)

- Universality class for such systems via AdS/CFT

- Which are the “good” ingredients in the bulk? Bottom-Up approach

?

QFT at µ

Bulk (IR) Boundary (UV)

z 0∞

lördag 7 juli 12

► Why these systems?

Non-Fermi liquid (NFL):

- they are not described by Landau-Fermi theory

- strange metallic behavior: resistivity ～ T (ex. in high-Tc superconductors)

(vs. Fermi liquid: resistivity ～ T2 )

interaction between two electronic quasiparticles

V (r, t) = �g2⌘m(r, t)s · s⌅ . (13)

Here s, s⌅ are spins, g is a coupling and ⌘m is the magnetic susceptibility. The susceptibility

becomes large at the onset of antiferromagnetism. When s and s⌅ form a singlet it turns

out that (13) is repulsive near the origin but then oscillates in sign with a period of order

the lattice spacing. Thus there is an attractive interaction between the quasiparticles when

a finite distance apart. This forces the resulting ‘Cooper pair’ operator to have a nonzero

angular momentum (� = 2), leading to a d-wave superconductor, as is observed.

The interaction (13) only makes sense if there are weakly interacting quasiparticles.

This picture seems to work at some level for the materials at hand. However, given the

nearby quantum critical point and the associated non-Fermi liquid behaviour, observed in

many heavy fermion compounds, it might be instructive to have a more nonperturbative

approach [17].

The cuprate high-Tc superconductors typically have the following phase diagram as a

function of temperature and hole doping (that is, reducing the number of conducting elec-

trons per Cu atom in the copper oxide planes by chemical substitution, e.g. La2�xSrxCuO4)

Figure 4: Schematic temperature and hole doping phase diagram for a high-Tc cuprate.

There are antiferromagnetic and a superconducting ordered phases. Figure taken from [18].

This phase diagram is obviously similar to that of the heavy fermion compounds in

figure 3. One important di�erence is that the antiferromagnetic phase is separated from

12

interaction between two electronic quasiparticles

V (r, t) = �g2⌘m(r, t)s · s⌅ . (13)

Here s, s⌅ are spins, g is a coupling and ⌘m is the magnetic susceptibility. The susceptibility

becomes large at the onset of antiferromagnetism. When s and s⌅ form a singlet it turns

out that (13) is repulsive near the origin but then oscillates in sign with a period of order

the lattice spacing. Thus there is an attractive interaction between the quasiparticles when

a finite distance apart. This forces the resulting ‘Cooper pair’ operator to have a nonzero

angular momentum (� = 2), leading to a d-wave superconductor, as is observed.

The interaction (13) only makes sense if there are weakly interacting quasiparticles.

This picture seems to work at some level for the materials at hand. However, given the

nearby quantum critical point and the associated non-Fermi liquid behaviour, observed in

many heavy fermion compounds, it might be instructive to have a more nonperturbative

approach [17].

The cuprate high-Tc superconductors typically have the following phase diagram as a

function of temperature and hole doping (that is, reducing the number of conducting elec-

trons per Cu atom in the copper oxide planes by chemical substitution, e.g. La2�xSrxCuO4)

Figure 4: Schematic temperature and hole doping phase diagram for a high-Tc cuprate.

There are antiferromagnetic and a superconducting ordered phases. Figure taken from [18].

This phase diagram is obviously similar to that of the heavy fermion compounds in

figure 3. One important di�erence is that the antiferromagnetic phase is separated from

12

picture by Hartnoll ’09, and by S. Kasahara et al. ’10

(in the 2nd picture: resistivity ～ Tα )

Introduction and Motivations

lördag 7 juli 12

► Why these systems?

