Part 1. March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 2

Transcription

1 MAR 5, 2014

2 Part 1 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 2

3 ! Examples of relativistic matter Electrons, protons, quarks inside compact stars (white dwarfs, neutron, hybrid or quark stars) Quark gluon plasma in heavy ion collisions (k B T ~ 200 MeV ~ K) Hot matter in the Early Universe (k B T ~ 100 GeV at EW transition) Quasiparticles in graphene (zero mass Dirac fermions) March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 3

4 ! Relativistic matter ( ) compare with nonrelativistic case ( ) High density (e.g., in stars) leads to occupation of states with large momenta: High temperature (e.g., heavy ion collisions) means energetic particles, Vanishing mass (e.g., graphene) works too March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 4

5 Part 2 Review: arxiv: March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 5

6 ! Strong magnetic fields exist inside compact stars to Gauss! In heavy ion collisions, positive ions generate short-lived (!t " s) magnetic fields to Gauss! Early Universe up to Gauss Illustration by Carin Cain! Graphene (High Magnetic Field Laboratory) 4.5! 10 5 Gauss March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 6

7 ! Fermions in magnetic field L = " i# µ D "+ (interactions) µ! Free energy spectrum E (3+1) ( p ) = ± 2n eb + p 2 n 3 3 where s = ± 1 2 (spin) n = s + k Occupied k = 0, 1, 2, (orbital) March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 7

8 ! Low-energy is due to n=0 Landau level n = 0 : E 0 (3+1) ( p 3 ) = ± p 3 # k = 0, s = " 1 & % ( $ 2'! This is (1+1)D spectrum! Occupied! Propagator looks (1+1)D as well: S( p ) " i e # p $ 2! p ˆ 2 + m ( p ˆ + m 1# i%1 % 2 ), where p ˆ = p 0 % 0 # p 3 % 3 s = " 1 2 spin projector March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 8

9 ! Low-energy regime is dimensionally reduced D " D # 2! Density of states at!"#"$ dn = eb N f de 4# 2 E " 0 (This may remind superconductivity ) [Gusynin, Miransky, Shovkovy, Phys. Rev. Lett. 73 (1994) 3499] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 9

10 ! n=0: particles & anti-particles! Bound states are energetically favorable (an energy gain of E b per pair)! Bound states are bosons! Bosons can (and will) occupy same zero momentum quantum state! Bose condensate forms! Symmetry breaking! energy (mass) gap [Gusynin, Miransky, Shovkovy, Phys. Rev. Lett. 73 (1994) 3499] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 10

11 ! While m 0 =0 originally, a nonzero dynamical mass m dyn is generated (2D) " # eb, and m dyn m dyn! This happens even at the weakest interaction ( catalysis ) (3D) " eb e #C /$! The phenomenon is universal (model details are irrelevant)! Dimensional reduction is the key ingredient (massless bound states form = symmetry breaking) March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 11

12 ! Input Spin-1/2 charged particles and B#0 Attractive particle-antiparticle interaction! Output Dimensional reduction D->D-2 (low energies) Bound states form and condense Symmetry breaks down Dynamical mass is generated March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 12

13 Part 3 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 13

14 ! It is a single atomic layer of graphite [Novoselov et al., Science 306, 666 (2004)]! 2D crystal with hexagonal lattice of carbon atoms! Interesting basic physics! Great promise for applied physics March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 14

15 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 15

16 ! Low energy quasiparticles are massless Dirac fermions ( ) v F = c /300! Spinor: % # KAs ' # KBs " s = ' ' # K $ Bs ' & # K $ As! Low-energy model with U(4) global symmetry: ( * * * * ) H 0 = v F " d 2 r # s ( $ 1 % x +$ 2 % y )# s [Wallace, Phys. Rev. 71, 622 (1947)] [Semenoff, Phys. Rev. Lett. 53, 2449 (1984)] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 16

17 ! Charge carriers are spin-$ fermions with m=0! m dyn #0 is expected in a strong magnetic field [Khveshchenko, PRL 87, (2001)] [Gorbar, Gusynin, Miransky, Shovkovy, PRB 66 (2002) ]! Possible complications: many types of Dirac masses in 2D competition with quantum Hall ferromagnetism nonzero electron/hole density impurities, lattice defects, ripples, etc.! How to test this experimentally? March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 17

18 ! General setup Current starts to run: Steady state: Hall conductivity: j x = " xy E y March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 18

19 4-fold degenerate levels E n = ± 2!v F 2 n eb n=3 n=2 n=1 E F n=0 DOS n=1 n=2 n=3 " xy = # e2 h = 4 $ n + 1 ' & % 2( ) e2 h!!%(#!!%&#!!$#!!"#!!'"#!!'$#!!'%&# [Gusynin, Sharapov, Phys. Rev. Lett. 95, (2005)] [Peres, Guinea, Castro Neto, Phys. Rev. B 73, (2006)] [Novoselov et al., Nature 438, 197 (2005)], [Zhang et al., Nature 438, 201 (2005)] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 19

