1. Dynamical instabilities and anomalous transport in gauge theories with chiral fermions
Pavel Buividovich
(Regensburg)

2. Some recent real-time studies
Real-time dynamics of chirality pumping [ ]
with Semen Valgushev
Chiral plasma instability and
inverse cascade [ ] with Maksim Ulybyshev
Evaporating black holes from holography and Super-Yang-Mills, with M. Hanada [in preparation]

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

4. 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

5. 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!!!

6. Chiral anomaly: brief intro
Charged particle in
magnetic field: 1D motion
(transverse motion “confined”,
spin || magnetic field)
Massless fermions have two “chiral states”
Spin (anti)parallel to momentum
Quantum field theory: Vacuum is full
…of virtual particles
Let’s switch electric field!!!

7. Chiral anomaly: brief intro
Right-moving particles
Left-moving anti-particles

8. Chiral anomaly: going 3D
In quantum theory
1D reduction = lowest Landau level
Number of LLLs/Area = B/(2π)
Anomaly equation γ π0 π0 γ

9. Chiral Magnetic Effect
Imbalance of left- and right-movers μA
-μA
Electric current
In 3D: direction of B +
Landau level degeneracy
… In fact, “rotation” of anomaly diagram
Here we turn the picture upside down…

10. How to magnetize and rotate chiral matter?
Relative motion
of two large charges (Z ~ 100)
Large magnetic field
in the collision region
URQMD simulations Au+Au
No backreaction
From [Skokov, Toneev,
ArXiv: ]
Weak energy dependence!!!

11. Before hydro / kinetics …
Glasma state E || B
Production of axial charge!
Magnetic field is maximal!
Key objectives:
Initial conditions for anomalous hydro (axial charge density, electric current, energy density)
Real-time shocks, plasma instabilities, etc.
Key tools:
Classical-statistical field theory:
fully quantum CHIRAL fermions+ classical gauge fields
Key innovation [PB+coauth, , ]
Before quark-gluon plasma thermalizes and hydrodynamics can be applied, the region between colliding nuclei can be best described as the so-called glasma state with strong parallel chromo-magnetic and chromo-electric fields. Here axial anomaly leads to rapid production of axial charge. In this early collision stage, also the magnetic field is maximal, and anomalous transport effects are most important. My objective here is to simulate the evolution of chiral fermions in this strongly non-equilibrium state. The final state of this evolution can be used as initial conditions for anomalous hydrodynamics. We will use the classical-statistical approximation with fully quantum fermions, our know-how here is the use of lattice chiral fermions. With this algorithm, one can also simulate more general nonlinear phenomena in chiral plasma, such as shock waves and plasma instabilities, and real-time responses of Dirac materials.

12. Interplay of gauge and fermion chirality
μA, QA- not “canonical” charge/chemical potential
“Conserved” charge:
Chern-Simons term
(Magnetic helicity)
Integral gauge invariant
(without boundaries)

13. What happens with this equation if gauge fields are dynamical?
Chirality pumping
[With S.
Valgushev]
What happens with this equation if gauge fields are dynamical?
[Anselm,
Iohansen’88]
Q5(t)
External
Dynamic t

14. 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)

15. 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)

16. 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

17. 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 ]

18. Anomalous Maxwell equations (now a coarse approximation…)
Maxwell equations + ohmic conductivity + CME
Chiral
magnetic
conductivity
Ohmic
conductivity
Conservation of energy in the presence of CME
Keeping chiral imbalance constant requires work

19. Anomalous Maxwell equations (now a coarse approximation…)
Assume dispersionless electric conductivity and CME response
Neglect nonlinear effects (Schwinger pair production, etc)
Plane wave solution

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

21. Chiral instability and Inverse cascade
Energy of large-wavelength modes grows
… at the expense of short-wavelength modes!
2D turbulence, from H. J. H. Clercx and G. J. F. van Heijst Appl. Mech. Rev 62(2),

22. 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?

23. 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!!!

24. 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!!!

25. 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!

26. 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, Wilson fermions, MPI parallelism
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 polarizations
Hamiltonian is
CP-symmetric,
State is not!!!
[No momentum separation]

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

28. Universal late-time scaling QA ~ t-1/2
[Yamamoto ], [Hirono,Kharzeev,Yin ]
QA ~ t-1/2

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

30. Power spectrum and inverse cascade
(L) Magnetic (R)
Small amplitude, QA ~ const
Electric
Chiral laser with circular polarization!