Non-Fermi liquid (NFL):

- they are not described by Landau-Fermi theory

- strange metallic behavior: resistivity ～ T (ex. in high-Tc superconductors)

(vs. Fermi liquid: resistivity ～ T2 )

TSDW Tc

T0

2.0

0

!"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF

Resistivity⇠ ⇢0 +AT↵

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)Tuesday, April 17, 2012

TSDW Tc

T0

2.0

0

!"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF

Resistivity⇠ ⇢0 +AT↵

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)Tuesday, April 17, 2012

picture by Hartnoll ’09, and by S. Kasahara et al. ’10

(in the 2nd picture: resistivity ～ Tα )

Introduction and Motivations

lördag 7 juli 12

Intro and Motivations: Intermezzo

► Recall: Landau-Fermi theory describes Fermi liquids (FL):

- Weakly interacting quasi-particles ( “dressed electrons”)

- Fermi surface

- very robust (up to BCS): IR free fixed point

G�1(! = 0, k = kF ) = 0 G : fermionic Green’s function

► But... not Non-Fermi liquids

- are not described by quasi-particles

- but still central role of Fermi surface

kF

► Recall: Fermi surface is a locus of points where the fermionic Green’s function has a pole in the momentum space:

lördag 7 juli 12

Intro and Motivations

- Quantum Critical Point (QCP) and Quantum Phase Transition (ex. FL can be tuned to a QCP by doping the material) [Broun ’08]

- What’s special about QCT? 1) strongly interacting 2) scale invariant ... it sounds good!

- RG flow from free fixed point (Landau-Fermi theory) to a non-trivial fixed point (Non-Fermi liquid)

- NFL: anisotropic scaling behavior:

where s = dynamical critical exponent

► What’s behind?

! ! �! , k ! �sk

interaction between two electronic quasiparticles

V (r, t) = �g2⌘m(r, t)s · s⌅ . (13)

Here s, s⌅ are spins, g is a coupling and ⌘m is the magnetic susceptibility. The susceptibility

becomes large at the onset of antiferromagnetism. When s and s⌅ form a singlet it turns

out that (13) is repulsive near the origin but then oscillates in sign with a period of order

the lattice spacing. Thus there is an attractive interaction between the quasiparticles when

a finite distance apart. This forces the resulting ‘Cooper pair’ operator to have a nonzero

angular momentum (� = 2), leading to a d-wave superconductor, as is observed.

The interaction (13) only makes sense if there are weakly interacting quasiparticles.

This picture seems to work at some level for the materials at hand. However, given the

nearby quantum critical point and the associated non-Fermi liquid behaviour, observed in

many heavy fermion compounds, it might be instructive to have a more nonperturbative

approach [17].

The cuprate high-Tc superconductors typically have the following phase diagram as a

function of temperature and hole doping (that is, reducing the number of conducting elec-

trons per Cu atom in the copper oxide planes by chemical substitution, e.g. La2�xSrxCuO4)

Figure 4: Schematic temperature and hole doping phase diagram for a high-Tc cuprate.

There are antiferromagnetic and a superconducting ordered phases. Figure taken from [18].

This phase diagram is obviously similar to that of the heavy fermion compounds in

figure 3. One important di�erence is that the antiferromagnetic phase is separated from

12

interaction between two electronic quasiparticles

V (r, t) = �g2⌘m(r, t)s · s⌅ . (13)

Here s, s⌅ are spins, g is a coupling and ⌘m is the magnetic susceptibility. The susceptibility

becomes large at the onset of antiferromagnetism. When s and s⌅ form a singlet it turns

out that (13) is repulsive near the origin but then oscillates in sign with a period of order

the lattice spacing. Thus there is an attractive interaction between the quasiparticles when

a finite distance apart. This forces the resulting ‘Cooper pair’ operator to have a nonzero

angular momentum (� = 2), leading to a d-wave superconductor, as is observed.

The interaction (13) only makes sense if there are weakly interacting quasiparticles.

This picture seems to work at some level for the materials at hand. However, given the

nearby quantum critical point and the associated non-Fermi liquid behaviour, observed in

many heavy fermion compounds, it might be instructive to have a more nonperturbative

approach [17].

The cuprate high-Tc superconductors typically have the following phase diagram as a

function of temperature and hole doping (that is, reducing the number of conducting elec-

trons per Cu atom in the copper oxide planes by chemical substitution, e.g. La2�xSrxCuO4)

Figure 4: Schematic temperature and hole doping phase diagram for a high-Tc cuprate.

There are antiferromagnetic and a superconducting ordered phases. Figure taken from [18].

This phase diagram is obviously similar to that of the heavy fermion compounds in

figure 3. One important di�erence is that the antiferromagnetic phase is separated from

12

k ⇠ !s

lördag 7 juli 12

Intro and Motivations

- Quantum Critical Point (QCP) and Quantum Phase Transition (ex. FL can be tuned to a QCP by doping the material) [Broun ’08]

- What’s special about QCT? 1) strongly interacting 2) scale invariant ... it sounds good!