20 ! New plateaus at "=0 "=±1 "=±3 "=±4!!'(#!"#$%&'(&#)*+&,-.&)$*#/01231&453316&!!(#!!'%#!!%#!!&#! Some Landau level degeneracy is lifted [Novoselov et al., Science 315, 1379 (2007)] [Abanin et al., Phys. Rev. Lett. 98, (2007)] [Checkelsky et al., Phys. Rev. Lett. 100, (2008)] [Xu Du et al., Nature 462, 192 (2009)] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 20

21 ! Charge carriers are massless Dirac fermions! Spectrum in magnetic field: E n = ± 2!v F 2 n eb! Degenerate E=0 level with particles & holes! Electron-hole (excitonic) pairing occurs! m dyn #0 is generated! In qualitative agreement with experiment [Gorbar, Gusynin, Miransky, Shovkovy, Phys. Rev. B 66 (2002) ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 21

22 ! Many order parameters may be generated (pairing from different valleys/sublattices)! Dirac masses [triplet under U(2) s ] " s : # P s # =$ + KAs $ KAs %$ + + KBs $ KBs +$ & (charge-density wave)! Haldane masses [singlet under U(2) s ] " s : # $ 3 $ 5 P s # =% + KAs % KAs &% + + KBs % KBs & (% '! + spin & pseudo-spin densities + K As $ K & As %$ K & Bs $ K & Bs + K As % K ' As &% K ' Bs % K ' Bs ) [Gorbar, Gusynin, Miransky, Shovkovy, Phys. Rev. B 78 (2008) ] [Gorbar, Gusynin, Miransky, Shovkovy, Phys. Scr. T 146 (2012) ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 22

23 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 23

24 ! Competition between Dirac & Haldane masses is subtle! Symmetry breaking lattice effects! Dynamical screening effects! Competition with quantum Hall ferromagnetism! Nonzero electron/hole density (%>0)! impurities, lattice defects, ripples, etc. March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 24

25 Part 4 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 25

26 ! Helicities of massless (or ultra-relativistic) particles are (approximately) conserved Righ-handed Left-handed! Conservation of chiral charge is a property of massless Dirac theory (classically)! The symmetry is anomalous at quantum level March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 26

27 ! Chiral charge is produced by topological QCD configurations d(n R " N L ) dt! Random fluctuations with nonzero chirality in each event! Driving electric current!! j = " e2 B = " g2 N f % d 3 µ$ x F 16# 2 a F a µ$ N R " N L # 0 $ µ 5 # 0 2# 2 µ 5 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 27

28 ! Dipole pattern of electric currents (or charge correlations) in heavy ion collisions March 5, 2014 [Kharzeev, McLerran, Warringa, Nucl. Phys. A 803, 227 (2008)] [Fukushima, Kharzeev, Warringa, Phys. Rev. D 78, (2008)] Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 28

29 [B. I. Abelev et al. [The STAR Collaboration], arxiv: ] [B. I. Abelev et al. [STAR Collaboration], arxiv: ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 29

30 ! Axial current induced by fermion chemical potential!! j = " e B 5 2# 2 µ (free theory!) [Vilenkin, Phys. Rev. D 22 (1980) 3067] [Metlitski & Zhitnitsky, Phys. Rev. D 72, (2005)] [Newman & Son, Phys. Rev. D 73 (2006) ]! Exact result (is it?), which follows from chiral anomaly relation! No radiative corrections expected March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 30

31 ! Start from a small baryon density and B#0! Produce back-to-back electric currents March 5, 2014 [Gorbar, Miransky, Shovkovy, Phys. Rev. D 83, (2011)] [Burnier, Kharzeev, Liao, Yee, Phys. Rev. Lett. 107 (2011) ] Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 31

32 ! Any radiative corrections to CSE?!! j = " e B 5 2# 2 µ +... (yes)! Any dynamical parameter " ( chiral shift ) associated with this condensate? L = L 0 + "# $ 3 $ 5 # (yes) [Gorbar, Miransky, Shovkovy, Phys. Rev. D 83 (2011) ]! Note: "=0 is not protected by any symmetry March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 32

33 ! Chirality is " well defined at Fermi surface! L-handed Fermi surface: ( k 3 >> m) n = 0 : k 3 = + (µ " s # $) 2 " m 2 n > 0 : k 3 = + ( µ 2 " 2n eb " s # $) 2 " m 2 k 3 = " ( µ 2 " 2n eb + s # $) 2 " m 2! R-handed Fermi surface: n = 0 : k 3 = " (µ " s # $) 2 " m 2 n > 0 : k 3 = " ( µ 2 " 2n eb " s # $) 2 " m 2 k 3 = + ( µ 2 " 2n eb + s # $) 2 " m 2 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 33

