Universe Heavy-ion collisions Compact stars Dirac semimetals, graphene, etc.

Transcription

1 NOV 23, 2015

2 MAGNETIC FIELDS EVERYWHERE [Miransky & Shovkovy, Physics Reports 576 (2015) pp ] Universe Heavy-ion collisions Compact stars Dirac semimetals, graphene, etc. November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 2

3 Universe Current galactic magnetic fields ~ 10-6 G Current magnetic fields in voids ~ G Problem of magnetogenesis in Early Universe Perhaps during the electro-weak phase transition to G (~ 1 GeV to 100 GeV) November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 3

4 Magnetic field/helicity Magnetic helicity evolution in the Early Universe d(n L n R ) = 2α 1 dt π V d 3 x ( E! B)! B! = σ E! + α π µ! B + E! 5 t! E =! B t [Vilenkin, Phys. Rev. D22, 3080 (1980)] [Joyce & Shaposhnikov, astro-ph/ ] [Giovannini & Shaposhnikov, hep-ph/ ] [Boyarsky et al., arxiv: ] [Tashiro et al., arxiv: ] [Manuel et al., arxiv: ] [Buividovich et al., arxiv: ] [Hirono et al., arxiv: ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 4

5 Magnetized QGP at RHIC/LHC B ~ to G (~ 100 MeV) v B Little Bangs Using Lienard-Wiechert potentials, v [Rafelski & Müller, PRL, 36, 517 (1976)], [Kharzeev et al., arxiv: ], [Skokov et al., arxiv: ], [Voronyuk et al., arxiv: ], [Bzdak &. Skokov, arxiv: ], [Deng & Huang, arxiv: ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 5

6 CME, CSE, etc. Chiral magnetic/separation effects, chiral magnetic waves (correlations of charged particle in HIC) Signs of local P-violation? [Kharzeev, McLerran, Warringa, Nucl. Phys. A 803, 227 (2008)] [Fukushima, Kharzeev, Warringa, Phys. Rev. D 78, (2008)] Review: Kharzeev, Liao, Voloshin, Wang, arxiv: [hep-ph] Signs of a chiral magnetic wave? [Yee, Kharzeev, Phys. Rev. D 83, (2011)] [Gorbar, Miransky, Shovkovy, Phys. Rev. D 83, (2011)] [Burnier, Kharzeev, Liao, Yee, Phys. Rev. Lett. 107 (2011) ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 6

7 Dirac/Weyl materials High magnetic field lab 10 5 G (~ 100 vf=c/300) Graphene 3D materials with Dirac/Weyl quasiparticles Bi 1-x Sb x alloy (at x 4%) Na 3 Bi Cd 3 As 2 ZrTe 5 [Z. K. Liu et al., arxiv: ] [M. Neupane et al., arxiv: ] [S. Borisenko et al., arxiv: ] [X. Li et al., arxiv: ] TaAs, NbAs, TaP, [arxiv: , arxiv: , arxiv: , arxiv: ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 7

8 MAGNETIC CATALYSIS [Miransky & Shovkovy, Physics Reports 576 (2015) pp ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 8

9 Landau spectrum at B 0 Dirac equation with massless fermions (e<0) iγ 0 i!! $ γ (! + ie A) & % 0 ' Ψ = 0 Energy spectrum E n (3+1) ( p 3 ) = ± 2n eb + p 3 2 s = ± 1 2 (spin) Occupied where n = s + k k = 0, 1, 2, (orbital) November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 9

10 Dimensional reduction Low energy theory: highly degenerate n=0 Landau level E 0 (3+1) ( p 3 ) = ± p 3 Occupied Low-energy excitations are (1 space +1 time )- dimensional Motion in xy-plane is restricted November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 10

11 Essence of magnetic catalysis Dimensional reduction D D 2 Nonzero density of states at E = 0 dn Symmetry breaking happens at arbitrarily weak interaction e.g., = eb N f de 4π 2 E 0 In other words, B-field acts as a catalyst [Gusynin, Miransky, Shovkovy, PRL 73 (1994) 3499] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 11