- RG flow from free fixed point (Landau-Fermi theory) to a non-trivial fixed point (Non-Fermi liquid)

- NFL: anisotropic scaling behavior:

where s = dynamical critical exponent

► What’s behind?

! ! �! , k ! �sk

QCT

T

ggc

h�i = 0 h�i 6= 0

QCT

T

ggc

h�i = 0 h�i 6= 0

k ⇠ !s

lördag 7 juli 12

Intro and motivations: Our work: Goals...

► Recall the Big Goal: geometries to holographically model Non-Fermi liquids

► Goal: - Fermi surface is our key but how is it encoded in a holographic geometry? - Which are the “good” ingredients in the bulk? Test holographic models on the market! - How do we test the holographic models? - When “good” features in the bulk stay “good” in the boundary (see Focus I)?

► Focus 1: - When do bulk Fermi features (FS) induce boundary Fermi features (FS)?

?

Bulk(IR)

Boundary(UV)

QFT at µ

z

?

Bulk(IR)

Boundary(UV)

QFT at µ

z

lördag 7 juli 12

Intro and motivations: Our work: Goals...

► Recall the Big Goal: geometries to holographically model Non-Fermi liquids

► Goal: - Fermi surface is our key but how is it encoded in a holographic geometry? - Which are the “good” ingredients in the bulk? Test holographic models on the market! - How do we test the holographic models? - When “good” features in the bulk stay “good” in the boundary (see Focus I)?

► Focus 1: - When do bulk Fermi features (FS) induce boundary Fermi features (FS)?

Bulk(IR)

Boundary(UV)

QFT at µ

zBulk(IR)

Boundary(UV)

QFT at µ

z

lördag 7 juli 12

Intro: Friedel oscillations► Strategy?

- isolate properties related to Fermi surface: Friedel oscillations- search for such a signal in holographic models

► What?

- oscillations in configuration space present in static response functions, like current current correlation functions, at very low temperature (also T=0)- due to the presence of a sharp Fermi surface- present in FL andNFL

► Example:

2 kf singularity!

sinx

x

! FT : ✓(k)sinx

x

! FT : ✓(k)

lördag 7 juli 12

Intro: Friedel oscillations► Strategy?

- isolate properties related to Fermi surface: Friedel oscillations- search for such a signal in holographic models

► What?

- oscillations in configuration space present in static response functions, like current current correlation functions, at very low temperature (also T=0)- due to the presence of a sharp Fermi surface- present in FL andNFL

► Example:

2 kf singularity!

Non relativistic degenerate fermions in (2+1) dimensions

1 2 3 4 5 6qêk f

0.05

0.10

0.15

c HqLm e2

h�⇢(k)i = �(k)�At(k) , �(k) ⇠ h⇢(�k)⇢(k)i

⇢ = density current

Non relativistic degenerate fermions in (2+1) dimensions

1 2 3 4 5 6qêk f

0.05

0.10

0.15

c HqLm e2

h�⇢(k)i = �(k)�At(k) , �(k) ⇠ h⇢(�k)⇢(k)i

⇢ = density current

lördag 7 juli 12

Things you have to remember about AdS/CFT

► Minimal Dictionary [Maldacena ’97] [Witten ’98] [Gubser et al ’98]

- Dynamical field with boundary value

- Maxwell field

Bulk Boundary (z = 0)

- Linear response - Linear response

Aµ(z) ⇠ A(0)µ + zA(1)

µ + . . . z ! 0

A(0)µ = µ , A(1)

µ =< J t >

Aµ

�0� - Operator sourced by

- U(1) global current with µ

O �0

- Partition Function - Expectation value

Zbulk[� ! �0] = hexp

✓i

Z�0O

◆i

lördag 7 juli 12

Electron star [Hartnoll et al. ’09] [Arsiwala et al. ’10] [Hartnoll&Tavanfar’10]

► Ingredients in the bulk:

- Maxwell gauge field:

- High density of fermions (Oppenheimer-Volkoff approx)

- T=0 perfect fluid of free charged fermions with mass m and charge e in the bulk

