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

◆!

<latexit sha1_base64="AhverLXR8qRcRADM5HCPEASI1YI=">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</latexit>

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=">AAACuHichVE9SyNRFD0ZP9aNX9FtBJvBkMUoCW9cQREE0cYyRpO4JCHOjG/iw/liZhJwQ2rBZkuLrVawEGs70WKb/QMW/gTZ0gUbi72ZDCwq6h3evHvPO+e+d7iaawo/YOw2JvX09vV/GPgYHxwaHhlNjI0Xfafh6bygO6bjbWuqz01h80IgApNvux5XLc3kJW1/rXNeanLPF469FRy4vGqpdVsYQlcDgmqJ9FrZqMrLciZfa1U8S1Y1vy0b8qwc1dwSfntayRjpWiLJsiwM+WWiREkSUeScxCUq2IUDHQ1Y4LARUG5ChU9fGQoYXMKqaBHmUSbCc4424qRtEIsTQyV0n/51qsoRalPd6emHap1uMWl5pJSRYjfsjN2z3+yc3bHHV3u1wh6dtxzQrnW13K2NHk1sPryrsmgPsPdf9YZCI3bI+zrDD8ulN90FMLAYuhLk0g2Rjl+926H57fh+cymfan1mJ+wPOf3Jbtkv8mo3/+qnGzz/A3EalfJ8MC+T4lxW+ZKd25hPrqxGQxvAJKYwTZNZwArWkUOB7v2OC1zhWlqSdqS6JLpUKRZpPuFJSN4/fHSg1g==</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.