Giant D5 Brane Holographic Quantum Hall States Gordon W. Semenoff - PowerPoint PPT Presentation

Giant D5 Brane Holographic Quantum Hall States Gordon W. Semenoff University of British Columbia Gauge/Gravity Duality 2013 Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Prelude • Quantized Hall effect occurs in two-dimensional electron gas “2DEG”. • Spectrum of weakly coupled charged particles in a constant magnetic field are flat bands called Landau levels. • The many-body state of Fermions completely filling a Landau level is incompressible ∂ρ ∂µ = 0 • A result of this (plus localization) is the integer quantum Hall effect. • Does this phenomenon persist in the strong coupling limit which is described by AdS/CFT holography, specifically top-down (probe-brane) holography? Ch.Kristjansen G.W.Semenoff arXiv:1212.5609 Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

D3 - Probe D5 brane System • N coincident D3 branes and N 5 coincident D5 branes oriented as 0 1 2 3 4 5 6 7 8 9 D 3 X X X X O O O O O O D 5 X X X O X X X O O O brane extends in directions X , sits at point in directions O • # ND = 4 system – preserves 1/2 of supersymmetries • ’t Hooft limit: N → ∞ , λ = 4 πg s N fixed: D3’s → AdS 5 × S 5 • probe limit N 5 << N embed D5’s in AdS 5 × S 5 √ • flat space ∼ strong coupling R 2 = λα ′ >> 1 • “2DEG” = D3-D5 strings hypermultiplet - fund. reps. of SU ( N ), U ( N 5 ) Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Probe D5 brane • Probe brane geometry from solving Dirac-Born-Infeld action plus Wess-Zumino terms ∫ [ − det( g + 2 πα ′ F ) + 2 πα ′ F ∧ ω (4) ] √ d 6 σ S 5 = N 5 T 5 − • ∃ a supersymmetric solution with SO (3) × SO (3) R-symmetry where worldsheet is AdS 4 × S 2 , √ r 2 ( − dt 2 + dx 2 + dy 2 ) + dr 2 [ ] F = 0 , ds 2 = λα ′ r 2 + d Ω 2 2 • AdS 5 × S 5 coordinates and 4-form dS 2 λα ′ = r 2 ( − dt 2 + dx 2 + dy 2 + dz 2 )+ dr 2 r 2 + dψ 2 +sin 2 ψd 2 Ω 2 +cos 2 ψd 2 ˜ Ω 2 √ ω (4) = λα ′ 2 r 4 dt ∧ dx ∧ dy ∧ dz + λα ′ 2 c ( ψ ) d Ω 2 ∧ d ˜ Ω 2 2 Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Dual to superconformal defect field theory • Field theory dual is bulk N = 4 Yang-Mills plus a hypermultiplet defect theory with SO(3) × SO(3) R-symmetry O.DeWolfe D.Z.Freedman H.Ooguri hep-th/0111135 J.Erdmenger Z.Guralnik I.Kirsch hep-th/0203020 { } ∫ − 1 2Tr F µν F µν + . . . d 4 x S = N 5 N [ ¯ ∫ ∑ ∑ d 3 x ψ σ α iγ µ ∂ µ ψ σ φ σ α ∂ µ φ σ ] + α + ∂ µ ¯ + interactions α σ =1 α =1 • Fermion ψ , scalar φ are SO(3) spinors (with different SO(3)’s), fundamental rep. of global U ( N 5 ) and fundamental rep. of SU(N) gauge group. • Holographic description introduces temperature T , U (1) ⊂ U ( N 5 ) charge density ρ , magnetic field B Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Weak Coupling N 5 N [ ¯ ∫ ∑ ∑ d 3 x ψ σ α iγ µ D µ ψ σ φ σ α D µ φ σ ] S = α + D µ ¯ + interactions α σ =1 α =1 External Magnetic field • D µ = ∂ µ + iA µ with a background magnetic field ⃗ ∇ × ⃗ A = B • Landau levels √ – Fermions E n = 2 Bn √ – Boson ω n = (2 n + 1) B B 2 π · 2 · N · N 5 – n = 0 , 1 , 2 , ... ; Landau level density is B • There exist 2 π 2 NN 5 fermion zero modes. • The lowest energy non-zero modes are scalars. Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Landau levels N 5 = 1, fermion is an SO (3) doublet In the charge neutral state, half of the zero modes are filled E (6B)^(1/2) (4B)^(1/2) (2B)^(1/2) 0 -(2B)^(1/2) -(4B)^(1/2) -(6B)^(1/2) Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Quantum Hall Ferromagnetism/Magnetic catalysis Arbitrarily weak repulsive interactions favor an asymmetric filling of the zero modes E (6B)^(1/2) (4B)^(1/2) (2B)^(1/2) E_int 0 -(2B)^(1/2) -(4B)^(1/2) -(6B)^(1/2) U (2) → U (1) × U (1) quantum Hall ferromagnetism/magnetic catalysis of chiral symmetry breaking:   ⟨ ¯  1 0 ̸ = 0 τ 3 =  both condensates ψ † τ 3 ψ ψτ 3 ψ ⟨ ⟩ ⟩ ̸ = 0 , − 1 0 nonzero Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Integer quantum Hall state Adding fermions to the neutral ground state eventually results in a charge gapped integer Hall state: E (6B)^(1/2) (4B)^(1/2) (2B)^(1/2) (B)^(1/2) E_int 0 -(2B)^(1/2) -(4B)^(1/2) -(6B)^(1/2) The next U(1) charged state is a scalar, no more Hall states. There are three incompressible states ν = 1 , 0 , − 1 Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Quantum Hall Ferromagnetism with many flavors, it is generally expected that the flavor degeneracy of zero modes is entirely removed E (6B)^(1/2) (4B)^(1/2) (2B)^(1/2) 0 -(2B)^(1/2) -(4B)^(1/2) -(6B)^(1/2) There are as many as 2 N 5 + 1 incompressible states ν = N 5 , ..., 1 , 0 , − 1 , ..., − N 5 G.W.S., Fei Zhou, arXiv:1104.4714 Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Integer Hall effect in Graphene K. Novoselov et. al. Nature 438, 197 (2005) Y. Zhang et. al. Nature 438, 201 (2005) σ xy = 4 e 2 n + 1 ( ) 2 h Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Graphene N 5 =4 U (4) → U (1) 4 seen in high field experiments Z.Jiang Y.Zhang Y.-W.Tan H.L.Stormer P.Kim S.S.Comm. 143, 14 (2007) Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Hall States The gapped states have charge densities and Hall conductivities ρ = B 2 π N · (0 , ± 1 , ± 2 , . . . , ± N 5 ) , σ xy = N · (0 , ± 1 , ± 2 , ± 3 , . . . , ± N 5 ) or filling fractions ν ≡ 2 π ρ B = 0 , ± 1 , ± 2 , . . . , ± N 5 N All other quantum Hall states are beyond the threshold for creating scalars. Do the quantum Hall states survive when we turn on the coupling? Do they survive at strong coupling? No renormalization for Chern-Simons term beyond one loop S eff = σ xy ∫ d 3 xϵ µνλ A µ ∂ ν A λ + . . . 4 π Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Probe D5 brane with a magnetic field • Introduce a magnetic field B ( m = 0 , ρ = 0 , T = 0) • V.Filev C.Johnson J.Shock arXiv:0903.5345 For any B , D5-brane is no longer AdS 4 × S 2 F = Bdx ∧ dy r 2 ( − dt 2 + dx 2 + dy 2 ) + dr 2 [ ] √ ds 2 = r 2 (1 + r 2 ˙ ψ ( r ) 2 ) + sin 2 ψ ( r ) d Ω 2 λα ′ 2 + < ¯ 2 + m = 0 ψ ( r → ∞ ) = π ψ⃗ τψ > + . . . ψ ( r = r 0 ) = 0 , r r 2 • Mass gap for D3-D5 strings • Spontaneously broken SO(3) chiral symmetry for any nonzero magnetic field (at zero temperature and density). • Quantum Hall Ferromagnetism/Magnetic catalysis at strong coupling, ρ = 0. Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