12 Magnetic catalysis in QCD Magnetic catalysis of chiral symmetry breaking & anisotropic confinement (vacuum QCD, T 0) T=0: catalysis T c (B): inverse catalysis [Miransky & I.S., Phys. Rev. D 66 (2002) ] [Bali et al., Phys. Rev. D86, (2012)], [Bali et al., JHEP 02, 044 (29012)] [Bali et al., PRD86, (2012)] [G. Endrodi, arxiv: ], November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 12

13 CHIRAL SHIFT GENERATION Dirac + B = Weyl November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 13

14 Dirac vs. Weyl (semi-)metal Low-energy Hamiltonian of Dirac/Weyl semimetal!! H = d 3 rψ# $ iv F γ p ( )! b! γ ( )γ 5 % & ψ + H int Dirac semimetal Weyl semimetal Broken T-symmetry November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 14

15 Landau spectrum µ What if LLL is partially filled? November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 15

16 Partially filled LLL LLL is spin polarized and chirally asymmetric states with p 3 <0 (and s= ) are R-handed states with p 3 >0 (and s= ) are L-handed This indeed implies axial current density R R R R p 3 p 3 p 3 p 3 s s s s! j5! j5 s s s s p 3 p 3 p 3 p 3 L L L L November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 16

17 Chiral shift at B 0 The axial current (CSE) in free theory ψγ 3 γ 5 ψ = eb 2π 2 µ with! B = (0,0,B) [Metlitski & Zhitnitsky, Phys. Rev. D 72, (2005)] + interactions should induce a chiral shift parameter Δ associated with the condensate, δl = Δψγ 3 γ 5 ψ This is a perturbative effect: there is no symmetry to protect Δ =0 [Gorbar, Miransky, Shovkovy, Phys. Rev. C 80, (R) (2009)] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 17

18 Chiral Fermi surface Chirality is well-defined at Fermi surface ( p 3 m) L-handed Fermi surface: p 3 n > 0 : p 3 = + ( µ 2 2n eb s Δ) 2 m 2 n p 3 = R-handed Fermi surface: ( µ 2 2n eb + s Δ) 2 m 2 p 3 n > 0 : p 3 = ( µ 2 2n eb s Δ) 2 m 2 n p 3 = + ( µ 2 2n eb + s Δ) 2 m 2 [Gorbar, Miransky, Shovkovy, Phys. Rev. D 83, (2011)] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 18

19 QED in weak field (B 0) The result has the form Σ (1) ( p) = γ 3 γ 5 Δ( p) +γ 0 γ 5 µ 5 ( p) Near Fermi surface (p 0 0, p p F ) Δ( p) α ebµ $ & m 2 ln π m 2 & % 2µ p p F µ 5 ( p) α eb µ π m 2 p 3 p F ( ) 1 & m 2 ( ln ' 2µ p p F November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 19 ' ) ) ( ( ) 1 [Gorbar, Miransky, Shovkovy, Wang, Phys. Rev. D 88, (2013)] ) + *

20 Dispersion relations in QED Let us use the condition (for a small B) Det[ i S 1 ( p) + Σ (1) (p)] = 0 [Gorbar, Miransky, Shovkovy, Wang, Phys. Rev. D 88, (2013)] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 20

21 QED in strong field Self-energy in the Landau-level representation: Σ = 2e k 2 l 2 ( 1) n m n +γ 3 γ 5 Δ n γ 0 γ 5 µ 5,n +... n=0 where m n, Δ n, µ 5,n, are projections of the self-energy on the nth Landau level, Δ n (k 0,k 3 ) = ( 1)n l 2 8π µ 5,n (k 0,k 3 ) = ( 1)n l 2 ( )' ( P + L n P L n 1 8π d 2 d 2 k e k 2 l 2 % L n + L ' & n+1 ( Tr % & γ 0 Σ(k) ' ( ) * +... k e k 2 l 2 % L n L ' & n+1 ( Tr % & γ 0 γ 5 Σ(k) ' ( [Gorbar, Miransky, Shovkovy, Wang, Phys. Rev. D 88, (2013)] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 21

22 Chiral shift B 0 Model fit: Δ n = α eb µ # eb n & % $ µ 2 ( ' [Xia, Gorbar, Miransky, Shovkovy, Phys. Rev. D 90, (2014)] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 22

