Gauge/Gravity Duality applied to Condensed Matter Systemsbajnok.web.elte.hu/JHW/Talks/Ammon.pdf ·...

Gauge/Gravity Duality applied toCondensed Matter Systems

Martin Ammon

University of California, Los Angeles

35th Johns Hopkins Workshop onAdS/CFT and its Applications

June 24th, 2011

Based on:

MA, J. Erdmenger, M. Kaminski, A. O’Bannon, 1003.1134

MA, Review on AdS/CMT from top-down approach,Fortschritte der Physik, 58 (2010) 1123-1250.

Martin Ammon (UCLA) AdS/CMT June 24, 2011 1 / 21

Outline

1 Motivation

2 Applying AdS/CFT to Condensed Matter Systems

3 AdS/CMT - the top-down approach

4 Fermi Surfaces in holographic p-wave superfluids

5 Conclusion


Outline, part II

Motivation

Can we use string theory to study experimental observations?

[String theory gr., Uppsala Universitet]

gauge/gravity

←→duals

[RHIC]

Model

N = 4 SYM coupled tohypermultiplets

at finite T and finite density

Results

Conductivity tensor

Superconducting state

emergent Fermi SurfacesMartin Ammon (UCLA) AdS/CMT June 24, 2011 3 / 21

Motivation

Condensed matter physics

is based on two cornerstones:Landau’s theory of Phase transitions

Fermi liquid theory


Motivation



- classifies different phases by their symmetries (order parameter),- Phase transitions are associated with changes in symmetry.

Fermi liquid theory


Motivation




Fermi liquid theory- treats properties of electrons as small perturbations of the ground

state (filled single-particle levels up to EF ).


Motivation






But there are also

- systems with topological phase transitions and

- strongly correlated electron systems.


Motivation






But there are also

- systems with topological phase transitions and

- strongly correlated electron systems.

⇒ New conceptional ideas are needed!


Motivation

Strongly correlated electron systems

Examples

High-Tc superconductors, e.g. cuprates like La2−xSrxCuO2,

Heavy fermion compounds such as CePd2Si2,

Fractional quantum Hall liquids,

Luttinger liquids in one-dimensional conducting systems.


Motivation

Strongly correlated electron systems

Examples

High-Tc superconductors, e.g. cuprates like La2−xSrxCuO2,

Heavy fermion compounds such as CePd2Si2,

Fractional quantum Hall liquids,

Luttinger liquids in one-dimensional conducting systems.

typical (schematic)

phase diagram


Motivation

Quantum Phase Transitions

Definition & Consequences

Quantum Phase Transition:Phase Transition at T = 0.

Caused by non-analyticity inground state energy,

Driven by quantum fluctuations!

Quantum critical region (QCR):Temperature is the only relevantscale: ⇒ Scale invariant!

~x → λ~x ⇒ t → λz t .

[Herzog]


Motivation

Quantum Phase Transitions

Definition & Consequences

Quantum Phase Transition:Phase Transition at T = 0.

Caused by non-analyticity inground state energy,

Driven by quantum fluctuations!

Quantum critical region (QCR):Temperature is the only relevantscale: ⇒ Scale invariant!

~x → λ~x ⇒ t → λz t .

[Herzog]

Effective Theories in QCR-Regionare difficult to find!for example: O(N) models (Wilson-Fisher fixed point)


Applying AdS/CFT to Condensed Matter Systems

Applying AdS/CFT to Condensed Matter Systems:Idea




Quantum Critical theory:




Quantum Critical theory:strongly coupled field theory with a global U(1) symmetry(conserved current Jµ).




Quantum Critical theory:strongly coupled field theory with a global U(1) symmetry(conserved current Jµ).described by (super-) gravity theory in asymptotically AdSspacetime with U(1) gauge fieldfinite temperature↔ (non-extremal) black hole solutionconserved current Jµ ↔ gauge field.





Charge carriers: scalars, vectors, fermions charged under U(1).




Goal

Build superconductors

Model fermi surfaces

Calculate conductivities

of the charge carriers.


Charge carriers: scalars, vectors, fermions charged under U(1).



What we can learn from AdS/CMT

What AdS/CMT can achieve:Identify new phenomena at strong coupling

→ e.g. holographic superconductors do not obey BCS theory!

→ energy gap ∆ of charged excitations: 2∆ ≈ 8.4Tc 6= 3.54Tc.

Can compare dynamics of strongly coupled system with weakcoupling

⇒ Construct counterexamples to intuitive weak-couplingarguments!Find universal behaviour

→ Not obvious!

