SMR 1646 - 12 ___________________________________________________________ Conference on Higher Dimensional Quantum Hall Effect, Chern-Simons Theory and Non-Commutative Geometry in Condensed Matter Physics and Field Theory 1 - 4 March 2005 ___________________________________________________________ Emergent gravity as the nematic dual of Lorentz-invariant Elasticity Johannes ZAANEN Stanford Univ., Stanford, U.S.A. ___________________________________________________________________ These are preliminary lecture notes, intended only for distribution to participants. EmergentEmergent gravitygravity asas thethe nematicnematic dualdual ofof LorentzLorentz--invariantinvariant ElasticityElasticityTalk Jan Zaanen Hagen Kleinert Sergei Mukhin Zohar Nussinov Vladimir Cvetkovic 1 Emergent Einstein Gravity c 3 S =− dV −g R + S 16πG ∫ matter Einstein’s space time = the Lorentz invariant topological nematic superfluid (at least in 2+1D) A medium characterized by: - emergent general covariance - absence of torsion- and compressional rigidity - presence of curvature rigidity (topological order) 2 Plan of talk 1. Plasticity (defected elasticity) and differential geometry 2. Fluctuating order and high Tc superconductivity 3. Dualizing non-relativistic quantum elasticity 3.a Quantum nematic orders 3.b Superconductivity: dual Higgs is Higgs 4. The quantum nematic world crystal and Einstein’s space time 3 Quantum-elasticity: basics Quantum elastic action, isotropic medium ν ⎡ 2 2 2 2 2 2⎤ S = µ u + 2u + u + ()u + u + ()∂τ u + ()∂τ u ⎣⎢ xx xy yy 1−ν xx yy x y ⎦⎥ Shear modulus µ, Poisson ratio ν, compression modulus κ = µ()1 + ν /1()− ν 2 = µ ρ = = = ()∂ + ∂ c ph 2 / 1, h 1, uab a ub b ua /2 Describes transversal- (T) and longitudinal (L) phonon, 2 2 ⎡⎛ q ⎞ ⎛ q ⎞ ⎤ S = µ ⎢⎜ + ω 2 ⎟ | uT |2 +⎜ + ω 2 ⎟ | uL |2⎥ ⎣⎝ 2 ⎠ ⎝ 1−ν ⎠ ⎦ 4 Elasticity and topology: the dislocation J.M. Burgers (Delft): Discovery of the War time needs (Peierls, Mott, topological excitation Friedel, …) 5 The Singularities Dislocation: Disclination: Restores translational invariance Restores rotational invariance Destroys curvature rigidity, like Destroys shear rigidity mass source in gravity (!) Topological charge: Burgers vector Topological charge: Franck ‘scalar’ Disclinations are ‘bad’ (difficult) Disclinations ‘liberate’ non-abelian nature of the euclidean group (diffeomorphism) 6 Engineering curvature Disclination in ‘buckystuff’: like conical singularity in 2+1 D gravity Fluctuating geometry = simplexes with fluctuating edge-lengths “Regge Calculus” 7 The mathematical machine Hagen Kleinert (FU Berlin) Abelian Higgs duality Theory of plasticity in 3D (classical): similarity with euclidean gravity! 8 Elasticity and general covariance General covariance: infinitisimal coordinate transformation µ = δ µ → δ µ − ∂ ξ µ ea a a a = a = δ → δ + ∂ ξ + ∂ ξ gµν eµ eaν µν µν ( µ ν ν µ ) General covariance = gauge invariance under elastic deformations = δ → δ + ∂ + ∂ gij ij ij ( iu j j ui ) 9 Plasticity and differential geometry Covariant derivative: Connection, linearized: µ µ µ λ Γ λ = ∂ ∂ ξ λ Dν v = ∂ν v +Γνλ v µν µ ν Curvature tensor: Rµνλκ = ()∂µ ∂ν −∂ν∂µ ∂λξκ λ λ Associate: ξ → u ==> Einstein tensor corresponds with disclination current! 