Chiral magnetic effect (

Chiral magnetic effect in two-band lattice model of Weyl semi-metal

Ming-Che Chang, Taiwan Normal University, Taiwan Min-Fong Yang*, Tunghai University, Taiwan

Refs: Phys Rev B 91, 115203 (2015) ; B 92, 205201 (2015).

181207@NTU

Chiral magnetic effect ⁽

)

• Quark-gluon plasma in heavy ion collision

• Relativistic plasma in astrophysics

• Weyl semimetal

• …

CME B

J = − α B

r r

Figs from Dobrin Nature 2017, Chernodub arXiv 1002.1473, Vazifeh PRL 2013

Note: need to break space-inversion symmetry

Monopole in Weyl semimetal

• Berry flux (or monopole charge) of a Weyl node is quantized

• Nielsen-Ninomiya theorem requires Weyl monopole (in a BZ) to appear in pairs with opposite chiralities

Weyl monopole

• The sign of a monopole charge Q_na depends on band-n and node-a

Dirac magnetic monopole

• A Weyl node is a monopole in momentum space

n=+

n=−

a = L R

Opposite Q

Opposite Q

Hall effect

2b 2b₀

Chiral magnetic effect in Weyl semimetal

(Zyuzin and Burkov, Phys Rev B 2012)

z

b

H = τ σ ⋅ + k σ ⋅ + τ b r r r r

momentum separation

energy separation

• Low-energy effective theory for a pair of Weyl nodes

( )

( )

2

2

2 2 0

1 2

2

t

J e E B

h

e E b B

h b

θ θ

π π

→ = ∇ × + ∂

= × −

r r r

r r

r

Chiral magnetic effect

• Effective electromagnetic action

• Current density

3

0

1

2

with the axion fiel

( , )

d

=2( )

S e dtd x r E

b

h t

b

B

r t

θ

π

θ

θ

= ⋅

⋅ −

∫

r

r

r

r r

(Relativistic covariance requires AHE and CME to both exist)

Results against ^(static) CME

• Semiclassical analysis

• Numerical work on lattice

• And more …

( )

2 3

3

2 2

2

( ) ( )

,

2 0

n na

2

B n n

n

a n

n

n

e d k

v f

e

h d Q

µ

f

ε ε π

α π

ε π

Ω Ω

Φ Φ =

=

Ω

= ⋅

=

∑∫

∑∫

r h

r

Opposite monopole charges

Opposite Berry curvatures

Berry flux over an iso-energy surface

(a-th node in n-th band) Energy-integrated

Berry flux

Berry curvature

CME B

J = − α B

r r

However,

Same Berry flux

(Basar, Kharzeev, and Yee, PRB 2014)

Argument against ^(static) CME

can be > 0 or < 0

Can extract energy out of equilibrium state!

J = − α

B

r r

• Work done by field on charges

To resolve this issue, we propose

• A minimum model with two bands

• Use linear response theory

• Consider both orders of taking the limits

What we found (for a clean and infinite system)

• Static limit:

• uniform limit:

( )

( )

0

0 0

0

static limit: lim lim , uniform limit: lim lim ,

q B

q B

q

ω

q

ω

α ω

α ω

→

→

→

→

r

r

Chang and Yang, Phys Rev B 2015 Goswami and Tewari, 1311.1506

If there are impurities, the conclusion might change (later)

B

0 α =

B

0

α ≠

3 3

3

3

3

1

1

1 (2 )

( )

( )

2

( )

n n H

n

n n F

n

n n n n n

n n

n

m

n F

n

n

d k f

d k v f

d k v f d f

d f

e

k m v

µ

π σ

ε ε

ε ε

π

ε ε

Ω

Ω

Ω =

 ∂ 

Ω ⋅  −  =

 ∂ 

Ω ⋅ = Φ

 ∂ 

⋅  −  =

∂

 

 

 

   

 

 

 

   

   

Φ

 Φ

∫

∫

∫ ∫

∫

r r

r r

r r

h

h

h

h

Quantities related to Berry curvature in Weyl semimetal

Hall conductivity

Berry flux through Fermi surface ( chiral anomaly)

Energy-integrated Berry flux ( static CME)

m-flux

( dynamic CME)

“Not related to topology”

•

•

•

• -

n

ΦΩ

( ) 2 2

na

ε π

π

ΦΩ = = ±

Magnetic moment of a Bloch electron

in 2-band model

k

m_± = ± Ωd^r _± e r r

h

CME coefficient: linear response theory

• Uniform limit

• static limit

( )

2 0 0

2

2 2 0

lim lim , [ ]

= [ ] 0

( )

n

n n k n n

q n n

n n n

n

L R

f

q e dk v f nd v

e dk v f

e h

ω

α ω

ε

µ µ

→ →

=±

=±

 ∂ 

=  ⋅ Ω − ⋅ Ω 

 ∂ 

⋅ Ω



=  −

∑∫

∑∫

r

r r r

r r r

h

r r r h

( )

2

0 0 0

lim lim , [ ]

= [ ]

