Zilch currents in CKT Pavel Mitkin MIPT&ITEP ICNFP, August - PowerPoint PPT Presentation

Zilch currents in CKT Pavel Mitkin MIPT&ITEP ICNFP, August 2019

Chiral effects CKT for fermions Vortical effect in the CKT Vortical effects for photons Definition of zilch Zilch in the CKT Conclusions

Chiral Effects

Chiral Anomaly ψ i γ µ D µ ψ − 1 L = ¯ 4 F 2  � ∂ µ ¯ ψγ µ γ 5 ψ = 0

ψ i γ µ D µ ψ − 1 L = ¯ 4 F 2  � ∂ µ ¯ ψγ µ γ 5 ψ = 0  � ψγ µ γ 5 ψ = e 2 ∂ µ ¯ 2 π 2 E · B

In chiral media anomaly results in transport phenomena J µ = σ B B µ + σ ω ω µ , J µ 5 = σ 5 , B B µ + σ 5 , ω ω µ σ B = µ 5 , σ ω = µµ 5 2 π 2 π 2 � µ 2 + µ 2 + T 2 � σ 5 , B = µ 5 , σ 5 , ω = 2 π 2 2 π 2 6 where B µ = ˜ F µν u ν and ω µ = 1 2 ǫ µναβ u ν ∂ α u β .

In chiral media anomaly results in transport phenomena J µ = σ B B µ + σ ω ω µ , J µ 5 = σ 5 , B B µ + σ 5 , ω ω µ σ B = µ 5 , σ ω = µµ 5 2 π 2 π 2 � µ 2 + µ 2 + T 2 � σ 5 , B = µ 5 , σ 5 , ω = 2 π 2 2 π 2 6 where B µ = ˜ F µν u ν and ω µ = 1 2 ǫ µναβ u ν ∂ α u β .

Chiral effects were studied in various approaches: ◮ Free Dirac gas, linear response and strong field limit; ◮ Holographic plasma; ◮ Collisionless kinetic theory; ◮ Hydrodynamics; appearing to be pretty robust and always proportional to the anomalous coefficient ∂ µ J µ 5 = C E · B  � σ B ∼ σ ω ∼ σ 5 , B ∼ σ 5 , ω − T 2 6 ∼ C

◮ Chiral effects are a macroscopic manifestation of quantum anomaly ◮ Time parity of ❇ and Ω → chiral effects are dissipationless ◮ The origin of vortical effect is less clear ◮ tCVE → connection with gravitational anomalies?

Anomaly from Berry curvature in CKT The semiclassical action of a single particle: � dt ( ♣ · ˙ ① + ❆ ( ① ) · ˙ ① − ❛ p · ˙ S = ♣ − H ( p , x )) A single left-/right-handed fermion satisfies the Weyl equation ( σ · ♣ ) u p = ±| ♣ | u p The intersection of energy levels produces Berry connection i ❛ p ≡ u † p ∇ p u p with a monopole-like curvature in momentum space p ˆ ❜ = ∇ × ❛ p = ± 2 | ♣ | 2

Poisson brackets for this action are ǫ ijk B k ǫ ijk Ω k { p i , x j } = δ ij + Ω i B j { p i , p j } = − { x i , x j } = 1 + ❇ · Ω 1 + ❇ · Ω 1 + ❇ · Ω ∂ A j ∂ a ♣ j where B i = − ǫ ijk ∂ x k , Ω i = − ǫ ijk ∂ x k . Using these brackets one can proceed to develop a kinetic theory 1 for Fermi-liquid and obtain kinetic equation which implies non-conservation of the particles current: k ∂ t n + ∇ ❥ = 4 π 2 ❊ · ❇ where k is the number of quanta of Berry curvature through the Fermi surface. 1 Son, Yamamoto, (2012)

Equations of motion can be written as √ ① = ∂ε G ˙ ∂ ♣ + ❊ × ❜ + ❇ (ˆ ♣ · ❜ ) √ ♣ = ❊ + ∂ε G ˙ ∂ ♣ × ❇ + ❜ ( ❊ · ❇ ) √ where G = (1 + ❇ · ❜ ) 2 . The factor of G plays role of a Jacobian in the phase space √ d 3 xd 3 p / (2 π ) 3 → Gd 3 xd 3 p / (2 π ) 3 and needed to have a measure satisfying Liouville equation 2 : √ √ √ ♣ ) = 2 π ❊ · ❇ δ (3) ( ♣ ) ∂ t G + ∇ x ( G ˙ ① ) + ∇ p ( G ˙ 2 M. Stephanov et al, PRL, 2012

While the modified Liouville equation already indicates the axial anomaly, we can evaluate the current √ � ❥ = G ˙ ① f ( ♣ , ① ) p The explicit expression involves the dispersion which should also include the magnetization term ε = | ♣ | (1 − ❜ · B ) Taking the equilibrium limit and setting E = 0 one finds the same CME current ❥ ± = ± µ ± ❥ el = µ 5 ⇒ 4 π 2 ❇ 2 π 2 ❇ as in other approaches.

