DEC 5, 2016
CHIRAL MATTER
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Chiral fermions
• Massless Dirac fermions:
! !
! !
Σ⋅ p
5
0
γ p0 − γ ⋅ p Ψ = 0 ⇒ ! Ψ = sign( p0 )γ Ψ
| p|
(
)
For particles (p0 > 0):
chirality = helicity
For antiparticles (p0 < 0):
chirality = - helicity
• Massive Dirac fermions in ultrarelativistic regime
- High temperature:
T >> m
- High density:
µ >> m
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
3
Chiral matter
• Matter made of chiral fermions with nL ≠ nR
• Unlike the electric charge (nR + nL), the chiral
charge (nR - nL) is not conserved
∂(nR + nL ) ! !
+∇⋅ j = 0
∂t
!
!
∂(nR − nL ) ! ! e 2 E ⋅ B
+ ∇ ⋅ j5 =
2
∂t
2π c
• The chiral symmetry is anomalous in quantum
theory
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
4
Magnetic Fields
• Strong magnetic fields are common inside
compact stars
– 1010 to 1015 Gauss
– 1018 to 1019 Gauss
• Early Universe
– up to 1024 Gauss
Illustration by Carin Cain
• In heavy ion collisions, positive ions generate
short-lived (Δt ≈ 10-24 s) magnetic fields
• High Magnetic Field Laboratory
– 4.5
December 5-8, 2016
105 Gauss
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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ROLE OF MAGNETIC FIELD
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
6
Landau spectrum at B≠0
• Dirac equation with massless fermions
!
!
!
$i γ ∂ − i γ ⋅ ∇ + i e A & Ψ = 0
0
%
'
(
0
)
• Energy spectrum
E
(3+1)
n
2
3
( p3 ) = ± 2n eB + p
where
€
December 5-8, 2016
s = ± 12 (spin)
Occupied
1
n = s+ k +
2
k = 0, 1, 2,… (orbital)
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
7
Chiral separation effect (µ≠0)
— Right-handed
— Left-handed
Spin (s=↓) polarized LLL:
• p3<0 states are R-handed
• p3>0 states are L-handed
µ
This leads to CSE:
[Vilenkin, Phys. Rev. D 22 (1980) 3067]
[Metlitski & Zhitnitsky, Phys. Rev. D 72, 045011 (2005)]
[Newman & Son, Phys. Rev. D 73 (2006) 045006]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
8
CSE vs. anomaly
• Slowly changing electric/chemical potential
µ(z) = eΦ(z) ⇒ eE z = −∂z (eΦ) = −∂z µ
• From the anomaly relation,
€
2
2
e
e
∂ z j53 =
Bz E z = − 2 Bz ∂ z µ
2
2π
2π
• Suggesting that for massless fermions,
!
!
eB
j5 =
µ
2
2π
• There are radiative correction when gauge fields are
dynamical [Gorbar, Miransky, Shovkovy, Wang, Phys. Rev. D 88, 025025 (2013)]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
9
Chiral magnetic effect (µ5≠0)
— Right-handed
— Left-handed
Spin (s=↓) polarized LLL:
• p3<0 states are R-handed
electrons
µ5
• p3>0 states are L-handed
positrons
This leads to CME:
[Fukushima, Kharzeev, Warringa, Phys. Rev. D 78, 074033 (2008)]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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CMW/Quadrupole CME
• Start from a small baryon density and B≠0
!
!
eB
j5 =
µ
2
2π
!
!
eB
j =
µ
2 5
2π
• Produce back-to-back electric currents
[Gorbar, Miransky, Shovkovy, Phys. Rev. D 83, 085003 (2011)]
[Burnier, Kharzeev, Liao, Yee, Phys. Rev. Lett. 107 (2011) 052303]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Credits: Borisenko et al., Phys. Rev. Lett. 113, 027603 (2014)
DIRAC & WEYL MATERIALS
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
12
Dirac vs. Weyl materials
• Low-energy Hamiltonian of a Dirac/Weyl
material
T
P
! ! 5
!
!
0 5⎤
⎡
H = ∫ d r ψ ⎣−iv F γ ⋅ p − b ⋅ γ γ + b0γ γ ⎦ψ
3
(
) (
Dirac
P, T
)
Weyl
P, T
P, T
!
2b
!
2b
€
December 5-8, 2016
€
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
2b0
€
13
Dirac materials
• Bi1-xSbx alloy (at x ≈ 4%)
• Na3Bi
[Z. K. Liu et al., Science 343, 864 (2014)]
• Cd3As2
[S. Borisenko et al., Phys. Rev. Lett. 113, 027603 (2014)]
[M. Neupane et al., Nature Commun. 5, 3786 (2014)]
[X. Li et al., Nature Physics 12, 550 (2016)]
• ZrTe5
E = v 2x k 2x + v 2y k 2y + v 2z k 2z , v x ≈ v y ≈ 3.74 ×105 m/s, v z ≈ 2.89 ×10 4 m/s
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Weyl materials
• TaAs (tantalum arsenide)
[S.-Y. Xu et al., Science 349, 613 (2015)]
[B. Q. Lv et al., Phys. Rev. X5, 031013 (2015)]
• NbAs (niobium arsenide)
[S.-Y. Xu et al., Nature Physics 11, 748 (2015)]
• TaP (tantalum phosphide)
[S.-Y. Xu et al., Science Adv. 1, 1501092 (2015)]
• NbP (niobium phosphide) [I. Belopolski et al. arXiv:1509.07465]
• WTe2 (tungsten telluride)
December 5-8, 2016
[F. Y. Bruno et al., Phys. Rev. B 94, 121112 (2016)]
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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PSEUDO-ELECTROMAGNETIC FIELDS
[Zubkov, Annals Phys. 360, 655 (2015)]
[Cortijo, Ferreiros, Landsteiner, Vozmediano. Phys. Rev. Lett. 115, 177202 (2015)]
[Grushin, Venderbos,Vishwanath, Ilan, arXiv:1607.04268]
[Cortijo, Kharzeev, Landsteiner, Vozmediano, arXiv:1607.03491]
[Pikulin, Chen, Franz, Phys. Rev. X 6, 041021 (2016)]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
16
Strain in Weyl materials
• Strains affect low-energy quasiparticles in Weyl
materials [see also talks by M. Franz & M. Vozmediano]
! ! ! 5
!
