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Chiral Currents from Anomalies ω2 α 2 ω1 α1 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 1 Chiral currents from Anomalies Michael Stone Institute for Condensed Matter Theory University of Illinois Santa-Fe July 18 2018 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 2 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 3 Experiment aims to verify: k ρσαβ r T µν = F µνJ − p r [F Rνµ ]; µ µ 384π2 −g µ ρσ αβ k µνρσ k µνρσ r J µ = − p F F − p Rα Rβ ; µ 32π2 −g µν ρσ 768π2 −g βµν αρσ Here k is number of Weyl fermions, or the Berry flux for a single Weyl node. Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 4 What the experiment measures Contribution to energy current for 3d Weyl fermion with H^ = σ · k µ2 1 J = B + T 2 8π2 24 Simple explanation: The B field makes B=2π one-dimensional chiral fermions per unit area. These have = +k Energy current/density from each one-dimensional chiral fermion: Z 1 d 1 µ2 1 J = − θ(−) = 2π + T 2 β(−µ) 2 −∞ 2π 1 + e 8π 24 Could even have been worked out by Sommerfeld in 1928! Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 5 Are we really exploring anomaly physics? Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 6 Yes! Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 7 Energy-Momentum Anomaly k ρσαβ r T µν = F µνJ − p r [F Rνµ ]; µ µ 384π2 −g µ ρσ αβ Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 8 Origin of Gravitational Anomaly in 2 dimensions Set z = x + iy and use conformal coordinates ds2 = expfφ(z; z¯)gdz¯ ⊗ dz Example: non-chiral scalar field '^ has central charge c = 1 and energy-momentum operator is T^(z) =:@z'@^ z'^: Actual energy-momentum tensor is c T = T^(z) + @2 φ − 1 (@ φ)2 zz 24π zz 2 z c T = T^(¯z) + @2 φ − 1 (@ φ)2 z¯z¯ 24π z¯z¯ 2 z¯ c T = − @2 φ zz¯ 24π zz¯ z z¯ Now Γzz = @zφ, Γz¯z¯ = @z¯φ, all others zero. So find z z¯ r Tzz + r Tzz¯ = 0 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 9 Chiral field Energy-momentum tensor for chiral field Tz¯z¯ = 0 c T = T^(z) + @2 φ − 1 (@ φ)2 zz 24π zz 2 z c T = − @2 φ zz¯ 48π zz¯ ij In these coordinates Ricci scalar R = R ij is given by. −φ 2 R = −4e @zz¯ φ Now we find c rzT + rz¯T = − @ R zz zz¯ 96π z 1+1 d Gravitational Energy-Momentum Anomaly Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 10 Gravitational Anomaly in 1+1 dimensions 1+1 Chiral Fermion Anomaly in Minkowski signature coordinates: 1 νσ r T µν = − p @ R µ 96π −g σ Apply to r, t Schwarzschild metric 1 ds2 = −f(r)dt2 + dr2 f(r) where f(r) ! 1 at large r, and vanishes linearly as r ! rH+. 00 Ricci Scalar R = −f . ν Timelike Killing vector η gives genuine (non)- conservation 1 @ p 1 νση r (T µνη ) ≡ p −gT r = p ν @ R; µ ν −g @r t 96π −g σ Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 11 Hawking radiation in 1+1 dimensions (Robinson, Wilczek 2005; Banerjee 2008;...) Have @ p 1 1 @ 1 ( −gT r ) = f@ f 00 = ff 00 − (f 0)2 ; @r t 96π r 96π @r 2 Integrate 1 p r 1 1 00 1 0 2 −gT t = ff − (f ) : rH 96π 2 rH Nothing is coming out of the black hole: the LHS is zero at rH . 0 2 The RHS is zero at infinity, and equal to (1=96π)(f ) =2 at the horizon. Thus 1 1 T r (r ! 1) ≡ J = κ2 = πT 2 ; t 48π 12 Hawking X 0 where κ = f (rH )=2 is the surface gravity and THawking = κ/2π. Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 12 Hawking radiation Euclidean Schwarzschild manifold is skin of a salami sausage with circumference β = 2π/κ = 1=THawking.(Gibbons, Perry 1976) Recall that Sommerfeld gives Z 1 d 1 µ2 1 J = − θ(−) = 2π + T 2 β(−µ) 2 −∞ 2π 1 + e 8π 24 1 ! πT 2; when µ = 0. 12 Energy-momentum anomaly ) Sommerfeld. Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 13 Aside: Hawking radiation in the Laboratory? M. Stone, Class. Quant. Gravity 30 085003 (2013) . y Hall bar with electrodes to anti-confine. Inflowing current divides at \event horizon" Exterior Region Solve lowest Landau-level x Horizon quantum scattering problem Black Hole Interior (parabolic-cylinder functions) Find that chiral fermion edge is in Hawking-temperature thermal state Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 14 Gravitational Axial Anomaly k µνρσ k µνρσ r J µ = − p F F − p Rα Rβ µ 32π2 −g µν ρσ 768π2 −g βµν αρσ Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 15 Chiral Vortical Effect In a (Born) frame rotating with angular velocity Ω, have on-axis current (Vilenkin 1979, Landsteiner, Megias, Pe~na-Benitez2011): 1 Z 1 1 1 J (r = 0) = 2 d − z 2 β(−(µ+Ω=2)) β(−(µ−Ω=2)) 4π −∞ 1 + e 1 + e µ2Ω Ω3 1 = + + ΩT 2: 4π2 48π2 12 µ2 1 = Ω + T 2 + O(Ω3): 4π2 12 CME with B ! 2Ω Coriolis force? (Stephanov, Son et al.) | but no 4 ! 2 dimensional reduction, and no T 2=12 this way 2 Pontryagin (R ) term in axial anomaly? (Landsteiner et al.) Yes! Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 16 Axial-anomaly origin of CVE Obtain CVE from the mixed axial anomaly by using the 4-d metric 2 dt − Ω r2dφ 1 r2 (dφ − Ω dt)2 ds2 = −f(z) + dz2 + dr2 + : (1 − Ω2r2) f(z) (1 − Ω2r2) 2 2 2 2 2 2 Looks complicated, but ds ! −dt + dz + dr + r dφ as f(z) ! 1. Horizon at f(zH ) = 0 rotates with angular velocity Ω 0 Again THawking = f (zH )=4π Pontryagin density: 1 @ µνρσRa Rb = Ω (r[f 0(z)]2) + O[Ω3] 4 bµν aρσ @z Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 17 Axial-anomaly origin of CVE Assume no current at horizon, and integrate up k µνρσ r J µ = − p Rα Rβ ; µ 768π2 −g βµν αρσ from horizon to infinity. Again, by a seeeming miracle, result depends only on value of 0 f (zH ) = 4πTHawking. Find contribution to current is Ω J = T 2 + O(Ω3): z 12 ) CVE consequence of gravitational axial anomaly Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 18 General anomaly-related currents µν µ ν µν µ ν ν µ µ ν ν µ T = ( + p)u u − pg + ξTB(B u + B u ) + ξT!(! u + ! u ) µ µ µ µ Jn = nu + ξJBB + ξJ!! number current µ µ µ µ JS = su + ξSBB + ξS!! entropy current No-entropy-production imposes relationships (Stephanov, Yee 2015) 2 2 ξJB = Cµ, ξJ! = Cµ + XBT 2 ξSB = XBT; ξS! = 2µT XB + X!T 1 ξ = Cµ2 + X T 2 ; TB 2 B 2 ξ = Cµ3 + 3X µT 2 + X T 3 T! 3 B ! For the ideal Weyl gas 1 1 C = ;X = ;X = 0: 4π2 B 12 ! We have confirmed that XB is gravitational anomaly Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 19 Curved space Dirac/Weyl equation Massless Dirac equation in curved space: a µ 1 bc D6 Ψ ≡ γ ea @µ + 2 σ !bcµ Ψ = 0 µ ea = ea @µ is a Minkowski-orthonormal vierbein µ !abµdx is spin connection 1-form ab 1 a b 1 ab σ = 4 [γ ; γ ]; Γµ = 2 σ !abµ: Decompose 0 D6 D6 = + D6 − 0 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 20 Gravitational Anomaly from Dirac . Work in frame rotating with horizon to order Ω Find " 0 1 @ p @ f iΩ 0 = − + f[z]σ3 + − pf[z] @t @z 4f 2 @ 1 1 @ @ +σ + + σ + rΩ Ψ 1 @r 2r 2 r @φ @t 0 Absorb the 1=2r and f =4f by setting 1 Ψ = Ψ~ r1=2f 1=4 Introduce tortoise coordinate z∗ by dz∗ = dz=f(z). Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 21 Gravitational Anomaly from Dirac Find @ @ i 0 = + σ3 − f(z)Ω @t @z∗ 2 @ 1 @ @ +pf(z) σ + σ + rΩ Ψ 1 @r 2 r @φ @t ~ Could absorb a z∗ dependent phase into Ψ to remove the if(z)Ω=2. Bad idea! | this is an axial transformation. Origin of spectral flow and anomaly Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 22 My Conclusions: Experiment does not directly involve gravity yet it highlights the intimate and fascinating connection between temperature and gravity Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 23 ...first seen by Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 24.
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