gauge/gravity duality: applications to condensed … · gauge/gravity duality: applications to...
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Gauge/Gravity Duality:
Applications to Condensed Matter Physics
Johanna Erdmenger
Julius-Maximilians-Universitat Wurzburg
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New Gauge/Gravity Duality group at Wurzburg University
Permanent members
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Gauge/Gravity Duality
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Gauge/Gravity Duality
Brings together fundamental and empirical aspects of physics
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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
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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:
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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
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Conformal field theory
Quantum field theory
in which the fields transform covariantly under conformal transformations
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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)
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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
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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
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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
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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
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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
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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?
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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
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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
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
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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
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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
)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
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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
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
![Page 51: Gauge/Gravity Duality: Applications to Condensed … · Gauge/Gravity Duality: Applications to Condensed Matter Physics ... ,Classical fields in gravity theory ... New imput for](https://reader031.vdocuments.mx/reader031/viewer/2022022012/5b1cc9f47f8b9a16788b7926/html5/thumbnails/51.jpg)
Conclusion
New imput for understanding quantum gravity
New methods for calculating observables on strongly correlated systems
There are successes, however also ...
Many unsolved issues!
![Page 52: Gauge/Gravity Duality: Applications to Condensed … · Gauge/Gravity Duality: Applications to Condensed Matter Physics ... ,Classical fields in gravity theory ... New imput for](https://reader031.vdocuments.mx/reader031/viewer/2022022012/5b1cc9f47f8b9a16788b7926/html5/thumbnails/52.jpg)
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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