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

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

33. 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. !!!
Dispersionless linear response
not so good

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

35. N=1 Supersymmetric Yang-Mills in D=1+9:
Real-time instabilities of gauge fields + fermions: interesting side development
Black hole dynamics
N=1 Supersymmetric Yang-Mills in D=1+9:
gauge bosons+adjoint Majorana-Weyl fermions
Reduce to a single point = BFSS matrix model
[Banks, Fischler, Shenker, Susskind’1997]
N x N hermitian matrices
Majorana-Weyl fermions, N x N hermitian matrices
Recent numerics: [Bergner,Liu,Wenger, ]

36. System of N D0 branes joined by open strings
BFSS model and “holography”
“Holographic” dictionary [Witten’96]:
Xiiμ = D0 brane positions
Xijμ = open string excitations
System of N D0 branes
joined by open strings
Eigenvalues of X2 = “radial coordinates”

37. Stringy interpretation: black hole formation
Motivation
Classical equations of motion are chaotic
[Saviddy; Berenstein;Susskind;Hanada;…]
Positive Lyapunov exponents
Xμ matrices forget initial conditions
Kolmogorov entropy produced
Stringy interpretation:
black hole formation

38. evaporation Expectation So far classical real-time simulations
Classical flat directions [Xμ,Xν]=0 entropically suppressed
Fermions enhance tunneling (repulsive force)
Quantum fluctuations of bosons cancel repulsion at large distances (SUSY)
D0 branes tunneling = (???)
Black hole
evaporation

39. Classical-statistical field theory (CSFT)
[Son, Aarts, Smit, Berges, Tanji, Gelis,…]
Schwinger pair production
Axial charge generation in glasma
(Chiral) plasma instabilities
Closed system of equations:
Numerical solution! Fermions are costly!
CPU time + RAM memory scaling ~ N4
Parallelization is necessary

40. Initial conditions: some excited state
… for nontrivial evolution (thinking about black holes, we still believe in quantum mechanics)
Our initial conditions:
Gaussian random,
dispersion f
dispersion 1
Fermions are in ground state at given Xai
Tricky for Majoranas, no Dirac sea!!!

41. Hawking radiation of D0 branes
Some results N = 8, f = 0.48
Hawking radiation of D0 branes X2
With
fermions
Classical t
Thermalization

42. Not all configurations evaporate
Some results N = 8, f = 0.48
Not all configurations evaporate X2
Classical t
With fermions

43. Constant acceleration at late timesX2 t
Boson kinetic energy grows without bound
Fermion energy falls down
Origin of const force – SUSY violation?

44. Fierz identity (cyclic shift of indices):
CSFT and SUSY
In full quantum theory
Fierz identity (cyclic shift of indices):
In CSFT approximation
Fermionic 3pt function seems necessary!

45. Going beyond CSFT Return to Heisenberg eqs of motion
VEV with more non-trivial correlators
First, test this idea on the simplest example
Tunnelling
between potential wells? Absent in CSFT!

46. Next step: Gaussian Wigner function Derive closed equations for
Assume Gaussian wave function at any t
Simpler: Gaussian Wigner function
For other correlators: use Wick theorem!
Derive closed equations for
x, p, σxx , σxp , σpp

47. Origin of tunnelling Positive force even at x=0 (classical minimum)
Quantum force
causes classical trajectory
to leave classical minimum

48. Some results N = 8, f = 0.48
σ2 = 0.0, 0.1, 0.2, no fermions

49. Some results N = 8, f = 0.48
σ2 = 0.0, 0.1, 0.2, no fermions

50. Evaporation threshold: N = 6, f = 0.50
σ2 = 0.0, 0.2, no fermions

51. Quantum fluctuations vs evaporation
Quantum fluctuations of X variables suppress evaporation!!!
Small σ2: evaporation becomes slower
Intermediate σ2: no evaporation
Large σ2: fermions negligible
Expected trend, but still no exact cancellation of bosons and fermions
Fine-tuning with realistic initial conditions?
What effect quantum fluctuations of X have on classical dynamics?

52. Thank you for your attention!!!
Brief summary
Chirality pumping: backreaction makes axial charge and CME current oscillating, QA~B1/2 scaling vs. QA ~ B
Chiral plasma instability stops earlier than chiral imbalance is depleted
Real-time simulations of
black hole formation
and evaporation
Thank you for your attention!!!