1. Numerical study of real-time chiral plasma instability
Pavel Buividovich
(Regensburg)

2. Why chiral plasma? Collective motion of chiral fermions
High-energy physics:
Quark-gluon plasma
Hadronic matter
Neutrinos/leptons in Early Universe
Condensed matter physics:
Weyl semimetals
Topological insulators
Liquid Helium [G. Volovik]

3. Anomalous transport: Hydrodynamics
Classical conservation laws for chiral fermions
Energy and momentum
Angular momentum
Electric charge No. of left-handed
Axial charge No. of right-handed
Hydrodynamics:
Conservation laws
Constitutive relations
Axial charge violates parity
New parity-violating
transport coefficients

4. Anomalous transport: CME, CSE, CVE
Chiral Magnetic Effect
[Kharzeev, Warringa,
Fukushima]
Chiral Separation Effect
[Zhitnitsky,Son]
Chiral Vortical Effect
[Erdmenger et al.,
Teryaev, Banerjee et al.]
Flow vorticity
Origin in
quantum anomaly!!!

5. Instability of chiral plasmas
μA, QA- not “canonical” charge/chemical potential
“Conserved” charge:
Chern-Simons term
(Magnetic helicity)
Integral gauge invariant
(without boundaries)

6. Real-time dynamics of chirality Two real-time setups in this talk:
is nontrivial with dynamical gauge fields
Two real-time setups in this talk:
Chirality pumping in parallel electric and magnetic fields
Spontaneous decay of
chirality into helical gauge fields

7. Chirality pumping in E || B
What happens with this equation if gauge fields are dynamical?
[Anselm,
Iohansen’88]
Q5(t)
External
Dynamic t

8. Chirality pumping in E || B Maxwell equations with
Bz=const, Ez(z,t), other comps. 0
Anomaly equation
Vector and axial currents
(Lowest Landau level
= 1D fermions)

9. Chirality pumping in E || B Combining everything
Massive mode in chiral plasma
(Chiral Magnetic Wave
mixing with plasmon)
Mass is the back-reaction effect
(Also corrections due to higher Landau levels)

10. Chirality pumping in E || B Now assume Ez(z,t)=Ez(t)
No backreaction vs. Backreaction
Max. axial
charge
~ B1/2 vs B
Times > B-1/2
No backreaction
(ext. fields)
Q5(t)
Backreaction
(dynamic fields) t

11. Chirality pumping in E || B Short pulse of electric field
[numerics with overlap, S. Valgushev]
Higher Landau
levels excited
Very small effect on anomaly

12. Chiral instability and Inverse cascade
Energy of large-wavelength modes grows
… at the expense of short-wavelength modes!
Generation of cosmological magnetic fields [Boyarsky, Froehlich, Ruchayskiy, ]
Circularly polarized, anisotropic soft photons in heavy-ion collisions [Hirono, Kharzeev,Yin ][Torres-Rincon,Manuel, ]
Spontaneous magnetization of topological insulators [Ooguri,Oshikawa, ]
THz circular EM waves from Dirac/Weyl semimetals [Hirono, Kharzeev, Yin ]

13. Anomalous Maxwell equations (now a coarse approximation…)
Maxwell equations + ohmic conductivity + CME
Chiral
magnetic
conductivity
Ohmic
conductivity
Assumption:
Plane wave solution

14. Chiral plasma instability
Dispersion relation
At k <  = μA/(2 π2): Im(w) < 0
Unstable solutions!!!
Cf. [Hirono, Kharzeev, Yin ]
Real-valued solution:

15. What can stop the instability?
Helical structure
Helical structure of unstable solutions
Helicity only in space – no running waves
E || B - ``topological’’ density
Note: E || B not possible for oscillating ``running wave’’ solutions, where E•B=0
What can stop the instability?

16. What can stop the instability?
For our unstable solution with μA>0:
Instability depletes QA
μA and chi decrease, instability stops
Energy conservation:
Keeping constant μA requires work!!!