⇒ Smoking gun: Homes’ Law?What AdS/CMT cannot achieve:

exact numerical values (central charge in holographic setup: c →∞)!Martin Ammon (UCLA) AdS/CMT June 24, 2011 8 / 21


Adding charge carriers

There are two different approaches to add charge carriers:





Bottom-Up

Use phenomenological model

Add Fermions, scalars, gaugefields by hand

Charges & masses not fixed⇒ can scan different models!

Field theory dual not known





Bottom-Up





Top-Down

Use string theory embedding

Add e.g. D-branes to modelfermions, scalars, gaugefields

Charges & masses fixed

Dual field theory is known





Bottom-Up





Top-Down

Use string theory embedding

Add e.g. D-branes to modelfermions, scalars, gaugefields

Charges & masses fixed

Dual field theory is known

Here I will focus on the top-down approach using probe branes in AdS.


AdS/CMT - the top-down approach


Reminder

AdS5/CFT4 can be derived from Nc D3-branes.




Reminder


How to add charge carriers in the top-down approach?




Reminder



Idea

Add to the Nc D3-branes another stack of Nf coincident Dp-branes!

Fundamental strings between Dp- and D3-branes are interpretedas quarks qa

i .

quarks qai are charged under

- SU(Nc) gauge group and- U(Nf ) flavour symmetry.




Reminder



Idea

Add to the Nc D3-branes another stack of Nf coincident Dp-branes!

Fundamental strings between Dp- and D3-branes are interpretedas quarks qa

i .

quarks qai are charged under

- SU(Nc) gauge group and- U(Nf ) flavour symmetry.

charge carriers in the top-down approach:

the quarks qai and their gauge-invariant bound states (Mesons,

Mesinos), charged under U(Nf ) flavour symmetry.Martin Ammon (UCLA) AdS/CMT June 24, 2011 10 / 21


AdS/CMT - the top-down approach II

Which Dp-branes can we add?

Possible Dp - brane stacks (1/2 BPS intersections)

0 1 2 3 4 5 6 7 8 9Nc D3 • • • • - - - - - -Nf D7 • • • • • • • • - -Nf D7 • • - - • • • • • •

Nf D5 • • • - • • • - - -Nf D5 • - - - • • • • • -Nf D3 • • - - • • - - - -

For all these configurations the field theory is explicitly known.



AdS/CMT - the top-down approach III

How to add branes on the gravity side?

0 1 2 3 4 5 6 7 8 9Nc D3 • • • • - - - - - -Nf D5 • • • - • • • - - -



AdS/CMT - the top-down approach III

How to add branes on the gravity side?

0 1 2 3 4 5 6 7 8 9Nc D3 • • • • - - - - - -Nf D5 • • • - • • • - - -

Probe Limit: Nf fixed, Nc →∞

Gravity side: Ignore back–reaction of Dp–branes, study probeDp–branes wrapping asymp. AdSP × SQ in AdS5 × S5.

Gauge theory side: Ignore quantum (loop) effects of flavourdegrees⇒ β ∼ +O(Nf/Nc) ≈ 0

for D5 flavor branes: field theory is conformal!



AdS/CMT - the top-down approach IV

Effective low-energy degrees of freedom of Dp-branes: scalars, vectors, fermions.





Low-energy effective action:

bosonic degrees of freedom: DBI- and WZ action

SDBI = −τDp

∫

dp+1ζ√

det (−P[g] + 2πα′F )

where F is the field strength tensor, A the corresponding gauge field.

fermionic degrees of freedom (to order α′ 2)

Sferm =τDp

2

∫

dp+1ζ√

− detP[g]Tr[

ˆΨP−ΓA(

DA +18

i2 ∗ 5!

F(5)Γ(5)ΓA

)

Ψ + . . .

]





Low-energy effective action:

bosonic degrees of freedom: DBI- and WZ action

SDBI = −τDp

∫

dp+1ζ√

det (−P[g] + 2πα′F )

where F is the field strength tensor, A the corresponding gauge field.

fermionic degrees of freedom (to order α′ 2)

Sferm =τDp

2

∫

dp+1ζ√

− detP[g]Tr[

ˆΨP−ΓA(

DA +18

i2 ∗ 5!

F(5)Γ(5)ΓA

)

Ψ + . . .

]

Generalizations:

Mapping: conserved flavour current Jµ ↔ gauge field Aµ.

For Nf coincident Dp-branes: A is a U(Nf ) gauge field, Ψ are in the adjointrepresentation of U(Nf ).



