Gauge/Gravity Duality: Applications to Condensed Matter Physics. Johanna Erdmenger Julius-Maximilians-Universität Würzburg 1
New Gauge/Gravity Duality group at Würzburg University Permanent members 2
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 3
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 4
Duality: Gauge/Gravity Duality: Foundations
Gauge/Gravity Duality: Foundations Duality: Gauge/Gravity Duality: Gauge Theory Quantum Field Theory Gravity theory in higher dimensions 5
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 6
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 7
I. Foundations: Anti-de Sitter Space Hyperbolic space of constant negative curvature, has a boundary Figure source: Institute of Physics, Copyright: C. Escher 8
Anti-de Sitter Space Embedding of (Euclidean) AdS d+1 into Mink d+2 : X X + X 2 0 +X 2 1 +X 2 2 + +X 2 d+1 = L 2 X 2 = 1 Isometries of Euclidean AdS d+1 : P 2 =0 SO(d + 1, 1) Metric on Poincaré patch: ds 2 = e 2r/L dx µ dx µ + dr 2 Source: Costa, Goncalves, Penedones,1404.5625 9
Quantum field theory Conformal 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 locally Symmetry SO(d + 1, 1)
Conformal field theory Quantum field theory in which the fields transform covariantly under conformal transformations Conformal coordinate transformations: preserve angles locally Symmetry SO(d + 1, 1) Correlation functions are determined up to a small number of parameters also 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 locally Symmetry SO(d + 1, 1) Correlation functions are determined up to a small number of parameters also 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 locally Symmetry SO(d + 1, 1) Correlation functions are determined up to a small number of parameters also 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! 10
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 11
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 12
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 13
String theory origin of the AdS/CFT correspondence D3 branes in 10d duality AdS x S5 5 near-horizon geometry Low energy limit Supersymmetric SU(N) gauge theory in four dimensions (N ) Supergravity on the space AdS 5 S 5 14
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 15
Generating Functional Field-operator correspondence: e d d x φ 0 ( x)o( x) CF T = Z sugra φ(0, x)=φ0 ( x) Generating functional for correlation functions of particular composite operators in the quantum field theory coincides with Classical tree diagram generating functional in supergravity 16
Gauge/Gravity Duality 17
Large N limit SU(N) gauge theory Degrees of freedom scale as N 2
Large N limit SU(N) gauge theory Degrees of freedom scale as N 2 t Hooft limit: λ = g 2 N fixed, N Only planar Feynman diagrams contribute 18
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 19
Examples for applications Low-energy QCD Chiral symmetry breaking, mesons Quark-gluon plasma Shear viscosity over entropy density, η/s = 1/(4π) /k B Kovtun, Son, Starinets 2004
Examples for applications Low-energy QCD Chiral symmetry breaking, mesons Quark-gluon plasma Shear viscosity over entropy density, η/s = 1/(4π) /k B 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... 20
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 asymptotically AdS near the boundary Extra dimension corresponds to RG scale 21
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 22
Schwarzschild metric Action: Metric: S[g] = 1 2κ 2 5 ds 2 = L2 z 2 ( d 5 x g ( R + 12 ) L 2 ) f(z)dt 2 + dz2 f(z) + d x2, with f(z) = 1 M ( z z h ) d
Schwarzschild metric Action: Metric: S[g] = 1 2κ 2 5 ds 2 = L2 z 2 ( d 5 x g ( R + 12 ) L 2 ) f(z)dt 2 + dz2 f(z) + d x2, with f(z) = 1 M ( z z h ) d Near the horizon, in Euclidean coordinates (z, τ = it) this metric looks like a 2d plane in polar coordinates Regularity requires τ to be periodic with period β = 4πz h /d Hawking temperature T H = d 4πz h 23
Causal structure of space-time x = ct Set c = 1 x = t Flat space: Black hole: 24
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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 u µ = (u 0, u 1, u 2, u 3 ), u µ u µ = 1 u 0 = 1/ 1 v 2, u = v/ 1 v 2 Consider energy-momentum tensor T µν Contains information about energy density, energy and momentum flux Hydrodynamic expansion to first order in derivatives: T µν (x) = T (0) (1) µν (x) + T µν (x) +... T (0) µν (x) = (ɛ + P )u µu ν P g µν, T (1) µν = η ( µ u ν + ν u µ 2 3 g µν λ u λ) + ζg µν λ u λ η shear viscosity, ζ bulk viscosity 27
Holographic calculation of shear viscosity Energy-momentum tensor T µν dual to graviton g µν Calculate correlation function T xy (x 1 )T xy (x 2 ) from propagation through black hole space Shear viscosity is obtained from Kubo formula: η = lim 1 ω ImGR xy,xy(ω) Shear viscosity η = πn 2 T 3 /8, entropy density s = π 2 N 2 T 3 /2 η s = 1 4π k B (Note: Quantum critical system: τ = /(k B T )) 28
Charge and chemical potential Action: S = d d+1 x g ( ) 1 1 (R 2Λ) 2κ2 4g 2F mn F mn, Solution: Reissner-Nordström (RN) charged black hole Metric: ds 2 = L2 z 2 ( ) f(z)dt 2 + dz2 f(z) + d x2, with f(z) = 1 M ( ) z d ( z + Q 2 z h z h ) 2(d 1) Finite horizon even for T = 0 Gauge field: A t (z) = µ ( 1 ( z z h ) d 2 ) 29
Near-horizon geometry of RN black hole Near the black-hole horizon, the RN metric becomes ds 2 = d(d 1) L2 z 2 ( dt 2 ) + z 4 h 1 L 2 d(d 1) z 2 d z2 + L2 d x 2. z 2 h Metric of AdS 2 IR d 1 with factor ds 2 = L 2 ζ 2 ( dt2 + dζ 2 ) + d x 2, This region corresponds to the IR limit of the dual quantum field theory. Finite entropy at T = 0! 30
SYK models Sachdev-Ye-Kitaev model: Gaussian random couplings J αβ,γδ Sachdev+Ye 1993, Kitaev 2015, Sachdev 2015 H = 1 (2N) 3/2 N α,β,γ,δ=1 J αβ,γδ χ αχ β χ γχ δ µ α χ αχ α
SYK models Sachdev-Ye-Kitaev model: Gaussian random couplings J αβ,γδ Sachdev+Ye 1993, Kitaev 2015, Sachdev 2015 H = 1 (2N) 3/2 N α,β,γ,δ=1 J αβ,γδ χ αχ β χ γχ δ µ α χ αχ α Finite zero-temperature entropy (see talk be M. Rozali) 31
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) 32
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! 33