- fermions are in a local Lorentz frame (LL) at each value of z:

- fluid thermodynamic variables: p(z), ρ(z) and σ(z) with

- asymptotically AdS metric:

µloc

(z) =Atp�gtt

µloc

(z)

ds

2 =L

2

z

2(�f(z)dt2 + g(z)dz2 + dx

2 + dy

2)

At =eL

h(z) lim

z!0At = µ|@

Bulk (IR) Boundary (UV)

QFT at µ

z

0∞

++ ++++

IR Lifshitz geometry

Bulk (IR) Boundary (UV)

QFT at µ

z

0∞

++ ++++

IR Lifshitz geometry

lördag 7 juli 12

Electron star [Hartnoll et al. ’09] [Arsiwala et al. ’10] [Hartnoll&Tavanfar’10]

► Ingredients in the bulk:

- Maxwell gauge field:

- High density of fermions (Oppenheimer-Volkoff approx)

- T=0 perfect fluid of free charged fermions with mass m and charge e in the bulk

- fermions are in a local Lorentz frame (LL) at each value of z:

- fluid thermodynamic variables: p(z), ρ(z) and σ(z) with

- asymptotically AdS metric:

µloc

(z) =Atp�gtt

µloc

(z)

ds

2 =L

2

z

2(�f(z)dt2 + g(z)dz2 + dx

2 + dy

2)

At =eL

h(z) lim

z!0At = µ|@

lördag 7 juli 12

Electron star [Hartnoll et al. ’09] [Arsiwala et al. ’10] [Hartnoll&Tavanfar’10]

► The action:

S = SHE + SM + Sfluid =1

22

Zd

4x

p�G

✓R+

6

L

2

◆� 1

4e2

Zd

4x

p�GFµ⌫ F

µ⌫ +

Zd

4x

p�Gp

[Hartnoll &Tavanfar’10] [Schutz ’70]

► Properties of ES: Why this model?

- back-reaction of the metric w.r.t. fluid in a controlled approx

- in the interior IR emergent Lifshitz scaling ⟹ in the boundary emergent critical exponent s

- s is determined by the eom:

f = z

�2s+2, g = g1 , h = h1 ) (z, x, y) ! �(z, x, y) , t ! �

st

Bulk (IR) Boundary (UV)

QFT at µ

z

0∞

++ ++++

IR Lifshitz geometry

Bulk (IR) Boundary (UV)

QFT at µ

z

0∞

++ ++++

IR Lifshitz geometry

s = s(e,m)

lördag 7 juli 12

Electron star [Hartnoll et al. ’09] [Arsiwala et al. ’10] [Hartnoll&Tavanfar’10]

► The action:

S = SHE + SM + Sfluid =1

22

Zd

4x

p�G

✓R+

6

L

2

◆� 1

4e2

Zd

4x

p�GFµ⌫ F

µ⌫ +

Zd

4x

p�Gp

[Hartnoll &Tavanfar’10] [Schutz ’70]

► Properties of ES: Why this model?

- back-reaction of the metric w.r.t. fluid in a controlled approx

- in the interior IR emergent Lifshitz scaling ⟹ in the boundary emergent critical exponent s

- s is determined by the eom:

f = z

�2s+2, g = g1 , h = h1 ) (z, x, y) ! �(z, x, y) , t ! �

st

2 4 6 8 10 z

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s, r, p

2 4 6 8 10 z

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s, r, p

s = s(e,m)

lördag 7 juli 12

Go ahead I!

► Summary so far:

- Friedel oscillations as a diagnostic in order to detect signals of Fermi surfaces in electron star

- Compute static current-current correlation functions, which are bosonic observables

- but we want to detect a fermionic structure... 1-loop diagram = (1/N corrections)!

► Focus 1I: - we will use “internal’’ d.o.f. to study the low-energy dynamics:

- very different from the probe fermion approximation [Liu,McGreevy,Vegh ’09],[Hartnoll, Hofman, Tavanfar ’11],[Cubrovic, Liu, Schalm, Sun, Zaanen ’11]

lördag 7 juli 12

► How do we proceed?

