Emergent Gravity As the Nematic Dual of Lorentz-Invariant Elasticity

SMR 1646 - 12

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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

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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!

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

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

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: ∂ a ∂ b = −1 δ δ − −1 −1 σ c σ d µ u ν u r C µµaa µν ab C µλac Cνκbd λ κ r p p 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 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

32 Superconductivity: photons vs. stress-photons

Charged elastic medium (bosonic Wigner crystal): couple in EM gauge fields

2 r r µν ⎛ 1 ⎞ S = dx dτ[]n e u • E + Fµν F E =±⎜∂ Aτ − ∂τ A ⎟ Fµν = ∂µ Aν −∂ν Aµ ∫ e x,y ⎝ x,y c x,y ⎠

Dualize in stress photons, focus on magnetic sector: 2 2 = + + 1 nee Ltot LAA LAB LBB = bare London penetration depth, λ2 ρ 2 L c superconductor?? ⎛ ω 2 ⎞ = 1 + + 2 2 + 2 L ⎜ q ⎟ | A− | ....| A+ | Effective EM action: integrate out stress AA λ2 c 2 1 1 ⎝ L ⎠ photons

nee T L T L =− B− A− + fA()+ ,B− ,B+ AB ρc 2 1 1 1 1 1 Shear length finite (fluid): ⎡ ⎤ 1 ⎛ 2 ⎞ Crystal: Meissner term eaten L = ⎢⎜ 2ω 2 + q2 + ⎟ | BT |2 +gB()L ,BT ⎥ compensation incomplete, BB µ ⎜ λ2 ⎟ −1 −1 +1 4 ⎣⎢⎝ shear ⎠ ⎦⎥ electromagnetic Meissner ‘liberated’!!

33 The dual Higgs boson and electron loss

Dual (shear) Plasmon Higgs photon ω ≈ Dual Higgs mass p 1eV Ω≈50meV!?

34 Superfluid hydrodynamics

Superfluid: quantized Euler fluid r r ∂ ur + (ur •∇ )ur = ∇p ,"mod(2π)" t Landau Feynman

Footnote (this talk): this is the zero temperature hydrodynamics of a solid which has lost its rigidity against long range shear forces.

35 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

36 Emergent gravity: general covariance Distances measured by hopping from lattice site to lattice site, metric:

gµν = δ µν + (∂ µ uν + ∂ν uµ ) General covariance: metric is defined modulo local translations

Dislocation condensate: Coherently delocalized dislocations ==> 3a distances defined modulo local translations!

Bonus: Dislocations represent torsion, dislocation condensate = Riemann space without torsion 4a (like general relativity)! 37 Emergent gravity: curvature

Represent curvature (e.g., in 2+1D literally like conical defects) Disclination current (2+1D): 1 Θ µν = εµκλ∂κ ∂ λω ν , ω µ = εµνλ ()∂ν uλ − ∂λ uν 4 corresponds with the (linearized) Einstein tensor: 1 Θµν ≅ Gµν = Rνµ − gνµ R 2

In “solid”: disclinations are confined (infinite energy) In nematic superfluid: deconfined but massive …

38 Dynamics: double curl gauge fields

Recall: ‘conventional’ stress photons ==> only sources are disclinations σ µν = ε µκλ ∂ κ B iB λν → µν J J µν , µν = ε µκλ ∂ κ ∂ λ u ν P Disclination currents: one ‘order’ higher in duality 1 Θ µν = εµκλ∂κ ∂ λω ν , ω µ = εµνλ ()∂ν uλ − ∂λ uν 4 Kleinert: double curl stress gauge fields (two forms)

σ µν = εµκλεναβ∂κ ∂α hλβ ⎡ ⎛ ν ⎞ ⎤ = 2 τ 1 σ 2 − σ 2 + η ==> SWC ∫ d xd ⎢ ⎜ µν µµ ⎟ ihµν µν ⎥ ⎣ 4µ ⎝ 1+ ν ⎠ ⎦ ∂ η = Sources are truely conserved ( µ µν 0 ) ‘defect currents’

η µν = θ µν + ελµκ ∂λ Jνκ

39 The nematic ‘Lorentz’ crystal

Dislocation melting of the 2+1D ‘Lorentz’ (space-time isotropic) crystal:

(a) Consider topological nematic ==> space-time isotropy. (b) No preferred time direction ==> glide constraint is impossible ==> Dislocations turn into sources of compressional ‘photons’ ==> sound acquires a Higgs mass!