1 3 2

3

n

n n k n n

n n

n

static k n n

n n

q

f

q e dk v f nd v

e dk nd v f

ω α ω

ε

α ε

→ =±

=±

→

 ∂ 

=  ⋅ Ω − ⋅ Ω 

 ∂ 

+ ⋅ Ω ∂

∂

∑∫

∑∫

r

r

r r r

r r r

h

r r r

2-band model

ε

σ

r r r r

• Not zero (for a clean and infinite system)

• Later, semiclassical analysis shows that this should be interpreted as dynamic CME, instead of static CME

← ΦΩ ^(energy-

integrated) Equilibrium

Non-equilibrium

( )

1cos '

(sin sin sin )

' 2 cos cos

z so

so so x x y y z z

x y z

H k H H

H t k k k

H k

t

m k

σ σ σ

σ

= + +

= + +

= + − −

A two-band model

2m 2t₁

Data from static limit

Saturation since at large t₁, ₁

1 1 z

z

v t t⁻

≈ Ω ≈

t

α ∝

• No energy separation (between nodes), no CME

• Filled bands (insulator) don’t have CME

• In this model, if no Weyl nodes, then no CME

AHE

CME

m=0.5

Chang and Yang, Phys Rev B 2015

Phase diagram and number of Weyl nodes However, in other 2-band models, we found

CME in the absence of Weyl node

Note: chiral anomaly still requires Weyl nodes.

Semiclassical analysis

• E and B can oscillate in space/time

• Easier to consider finite q, ω, and include relaxation τ

Non-Abelian generalization:

J.W. Chen et al, Phys Rev D 2014 (quantities with ~ are

modified by a m.B term)

AHE static CME

Chiral anomaly Density of (x,k)

phase space

( h ω << ε

)

Equations of motion:

Boltzmann equation (with relaxation)

δ n

− τ ^k

• Consider dynamic electromagnetic field

ωτ <<1

• Finite τ removes the non-analyticity of α(0,0)

• No static CME under both limits (in equilibrium)

( ) [ ]

ij static ij ni nj

n n

e dk v m f i

α ω

ε τ

ω α δ ωτ

+

= + ∂

∑∫ ^r ∂

( ) ( )

0 0 0

2

lim lim , = lim lim

, [ ]

q q

n n n

n

q q

e dk v f

ω

ω

α ω α ω

→ →

→ →

= ∑∫ ⋅ Ω

r r

r r r h

Chiral magnetic effect ( E=0, finite τ):

← Φ

Intraband (dominated by intravalley)

ωτ >> 1

Dynamic CME, or Gyrotrpic Magnetic Effect

( ) 1 [ ]

3

n static

n n

n n

e dk v m f

α ω α

ε

= + ⋅ ∂

∑∫

← Φ

Magnetic moment of a Bloch electron

2/3 in LRT

• Dynamic B field induces an E field

(Need to put E back, and redo the semiclassical calculation)

( )

, 1

= ,

J i P B

M E

i J iq M

iq E q E B

B

ω α

β β α

ω

β ω

α

= − = −

= = −

→ = ×

× × =

= −

B B

E

E E

r r r

r r

r r r

r r r

r r

r

magnetoelectric effect

Thus, current is doubled.

Semiclassical analysis gives

Also, see Kharzeev et al, Phys Rev D 2017

Summary: Different versions of CME

Basar et al, Phys Rev B 2014 Zhong et al, Phys Rev Lett 2016

• Same chemical potential

• Static B field: no current

• Dynamic B field (non-equilibrium):

can have CME current

(related to natural gyrotropic effect)

2 2

J e B

h

µ

= ∆

r r

Negative magneto-resistance

Magnitude: J ~ 0.01 (A/mm²)

if ∆µ=0.01 meV, B=0.1 T

(Zhang et al, Nature Comm 2015)

• Different chemical potentials µ

µ

ε

ε

Dynamic CME and natural optical gyrotropy

(Ma and Pesin, Phys Rev B 2015; Zhong et al, PRL 2016)

Tsirkin, Puente, and Souza, Phys Rev B 2018

Landau and Lifshitz, Electrodynamics of continuous media

• Antisymmetric part

Π

( , ) ω q r = Π

( , 0 ω ) + Π

( ω ) q

TR odd TR even

(Faraday rotation) (natural optical rotation)

• Cubic symmetry, or higher

2 0

1 ,

( , ) ( )

ij

q

q

ε ω δ

ε ω

ω ^Π

= + r

r

due to A

A

( )

ij

ij ij

i

ω δ

α α α

ε

=

=

Π

• Dielectric function

(totally antisymmetric)

• Rotary power

2 0

Re( )

1 rad

Re 0.4

mm

n n

c

γ π

λ ε α

− +

≡ −

 

= ≈  

 

for 2 Weyl nodes with

∆ε=0.1 eV (e.g. SrSi₂) (dynamical CME)

• No CME at static B field

• CME from dynamic B field

• Dynamic CME: No Weyl node required, but need Fermi surface

(also, need to break space-inversion symmetry)

• Connection between dynamic CME and optical gyrotropy

( 1)

( 1)

ωτ ωτ

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