1 κ Θµν ≡ Gµν = Rµν − gµν Rκ 2 Crystal = geometry ‘Bad’ news: torsion = dislocation currents with curvature and 1 Sµνλ = ()∂µ∂ν −∂ν∂µ ξλ Jµ = εµνλSνλ torsion. 2 a a 10 Non-linear plasticity Use gravity techniques to solve problems associated with large defect densities/plastic deformations ( K. Kondo, 1952 ….) Substitute covariant derivatives for derivatives µ µ µ µ λ ∂ν v → Dν v = ∂ν v +Γνλ v Use full curvature, torsion tensors instead of linearized tensors Program not greatly successful when it matters (e.g. glasses). 11 The universe as a strange crystal … Special ‘crystal’: isotropic elastic medium in space and time directions (Euclidean signature) (a) Lorentz-invariance (b) Curvature is not quantized, Franck vectors are … 2π disclination in a triangular lattice 3 Deep geometrical/topological similarity with space-time (also: Bais et al, ‘quantum doubles’: crystal ==> Witten’s 2+1 D gravity) 12 Why the universe is not a ‘Lorentz’ crystal (a) Crystal ‘geometry’ is characterized by torsion, the universe is not … (b) Disclinations are (quadratically) confined: curvature costs infinite energy! (c) Deadly: crystals have no general covariance, the flat metric is preferred !!! Elastic deformation costs energy: action is not gauge invariant … These problems seem to be cured in the quantum NEMATIC world crystal … (at least in 2+1D) 13 Plan of talk 1. Plasticity (defected elasticity) and differential geometry 2. Fluctuating order and high Tc superconductivity 3. Dualizing non-relativistic quantum elasticity 3.a Quantum nematic orders 3.b Superconductivity: dual Higgs is Higgs 4. The quantum nematic world crystal and Einstein’s space time 14 Correlated superconductors Ideal Bose-Einstein gas BEC cold atomic gas, BCS metallic superconductor Helium 4 superfluid High Tc superconductors Roton Strongly correlated fluid: locally like a solid. 15 Stripe order: charge, spin, domain walls -- Charge order (e.g. STM) -- Spin order (neutrons,NMR) -- ‘Topological order’, ‘domainwall-ness’, ‘antiphaseboundariness’ 16 Electrons coming to a standstill STM Kapitulnik et al (Stanford) Davis et al (Cornell) These ‘stripes’ are ubiquitous in doped Mott insulator (nickelates, manganites, …) Cuprates: stripes are extremely quantum mechanical … 17 Quantum Fluctuating stripe order Neutron scattering: spin Fourier transform fluctuations, subpicosecond! Stripe spin wavevectors YBCO Tc=60K: Mook et JZ, Science 286, 251 al, Oak Ridge (1999) 18 Nematic superfluids Kivelson,Fradkin, Emery, Nature 393, 550,1998 Nematic order Bose condensation 19 The Singularities Dislocation: Disclination: Restores translational invariance Restores rotational invariance Destroys curvature rigidity, like Destroys shear rigidity mass source in gravity (!) Topological charge: Burgers vector Topological charge: Franck ‘scalar’ Disclinations are ‘bad’ (difficult) Disclinations ‘liberate’ non-abelian nature of the euclidean group (diffeomorphism) 20 Dislocations: example 1 2 3 1 4 2 5 6 3 7 4 8 5 6 21 Plan of talk 1. Plasticity (defected elasticity) and differential geometry 2. Fluctuating order and high Tc superconductivity 3. Dualizing non-relativistic quantum elasticity 