- Insert a “disturbance” in the system: shear modes

- Compute the correlators with 1/N corrections (induced magnetic effect):

- introduce an effective term in the action to take into account loop effects: polarization tensor Π

Things you have to remember about AdS/CFT II

O(x)O(y)

QFT

Gravity

hO(x)O(y)i

► Minimal Dictionary II [Maldacena ’97] [Witten ’98] [Gubser et al ’98] [Son et al. ‘02]

⟺ compute the on-shell action and run to the boundary (bulk-to-bdry propagator)

Zbulk[� ! �0] = hexp✓i

Z�0O

◆i

(�Ax

, �gtx

, �gyx

, �ux

)e�i!t+ikyy

hJx

(ky

)Jx

(ky

)i

lördag 7 juli 12

Go ahead II!

► Set-up

- the effective action

SPol

=e2

2

Zdzdz0d2k

p|g(z)||g(z0)|�A

µ

(z,�k)⇧µ⌫

CG,EC

(z, z0, k)�A⌫

(z0, k)

⇧µ⌫CG,EC is the coarse grained polarization tensor

- the fermionic loop: Recall the fermions live in a local Lorentz frame!

⇧µ⌫CG,flat(z, z

0, kL.L.) = �(z � z0)⇧µ⌫(0, kL.L.)

- we need to project all the quantities entering in Π into the LL frame: kL.L. =k

pgyy

= k z

- use for Π the expression for (3+1) QED

singular part: ⇧µ⌫(0, kL.L.) ⇠ Nµ⌫(kL.L.) ln

✓kL.L. � 2kfkL.L. + 2kf

◆+ ...

crucial point here: the quantities entering in the singular part

kL.L. = z k , kF (z) =p

µloc

(z)2 �m2 =

s(z At(z))2

f(z)�m2

⟹ the Fermi momentum depends on z: there is a different bulk FS for each point in the radial direction!

lördag 7 juli 12

(Absence of!) Friedel oscillations in the electron star

► Results we do NOT have boundary Fermi surfaces! :(

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

B. H. HTêTc=.51LE. C. HTêTc=.51LB. H. HTêTc=.03LE. C. HTêTc=.03L

► Key-points:

- continuum of bulk Fermi surfaces, each at each point z

- each bulk Fermi surface is different: set by different value of the local

- the bulk FS are very localized:... this is the ultimate meaning of the OV approx!

► Lesson:

each different FS at a different radius will not act coherently ⇒ summing from the deep interior (IR) to the boundary (UV) they are just smeared out! cf. [Kulaxizi, Parnachev ’08] for D4 − D8 − D8

µloc

(z)

lördag 7 juli 12

Friedel oscillations in the AdS hard wall geometry

► Ingredients in the bulk: [Sachdev ’11]

- Maxwell gauge field:

- low density of fermions = bulk single-particle wave function

- AdS metric frozen and truncated at :

At =eL

h(z) lim

z!0At = µ|@

ds

2 =L

2

z

2(�dt

2 + dz

2 + dx

2 + dy

2)zm

?

Bulk(IR)

Boundary(UV)

QFT at µ

z

?

Bulk(IR)

Boundary(UV)

QFT at µ

z

► The action

S = SM + SD = � 1

4e2

Zd

4x

p�GFµ⌫ F

µ⌫ +

Zd

4x

p�Gi

� �µDµ +m

�

lördag 7 juli 12

Friedel oscillations in the AdS hard wall geometry

► Ingredients in the bulk: [Sachdev ’11]

- Maxwell gauge field:

- low density of fermions = bulk single-particle wave function

- AdS metric frozen and truncated at :

At =eL

h(z) lim

z!0At = µ|@

ds

2 =L

2

z

2(�dt

2 + dz

2 + dx

2 + dy

2)zm

► The action

S = SM + SD = � 1

4e2

Zd

4x

p�GFµ⌫ F

µ⌫ +

Zd

4x

p�Gi

� �µDµ +m

�

lördag 7 juli 12

Friedel oscillations in the AdS hard wall geometry

► Ingredients in the bulk: [Sachdev ’11]

- Maxwell gauge field:

- low density of fermions = bulk single-particle wave function

- AdS metric frozen and truncated at :

At =eL

h(z) lim

z!0At = µ|@

ds

2 =L

2

z

2(�dt

2 + dz

2 + dx

2 + dy

2)zm

► The action

S = SM + SD = � 1

4e2

Zd

4x

p�GFµ⌫ F

µ⌫ +

Zd

4x

p�Gi

� �µDµ +m

�

lördag 7 juli 12

► Results: we do have boundary Fermi surface! :)

1 2 3kxêk f

-2.005

-2.035

-2.065

XrH-kxLrHkxL\Tot

1 2 3kxêk f

-2.005

-2.035

-2.065

XrH-kxLrHkxL\Tot

Friedel oscillations in the AdS hard wall geometry

► Key-points:

- deconfining geometry

- different role of the fermions: here treated QM!