E (6B)^(1/2) (4B)^(1/2) (2B)^(1/2) E_int 0 -(2B)^(1/2) -(4B)^(1/2) -(6B)^(1/2) Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Probe D5 brane with a magnetic field and density • Introduce a magnetic field B and density ρ ( m = 0 , T = 0) F = A ′ t ( r ) dr ∧ dt + Bdx ∧ dy r 2 ( − dt 2 + dx 2 + dy 2 ) + dr 2 [ ] √ ds 2 = r 2 (1 + r 2 ˙ ψ ( r ) 2 ) + sin 2 ψ ( r ) d Ω 2 λα ′ 2 + < ¯ 2 + m = 0 ψ ( r → ∞ ) = π ψ⃗ τψ > + . . . r r 2 • Probe D5 must reach Poincare horizon at r = 0 → all finite density states are ungapped (compressible) . • Chiral symmetry restored at critical density K.Jensen A.Karch D.T.Son E.G.Thompson arXiv:1002.3159 √ ν ≡ 2 πρ , ν crit . = 1 . 68 N 5 / λ NB Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Hall states of the D5 brane • As N 5 D5 branes enter the bulk of AdS 5 , they blow up to D7 brane with magnetic flux − det( g + 2 πα ′ F ) + (2 πα ′ ) 2 [ ] ∫ d 8 σ √ F ∧ F ∧ ω (4) S 7 = T 7 − 2 √ r 2 ( − dt 2 + dx 2 + dy 2 ) + dr 2 [ ds 2 = 1 + r 2 ψ ′ ( r ) 2 ) λα ′ ( + r 2 + sin 2 ψd 2 Ω 2 + cos 2 ψd 2 ˜ ] Ω 2 F = d t ( r ) dr ∧ dt + Bdx ∧ dy + N 5 2 d ˜ drA 7 Ω 2 (1) • F ∧ F ∧ c (4) ( r ) term in D7 brane action dissolves electric charge – completely only when ν = 1 • For ν = 1, D7 brane has Minkowski embedding and incompressible charge gapped state. Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

• For ν = 2 , 3 , ..., N 5 , N 5 D5 branes blow up into 2 , 3 , ..., N 5 D7 branes, each with ν = 1 which subsequently have Minkowski embeddings and incompressible charge gapped state. • How many of the states ν = 0 , ± 1 , ± 2 , ..., ± N 5 are stable still open question. Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Entropy at zero temperature Entropy of free fermions S ( T = 0) = − BN 2 π [ ν ln ν + (1 − ν ) ln(1 − ν )] Entropy of probe brane A.Karch M.Kulaxi A.Parnachev arXiv:0908.3493 √ S ( T = 0) = BN λ D 5 : 2 ν 2 π √ S ( T = 0) = BN λ 2 | 1 − ν | D 7 : 2 π Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.

Gauge/Gravity Duality 2013, Max Planck Institute, Munich, July 30,2013.