3.a Quantum nematic orders 3.b Superconductivity: dual Higgs is Higgs 4. The quantum nematic world crystal and Einstein’s space time 22 XY-electromagnetism duality in 2+1D Relativistic short hand (magnon), Consider quantum phase dynamics (XY in 2+1D), = 1 ∂ ϕ 2 π = 2 − ϕ − ϕ S ( µ ) mod(2 ) H ∑ (n i ) J ∑ cos( i j ) g i < ij > Hubbard-Stratanovich auxiliary field ξµ S = gξµξµ + iξµ∂µϕ ϕ = ϕ + ϕ Divide in smooth and multivalued field configurations, sm MV ξ ξ + ξ ∂ ϕ + ξ ∂ ϕ ξ ξ + ξ ∂ ϕ − ϕ ∂ ξ S = g µ µ i µ µ MV i µ µ sm g µ µ i µ µ MV i sm ( µ µ ) ϕ ξ sm acts like Lagrange multiplier ==> conserved==>imposedµ by gauge field Aµ ∂ µξ µ = 0 ⇒ ξ µ = εµνλ∂ν Aλ = µν + µ Dual action: S gFµν F iAµ JV , µ = ε ∂ ∂ ϕ ⇔ ϕ = π Vortex current: JV µνλ ν λ ∫ d 2 n 23 Disorder field theory Theory describing vortex tangle: Long range interaction Core vs kinetic energy Hard cores = d τ ∂ − Ψ 2 + 2 Ψ 2 + Ψ 4 + µν S ∫ dx d []| ()µ iAµ | m | | w | | Fµν F 24 Dualize elasticity: stress gauge fields (Kleinert) Quantum elasticity: = ∂ a∂ b S C µνab µ u ν u a ∂ a Hubbard-Stratanovich = stress (σ µ ) - strain ( µ u ) duality, = σ a −1 σ b + σ a∂ a + σ a ∂ a S µCµνab ν i µ µusm i µ µ uMV a Integrate over smooth displacement fields usm a a a ∂µσ µ = 0 ⇒ σ µ = εµνλ∂ν Bλ a Conservation of stress imposed by ‘stress photons’Bµ (pseudo tensors) −1 a a Dual action: S = [](ε∂B) C (ε∂B) + iBµ J µ a = ε ∂ ∂ a ⇔ a = a Jµ µνλ ν λ uMV ∫ du b a r = Jµ are dislocation currents, b ( b x , b y ) Burgers vector. 25 Phonons as stress photons ‘Phonon’ action (elastic medium) in stress photons: hc ph 2 ⎡ 2 2 T 2 1 2 2 T 2 2 2 L 2⎤ S = dq dω 2( ω + q /2)|B− | + ()()1−ν ω + q | B+ | +()ω + q | B− | 4µ ∫ ⎣⎢ 1 1+ ν 1 1 ⎦⎥ Transversal phonon Longitudinal phonon Instanteneous stress Strain (phonon) propagator recovered through dual relation: ∂ u a ∂ u b = C −1 δ δ − C −1 C −1 σ c σ d µ ν pr µµaa µν ab µλac νκbd λ κ pr a a σ µ = εµνλ∂ν Bλ 26 Nelson-Halperin-Young + hbar Nematic (‘hexatic’) quantum order: Particle transport at Dislocations bose condense dislocation collsions Disclinations stay massive Idealization: interstitials non-existent. 27 The dual (dislocation) condensate Ψ=Ψ iψ Ψ 2 ≠ = + + Dislocation Meissner Phase: e ,| | 0 ⇒ Leff LMeissner LMaxwell LDirector 2 2 ⎡ ωˆ 2 ωˆ −ωˆ 2 ⎤ = q0 | nT (Qab )| T 2 + L 2 − (1 ) T * L + LMeissner ⎢| B−1 | 4 2 | B−1 | 2 2 ()B+1 B−1 h.c. ⎥ 2µ ⎣⎢ 1+ ωˆ 1+ ωˆ ⎦⎥ =" ε∂ −1 ε∂ " =" ∂ 2 + 2 2 + 4" LMaxwell ( B)C ( B) LDirector ()Q mQ Q wQ Q λ = shear 1/q0 : shear penetration depth “Topological Nematic”(Toner- Lammert-Rokshar) Burgers vectors disordered Isotropic, but massive disclinations “Glide nematic” Burgers vectors disordered Broken rotational symmetry λ Anisotropic shear 28 Dislocations and shear 29 Dynamics: glide principle ‘topological dynamics’: propagation only possible along Burgers vector Field theory: requirement for finite compression modulus in liquid 30 Topological nematic superfluid: excitations Superfluid hydrodynamics: Isolated massless compression, massive shear: Euler fluid Periodicity (vortex quantization): inherited from dislocation condensate 31 Plan of talk 1. Plasticity (defected elasticity) and differential geometry 2. Fluctuating