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Chiral transport and turbulence
in core-collapse supernovae
Naoki Yamamoto (Keio University)
“Multi-dimensional Modeling and Multi-Messenger observation from Core-Collapse Supernovae,” October 24, 2019
Main topics
• Chiral transport phenomena
• Chiral plasma instability
• Chiral turbulence in supernovae
• Chiral radiation transport for neutrinos
Units: ~ = c = kB = e = 1
• One of the most energetic phenomena in the Universe
• But explosion is difficult in conventional 3D hydrodynamic theory
Core-collapse supernova explosions
One of the puzzles in astrophysics
http://www.riken.jp/pr/press/2009/20091211/
Parity
Chirality of fermionss
p
left-handed right-handed
mirror
s
p
Why is “God” left-handed?
W. Pauli
“God is just a weak left-hander.”
The laws of physics are left-right symmetric except for the weak interaction that acts only on left-handed particles.
From micro to macro
Macro Evolution of core-collapse supernovae (giant P violation)
Chiral kinetic theorySon, Yamamoto (2012); Stephanov, Yin (2012), …
Micro
Microscopic parity violation is reflected in macroscopic behavior:
Chirality of fermions (e, ν) in Standard Model
Yamamoto (2016)
Supernova = Giant Parity Breaker
p+ eL ! n+ ⌫Le
eLeReLeR
supernovae
νLνRνLνR
Ohnishi, Yamamoto (2014); Grabowska, Kaplan, Reddy (2015); Sigl, Leite (2016), …
supernovaeme
Chiral transport phenomena
Transport phenomena
• Classical and familiar examples:
• Ohm’s law:
• Fourier’s law:
je = �E
jQ = (�rT )
Quiz
• Is it possible to have the following current in metals?
je ⇠ B
Parity
• Assume the relation:
• Under the parity, (∵ B is axial-vector)
• It does not occur in our daily life.
je = B
�je = B
• Possible in chiral matter:
• This is called the chiral magnetic effect (CME).
Relativistic systems are different
je ⇠ (µR � µL)B
right-handed
left-handed
B
Chiral magnetic effect
Vilenkin (1980); Nielsen, Ninomiya (1983); Fukushima, Kharzeev, Warringa (2008), …
jR =µR
4⇡2B
jL = � µL
4⇡2B
s p
j =µR � µL
4⇡2B ⌘ µ5
2⇡2B
Chiral Vortical Effect
s p
Left-handed neutrinov
! = r⇥ v
Vilenkin (1979); Erdmenger et al. (2009); Banerjee et al. (2011); Son, Surowka (2009); Landsteiner et al. (2011)
j = �✓
µ2
8⇡2+
T 2
24
◆!
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Chiral Matter
• Electroweak plasma in early Universe
• Quark-gluon plasma in RHIC/LHC
• Weyl semi-metals (“3D graphene”)
• Lepton matter in supernovae
Electromagnetic Response of Weyl Semimetals
M.M. Vazifeh and M. FranzDepartment of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1
(Dated: March 26, 2013)
It has been suggested recently, based on subtle field-theoretical considerations, that the electro-
magnetic response of Weyl semimetals and the closely related Weyl insulators can be characterized
by an axion term ✓E · B with space and time dependent axion angle ✓(r, t). Here we construct a
minimal lattice model of the Weyl medium and study its electromagnetic response by a combina-
tion of analytical and numerical techniques. We confirm the existence of the anomalous Hall e↵ect
expected on the basis of the field theory treatment. We find, contrary to the latter, that chiral mag-
netic e↵ect (that is, ground-state charge current induced by the applied magnetic field) is absent in
both the semimetal and the insulator phase. We elucidate the reasons for this discrepancy.
When a three-dimensional topological insulator (TI)[1–3] undergoes a phase transition into an ordinary bandinsulator, its low-energy electronic spectrum at the crit-ical point consists of an odd number of 3D masslessDirac points. Such 3D Dirac points have been experi-mentally observed in TlBi(S1�xSex)2 crystals [4] and in(Bi1�xInx)2Se2 films [5]. In the presence of the time re-versal (T ) and inversion (P) symmetries the Dirac pointsare doubly degenerate and occur at high-symmetry po-sitions in the Brillouin zone. When T or P is broken,however, each Dirac point can split into a pair of ‘Weylpoints’ separated from one another in momentum k orenergy E, as illustrated in Fig. 1. The resulting Weylsemimetal constitutes a new phase of topological quan-tum matter [6–14] with a number of fascinating physicalproperties including protected surface states and unusualelectromagnetic response.