23 µ 5,n in B 0 Model fit: µ 5,n = α eb µ " 1 $ # 2n eb µ 2 % ' & 2 [Xia, Gorbar, Miransky, Shovkovy, Phys. Rev. D 90, (2014)] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 23

24 QED in strong field: δp 3 Model fit: p 3 p 3 (0) = ± α eb µ " eb n % $ # µ 2 ' & p 3 n p 3 n [Xia, Gorbar, Miransky, Shovkovy, Phys. Rev. D 90, (2014)] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 24

25 In QED: In QCD: How large is the asymmetry? α eb " µ 0.4 B $ # G %" ' 100MeV % $ &# µ ' MeV/c & α s eb µ 10 " B % $ # G " ' $ 400MeV &# µ % ' MeV/c & may have some observable consequences November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 25

26 SOME PROBLEMS TO ADDRESS Effects of finite size Role of inhomogeneities November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 26

27 Magnetic field + electric chemical potential = chiral current Effects of finite size Positive chiral charge? B µ 0 j 5! j5 = e! B 2π 2 µ Is the chiral charge truly separated? Negative chiral charge? November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 27

28 Model of Dirac semimetal with a slab geometry where Boundary conditions: Dirac semimetal and m(z) = M θ(z 2 a 2 ) + mθ(a 2 z 2 ), with vacuum band gap: M (broken chiral symmetry) a -a z y B x [Gorbar, Miransky, Shovkovy, Sukhachov, arxiv: ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 28

29 Wave functions Wave functions are standing waves, e.g., where the wavevector p z is determined by the spectral equation [Gorbar, Miransky, Shovkovy, Sukhachov, arxiv: ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 29

30 Only LLL contributes Discretized CSE For m 0: m=0 v F =2.5 ev Å a=100 Å m=0.05 ev v F =2.5 ev Å a=100 z=0 [Gorbar, Miransky, Shovkovy, Sukhachov, arxiv: ] November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 30

31 Axial current vs. µ and z Axial current density is non-uniform when m 0 m=0 v F =2.5 ev Å a=100 Å m=0.05 ev v F =2.5 ev Å a=100 Å Note that axial charge density vanishes: [Gorbar, Miransky, Shovkovy, Sukhachov, arxiv: ] j 5 0 = 0 November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 31

32 Axial current as a standing wave? Recall that LLL is spin polarized B R-handed L-handed B A perfect chirality flip at the boundary November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 32

33 Key observations Chiral current in the CSE is discretized m 0: chiral current density is non-uniform m=0: chiral current density is uniform Chiral current is not necessarily connected with a flow of chiral charge Chiral current need not lead to chiral charge accumulation on the boundary November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 33

34 Role of inhomogeneities? When µ 5 0, there is no CME in a slab What about chiral magnetic waves? What about inverse cascade in cosmology? Reality Inhomogeneous Non-static What is the systematic approach to use? November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 34

35 ANOMALOUS MAXWELL EQUATIONS November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 35

36 Anomalous Maxwell equations Common homogeneous approximation: How to generalize these equations to include inhomogeneities? November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 36

37 Starting point: Chiral kinetic theory where is an expansion in powers of E-M field and November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 37

38 Equations for chemical potentials Resulting equation of motion for µ λ : The corresponding equations for the currents: CME drift & diffusion Hall type CSE diffusion CESE November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 38

39 New types of currents New contribution to the electric current:! jnew = 2τ µ µ 5 3π 2 c µ 5 x! eτ 2 µ 5 µ 5 6π 2 x! B! eτ! µ E 5 6π 2 x! New contribution to the chiral current:! j5,new = eτ 2 µ µ 5 6π 2 x! B! + eτ 2 µ 5 e E! µ 6π 2 x! B! eτ! µ E 6π 2 x! November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 39

40 Instead of summary We now have Anomalous Maxwell equations for inhomogeneous chiral plasmas All previously known currents are reproduced A whole set of new currents obtained Observational signatures?! e.g., how about j5,new = eτ! µ E? 6π 2 x! November 23, 2015 Magnetic Fields in Strongly Interacting Matter, Utrecht University, The Netherlands 40