### 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 (BNLAPCTP), Ho-Ung Yee (U. Illinois at Chicago), and Yi Yin (MIT),
**arXiv:1711.08450** **[hep-th] **

### 𝑛𝑛

_{𝑅𝑅}

### − 𝑛𝑛

_{𝐿𝐿}

### ≠ 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 B_{0}

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 ω^{2 }

--- 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

_{A}

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.

^{Hirono}

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

_{Ohm}

### = J

_{CME}

### 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