Chirality and Macroscopic Helicities

Transcription

1 Chirality and Macroscopic Helicities Andrey V. Sadofyev MIT UCLA, March, 2017 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

2 Introduction µ ψγ µ γ 5 ψ = 1 2π 2E B Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

3 Introduction Chiral medium is a system consisting of massless fermions. This theory is subject to the axial anomaly meaning that at the quantum level there is a violation of the chiral symmetry J 5 E B. However the axial charge could be redefined Q 5 = Q (0) π 2 A B and we may think about its conservation. In medium the anomaly has macroscopic manifestations J a µ = σ a B B µ + σ a ωω µ x Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

4 Introduction In more details J V µ = σ V B B µ + σ V ω ω µ, J A µ = σ A BB µ + σ A ωω µ σ V B = µ A 2π 2, σ V ω = µ V µ A σ A B = µ V 2π 2, σ A ω = π 2 ( µ 2 V + µ 2 A + T 2 2π 2 6 ) and the conductivities appear to be pretty universal. Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

5 Chiral instabilities The dynamics of the system could be considerably influenced by the anomalous transport chiral instability 1 ( ) ) 2 B = σv (k B B = 2 σv B B k = 0 More accurately, considering helical modes B = KB one finds t B = 1 B + σv B B = t B = ( K 2 + σv B K ) B σ E σ E σ E σ E It results in the transfer of the chiral asymmetry to the macroscopic helicity N A H mh = A Bd 3 x 1 M. Joyce, M. Shaposhnikov, 1997; Y. Akamatsu, N. Yamamoto, 2013; Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

6 Chiral instabilities On the other hand, in the presence of dissipation H mh is not conserved due to reconnections and there is an opposite process 1 H mh N A The full axial charge tends to get redistributed among various degrees of freedom and the final distribution strongly depends on the details of the dynamics 2. Although, in the absence of intrinsic large parameters, it is natural to expect 3 N A H mh 1 Y. Hirono et al, 2015; Y. Hirono et al, 2016; V. Kirilin, AS, 2017; 2 Y. Hirono et al, 2015; V. I. Zakharov, 2016; K. Tuchin, V. I. Zakharov, 2016 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

7 Chiral instabilities Now concentrate on the CVE which gives an additional contribution to the axial charge: J µ 5 = σa ωω µ, Q fh σω A (v Ω) d 3 x. It survives in the absence of the EM fields µ J µ A = C ae B µ J µ A = 0 E,B 0 J A µ = σ A ωω µ + σ A B B µ J A µ = σ A ωω µ and we naively allow Q fh Q (0) 5. Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

8 Chiral instabilities A simple explanation may be obtained in an EFT description 1 δs = µ ψγ 0 ψd 4 x µu µ ψγ µ ψd 4 x where we introduced a slowly varying boost field. As a result one expects the following shift A A + µu Then, the axial charge conservation is modified ( t Q (0) 5 + C a A B + σb A 2 v B + x x x ) σσ A v Ω = 0. where helicities are known to correspond to the linkage of the flow and/or field lines. 1 AS, V.I. Shevchenko, V.I. Zakharov, 2011 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

9 Chiral instabilities If the generalized axial charge picture is valid it may considerably enrich the dynamics, for instance through the new instabilities 1 Q (0) 5 H mh, H fmh, H fh However it is NOT obvious that there is a unified conservation of the axial charge with helicities. Indeed, in the ideal hydrodynamics all helicities conserved separately σ E, η 0 = t H = 0 while one has to suggest a microscopic mechanism to transfer the medium motion to fermionic modes in the presence of the dissipation. 1 A. Avdoshkin, V. Kirilin, AS, V.I. Zakharov, 2014 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

10 Chiral vortical effects Recently we found an example of such a mechanism 1 with an intermediate generation of H mh σω A v Ω d 3 x A Bd 3 x N A Moreover, the first transition may be seen as a consequence of a novel chiral effect CVE for photons: K µ T 2 ω µ Here K 0 d 3 x corresponds to the full helicity of photons and K µ is known to be tied with their spin current 2. 1 A. Avkhadiev, AS, A.D. Dolgov et al., 1989 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

