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 ∇ T µν = F µνJ − √ ∇ [F Rνµ ], µ µ 384π2 −g µ ρσ αβ k µνρσ k µνρσ ∇ J µ = − √ F F − √ 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 Hˆ = 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 ∞ 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 ∇ T µν = F µνJ − √ ∇ [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 = exp{φ(z, z¯)}dz¯ ⊗ dz

 Example: non-chiral scalar field ϕˆ has central charge c = 1 and energy-momentum operator is Tˆ(z) =:∂zϕ∂ˆ zϕˆ:  Actual energy-momentum tensor is c T = Tˆ(z) + ∂2 φ − 1 (∂ φ)2
zz 24π zz 2 z c T = Tˆ(¯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¯ ∇ Tzz + ∇ 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 = Tˆ(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 ∇zT + ∇z¯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 ∇ T µν = − √ ∂ 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 ∂ √ 1 νση ∇ (T µνη ) ≡ √ −gT r = √ 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 ∂ √ 1 1 ∂  1  ( −gT r ) = f∂ f 00 = ff 00 − (f 0)2 , ∂r t 96π r 96π ∂r 2

 Integrate   ∞ √ r ∞ 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 → ∞) ≡ 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 ∞ 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 ∇ J µ = − √ F F − √ 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  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 ∇ J µ = − √ 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 ω(ω u + ω u ) µ µ µ µ Jn = nu + ξJBB + ξJωω number current µ µ µ µ JS = su + ξSBB + ξSωω entropy current No-entropy-production imposes relationships (Stephanov, Yee 2015) 2 2 ξJB = Cµ, ξJω = Cµ + XBT 2 ξSB = XBT, ξSω = 2µT XB + XωT 1 ξ = Cµ2 + X T 2
, TB 2 B 2 ξ = Cµ3 + 3X µT 2 + X T 3
T ω 3 B ω For the ideal Weyl gas 1 1 C = ,X = ,X = 0. 4π2 B 12 ω

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 ωbcµ Ψ = 0

µ  ea = ea ∂µ is a Minkowski-orthonormal vierbein µ  ωabµdx is spin connection 1-form ab 1 a b 1 ab  σ = 4 [γ , γ ], Γµ = 2 σ ωabµ.

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