- discreteness of the bulk Fermi surfaces (due to the gapped spectrum)

- delocalization of bulk Fermi surfaces: wave-functions fill all the way the space-time, they know all the geometry!

lördag 7 juli 12

► Results: we do have boundary Fermi surface! :)

1 2 3kxêk f

-0.002

-0.0035

-0.005

XrH-kxLrHkxL\Cor

1 2 3kxêk f

-0.002

-0.0035

-0.005

XrH-kxLrHkxL\Cor

Friedel oscillations in the AdS hard wall geometry

► Key-points:

- deconfining geometry

- different role of the fermions: here treated QM!

- discreteness of the bulk Fermi surfaces (due to the gapped spectrum)

- delocalization of bulk Fermi surfaces: wave-functions fill all the way the space-time, they know all the geometry!

lördag 7 juli 12

Summary and Future

► Summary:

- introduction to Non-Fermi liquid theory and Friedel oscillations

- review of electron star geometry

- strategy to compute current-current correlators including 1-loop effect (effective action with Π)

- results: static current-current correlators do NOT show boundary FS in the electron star geometry (yes in the AdS hard wall geometry)

- lesson: in order that bulk FS induces boundary FS in all correlation functions, it is necessary for the bulk FS to be non-local and discrete

► Future:

- better “stringy” embedding for AdS hard wall geometry? Yes: AdS soliton! It works the same!

- Gravitational back-reaction in AdS hard wall?

- Non-Fermi liquid?

lördag 7 juli 12

► Thanks!

lördag 7 juli 12

� � � � � �

*VMIHIP�SWGMPPEXMSRW�MR�%H7 ,EVH�;EPP

! 7IX�YT

� MRXVSHYGI�ER�IJJIGXMZI�XIVQ�MR�XLI�EGXMSR

74SP =I2

2

!H^H^!H!\ %µ(^,!!O)!µ!(^, ^!,!O)%!(^!,!O) .

� [MXL�XLI 1�PSST�ZEGYYQ TSPEVM^EXMSR XIRWSV

!µ!(^, ^!, ",!O) = !!

H#H2T

(2$)3"1µ+(#, T, ^, ^!)1!+(# + ", O + T, ^!, ^)

#,

[LIVI 1µ = !U"0"µ� ;I�[ERX�XS�GSQTYXI�MX�JSV " = 0�

� + MW�XLI�JIVQMSRMG�+VIIR W�JYRGXMSR��MR�0IQLERR�VITVIWIRXEXMSR �

+(#, O, ^, ^!) =$

" "=0

%1

# ! )"(O) + M % ()"(O))

&&",O(^)&

†",O(^

!) ,

! 'SRGVIXI�GSQTYXEXMSRW

� 8LI�KET HMWGVIXM^IW XLI�WTIGXVYQ� [I�GER�MWSPEXI�SRI�WMRKPI FYPO *IVQM�WYVJEGI

� ETTVS\MQEXI�XLI�HMWTIVWMSR�VIPEXMSR )"(O) [MXL�E�RSR�VIPEXMZMWXMG�JIVQMSR

� YWI�JSV�XLI ! XLI�I\TVIWWMSR�JSV�XLI�����RSR�VIPEXMZMWXMG�JIVQMSRW

!µ!VIP (^, ^

!,!T) !!"µ""†

|O"!""†

|O+T

"

|^,^!" P

#H2O

(2#)2

$$(|O + T| # OJ)$(OJ # |O|)

)1(O + T) # )1(O)

%

lördag 7 juli 12

� � � � � �

&SRYW�8VEGO�-�

! 'PEWWMGEP� &YPO�XS�&SYRHEV]�TVSTEKEXSV�MR�%H7 LEVH�[EPP

� 8LI�GPEWWMGEP�ISQ�JSV�XLI�KEYKI�½IPH�MR�%H7 LEVH�[EPP�KISQ�!!2^ ! O2\

"%0(O\, ^) = 0 .

