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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 2 / 21
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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 4 / 21
Motivation
Condensed matter physics
is based on two cornerstones:Landau’s theory of Phase transitions
- classifies different phases by their symmetries (order parameter),- Phase transitions are associated with changes in symmetry.
Fermi liquid theory
Martin Ammon (UCLA) AdS/CMT June 24, 2011 4 / 21
Motivation
Condensed matter physics
is based on two cornerstones:Landau’s theory of Phase transitions
- classifies different phases by their symmetries (order parameter),- Phase transitions are associated with changes in symmetry.
Fermi liquid theory- treats properties of electrons as small perturbations of the ground
state (filled single-particle levels up to EF ).
Martin Ammon (UCLA) AdS/CMT June 24, 2011 4 / 21
Motivation
Condensed matter physics
is based on two cornerstones:Landau’s theory of Phase transitions
- classifies different phases by their symmetries (order parameter),- Phase transitions are associated with changes in symmetry.
Fermi liquid theory- treats properties of electrons as small perturbations of the ground
state (filled single-particle levels up to EF ).
But there are also
- systems with topological phase transitions and
- strongly correlated electron systems.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 4 / 21
Motivation
Condensed matter physics
is based on two cornerstones:Landau’s theory of Phase transitions
- classifies different phases by their symmetries (order parameter),- Phase transitions are associated with changes in symmetry.
Fermi liquid theory- treats properties of electrons as small perturbations of the ground
state (filled single-particle levels up to EF ).
But there are also
- systems with topological phase transitions and
- strongly correlated electron systems.
⇒ New conceptional ideas are needed!
Martin Ammon (UCLA) AdS/CMT June 24, 2011 4 / 21
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.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 5 / 21
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.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 5 / 21
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.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 5 / 21
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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 5 / 21
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]
Martin Ammon (UCLA) AdS/CMT June 24, 2011 6 / 21
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)
Martin Ammon (UCLA) AdS/CMT June 24, 2011 6 / 21
Applying AdS/CFT to Condensed Matter Systems
Applying AdS/CFT to Condensed Matter Systems:Idea
Martin Ammon (UCLA) AdS/CMT June 24, 2011 7 / 21
Applying AdS/CFT to Condensed Matter Systems
Applying AdS/CFT to Condensed Matter Systems:Idea
Quantum Critical theory:
Martin Ammon (UCLA) AdS/CMT June 24, 2011 7 / 21
Applying AdS/CFT to Condensed Matter Systems
Applying AdS/CFT to Condensed Matter Systems:Idea
Quantum Critical theory:strongly coupled field theory with a global U(1) symmetry(conserved current Jµ).
Martin Ammon (UCLA) AdS/CMT June 24, 2011 7 / 21
Applying AdS/CFT to Condensed Matter Systems
Applying AdS/CFT to Condensed Matter Systems:Idea
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.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 7 / 21
Applying AdS/CFT to Condensed Matter Systems
Applying AdS/CFT to Condensed Matter Systems:Idea
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).
Martin Ammon (UCLA) AdS/CMT June 24, 2011 7 / 21
Applying AdS/CFT to Condensed Matter Systems
Applying AdS/CFT to Condensed Matter Systems:Idea
Goal
Build superconductors
Model fermi surfaces
Calculate conductivities
of the charge carriers.
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).
Martin Ammon (UCLA) AdS/CMT June 24, 2011 7 / 21
Applying AdS/CFT to Condensed Matter Systems
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
Applying AdS/CFT to Condensed Matter Systems
Adding charge carriers
There are two different approaches to add charge carriers:
Martin Ammon (UCLA) AdS/CMT June 24, 2011 9 / 21
Applying AdS/CFT to Condensed Matter Systems
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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 9 / 21
Applying AdS/CFT to Condensed Matter Systems
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
Top-Down
Use string theory embedding
Add e.g. D-branes to modelfermions, scalars, gaugefields
Charges & masses fixed
Dual field theory is known
Martin Ammon (UCLA) AdS/CMT June 24, 2011 9 / 21
Applying AdS/CFT to Condensed Matter Systems
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
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.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 9 / 21
AdS/CMT - the top-down approach
AdS/CMT - the top-down approach
Reminder
AdS5/CFT4 can be derived from Nc D3-branes.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 10 / 21
AdS/CMT - the top-down approach
AdS/CMT - the top-down approach
Reminder
AdS5/CFT4 can be derived from Nc D3-branes.