The low energy theory of an isolated Weyl point isgiven by the Hamiltonian
hW (k) = b0 + v� · (k � b), (1)
where v is the characteristic velocity, � a vector of thePauli matrices, b0 and b denote the shift in energy andmomentum, respectively. Because all three Pauli matri-ces are used up in hW (k), small perturbations can renor-malize the parameters, b0, b and v, but cannot open agap. This explains why Weyl semimetal forms a stablephase [6]. Although the phase has yet to be experimen-tally observed there are a number of proposed candidatesystems, including pyrochlore iridates [7, 8], TI multilay-ers [9–12], and magnetically doped TIs [13, 14].
The purpose of this Letter is to address the remarkableelectromagnetic properties of Weyl semimetals. Accord-ing to the recent theoretical work [15–18], the universalpart of their EM response is described by the topological✓-term,
S✓ =e2
8⇡2
Zdtdr✓(r, t)E ·B, (2)
(using ~ = c = 1 units) with the ‘axion’ angle given by
✓(r, t) = 2(b · r � b0t). (3)
� � � �
�
�
FIG. 1: Low energy spectra in Dirac and Weyl semimetals.
a) Doubly degenerate massless Dirac cone at the transition
from a TI to a band insulator. Weyl semimetals with the
individual cones shifted in b) momenta and c) energy. Panel
d) illustrates the Weyl insulator which can arise when the
excitonic instability gaps out the spectrum indicated in c). In
all panels two components of the 3D crystal momentum k are
shown.
This unusual response is a consequence of the chiralanomaly [19–21], well known in the quantum field the-ory of Dirac fermions. The physical manifestations of the✓-term can be best understood from the associated equa-tions of motion, which give rise to the following chargedensity and current response,
⇢ =e2
2⇡2b ·B, (4)
j =e2
2⇡2(b⇥E � b0B). (5)
Eq. (4) and the first term in Eq. (5) encode the anoma-lous Hall e↵ect that is expected to occur in a Weylsemimetal with broken T [7–10]. The second term in Eq.(5) describes the ‘chiral magnetic e↵ect’ [22], whereby aground-state dissipationless current proportional to theapplied magnetic field B is generated in the bulk of aWeyl semimetal with broken P.The anomalous Hall e↵ect is known to commonly oc-
cur in solids with broken time-reversal symmetry. In thepresent case of the Weyl semimetal its origin and magni-tude can be understood from simple physical arguments[7–10] applied to the bulk system as well as in the limitof decoupled 2D layers [18]. Understanding the chiralmagnetic e↵ect (CME) in a system with non-zero en-
arX
iv:1
303.
5784
v1 [
cond
-mat
.mes
-hal
l] 2
2 M
ar 2
013
Nielsen, Ninomiya (1983), …
Kharzeev, Mclerran, Warringa (2008), …
Joyce, Shaposhnikov (1997), …
Yamamoto (2016), …
http://www0.bnl.gov/rhic/news2/QGP SupernovaeWeyl semimetals
Chiral plasma instabilityElectroweak theory Redlich, Wijewardhana (1985); Rubakov (1986); Laine (2005)
Early Universe Joyce, Shaposhnikov (1997); Boyarsky et al. (2012); Tashiro et al. (2012)
Quark-gluon plasma Akamatsu, Yamamoto (2013); Manuel, Torres-Rincon (2015), …
Neutron stars Ohnishi, Yamamoto (2014), …
δB
Assume homogeneous initially
Chiral plasma instability
µ5 ⌘ µR � µL
δj ~ µ5δB Chiral magnetic effect
Chiral plasma instability