!
0 5⎤
⎡
H = ∫ d r ψ ⎣−iv F γ ⋅ p − b + A5 ⋅ γ γ + b0 + A5,0 γ γ ⎦ψ
3
(
) (
)
(
)
where the components of the chiral gauge fields are
!
A5,0 ∝ b0 b ∂||u||
!
A5,⊥ ∝ b ∂||u⊥
!2
!
A5,|| ∝ α b ∂||u|| + β ∑∂i ui
2b
2b0
i
The associated pseudo-EM fields are
!
! ! !
!
!
B5 = ∇ × A5 and E5 = −∇A0 − ∂t A5
December 5-8, 2016
€
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
€
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Chiral effects in Weyl materials
• Any qualitative properties
of Weyl materials directly
!
sensitive to b0 and b ?
• Some proposals:
– Anomalous Hall effect
– Anomalous Alfven waves
– Strain/torsion induced CME
– Strain/torsion induced quantum oscillations
– Strain/torsion dependent resistance
– etc.
• Spectrum of chiral (pseudo-)magnetic plasmons
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1610.07625]
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
18
General question
• What are the properties of plasmons
in magnetized
!
chiral material with b0 ≠ 0 and b ≠ 0 ?
• Chiral matter (µR≠µL)
– This is the case in equilibrium when b ≠ 0 (µ = −e b )
0
5
0
• Magnetic or pseudomagnetic field is present
• In general,
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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MODEL & METHOD
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1610.07625]
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
20
Chiral kinetic theory
• Kinetic equation:
where
[Son and Yamamoto, Phys. Rev. D 87, 085016 (2013)]
[Stephanov and Yin, Phys. Rev. Lett. 109, 162001 (2012)]
,
,
[see talks by Q. Wang
& M. Stephanov]
and
December 5-8, 2016
is the Berry curvature
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Current and chiral anomaly
• The definitions of density and current are
They satisfy the following anomalous relations:
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Consistent definition of current
• Additional Bardeen-Zumino term is needed,
• In components,
• Its role and implications:
– Electric charge is conserved locally (!" $" = 0)
– Anomalous Hall effect is reproduced
– CME vanishes in equilibrium ('( = −*+, )
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Collective modes
We search for plane-wave solutions with
and the distribution function
where
,
The polarization vector & susceptibility tensor:
The plasmon dispersion relations follow from
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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RESULTS
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1610.07625]
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Chiral magnetic plasmons
Non-degenerate plasmon frequencies @ k=0:
where the Langmuir frequency is
and
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1610.07625]
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Plasmon frequencies, - ⊥ +
+/ = 0.2ℏΩ4 /*
9
678
−
:
678
2*<=> +/
≈
?@ℏ
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1610.07625]
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Plasmon frequencies, - ∥ +
+∥ = 0.2ℏΩ4 /*
-,⋆ ∝ <'+∥ @E = 0
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1610.07625]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Plasmons with F ≠ 0, F ∥ -, -(
• The longitudinal mode is sensitive to -(
F∥ ∝ <*-( /ΩI4
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1610.07625]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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The transverse modes split (in different ways) when (i) - ≠ 0&' ≠ 0,
or (ii) -( ≠ 0&'( ≠ 0, or (iii) +∥ ≠ 0, or (iv) +/ ≠ 0.
For example,
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
Plasmons with F ≠ 0, F ∥ -, -(
30
Plasmons with F ≠ 0, F ⊥ -, -(
The transverse modes split (in different ways) when (i) - ≠ 0&' ≠ 0,
or (ii) -( ≠ 0&'( ≠ 0
+∥ = 0.2ℏΩ4 /*
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Plasmons with F ≠ 0, F ⊥ -, -(
The transverse modes split (in different ways) when (i) - ≠ 0&' ≠ 0,
or (ii) -( ≠ 0&'( ≠ 0
+/ = 0.2ℏΩ4 /*
[Gorbar, Miransky, Shovkovy, Sukhachov, arXiv:1611.05470]
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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Summary
• Consistent chiral kinetic theory is needed
• Chiral magnetic plasmons (KMPs) are sensitive
to local charge (non-)conservation
• Properties of KMPs carry information about
!
b0 and b
• KMPs are not only due to the oscillation of
electric charge, but also chiral charge
December 5-8, 2016
Chiral Matter: From quarks to Dirac semimetals, RIKEN, Japan
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