The simplest intuitive approach to describe vorticity via CKT 3 relies on the substitution ❇ → 2 | ♣ | Ω transforming the Lorentz force into the Coriolis force ♣ = ❊ eff + 2 | ♣ | ˙ ˙ ① × Ω Concentrating on the polarization currents we finally find � µ 2 � µ 2 + µ 2 4 π 2 Ω + T 2 + T 2 � � ± 5 ❥ ± = ± ⇒ ❥ 5 = Ω 2 π 2 12 6 which agrees with other derivations of chiral effects. 3 M. Stephanov et al, PRL, 2012

◮ One may be interested in the response of the helicity current of massless particles of arbitrary spin - say, photons ◮ Vortical effect for photons can indeed be found via Kubo formula 4 for the helicity current K µ = T 2 K µ = ǫ µναβ A ν ∂ α A β 6 ω µ ◮ The approach in CKT is also applicable for theory with constituents of an arbitrary spin 5 4 A. Avkhadiev, A. Sadofyev, PRD, 2017 5 X. G. Huang and A. V. Sadofyev, JHEP (2019)

Zilch currents

In 1964 Lipkin pointed out 6 that there is additional conserved current in the free electrodynamics ζ = ❍ · ❇ + ● · ❊ J ζ = − ❍ × ❊ + ● × ❇ , with ❍ = ∇ × ❇ , ● = ∇ × E 6 H. Lipkin, Journal of Mathematical Physics 5, (1964)

Later it was found 7 that there is an infinite number of related currents. In the covariant form they can be written as Z µ = F µν ∂ 2 n +1 F 0 ν − ˜ ˜ F µν ∂ 2 n +1 F 0 ν 0 0 In a fixed guage after quantization one can see that corresponding charge can serve as a specific measure of helicity � a † d 3 x : h := � ( − 1) λ ˆ Q h = λ ( J )ˆ a λ ( J ) J ,λ � d 3 x : ζ ( n ) := 2( − 1) n � ( − 1) λ ω (2 n +2) ˆ a † Q ζ = λ ( J )ˆ a λ ( J ) J ,λ 7 T.W.B. Kibble, Journal of Mathematical Physics 6, (1965).

Vortical effect for such a current of spin 3 was recently calculated explicitly 8 ❏ ζ (0) = 8 π 2 T 4 Ω 45 8 M. N. Chernodub, A. Cortijo and K. Landsteiner, Phys. Rev. D (2018)

Chiral kinetic theory We would like to obtain this result in CKT in order to connect ZVE with Berry phase. We need to construct a current of a charge � d 3 x : ζ ( n ) := 2( − 1) n � a † ( − 1) λ ω (2 n +2) ˆ Q ζ = λ ( J )ˆ a λ ( J ) J ,λ However, abundance of symmetries in a free electrodynamics implies abundance of conserved currents of the same charge. Therefore we redefine zilch of spin 2 n + 3 as Z i = ˜ F µ F 0) µ − F µ ˜ ( i ∂ 2 n +1 ( i ∂ 2 n +1 F 0) µ 0 0 The net value on the axis calculated in field theory is 2( − 1) n ❏ ζ (0) = (2 n + 5) Ω T 2 n +4 (2 n + 4)! ζ (2 n + 4) 3 π 2 (2 n + 3)

We expect that the current in kinetic theory for a particle of certain helicity is � Z i = 2( − 1) n p 2 n +2 j i ) (0 p In order for j i to be genuine vector it has to include a magnetization current 9 j µ = p µ f + S µν ∂ ν f 9 J. Y. Chen, D. T. Son, M. A. Stephanov, H. U. Yee, Y. Yin, “Lorentz Invariance in Chiral Kinetic Theory,” PRL (2014)

p µ u µ − 1 2 S µν ω µν � � Here f = n is a distribution function, p and ω ij = ǫ ijk ω k is a vorticity tensor. S ij = ǫ ijk p k The measure of integration is d 4 p � � (2 π ) 3 2 δ ( p 2 ) θ ( u · p ) = p

2( − 1) n Z i (0) = (2 n + 5) Ω i T 2 n +4 (2 n + 4)! ζ (2 n + 4) 3 π 2 (2 n + 3) The final expression for the current on the axis coincides with results obtained in the field theory

Conclusions ◮ CKT allows to calculate CVE for particles of an arbitrary spin ◮ ZVE - as a vortical effect for gauge invariant measure of helicity of photons - can be reproduced as well ◮ To do so one has to introduce a notion of the zilch current in CKT ◮ Strinkingly, the vortical effect in the zilch current is related to the Berry phase and the topological properties of the system in analogy with other chiral effects