34 d 4 k " (1) (p) = #4i$ ' ( 2$ ) 4 % µ S (1) (k)% & D µ& (k # p)! The result has the form " (1) (p) = # 3 # 5 $ +# 0 # 5 µ 5 (p) +! where, in the small B limit, " # $ eb µ ' m 2 ln % m 2 ) ( 2 µ p & p F µ 5 ( p) " # $ eb µ % m 2 p 3 p F ( ) &1 & m 2 ( ln ' 2µ p # p F March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 34 *, + ( ) #1 ) + *

35 ! Let us use the condition (for a small B) Det[ i S "1 ( p) + # (1) (p)] = 0 [Gorbar, Miransky, I.S., Wang, PRD 88 (2013) ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 35

36 ! Lagrangian density L = " 1 4 F µ# F µ# +$ ( i% µ D µ + µ% 0 " m)$ + (counterterms)! Axial current j 5 3 = "Z tr [# 3 # 5 G(x, x) ] 2! To leading order in coupling &=e 2 /(4') G(x, y) = S(x, y) + i # d 4 ud 4 v S(x,u)"(u,v) S(v, y) [Gorbar, Miransky, Shovkovy, Wang, PRD 88 (2013) ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 36

37 ! Expand S(x,y) in powers of gauge field A µ ext! To leading order in coupling, j 5 3 ext = A µ 0! Next-order radiative corrections are j 5 3 ext ext A µ A µ " = A µ ext March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 37

38 ! Final result (loops+counterterms) j 5 3 = # " eb µ & ln 2µ " 2$ 3 m + ln m 2 % m + 4 ) ( + # ' 2 3* " eb & m2 2$ 3 µ ln 23 / 2 µ # 11 ) ( + ' m % 12*! Unphysical dependence on photon mass because infrared physics with m " # k 0, k 3 # eb not captured properly! Note: similar problem exists in calculation of Lamb shift [Gorbar, Miransky, Shovkovy, Wang, PRD 88 (2013) ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 38

39 ! Perpendicular momenta cannot be defined with accuracy better than (In contrast to the tacit assumption in using expansion in powers of B-field)! Screening effects provide a natural infrared regulator "k # min ~ eb m " # $µ (Formally, this goes beyond the leading order in coupling) March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 39

40 Part 5 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 40

41 ! Solid state materials with Dirac quasiparticles: Bi 1-x Sb x alloy! New 3D Dirac materials (ARPES): Na 3 Bi [Z. K. Liu et al., arxiv: ] Cd 3 As 2 [M. Neupane et al., arxiv: ] [S. Borisenko et al., arxiv:130 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 41

42 n-doped p-doped [S. Borisenko et al., arxiv: ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 42

43 In the vicinity of 3D Dirac points: [Z. K. Liu et al., arxiv: ] E = v x k x + v y k y + v z k z March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 43

44 ! Hamiltonian of a Dirac semimetal! H (D ) = ' d 3 r" [#iv ( F $ % &! ) # µ 0 $ 0 ]" + H int cf. Weyl semimetal! H (W ) = ' d 3 r" [#iv F $ % &! ( b %! $ )$ 5 # µ 0 $ 0 ]" + H int! b ( ) #!! In a Dirac semimetal, a nonzero chiral shift will be induced when B#0, i.e.,! b " # g v F 2 c µ 0 e! B chiral shift [Gorbar, Miransky, Shovkovy, Phys. Rev. B 88, (2013)] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 44

45 ! ( 33 is expected to decrease with B because " 33 # B 2 (weak B) " 33 # B (strong B) [Son & Spivak, Phys. Rev. B 88, (2013)] [Nielsen & Ninomiya, Phys. Lett. 130B, 390 (1983)]! Experimental confirmation? [Kim, et al., PRL 111, (2014)] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 45

46 [Gorbar, Miransky, Shovkovy, Phys. Rev. B 89 (2014) ] ( HLL " ) 33 = 1 ( HLL ) # 33 " 33 = 1 # 33 (LLL " ) 33 = 1 (LLL ) # 33 Note: (LLL " 33 = " ) ( HLL 33 +" ) (LLL 33, where " ) 33 = e2 v F eb 4# 2 c$ 0 March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 46

47 [Gorbar, Miransky, Shovkovy, Phys. Rev. B 89 (2014) ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 47

48 (b " 12 = " =0) 12 +" 12,anom where " 12,anom = e2 b 2# 2 [Gorbar, Miransky, Shovkovy, Phys. Rev. B 89 (2014) ] March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 48

49 ! Studies of relativistic matter in magnetic field are relevant for many branches of physics! The underlying physics is conceptually rich! Recent developments include Magnetic catalysis Chiral magnetic/separation effect Chiral shift Chiral magnetic spiral and many others March 5, 2014 Quantum Hadron Physics Laboratory, RIKEN, Wako, Japan 49