Symmetric = simple version of non-relativistic nematics

40 The nematic ‘Lorentz’ crystal

ν 2 ⎡ 1 ⎛ 2 2 ⎞ ⎤ = τ σ − σ + η σ µν = εµκλεναβ ∂κ ∂α hλβ Input: SWC ∫ d xd ⎢ ⎜ µν µµ ⎟ ihµν µν ⎥ ⎣ 4µ⎝ 1+ ν ⎠ ⎦ ηµν = θµν + ελµκ ∂λJνκ ⎡ m2 ⎤ = 2 τ d 2 + ε ∂ Condense dislocations ‘isotropically’, Sdislo ∫ d xd ⎢ Jµν iJµν µκλ κ hλν ⎥ ⎣ 2 ⎦

⎡ ν ⎤ 2 1 ⎛ 2 2 ⎞ 1 1 S = d xdτ ⎜ σ µν − σ µµ ⎟ + σ µν σ µν eff ,space ∫ ⎢ µ + ν 2 ∂ 2 ⎥ ⎣ 4 ⎝ 1 ⎠ 2m d ⎦

⎡ 2 ⎤ = 2 τ mθ θ 2 + θ Sdiscl ∫ d xd ⎢ µν ihµν µν ⎥ ⎣ 2 ⎦

41 Geometrical meaning

Consider dual ‘stress’ geometry: h µν = g µν − δ µν

Stress tensor turns into Einstein tensor: σ µν ≡ Gµν ⎡ ν ⎤ 2 1 ⎛ 2 2 ⎞ 1 1 S = d xdτ ⎜ G µν − Gµµ ⎟ + Gµν Gµν eff ,space ∫ ⎢ µ + ν 2 ∂ 2 ⎥ ⎣ 4 ⎝ 1 ⎠ 2m d ⎦

Non-linear generalization ∂ → , etc; identify 2 = π µ Dµ m d 8 G 1 1 c 3 → d 2 xdt −g G G →− d 2 xdt −g R ∫ 2 µν − 2 µν π ∫ 2md D 16 G At long distances this becomes exactly the Einstein action Incompressible (2+1 D): shear and compression massive, curvature rigidity is still present. 42 Gravitating matter

Normal matter: gravity is uniformly attractive = 2 τ Smatter ∫ d xd []hµν Tµν T - symmetric Belinfante energy-momentum tensor No condensed matter analogy !

Disclinations: massive excitations ‘guarding’ the curvature rigidity of space ⎡ 2 ⎤ = 2 τ mθ θ 2 + θ Sdiscl ∫ d xd ⎢ µν ihµν µν ⎥ ⎣ 2 ⎦ Anti-dislocations do antigravity … Startrek!

43 Emergent Einstein Gravity

c 3 S =− dV −g R + S 16πG ∫ matter

44 Plastic Cosmology: the competing phases

The Lorentz-crystal: non- Space-time does not resist gravitating torsion and curvature: matter causes vacuum pressure catastrophic deflation dominated universe (curvature confined)

Our place Fortunately!

45 Conclusions

• Condensed matter physics: strongly correlated superconductors The quantum nematic orders The Meissner phase as the dual dislocation condensate • Waiting for experiments

• Relativistic generalization: emergent gravity. Makes sense in 2+1D, 3+1D generalization? • Allegory or the holy truth? Subject for the Theology Department!

J. Zaanen, Z. Nussinov and S.I. Mukhin, Ann. Phys. (NY) 310, 181 (2004) (cond-mat/0309397) ; H. Kleinert and J. Zaanen, Phys. Lett. A 324, 361 (2004) (cond-mat/0309379); V. Cvetkovic, S.I. Mukhin, J. Zaanen, in preparation. 46