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Gauge/Gravity Duality:
Applications to Condensed Matter Physics
Johanna Erdmenger
Julius-Maximilians-Universitat Wurzburg
1
New Gauge/Gravity Duality group at Wurzburg University
Permanent members
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Gauge/Gravity Duality
Gauge/Gravity Duality
Brings together fundamental and empirical aspects of physics
Gauge/Gravity Duality
Brings together fundamental and empirical aspects of physics
Fundamental:
String theory: Unification of interactions, quantization of gravity
Empirical:
New method for describing strongly correlated systems
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Gauge/Gravity Duality
Duality:
A physical theory has two equivalent formulations
Gauge/Gravity Duality
Duality:
A physical theory has two equivalent formulations
Same dynamics
One-to-one map between states
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Gauge/Gravity Duality: Foundations
Duality:
Gauge/Gravity Duality: Foundations
Duality:
Gauge/Gravity Duality:
Gauge TheoryQuantum Field Theory ⇔ Gravity theory
in higher dimensions
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Gauge/Gravity Duality
Conjecture which follows from a low-energy limit of string theory
Duality:
Quantum field theory at strong coupling⇔ Theory of gravitation at weak coupling
Holography:
Quantum field theory in d dimensions⇔ Gravitational theory in d+ 1 dimensions
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Gauge/gravity duality
I. Foundations
Origin and tests of gauge/gravity duality
AdS/CFT correspondence
II. Generalizations towards applications
Breaking conformal symmetry: RG flows
Finite temperature
Finite charge density and chemical potential
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I. Foundations: Anti-de Sitter Space
Hyperbolic space of constant negative curvature, has a boundaryFigure source: Institute of Physics, Copyright: C. Escher
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Anti-de Sitter Space
Embedding of (Euclidean) AdSd+1
into Minkd+2:
−X20+X
21+X
22+· · ·+X2
d+1 = −L2
Isometries of Euclidean AdSd+1:
SO(d+ 1, 1)
Metric on Poincare patch:ds2 = e2r/Ldxµdx
µ + dr2
X+X�
X2 = �1
P 2 = 0
Source: Costa, Goncalves, Penedones,1404.5625
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Conformal field theory
Quantum field theory
Conformal field theory
Quantum field theory
in which the fields transform covariantly under conformal transformations
Conformal field theory
Quantum field theory
in which the fields transform covariantly under conformal transformations
Conformal coordinate transformations: preserve angles locallySymmetry SO(d+ 1, 1)
Conformal field theory
Quantum field theory
in which the fields transform covariantly under conformal transformations
Conformal coordinate transformations: preserve angles locallySymmetry SO(d+ 1, 1)
⇒ Correlation functions are determined up to a small number of parametersalso for more than two dimensions
Conformal field theory
Quantum field theory
in which the fields transform covariantly under conformal transformations
Conformal coordinate transformations: preserve angles locallySymmetry SO(d+ 1, 1)
⇒ Correlation functions are determined up to a small number of parametersalso for more than two dimensions
In AdS/CFT correspondence: Conformal field theory in 3+1 dimensions:N = 4 SU(N) Super Yang-Mills theory
Conformal field theory
Quantum field theory
in which the fields transform covariantly under conformal transformations
Conformal coordinate transformations: preserve angles locallySymmetry SO(d+ 1, 1)
⇒ Correlation functions are determined up to a small number of parametersalso for more than two dimensions
In AdS/CFT correspondence: Conformal field theory in 3+1 dimensions:N = 4 SU(N) Super Yang-Mills theory
Symmetries of AdS and CFT coincide!
