Gauge/Gravity Duality: Applications to Condensed … · Gauge/Gravity Duality: Applications to Condensed Matter Physics ... ,Classical ﬁelds in gravity theory ... New imput for

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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


Brings together fundamental and empirical aspects of physics


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

3


Duality:

A physical theory has two equivalent formulations


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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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



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






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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‘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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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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Important conceptional questions:

Understanding the foundations of gauge/gravity duality, proof?


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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25

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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



There are successes, however also ...

Many unsolved issues!

Conclusion



There are successes, however also ...

Many unsolved issues!

A wealth of important work to be done!

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Gauge/Gravity Duality: Applications to Condensed … · Gauge/Gravity Duality: Applications to Condensed Matter Physics ... ,Classical ﬁelds in gravity theory ... New imput for