11 Chiral vortical effects But first let s shortly review the thermal contribution to the CVE for fermions (tcve): The first ancestor of this effect is a polarized counterpart of the Hawking radiation for neutrinos by a rotating BH 1 : dn ν dtdɛdo ɛ2 M 2 (1 2πLMΩ cos θ) It can be extended to the rotating thermal radiation giving the common tcve 2 J ν µ = T 2 12 ωµ But WHY do people consider this transport to be anomalous? 1 A. Vilenkin, A. Vilenkin, 1979 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

12 Chiral vortical effects The tcve is shown to be related to the gravitational cousin of the axial anomaly in a holographic consideration 1, the dictionary gives d = 5 d = 4 χ ɛ MNPQR A M RA BNP R AQR B = µ J µ 5 χr R where χ controls the anomalous coefficient. J µ 5 χt 2 ω µ 1 K. Landsteiner et al., 2011 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

13 Chiral vortical effects However the relation is less clear on the field theory side. For instance, all other chiral effects are fixed by the axial anomaly even in the hydrodynamic limit 1 but NOT the tcve 2. Indeed, the gravitational anomaly appears to be of higher order in derivatives: µ J µ 5 = CE B O( 2 ) µ J µ 5 = C g R R O( 4 ) J µ 5 µ2 ω µ O( ) J µ 5 T 2 ω µ O( ) and cannot play the same role as the axial anomaly in EM fields. 1 D. T. Son, P. Surowka, Y. Neiman, Y. Oz, 2011 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

14 Chiral vortical effects In the weakly interacting limit, the tcve can be obtained by the Kubo formula with gravitational perturbation g 0i v i : J µ A (x) = σa ωω µ, σω A i = lim ɛ ijk J i q 0 q A T 0j ω=0 k q+k J A i T 0j > > k k q K. Landsteiner et al., 2011 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

15 Chiral vortical effects Less direct relation to the anomalies: The coefficient in front of the gravitational anomaly and the coefficient in the conductivity coincide 1 µ J µ 5 = # 768π 2 ɛµνρλ R α βµν Rβ αρλ, Jµ 5 = # 12 T 2 ω µ The tcve is (non)-renormallizable and given by the sum of the quantized 3d contributions 2 σω A 1 = lim ɛ ijk JA i q 0 q T 0j = 2T 2 k m = T 2 6 m=1 this picture can be used to show its relation to the global anomalies 3. 1 K. Landsteiner et al., S. Golkar, D.T. Son, S. Golkar, S. Sethi, 2015 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

16 Chiral vortical effect for bosons Let s summarize: The tcve can be seen as a polarization effect in a rotating thermal radiation. It has an analog in the Hawking radiation of a rotating BH. It is related to the gravitational anomaly in the holographic considerations. There is a connection of the tcve and the global anomalies. Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

17 Chiral vortical effect for bosons Let s summarize: The tcve can be seen as a polarization effect in a rotating thermal radiation. One expects polarization effects for any non-zero spin. It has an analog in the Hawking radiation of a rotating BH. It is related to the gravitational anomaly in the holographic considerations. There is a connection of the tcve and the global anomalies. Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

18 Chiral vortical effect for bosons Let s summarize: One expects polarization effects for any non-zero spin. It has an analog in the Hawking radiation of a rotating BH. The spin-gravity interaction is trivial and it is natural to expect the same effect for s 1 2 It is related to the gravitational anomaly in the holographic considerations. There is a connection of the tcve and the global anomalies. Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

19 Chiral vortical effect for bosons Let s summarize: One expects polarization effects for any non-zero spin. The spin-gravity interaction is trivial and it is natural to expect the same effect for s 1 2 It is related to the gravitational anomaly in the holographic considerations. There are well known examples of gravitational anomalies for theories with s > 1 2 There is a connection of the tcve and the global anomalies. Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