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Foundations: String theory
String theory provides framework for gauge/gravity duality
Foundations: String theory
String theory provides framework for gauge/gravity duality
Two types of degrees of freedom: open and closed strings
Open strings : Gauge degrees of freedom of the Standard Model
Closed Strings: Gravitation
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D-Branes
D-branes are surfaces embedded into 9+1 dimensional space
D3-Branes: (3+1)-dimensional surfaces
Open Strings may end on these surfaces⇔ Dynamics
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D-Branes
Low-energy limit (Strings point-like)⇒
Open Strings⇔ Dynamics of gauge fields on the brane
D-Branes
Low-energy limit (Strings point-like)⇒
Open Strings⇔ Dynamics of gauge fields on the brane
Second interpretation of D-branes:
Solitonic solutions of ten-dimensional supergravity
Heavy objects which curve the space around them
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String theory origin of the AdS/CFT correspondence
near-horizon geometryAdS x S
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D3 branes in 10d
duality
⇓ Low energy limit
Supersymmetric SU(N) gau-ge theory in four dimensions(N →∞)
Supergravity on the spaceAdS5 × S5
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Gauge/Gravity Duality
‘Dictionary’ Gauge invariant field theory operators⇔ Classical fields in gravity theory
Symmetry properties coincide, generating functionals are identified
Test: (e.g.) Calculation of correlation functions
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Generating Functional
Field-operator correspondence:
〈e∫ddxφ0(~x)O(~x)〉CFT = Zsugra
∣∣∣φ(0,~x)=φ0(~x)
Generating functional for correlation functions of particular composite operatorsin the quantum field theory
coincides with
Classical tree diagram generating functional in supergravity
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Gauge/Gravity Duality
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Large N limit
SU(N) gauge theory
Degrees of freedom scale as N2
Large N limit
SU(N) gauge theory
Degrees of freedom scale as N2
’t Hooft limit: λ = g2N fixed, N →∞
Only planar Feynman diagrams contribute
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Gauge/gravity duality
Important conceptional questions:
Understanding the foundations of gauge/gravity duality, proof?
Gauge/gravity duality
Important conceptional questions:
Understanding the foundations of gauge/gravity duality, proof?
New input for the description of strongly coupled systems in
Elementary particle physics and condensed matter physics
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Examples for applications
Low-energy QCD Chiral symmetry breaking, mesons
Quark-gluon plasmaShear viscosity over entropy density, η/s = 1/(4π)~/kB
Kovtun, Son, Starinets 2004
Examples for applications
Low-energy QCD Chiral symmetry breaking, mesons
Quark-gluon plasmaShear viscosity over entropy density, η/s = 1/(4π)~/kB
Kovtun, Son, Starinets 2004
Condensed matter physics
– Quantum phase transitions– Non-Fermi liquids, strange metals– Transport properties– Universal behaviour– Superconductivity– Interactions with magnetic impurities– Disorder– . . .
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Generalizations of AdS/CFT
to less symmetric examples of gauge/gravity duality
Generalizations of AdS/CFT
to less symmetric examples of gauge/gravity duality
Consider gravity solutions with less symmetry
Break conformal symmetry by considering spaces which are only asymptoticallyAdS near the boundary
Extra dimension corresponds to RG scale
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Finite temperature
Quantum field theory at finite temperature:
Dual to gravity theory with black hole
Hawking temperature identified with temperature in the dual field theory
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Schwarzschild metric
Action:S[g] =
1
2κ25
∫d
5x√−g
(R +
12
L2
)Metric:
ds2
=L2
z2
(−f(z)dt
2+
dz2
f(z)+ d~x
2
),
with f(z) = 1−M(z
zh
)d
Schwarzschild metric
Action:S[g] =
1
2κ25
∫d
5x√−g
(R +
12
L2
)Metric:
ds2
=L2
z2
(−f(z)dt
2+
dz2
f(z)+ d~x
2
),
with f(z) = 1−M(z
zh
)dNear the horizon, in Euclidean coordinates (z, τ = it) this metric looks like a 2d plane in polarcoordinates
Regularity requires τ to be periodic with period β = 4πzh/d
Hawking temperature
TH =d
4πzh
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Causal structure of space-time
x = ct Set c = 1⇒ x = t
Flat space:
Black hole:
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Shear viscosity
Hydrodynamics: Long wavelength, low-frequency fluctuations in fluids
Expand physical quantities in derivatives of the fluid velocity: ~v, ∇~v, ∇∇~v . . .
