Electromagnetic duality in AdS/QHE: magnetic monopoles and the - PowerPoint PPT Presentation

Electromagnetic duality in AdS/QHE: magnetic monopoles and the quantum Hall effect Brian Dolan National University of Ireland, Maynooth and Dublin Institute for Advanced Studies Allan Bayntun 1 , Cliff Burgess 1 , 2 and Sung-Sik Lee 1 1 Perimeter Institute, 2 McMaster University, Canada AdS4/CFT3 and the Holographic States of Matter Galileo Galilei Institute for Theoretical Physics 3rd November 2010 Brian Dolan 0

Outline The Quantum Hall effect Review of QHE Modular symmetry Quantum phase transitions and temperature flow Selection rule AdS/CFT correspondence Duality in bulk theory Schwinger-Zwanziger quantisation Bulk solution and scaling exponents Brian Dolan 0

The Classical Hall Effect Edwin Hall (1879) B I I W L p-type n-type ◮ J α = σ αβ E β ( σ xx = σ yy ). ◮ z = x + iy ⇒ ρ := ρ xy + i ρ xx , σ = σ xy + i σ xx = − 1 /ρ . ◮ Classically: ρ cl xy = − B en , ρ cl m xx = e 2 n τ c , τ c = collision time. Im ( ρ ) ≥ 0 ⇔ Im ( σ ) ≥ 0. ◮ Brian Dolan 1

The Quantum Hall Effect von Klitzing (1980); Tsui + Störmer (1982) ρ xy 1 1/2 1/3 1/4 1/5 B ◮ For low T , high purity and high particle density, resistance is quantised: R H = h / e 2 = 25 . 812807449 ( 86 ) k Ω . � h � � e 2 � ◮ ρ = 1 , p ∈ Z ; σ = p , von Klitzing (1980). p e 2 h Integer QHE � e 2 � ◮ σ = p , p , q ∈ Z , q odd, Tsui + Störmer (1982). q h Fractional QHE Brian Dolan 2

Stormer (1992) Brian Dolan 3

The Quantum Hall Effect Energy 4 ν=3 ω c = eB m h ω c 2 1 ◮ Free particles in transverse B ⇒ Harmonic Oscillator. � � � �� e 2 � � � eB � = � B ◮ Degeneracy/unit area: g = . h e h � h � � e 2 � ◮ Filling factor, ν := n / g = ne ⇒ | σ cl xy | = ν , ( σ xx = 0). e 2 B h ◮ Filled Landau Levels inert ⇒ pseudo-particle excitations are � e 2 � the same for σ → σ + 1, h = 1 . ◮ Particle-hole symmetry (one-third full = one-third empty): σ → 1 − ¯ σ . Brian Dolan 4

The Law of Corresponding States Kivelson, Lee and Zhang (1992); Lütken+Ross (1992) ◮ Physics of pseudo-particle excitations is symmetric under Landau level addition: σ → σ + 1 − 1 − 1 → Flux attachment: σ + 2 σ Particle-Hole Interchange: σ → 1 − σ Modular Group: σ → a σ + b Γ 0 ( 2 ) ⊂ Γ( 1 ) : c σ + d a , b , c , d ∈ Z , ad − bc = 1 with c even. � a � b Γ 0 ( 2 ) : γ = ∈ Sl ( 2 , Z ) , det γ = 1; c even. c d Γ( 1 ) , Fradkin+Kivelson (1996); Γ( 2 ) , Georgelin et al (1996) Witten [hep-th/0307041]; Leigh+Petkou [hep-th/0309171]. Brian Dolan 5

Quantum Phases ◮ Hall Plateaux ⇔ Phases of 2-D “Electron” Gas. ◮ Law of Corresponding States: maps between phases. ◮ σ xy : p / q → p ′ / q ′ is a Quantum Phase Transition, Fisher (1990). ◮ For σ xy = 1 / q , quasi-particles have electric charge e / q , Laughlin (1983). ◮ Second order phase transition between phases: ξ ≈ | ∆ B | − ν ξ , ∆ B = B − B c . ◮ Simple scaling ⇒ σ ( T , ∆ B , n , . . . ) = σ (∆ B / T κ , n / T κ ′ , . . . ) ◮ Superuniversality: κ and κ ′ are the same for all transitions. ◮ σ flows as T is varied. ◮ Experimentally: κ = 0 . 42 ± 0 . 01 (Wanli et al (2009)) Brian Dolan 6