� XLI�GPEWWMGEP�FYPO�XS�FSYRHEV]�+VIIR W�JYRGXMSR��M�I� XLI�WSPYXMSR�[LMGL�KSIW�XS�SRI�EX�XLIFSYRHEV] �

+&!0 (O\, ^) = GSWL(O\^)! XERL(O\^Q) WMRL(O\^) " #"(!O\)"(O\)$0 = !O\ XERL(O\^Q)

! 3RI�PSST�

� XLI�SRI�PSST�GSVVIGXIH�ISQ�JSV�XLI�KEYKI�½IPH�

!!2^ ! O2\

"%0(O\, ^) = !I2

#H^!

$!00

VIP (^, ^!, O\)%0(O\, ^!) +!0\

VIP (^, ^!, O\)%\(O, ^!)

%

� [MXL�E�LMKL�GSR½RIQIRX�WGEPI�ERH�E�WQEPP�*IVQM�WYVJEGI�ZSPYQI� XLI�HMJJIS�MRXIKVEP�IUYEXMSRGER�FI�WSPZIH�TIVXYVFEXMZIP]�MR # % 1

Q2!

JSV�XLI�FYPO�XS�FSYRHEV]�+VIIR W�JYRGXMSRW

+&!(O\, ^) = +&!0 (O\, ^) + #+&!

1 (O\, ^) + .... (X ! )

0 =!!2^ ! O2\

"+&!0 (O\, ^) ,

0 =!!2^ ! O2\

"+&!1 (O\, ^) +

I2

#

#H^!!00

VIP (^, ^!, O\)+&!

0 (O\, ^!)

lördag 7 juli 12

� � � � � �

&SRYW�8VEGO�--� 4SPEVM^EXMSR�XIRWSV�MR�%H7 LEVH�[EPP

��;LEX�[I�EVI�GSQTYXMRK�!µ! = !µ! |ZEG + !µ!

VIP .

!µ!VIP (^, ^!,!O) = !

!H" H2T

(2#)3

"1µ+(",!T, ^, ^!)1!+(",!O +!T, ^!, ^)

#

!"1µ+0(",!T, ^, ^!)1!+0(",!O +!T, ^!, ^)

#.

+(", O, ^, ^!) =$

""=0

%1

" ! )"(O) + M $ ()"(O))

&%",O(^)%

†",O(^

!) ,

��-WSPEXMRK�SRI�WMRKPI�*IVQM�WYVJEGI�

!µ!VIP (^, ^!,!T) = !µ!

EREP]XMG + 1µ %1,O(^)%†1,O(^

!)1! %1,O+T(^!)%†

1,O+T(^) "!

H2O

(2#)2

%&(|O + T| ! OJ)&(OJ ! |O|)

)1(O + T) ! )1(O) + M$ ()1(O))!

&(OJ ! |O + T|)&(|O| ! OJ)

)1(O + T) ! )1(O) ! M$ ()1(O))

&

��2SR�VIPEXMZMWXMG�ETTVS\MQEXMSR� )1(O) # E1 + F1O2

!µ!VIP (^, ^!,!T) =

=1

TF1

!H2O

(2#)2

" &()1(O + T))&(!)1(O))

T + 2 GSW(&)O + M$ ()1(O))"

'1µ %1,O(^)%

†1,O(^

!)1! %1,O+T(^!)%†

1,O+T(^)(

!&()1(O ! T))&(!)1(O))

T ! 2 GSW(&)O ! M$ ()1(O))

'1µ %1,O#T(^)%

†1,O#T(^

!)1! %1,O(^!)%†

1,O(^)( #

lördag 7 juli 12

� � � � � �

&SRYW�8VEGO�--� 4SPEVM^EXMSR�XIRWSV�MR�%H7 LEVH�[EPP

��8LI�WXEXMG�TSPEVM^EXMSR�MW�TYVIP]�VIEP�

!µ!VIP (^, ^

!,!T) =1

4F1"2TP

!HOH#

O #(OJ ! O)