Results of the Top-Down approach

Calculation of conductivitiesDC conductivities: the method [A. Karch, A. O’Bannon, ’07]

DC conductivities for arbitrary electric and magnetic fields[M.A., H. Ngo, A. O’Bannon, ’09]

DC & AC conductivities for QCP with z 6= 1 (Lifshitz symmetry)[S. Hartnoll, J. Polchinski, E. Silverstein, D. Tong, ’09]

DC & AC conductivities for QCP with z = 2 (Schrödinger symmetry)[M.A., C. Hoyos, A. O’Bannon, J. Wu, ’10]

Holographic p-wave Superconductors[M.A., J. Erdmenger, M. Kaminski, P. Kerner, ’08, ’09 + many other groups afterwards]

Holographic fermi surfaces [M.A., J. Erdmenger, M. Kaminski, A. O’Bannon, ’10]

Effective action, Dual Field Theory operators, Fermi Surfaces in p-wave

superconductors

Identifying Quantum critical points[Karch et al., Evans et al. , ’10]


Fermi Surfaces in holographic p-wave superfluids

The p-wave superfluid

Gauge theory side

N = 4 SYM coupled to two flavour fields (in the fundamentalrepresentation of the gauge group)

at finite temperature T and at finite isospin chemical potential µI

above a critical isospin chemical potential (or equiv. below acritical temperature Tc):

ρ-Meson condensation

breaks rotational symmetry (p-wave symmetry)

Gravity side

Consider probe Dp-branes in AdS5(BH)× S5 (here: D5-branes).

Nonzero A3t and A1

x (see Johanna Erdmenger’s talk)!

Note: Competing s-wave phase [Wapler, 11]



Spectral function in AdS/CFT

Spectral function R

is given by the retarded Green function GR,

R = −2 Im GR

with GR(k) = −i∫

d4xeikxθ(x0)[

J (x),J (0)]

±,

[Wikipedia]



Spectral function in AdS/CFT

Spectral function R

is given by the retarded Green function GR,

R = −2 Im GR

with GR(k) = −i∫

d4xeikxθ(x0)[

J (x),J (0)]

±,

[Wikipedia]

Determination in gravity dual

by on-shell DBI action SDBI,on-shell for fluctuation a around thebackground gauge field A [Son, Starinets, ’02]

GR =δ2SDBI,on-shell

δa2bdy



Fermi surfaces: ARPES measurements

ARPES experiment

[Wikipedia]

Measurement

[ARPES]

Result:

The Fermi surface collapses to points in high Tc-superconductors!



Holographic Fermi surfaces

Fermionic fluctuations⇒ Calculate fermionic spectral function R(ω, k)

action:

Sferm =τDp

2

∫

dp+1ζ√

− detP[g]Tr[

ˆΨP−ΓA(

DA +18

i2 ∗ 5!

F(5)Γ(5)ΓA

)

Ψ + . . .

]

F(5) term induces bulk mass for fermions.

⇒ bulk mass and charge completely fixed!

technical problems (solved in 1003.1134)

Identify dual fermionic operators!

Holographic renormalization for fermions in AdS!

Effective way to solve numerically coupled fermionic EOMs!



Holographic Fermi surfaces II

Properties of R(ω, k)

Massless excitations for k ∼ kF .

- Emergent Fermi surface- kF : Fermi momentum







0 1 2 3 4 50

2

4

6

8

R

kx/πT







Exponents of the (non-) Fermi liquid

Definition of α and z:

ω⋆ ∼ (k − kF )z

R ∼ (k − kF )−α ,

0 1 2 3 4 50

2

4

6

8

R

kx/πT







Exponents of the (non-) Fermi liquid

Definition of α and z:

ω⋆ ∼ (k − kF )z

R ∼ (k − kF )−α ,

Fermi liquid: α = z = 1AdS/CFT: z = 1, α = 2⇒ Non-Fermi liquid!

0 1 2 3 4 50

2

4

6

8

R

kx/πT



Holographic Fermi surfaces III

What about fermions in the superconducting state?


Conclusion

Conclusion

Results

We can embed superfluids & superconductors and Non-Fermiliquids into a top-down approach using probe branes in AdS.

Dual field theory is known explicitly. Comparison to a perturbativeanalysis possible!


Conclusion

Conclusion

Results

We can embed superfluids & superconductors and Non-Fermiliquids into a top-down approach using probe branes in AdS.

Dual field theory is known explicitly. Comparison to a perturbativeanalysis possible!

OutlookNew insights into high-Tc superconductors andnon-Fermi liquids possible?

Calculate interesting quantities also at weakcoupling and compare to results from thetop-down approach.

No Universal behavior found so far!

What is the holographic description of topologicalphase transitions?


Gauge/Gravity Duality applied to Condensed Matter Systemsbajnok.web.elte.hu/JHW/Talks/Ammon.pdf ·...

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