Magnetohydrodynamics with chiral anomaly: formulation and phases of collective excitations and instabilities

Koichi Hattori

Fudan University  Yukawa Institute on Dec. 1

Workshop on Recent Developments in Chiral Matter and Topology

@ National Taiwan University, Dec. 6-9, 2018

Magnetohydrodynamics with chiral anomaly:

formulation and phases of collective excitations and instabilities

KH, Yuji Hirono (BNLAPCTP), Ho-Ung Yee (U. Illinois at Chicago), and Yi Yin (MIT), arXiv:1711.08450 [hep-th]

𝑛𝑛

− 𝑛𝑛

≠ 0, B ≠ 0

Chiral fluid

R

R

R R R

R

R L

L L

L R

R

Anomaly-induced transports in a magnetic OR vortex field

Non-dissipative transport phenomena with

time-reversal even and nonrenormalizable coefficients.

Anomaly relation:

Cf., An interplay between the B and ω leads to a new nonrenormalizable transport coefficient for the magneto-vorticity coupling.

KH and Y.Yin, Phys.Rev.Lett. 117 (2016) 152002 [1607.01513 [hep-th]]

Low-energy effective theory

of the chiral fluid in a dynamical magnetic field

R

R

R R R

R

R L

L L

L R

R

Chiral magnetohydrodynamics

(Chiral MHD, or anomalous MHD)

Strong magnetic fields

induced by relativistic heavy-ion collisions

Kharzeev, PPNP

Z ~ 80, v > 0.99999 c, Length scale ~ 1/Λ_QCD

W.-T. Deng & X.-G. Huang KH and X.-G. Huang

One can study the interplay btw QCD and QED.

Besides,

‣ Weyl & Dirac semimetals

‣ Strong B field by lattice QCD simulations

‣ Neutron stars/magnetars

‣ High intensity laser fields

‣ Cosmology

1. Formulation of the chiral magnetohydrodynamics (chiral MHD) --- Finite chirality imbalance (𝑛𝑛_𝑅𝑅 ≠ 𝑛𝑛_𝐿𝐿)

--- Dynamical magnetic field

2. Collective excitations with the linear analysis wrt δv and δB.

(MHD has a fluctuation of dynamical magnetic field δB.)

3. Summary

Plan for the rest of talk

Formulating the chiral MHD

-- Anomalous hydrodynamics

Son & Surowka

Anomalous hydrodynamics

in STRONG & DYNAMICAL magnetic fields

-- Anomalous magnetohydrodynamics (MHD)

This work.

and external

and dynamical

n_A: # density of axial charge Neutral plasma (n_V = 0)

No E-field in the global equilibrium

EoMs:

Constitutive eqs. in the ideal order (zeroth order in derivative)

From EoM + thermodynamic relation

Therefore,

Constitutive eqs. and the entropy generation in the first order

Computing the entropy current,

Insuring the semi-positivity with bilinear forms

Positivity is insured by a bilinear form:

Therefore, we get a “constitutive eq.” of the E-field:

KH, Hirono, Yee, Yin

provides 5 dissipative and

2 non-dissipative (Hall) viscous coefficients

de Groot; Landau & Lifshitz; Huang, Sedrakian, & Rischke; Hernandez & Kovtun; …

Similarly,

3 diffusion coefficients

Conductivities: CME and dissipative terms

There appear the longitudinal and transverse Ohmic conductivities due to the breaking of the rotational symmetry.

The CME current is completely fixed by C_A, and is necessary for insuring the semi-positive entropy production.

The CME has the universal form in the MHD regime as well.

Conductivities and viscosities in strong B fields

In the LLL, charged fermions transport charges and momenta only along the B.

 Effective dimensional reduction to (1+1) D in the fermion sector.

Longitudinal conductivity

KH, S.Li, D.Satow, H.-U. Yee, 1610.06839 [hep-ph];

KH, D.Satow, 1610.06818 [hep-ph].