Relativistically: Four-velocity uµ = (u0,u1,u2,u3), uµuµ = 1
u0 = 1/√1− ~v2, ~u = ~v/
√1− ~v2
Consider energy-momentum tensor Tµν
Contains information about energy density, energy and momentum flux
Hydrodynamic expansion to first order in derivatives:
Tµν(x) = T(0)µν (x) + T
(1)µν (x) + . . .
T(0)µν (x) = (ε+ P )uµuν − Pgµν , T
(1)µν = η
(∂µuν + ∂νuµ − 2
3gµν∂λuλ)
+ ζgµν∂λuλ
η shear viscosity, ζ bulk viscosity
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Holographic calculation of shear viscosity
Energy-momentum tensor Tµν dual to graviton gµν
Calculate correlation function 〈Txy(x1)Txy(x2)〉 from propagation throughblack hole space
Shear viscosity is obtained from Kubo formula:
η = −lim 1
ωImGRxy,xy(ω)
Shear viscosity η = πN2T 3/8, entropy density s = π2N2T 3/2
η
s=
1
4π
~kB
(Note: Quantum critical system: τ = ~/(kBT ))
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Charge and chemical potential
Action:
S =
∫dd+1x√−g(
1
2κ2(R− 2Λ)− 1
4g2FmnFmn
),
Solution: Reissner-Nordstrom (RN) charged black hole
Metric:
ds2
=L2
z2
(−f(z)dt
2+
dz2
f(z)+ d~x
2
),
with f(z) = 1−M(z
zh
)d+Q
2
(z
zh
)2(d−1)
Finite horizon even for T = 0
Gauge field:
At(z) = µ
(1−
(z
zh
)d−2)
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Near-horizon geometry of RN black hole
Near the black-hole horizon, the RN metric becomes
ds2 = d(d− 1)L2z2
z4h
(−dt2) + 1
d(d− 1)
L2
z2dz2 +
L2
z2h
d~x2 .
Metric of AdS2 × IRd−1 with factor
ds2 =L2
ζ2(−dt2 + dζ2) + d~x2 ,
This region corresponds to the IR limit of the dual quantum field theory.
Finite entropy at T = 0!
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SYK models
Sachdev-Ye-Kitaev model:
Gaussian random couplings Jαβ,γδ Sachdev+Ye 1993, Kitaev 2015, Sachdev 2015
H =1
(2N)3/2
N∑α,β,γ,δ=1
Jαβ,γδ χ†αχβχ
†γχδ − µ
∑α
χ†αχα
SYK models
Sachdev-Ye-Kitaev model:
Gaussian random couplings Jαβ,γδ Sachdev+Ye 1993, Kitaev 2015, Sachdev 2015
H =1
(2N)3/2
N∑α,β,γ,δ=1
Jαβ,γδ χ†αχβχ
†γχδ − µ
∑α
χ†αχα
Finite zero-temperature entropy
(see talk be M. Rozali)
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Kondo models
Magnetic impurities in gauge/gravity duality
J.E., Flory, Hoyos, Newrzella, O’Bannon, Papadimitriou, Wu 2013-16
(See talk be A. O’Bannon)
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Conclusion
New imput for understanding quantum gravity
New methods for calculating observables on strongly correlated systems
Conclusion
New imput for understanding quantum gravity
New methods for calculating observables on strongly correlated systems
There are successes, however also ...
Many unsolved issues!
Conclusion
New imput for understanding quantum gravity
New methods for calculating observables on strongly correlated systems
There are successes, however also ...
Many unsolved issues!
A wealth of important work to be done!
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