T/2 + O GSW(#)

"1µ $1,O(^)$

†1,O(^

!)1! $1,O+T(^!)$†

1,O+T(^)#

��JYVXLIV�ETTVS\� OJ MW�WQEPP�[LMGL�QIERW�XLEX�XLI�QEKRMXYHI�SJ�XLI�PSST�QSQIRXYQ�MW�RIZIV�PEVKI�ERH�XLEXXLI�[EZI�JYRGXMSRW�EVI�WPS[P]�ZEV]MRK�JYRGXMSRW�SJ�XLI�QSQIRXE�

"10 $1,O(^)$

†1,O(^

!)10 $1,O+T(^!)$†

1,O+T(^)#

""10 $1,0(^)$

†1,0(^

!)10 $1,T(^!)$†

1,T(^)#

.

8LMW�ETTVS\MQEXMSR�PSWIW�WSQI�SJ�XLI�ERKYPEV�MRJSVQEXMSR�MR�XLI�[EZI�JYRGXMSRW� FYX�MW�WYJ½GMIRX�XS�HMWTPE]XLI�IWWIRXMEP�JIEXYVIW�

��8LI�VIWYPX�

!µ!VIP (^, ^

!,!T) =1

4F1"2T

"1µ $1,0(^)$

†1,0(^

!)1! $1,T(^!)$†

1,T(^)#P

!HOH#

O #(OJ ! O)

T/2 + O GSW(#)

��-RXVSHYGMRK�XLI�IJJIGXMZI�I\TERWMSR�TEVEQIXIV� % # 14"F1

�

!µ!VIP (^, ^

!,!T) = !%"1µ $1,0(^)$

†1,0(^

!)1! $1,T(^!)$†

1,T(^)#$

%1 !

&

1 !'

2OJT

(2

#(T ! 2OJ)

)

*

8LMW�MW�XLI�JSVQ�SJ�XLI�TSPEVM^EXMSR�YWIH�MR�XLI�RYQIVMGW�

lördag 7 juli 12

the local chemical potential in ES is smaller than the local chemical potentialfor the AdS-RN (just the ratio h/sqrtf). This means that it will be populateda density of fermions before to reach the horizon. But this is true only fora mass less than 1! Only in this regime the bh is never a solution (it is anunstable vacuum). Of course, the extremal bh emerges when z-¿ infinity andm goes to 1.

- third oder phase transition

- outside a AdS-RN metric

understand better the meaning of the critical exponent in field theory.

About the parameters Let us check the validity of the approximationsmade. For let’s write what the rescaled parameters are:

m =

e

m , � =e

4

L

2

2

�

� =1

eL

2

� , ⇢ =1

L

2

2

⇢ , p =1

L

2

2

p ,

- classical gravity: L << 1

- � is of order 1 free parameter (e.g. for the electrons is 1

⇡2 , then

� ⇠ 1 ! 1 ⇠ � =e

4

L

2

2

� ! e

2

<< 1 , and , e2 ⇠

L

<< 1

open/closed string coupling!! in the second step I used the fact that we are inclassical gravity.

- the other rescaled parameter

m

2 =

2

e

2

m

2 ! m

2

L

2

L

2

=

2

L

2

e

2

L

2

m

2 = L

2

e

2

m

2 ! m ⇠ emL

- For a dual operator � ⇠ mL >> 1

� ⇠ mL ⇠ mL

e

2

e

2

⇠ m

e

2

=m

e

>> 1 and e

2

<< 1 ! m ⇠ 1

- high density compared to the scale of curvature

�L

3 ⇠ (�µ3)L3 ⇠

2

e

4

L

2

�(e

)3L3 ⇠ L

e

⇠ 1

e

3

>> 1 (21)

where I have used that � ⇠ 1 and � ⇠ �(E2 �m

2)3/2|µ and the definition of µ,

µ = eL hv/(L

pf).

- The last equation (21) is also consistent with the Compton wavelength, since:

mL >> 1 ! �C

L

<< 1 with �C =h

mc

⇠ eL

1

m

Notice that the control of the loop parameter is really

h ⇠ � ⌘ eL

it controls also the WKB approx and the localization approx. So � is large!

12lördag 7 juli 12