Cf., Landau-level resummation, Fukushima, Hidaka.

Longitudinal bulk viscosity

KH, X.-G.Huang, D.Satow, D.Rischke, 1708.00515 [hep-ph].

Computation by the perturbation theory at finite T and B

Strong B

Quarks live in (1+1) D Gluons live in (3+1) D

“Mismatched dimensions”

Phases of the collective excitations

and instabilities from a linear analysis

Collective excitations in MHD without anomaly

Tension of B-field  Restoring force

Fluid energy (mass) density  Inertia Transverse Alfven wave Oscillation

* Magnetic lines move together with the fluid volume.

0. Stationary solutions

Alfven wave velocity

2. Wave equation Transverse wave propagating along background B₀

1. Transverse perturbations

Linearlize the set of hydrodynamic eqs.

with respect to the perturbation.

Alfven wave from a linear analysis

Same wave equation for δv

 Fluctuations of B and v propagate together.

How does the CME change the

hydrodynamic waves in chiral fluid?

--- Drastic changes by only one term in the current

Eigenvalues V: Dispersion relations Eigenvectors ψ: Polarizations

Eigenmodes of chiral MHD

6 degrees of freedom

M_A: Modification by a finite μA

6 × 6 matrix from the linearlized EoMs

“Phase diagram” of the eigenmodes

Secular eq. is a cubic eq. of ω²

--- 3 modes propagating in the opposite directions (6 solutions in total)

Stability of the waves from classification of solutions

1 real and 2 pure imag. sols.

1 real and 2 complex sols.

3 real solutions

Alfven and magneto-sonic waves

B θ k

Direction of wave wrt B (θ)

Real part of V Imaginary part of V

Dispersion relations of the waves

There is a pair of modes (green) which are stable in any phase.

[Will not be focused hereafter.]

Real part of V Imaginary part of V

Dispersion relations of the waves

Real part of V Imaginary part of V

Dispersion relations of the waves

R Helicity

L Helicity

Polarizations on the Poincare sphere with a varying μ

Stokes vector

Linear polarizations (μ_A = 0)

Linear polarizations (μ_A = 0)

The unstable modes have helical nature.

Equator: Linear polarizations

Upper and lower hemispheres: R and L polarizations (Poles: R and L circular polarizations)

Helicity decomposition

(Circular R/L polarizations) LH mode RH mode Signs of the imaginary parts

(Damping/growing modes in the hydrodynamic time evolution)

Positive

(Damping) Negative

(Growing)

New hydrodynamic instability in a chiral fluid

A helicity selection, depending on the sign of μ_A.

Magnetic helicity

Chiral Plasma Instability (CPI)

Akamatsu&Yamamoto

Helicity conversions

as the topological origin of the instability

Fluid helicity (structures of vortex strings)

Real-time & beyond-linear analysis demanded.

Difference btw the # of R and L fermions:

“Chiral imbalance ”

Summary

The CME drastically changes the time evolution of the chiral fluid in a B-field.

- Chiral fluid is not stable against a small perturbation on v and B.

- One of the helicities is strongly favored against the other due to a finite μA.

Helical excitations

Second law of thermodynamics determines the form of the CME current, reproducing the universal form.

Stay tuned for a microscopic derivation of MHD. Hongo & KH Formulation

Phases of the collective excitations and instabilities

Backup slides

Hydrodynamic variables when μV = 0

We work in the world after the E-field is damped.

Steady state: J

= J

Estimate of the relaxation time of n_A

(Relaxation time of E ~ 1/σ) << (Our time scale) << (Relaxation time of nA ~ σ)

Collective excitations in chiral MHD

Alfven wave, fast and slow magneto-sonic waves, when ε_A = 0.

Effects of anomaly

Dotted: Without anomaly effects

[Alfven (red), fast sonic (blue), slow sonic (green)]

Solid: With anomaly effects which mix the waves