17. Real-time dynamics of chiral plasma
Approaches used so far:
Anomalous Maxwell equations
Hydrodynamics (long-wavelength)
Holography (unknown real-world system)
Chiral kinetic theory (linear response, relaxation time, long-wavelength…)
What else can be important:
Nontrivial dispersion of conductivities
Developing (axial) charge inhomogeneities
Nonlinear responses
Let’s try to do numerics!!!

18. Bottleneck for numerics!
Real-time simulations: classical statistical field theory approach [Son’93, Aarts&Smit’99, J. Berges&Co]
Full real-time quantum dynamics of fermions
Classical dynamics of electromagnetic fields
Backreaction from fermions onto EM fields
Vol X Vol matrices,
Bottleneck for numerics!

19. Algorithmic implementation [ArXiv:1509.02076, with M. Ulybyshev]
Wilson-Dirac fermions with zero bare mass Fermi velocity still ~1 (vF << 1 in progress)
Dynamics of fermions is exact, full mode summation (no stochastic estimators)
Technically: ~ 10 Gb / (200x40x40 lattice), MPI
External magnetic field from external source (rather than initial conditions )
Anomaly reproduced up to ~5% error
Energy conservation up to ~2-5%
No initial quantum fluctuations in EM fields
(to avoid UV catastrophe)

20. Chirality pumping, E || B
Still quite large
finite-volume effects
in the anomaly coeff.
~ 10%
Late-time: QA ~ Sin[E t/2]

21. Chirality decay: options for initial chiral imbalance
Excited state with
chiral imbalance
Chiral chemical potential
Hamiltonian is
CP-symmetric,
State is not!!!
Equilibrium
Ground state

22. Decay of axial charge and inverse cascade [PB, Ulybyshev 1509.02076]
Excited initial
state with
chiral imbalance
μA < 1 on the lattice
To reach k < μA/(2 π2):
200x20x20 lattices
Translational invariance in 2 out of 3 dimensions
To detect instability and inverse cascade:
Initially n modes of EM fields with equal energies and random linear polarizations
Hamiltonian is
CP-symmetric,
State is not!!!
[No momentum separation]

23. If amplitude small, no decay
Axial charge decay
200 x 20 x 20 lattice, μA= 0.75
If amplitude small, no decay

24. Axial charge decay
200 x 20 x 20 lattice, μA= 1.5

25. Is axial charge homogeneous?
… yes, deviations only ~ 20% at late times

26. Universal late-time scaling
QA ~ t-1/2

27. Power spectrum and inverse cascade short-scale fluctuations
Fourier transform
the fields
Basis of helical
components
Smearing the
short-scale fluctuations

28. Power spectrum and inverse cascade
(L) Magnetic (R)
Electric
Small amplitude, QA ~ const

29. Power spectrum and inverse cascade Large amplitude, QA decays
(L) Magnetic (R)
Electric
Large amplitude, QA decays

30. Power spectrum and inverse cascade
(L) Magnetic (R)
Electric
Large amplitude, large μA, QA decays

31. Power spectrum and inverse cascade
(L) Magnetic (R)
Electric
Smaller vF, QA decays

32. Inverse cascade: correllation lengths
Average wavelength
of electric
or magnetic fields
Electric
Magnetic

33. Overall transfer of energy
Amplitude f = 0.2

34. Overall transfer of energy
Amplitude f = 0.05

35. Discussion and outlook
Axial charge decays with time
(nature doesn’t like fermion chirality)
Large-scale helical EM fields
Short EM waves decay
Non-linear mechanism!
Instability stops much
earlier than predicted by
anomalous Maxwell eqns. !!!

36. Chiral instability: discussion and outlook
Premature saturation of instability: physics or artifact of Wilson fermions?
Overlap fermions in CSFT?

37. Chiral instability: discussion and outlook
Non-Abelian gauge fields – much stronger instability
Topology more complicated
Numerics more expensive
Quantum fluctuations in
gauge sector???