Temperature flow Burgess+Lütken (1997), BD (1999), Lütken+Ross (2009). ◮ Attractive fixed points at σ xy = p / q , q odd; repulsive points for q even. ◮ Fractal structure near real axis (no true fractals in Nature). (Wigner crystal for σ xy < 1 7 ; � ω c < k B T ( σ xy >> 1).) Brian Dolan 7

σ (∆ B / T κ , n / T κ ′ ) S.S. Murzin et al (2002) Brian Dolan 8

0.2 (a) 2 /h) 0.1 σ xx (e 0.0 1/5 1/3 2/5 1/2 3/5 2/3 0 1/4 0.2 (b) 2 /h) 0.1 σ xx (e 0.0 1/5 1/4 1/3 2/5 1/2 3/5 2/3 0 2 /h) σ xy (e S.S. Murzin et al., (2005) Brian Dolan 9

Selection Rule σ 1+i 3+i 2 2 3+i 10 0 1 2 ◮ Any σ xy : p q → p ′ q ′ can be obtained from σ : 0 → 1 by some γ ∈ Γ 0 ( 2 ) , � p ′ − p � p q , γ ( 1 ) = p ′ γ ( 0 ) = p q ′ ⇒ γ = , det γ = 1 ⇒ q ′ − q q ◮ Selection Rule: p ′ q − pq ′ = 1 BPD (1998). Brian Dolan 10

Stormer (1992) Brian Dolan 11

AdS/CFT Correspondence ◮ AdS/CFT: ( 2 + 1 ) -d sample is boundary of ( 3 + 1 ) -d gravity coupled to matter. ◮ QHE: strongly interacting electrons in 2 + 1 dimensions. ◮ Conductivity is dimensionless ⇒ CFT in ( 2 + 1 ) -d. ◮ Use classical gravity + matter in ( 3 + 1 ) -d bulk. ◮ Bulk theory: AdS 4 -black-hole-dyon (AdS 4 -Reissner-Nordström) coupled to U ( 1 ) gauge theory with charged matter. Hartnoll+Kovtun [0704.1160]; Keski-Vakkuri+Per Kraus [0905.4538]. Brian Dolan 12

Electromagnetic duality in bulk ◮ Include dilaton φ and axion χ in bulk: � � 1 � �� � R − 2 Λ − 1 ∂φ.∂φ + e 2 φ ∂χ.∂χ S = 2 κ 2 2 � √− gd 4 x − 1 2 e − φ F 2 − χ F µν = 1 2 F � ( � ǫ µνρσ F √− g F ρσ ) 2 ◮ Constitutive relations: D i = G i 0 , H i = 1 2 ǫ ijk G jk � D = e − φ E + χ B ∂ L 2 G µν := − √− g ⇒ H = e − φ B − χ E ∂ F µν ◮ Define τ := χ + ie − φ ; F = F − i � F and G = − � G − iG ◮ Equations of motion invariant under Sl ( 2 , R ) , ( ad − bc − 1 ) . � G � � a � � G � τ → a τ + b b Gibbons+ c τ + d , → F c d F Rasheed (1995) � E � � cos α � � E � − sin α ◮ Generalises EM duality: → . B sin α cos α B Brian Dolan 13

Modular symmetry ◮ Dyons { Q , M } , { Q ′ , M ′ } ⇒ Q ′ M − M ′ Q = 2 π N � n m n ′ e − n ′ Q = n e e , M = n m ( h / e ) ⇒ m n e = N ◮ Dirac-Schwinger-Zwanziger quantisation condition: semi-classically, Sl ( 2 , R ) → Sl ( 2 , Z ) . ◮ Generalises Dirac quantisation condition: for { Q ′ , 0 } and { 0 , M } , Q ′ M = 2 π N � , SO ( 2 ) → Z 2 . ◮ In full quantum theory expect a sub-modular group, e.g. Γ( 2 ) for N = 2 SUSY Yang-Mills, Seiberg+Witten (1994). Brian Dolan 14