How to add charge carriers in the top-down approach?
Martin Ammon (UCLA) AdS/CMT June 24, 2011 10 / 21
AdS/CMT - the top-down approach
AdS/CMT - the top-down approach
Reminder
AdS5/CFT4 can be derived from Nc D3-branes.
How to add charge carriers in the top-down approach?
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.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 10 / 21
AdS/CMT - the top-down approach
AdS/CMT - the top-down approach
Reminder
AdS5/CFT4 can be derived from Nc D3-branes.
How to add charge carriers in the top-down approach?
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
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.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 11 / 21
AdS/CMT - the top-down approach
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 • • • - • • • - - -
Martin Ammon (UCLA) AdS/CMT June 24, 2011 12 / 21
AdS/CMT - the top-down approach
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!
Martin Ammon (UCLA) AdS/CMT June 24, 2011 12 / 21
AdS/CMT - the top-down approach
AdS/CMT - the top-down approach IV
Effective low-energy degrees of freedom of Dp-branes: scalars, vectors, fermions.
Martin Ammon (UCLA) AdS/CMT June 24, 2011 13 / 21
AdS/CMT - the top-down approach
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 + . . .
]
Martin Ammon (UCLA) AdS/CMT June 24, 2011 13 / 21
AdS/CMT - the top-down approach
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 + . . .
]
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 ).
Martin Ammon (UCLA) AdS/CMT June 24, 2011 13 / 21
AdS/CMT - the top-down approach
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]
Martin Ammon (UCLA) AdS/CMT June 24, 2011 14 / 21
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]
Martin Ammon (UCLA) AdS/CMT June 24, 2011 15 / 21
Fermi Surfaces in holographic p-wave superfluids
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]
Martin Ammon (UCLA) AdS/CMT June 24, 2011 16 / 21
Fermi Surfaces in holographic p-wave superfluids
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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 16 / 21
Fermi Surfaces in holographic p-wave superfluids
Fermi surfaces: ARPES measurements
ARPES experiment
[Wikipedia]
Measurement
[ARPES]
Result:
The Fermi surface collapses to points in high Tc-superconductors!
Martin Ammon (UCLA) AdS/CMT June 24, 2011 17 / 21
Fermi Surfaces in holographic p-wave superfluids
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!
Martin Ammon (UCLA) AdS/CMT June 24, 2011 18 / 21
Fermi Surfaces in holographic p-wave superfluids
Holographic Fermi surfaces II
Properties of R(ω, k)
Massless excitations for k ∼ kF .
- Emergent Fermi surface- kF : Fermi momentum
Martin Ammon (UCLA) AdS/CMT June 24, 2011 19 / 21
Fermi Surfaces in holographic p-wave superfluids
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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 19 / 21
Fermi Surfaces in holographic p-wave superfluids
Holographic Fermi surfaces II
Properties of R(ω, k)
Massless excitations for k ∼ kF .
- Emergent Fermi surface- kF : Fermi momentum
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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 19 / 21
Fermi Surfaces in holographic p-wave superfluids
Holographic Fermi surfaces II
Properties of R(ω, k)
Massless excitations for k ∼ kF .
- Emergent Fermi surface- kF : Fermi momentum
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
Martin Ammon (UCLA) AdS/CMT June 24, 2011 19 / 21
Fermi Surfaces in holographic p-wave superfluids
Holographic Fermi surfaces III
What about fermions in the superconducting state?
Martin Ammon (UCLA) AdS/CMT June 24, 2011 20 / 21
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!
Martin Ammon (UCLA) AdS/CMT June 24, 2011 21 / 21
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?
Martin Ammon (UCLA) AdS/CMT June 24, 2011 21 / 21
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?
Martin Ammon (UCLA) AdS/CMT June 24, 2011 21 / 21