δj δBind
r⇥B = j
Ampere’s law
Chiral plasma instability
δj δjind ~ µ5δBind
Chiral magnetic effect
Chiral plasma instability
δjind δB + δB’ind r⇥B = j
Ampere’s law
Positive feedback: instability
Chiral plasma instability
Chiral MHD turbulencein supernovae
Chiral MHD for supernovae
Proto-neutron star (PNS)
Masada, Kotake, Takiwaki, Yamamoto, arXiv:1805.10419
Chiral MHD for supernovae
CME
chiral anomaly
• Chiral MHD w/o vorticity at the core (proton, eR, eL):
@tB = r⇥ (v ⇥B) + ⌘r2B + ⌘r⇥ (⇠BB)
• Setup for proto-neutron stars (100 MeV = 1):
@tn5 =⌘
2⇡2(r⇥B � ⇠BB) ·B
⇢0 = 5.0, P0 = 1.0, ⇠B0 = 4.2⇥ 10�3, ⌘ = 100.0
+(di↵usion)(dissipation)
Masada, Kotake, Takiwaki, Yamamoto, arXiv:1805.10419
Movies of 3D simulations are available at:
http://www.kusastro.kyoto-u.ac.jp/~masada/movie.mp4
Masada, Kotake,Takiwaki,Yamamoto, arXiv:1805.10419
Energy spectra
• As time passes, energy in small-k and large-k regions grows
• Eventually, εM~k-2, εK~k-5/3
Masada et al., arXiv:1805.10419; see also Brandenburg et al., arXiv:1707.03385
εM
εK
Chiral radiation transport for neutrinos
(mostly in 2D)
Topology
rubber band surface of a plastic bottle (from the top)
⇡
M and N are topologically the same: S1
M N
Topological invariant
Mapping from S1 (rubber) to S1 (bottle): winding number n
Chirality and topology
momentum space spin space
p s
Right-handed fermions
Mapping: S2 (p-space) → S2 (spin space) winding number +1
Chirality and topology
momentum space spin space
s
Left-handed fermions
p
Neutrino matter = 3D topological matter
Mapping: S2 (p-space) → S2 (spin space) winding number -1
Yamamoto (2016)
Conventional ν radiation transfere.g., Mihalas, Mihalas (1984)
• In comoving frame for spherically symmetric metric,
for the distribution function µ ⌘ cos ✓̄where
E
"@tc
+ µ@r + E⇣µ2
c@t ln ⇢� (1� 3µ2)
v
cr� µ@tv
c2
⌘@E
+ (1� µ2)
✓1
r
⇣1 +
3µv
c
⌘� @tv
c2+
µ
c@t ln ⇢
◆@µ
#f = C[f ]
f(t, r, E, µ)
Chiral ν radiation transferYamamoto, Yang, to appear
• In comoving frame for the distribution function f(t, r, ✓,�, E, µ, �̄)
Chiral effects necessarily break spherical symmetry
E
"@tc
+ µ@r + E⇣µ2
c@t ln ⇢� (1� 3µ2)
v
cr� µ@tv
c2
⌘@E
+ (1� µ2)
✓1
r
⇣1 +
3µv
c
⌘� @tv
c2+
µ
c@t ln ⇢
◆@µ
+~p
1� µ2
2Er2
✓r⇣µ@t ln ⇢�
@tv
c
⌘+ 3µv
◆⇣sin �̄@✓ �
cos �̄
sin ✓@�⌘
� ~Er2
⇣r@t ln ⇢+ 2v
⌘+ 4⇡r2⇢cµ
⇣1� 8⇡r2⇢+ r(1� 4⇡r2⇢)@r ln ⇢
⌘!@�̄ + · · ·
#f = C[f ]
Chiral ν radiation transferYamamoto, Yang, to appear
• In inertial frame with matter in local thermal equilibrium,
New collision terms lead to finite fluid/cross helicities
C[f ] = �Rabsf +Remis(1� f)<latexit sha1_base64="6Xg4O/xh653kV2dAPwbYqGklc+0=">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</latexit>
Rabs = R0+~q · [R1! + v ⇥rR2 +R3B]<latexit sha1_base64="rtKuAIKkzQznSiKxx3RxTxljXC8=">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</latexit>
Ri = Ri(T, µ,v, q) (i = 1, 2, 3)<latexit sha1_base64="P7YqhsL0iJOOQnyuXFwJgeeyUmQ=">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</latexit>
Computable in QFT
Conclusion
• Neutrino matter = 3D topological matter
• Chiral effects can reverse the turbulent behavior from direct to inverse cascade.
• Chiral radiation transport of neutrinos should be applied in future numerical simulations.