Bulk theory ◮ Bulk metric: ( Λ = − 3 L 2 , v = L r ) � � f ( v ) v 2 + dx 2 + dy 2 − f ( v ) dt 2 dr 2 ds 2 = L 2 λ 2 v 2 z + v 2 ◮ v → 0 ( r → ∞ ) is UV-limit of ( 2 + 1 ) -d theory. ◮ z : Lifshitz scaling exponent ( x → ℓ x , y → ℓ y , t → ℓ z t ). | f ′ ( v h ) | ◮ f ( v h ) = 0 ⇒ finite temperature, T = L . 4 π v z − 1 h ◮ Matter: classical Sl ( 2 , R ) symmetry � Maxwell Gibbons+ Einstein-dilaton-axion- DBI Rasheed (1995) ◮ � �� � � � − √− g d 4 x g µν + ℓ 2 e − φ/ 2 F µν S U ( 1 ) = −T − det � d 4 x √− g χ F µν � − 1 F µν . 4 Brian Dolan 15

Scaling exponents from AdS/CFT ◮ Calculate CFT conductivity using probe brane ⇒ � � B n σ z , 2 2 T T z Karch+O’Bannon [0705.3870]; O’Bannon [0708.1994]; Hartnoll et al [0912.1061]; Goldstein et al [0911.3589; 1007.2490]. ◮ Bulk solution (Taylor [0812.0530]) : χ = 0, � v � z + 2 F vt = Q v 2 e − φ = v 4 , f ( v ) = 1 − , L 2 v h ⇒ Sl ( 2 , R ) z = 5. Scaling dimension (Bayntum, Burgess, Lee+BPD, arXiv:[1007.1917]) κ = 2 ⇒ z = 5 z = 0 . 4 Brian Dolan 16

Summary ◮ Modular transformations, σ → a σ + b c σ + d , map between phases of the QHE ( σ = σ xy + i σ xx ). ◮ The map is a symmetry of QHE vacua. ◮ Fractional charges in the quantum Hall effect are analogous to the Witten effect in 4-dimensions. ◮ 4-d bulk theory with electromagnetic duality, Sl ( 2 , R ) → Γ ⊂ Sl ( 2 , Z ) / Z 2 can give AdS/CFT with QHE in 2 + 1-d. ◮ Probe brane in bulk gives information about conductivity in CFT. ◮ Parameters in bulk solution ⇔ exponents in CFT, κ = 2 / 5. Brian Dolan 17

The symmetries of the modular group are beautifully exhibited by transforming to z = 1 + i σ 1 − i σ , (Poincaré map): ⌊ z ⌊ σ − →

Maxwell - Chern - Simons Theory ◮ Classical relation ( J 0 = en and σ xx = 0) B = − en ρ cl xy ⇒ σ cl xy B = J 0 from − σ xy A 0 B + A 0 J 0 + · · · ⇒ L eff [ A 0 ] = − σ xy 2 ǫ µνρ A µ ∂ ν A ρ + A µ J µ + · · · . L eff [ A ] = ◮ Include Ohmic conductivity, σ xx = i lim ω → 0 ( ωǫ ( ω )) , L eff [ A ] = − ǫ 4 F 2 − σ xy 2 ǫ µνρ A µ ∂ ν A ρ + A µ J µ + · · · , L eff [ A ] ≈ i σ xx 4 ω F 2 − σ xy 4 ǫ µνρ A µ F νρ + A µ J µ + · · · . Brian Dolan 21

Statistical Gauge Field x i − x j x i x j i ϑ � π (Σ i < j φ ij ) Ψ( x 1 , . . . , x N ) Ψ( x 1 , . . . , x N ) = e φ ij O ◮ Interchange i ↔ j , φ ij → φ ij + π ⇒ phase changes by ϑ . ◮ ϑ = 2 π k , identity; ϑ = π ( 2 k + 1 ) , Fermions ↔ Bosons. ◮ In Hamiltonian, − i � ∇ − e A → − i � ∇ − e ( A + a ) : � � a α ( x i ) = � ϑ β a α ( x i ) = 2 � ϑ α φ ij ⇒ ǫ βα ∇ ( i ) ∇ ( i ) δ ( x i − x j ) . e π e j � = i j � = i b ( x ) := ǫ βα ∇ β a α ( x ) = 2 � ϑ e n ( x ) . ◮ Brian Dolan 22

Composite fermions and flux attachment B ϑ = − 2 π � h � � h � ( ϑ = − 2 π ) n = ϑ ◮ b := ǫ βα ∇ β a α ⇒ b − 2 . = π e e ◮ A µ → A ′ µ = A µ + a µ . � 1 � ν ⇔ 1 ◮ ν = 1 / 3 ⇔ ν CF = 1, ν + 2 . Fractional QHE = Integer QHE for composite Fermions, Jain (1990). Brian Dolan 23