20 Chiral vortical effect for bosons Gravitational anomaly for photons 1 : F µν = F µν cos α + F µν sin α K µ = 1 g ɛ µναβ A ν α A β, µ K µ = 1 2 F F, µ K µ naive = 0 µ K µ triangle = 1 96π 2 R R 1 A.D. Dolgov et al., 1988 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

21 Chiral vortical effect for bosons Let s summarize: One expects polarization effects for any non-zero spin. The spin-gravity interaction is trivial and it is natural to expect the same effect for s 1 2 There are well known examples of gravitational anomalies for theories with s > 1 2 There is a connection of the tcve and the global anomalies. The relation of some chiral effects to the global anomalies could be extended to s 1 (some examples for d 4 in the 2 literature). Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

22 Chiral vortical effect for bosons Let s summarize: One expects polarization effects for any non-zero spin. The spin-gravity interaction is trivial and it is natural to expect the same effect for s 1 2 There are well known examples of gravitational anomalies for theories with s > 1 2 The relation of the tcve to the global anomalies could be extended to s 1 2 (some examples for d 4 in the literature1 ). 1 S.D. Chowdhury, J.R. David, 2016 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

23 Chiral vortical effect for bosons J µ A = Cɛµναβ A ν α A β ( ) J µ A = T e2 6 4π 2 ω µ S. Golkar, D. T. Son, 2012 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

24 Chiral vortical effect for bosons Combining these insights one expects the bosonic CVE (bcve) K µ T 2 ω µ and it is supported by the direct calculation 1. 1 A. Avkhadiev, AS, 2017 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

25 Chiral vortical effect for bosons Combining these insights one expects the bosonic CVE (bcve) i σ b = lim ɛ ijk K i q 0 q A T 0j ω=0 k K µ = ɛ µναβ A ν α A β T µν = F µλ F ν λ g µν ( 1 4 F m2 A 2 ) K µ T 2 ω µ and it is supported by the direct calculation 1. 1 A. Avkhadiev, AS, 2017 Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

26 Summary and Directions This effect can be used as an example of H fh H mh t H mh µ K µ µ T 2 ω µ Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

27 Summary and Directions This effect can be used as an example of H fh H mh t H mh µ K µ µ T 2 ω µ While the tcve relation to the anomalies is under discussion, it is clear that the bcve and tcve have the same origin (polarization by Ω, BH radiation, g. anomaly, global anomalies, etc.). Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

28 Summary and Directions This effect can be used as an example of H fh H mh t H mh µ K µ µ T 2 ω µ While the tcve relation to the anomalies is under discussion, it is clear that the bcve and tcve have the same origin (polarization by Ω, BH radiation, g. anomaly, global anomalies, etc.). Photons can have non-trivial topological phase 1 and one may think about a generalization of the chiral kinetic theory 2. 1 V. Liberman et al., 1992; K. Bliokh et al., 2004; M. Onoda et al., 2004; 2 N. Muller, R. Venugopalan, 2017; N. Yamamoto, 2017; Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29

29 Summary and Directions This effect can be used as an example of H fh H mh t H mh µ K µ µ T 2 ω µ While the tcve relation to the anomalies is under discussion, it is clear that the bcve and tcve have the same origin (polarization by Ω, BH radiation, g. anomaly, global anomalies, etc.). Photons can have non-trivial topological phase 1 and one may think about a generalization of the chiral kinetic theory 2. Possible phenomenological applications in condensed matter, primordial plasma, QGP, etc. 1 V. Liberman et al., 1992; K. Bliokh et al., 2004; M. Onoda et al., 2004; 2 N. Muller, R. Venugopalan, 2017; N. Yamamoto, 2017; Andrey V. Sadofyev (MIT) Chirality and Macroscopic Helicities UCLA, March, / 29