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Chiral magnetic effect without chirality source in asymmetric Weyl semimetals Yuta Kikuchi Stony Brook University & Kyoto University collaborators: Dima Kharzeev (Stony Brook U. & BNL & RIKEN-BNL), Rene Meyer (W¨ urzburg U.) arXiv:1610.08986

Stony Brook University, February 20, 2017

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Outline

Introduction Chiral kinetic theory Chirality imbalance without chirality source Candidate material: SrSi2 Summary

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Outline

Introduction Chiral kinetic theory Chirality imbalance without chirality source Candidate material: SrSi2 Summary

3 / 23

Weyl semimetals One dimensional case E(p)

Inversion (reflection) symmetric I : (px , py , pz ) → (−px , −py , −pz ) R : (px , py , pz ) → (px , py , −pz )

I

There must be both leftand right-moving particles.

I

Symmetric dispersion for left and right.

−π/a

left moving

right moving

π/a

p

symmetric

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Weyl semimetals One dimensional case E(p)

Inversion (reflection) symmetric I : (px , py , pz ) → (−px , −py , −pz ) R : (px , py , pz ) → (px , py , −pz )

I

There must be both leftand right-moving particles.

I

Symmetric dispersion for left and right.

−π/a

left moving

I

There still must be both leftand right-moving particles.

π/a

p

symmetric

Inversion (reflection) broken I

right moving

E(p)

−π/a

left moving

right moving

π/a

p

Asymmetric dispersion for left and right. asymmetric 4 / 23

Weyl semimetals Three dimensional case I

Weyl Hamiltonian HL/R = ±vF (p ± b/2) · σ There must be both left- and right-handed chiral fermions.

symmetric WSM

[Nielsen, Ninomiya (1984)]

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Weyl semimetals Three dimensional case I

Weyl Hamiltonian HL/R = ±vF (p ± b/2) · σ There must be both left- and right-handed chiral fermions.

symmetric WSM

[Nielsen, Ninomiya (1984)]

asymmetric WSM (aWSM) 5 / 23

Weyl semimetals Three dimensional case I

Weyl Hamiltonian HL/R = ±vF (p ± b/2) · σ There must be both left- and right-handed chiral fermions.

symmetric WSM

[Nielsen, Ninomiya (1984)]

I

Berry curvature Ω≡∇×A=±

p ± b/2 2|p ± b/2|3

A ≡ −iu † ∇p u

asymmetric WSM (aWSM) 5 / 23

Weyl semimetals Three dimensional case I

Weyl Hamiltonian HL/R = ±vF (p ± b/2) · σ There must be both left- and right-handed chiral fermions.

symmetric WSM

[Nielsen, Ninomiya (1984)]

I

Berry curvature Ω≡∇×A=±

p ± b/2 2|p ± b/2|3

A ≡ −iu † ∇p u I

The monopole charges sum up to zero globally Z k ≡ dS · Ω = ±1.

asymmetric WSM (aWSM) 5 / 23

Chiral magnetic effect B

[Kharzeev, McLerran, Warringa (2008); Fukushima, Kharzeev, Warringa (2008)]

I

For a massless fermion, there is one-to-one correspondence between its chirality and helicity ˆ · σ. h≡p

right handed spin

ˆR p

ˆL p left handed

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Chiral magnetic effect B

[Kharzeev, McLerran, Warringa (2008); Fukushima, Kharzeev, Warringa (2008)]

I

I

For a massless fermion, there is one-to-one correspondence between its chirality and helicity ˆ · σ. h≡p Spins are aligned by an external magnetic field.

right handed spin

ˆR p

ˆL p left handed

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Chiral magnetic effect B

[Kharzeev, McLerran, Warringa (2008); Fukushima, Kharzeev, Warringa (2008)]

I

I

I

For a massless fermion, there is one-to-one correspondence between its chirality and helicity ˆ · σ. h≡p

right handed spin

ˆL p left handed

Spins are aligned by an external magnetic field.

Chirality imbalance induces a current – Chiral magnetic effect (CME): J CME ∝ µ5 B. µ5 ≡

µ L − µR 2

ˆR p

J

More Red (left-handed) particles than Blue (right-handed) 6 / 23

Outline

Introduction Chiral kinetic theory Chirality imbalance without chirality source Candidate material: SrSi2 Summary

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Chiral kinetic theory I

Chiral kinetic theory = Boltzmann equation + Berry phase [Son, Yamamoto (2012,2013); Stephanov, Yin (2012)]

Z J=

d 3p [v fp + (v · Ω)fp B − (E × Ω)fp ] , (2π)3

v ≡ ∂/∂p, Ω ≡ ∇ × A,

fp : distribution function

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Chiral kinetic theory I

Chiral kinetic theory = Boltzmann equation + Berry phase [Son, Yamamoto (2012,2013); Stephanov, Yin (2012)]

Z J=

d 3p [v fp + (v · Ω)fp B − (E × Ω)fp ] , (2π)3

v ≡ ∂/∂p, Ω ≡ ∇ × A, I

fp : distribution function

The current is anomalous: k ∂n + ∇ · J = 2E · B ∂t 4π k: monopole charge of each Weyl node

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Chiral kinetic theory I

Chiral magnetic effect

J CME = σCME B, Z d 3p σCME ≡ (v · Ω)fp (2π)3

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Chiral kinetic theory I

Chiral magnetic effect

J CME = σCME B, Z d 3p σCME ≡ (v · Ω)fp (2π)3 I

Consider the contribution from the left-handed chiral fermions: p ˆ ( = |p|) Ω≡∇×A= , v = ∂/∂p = vF p 2|p|3 Near equilibrium state: fp =

1 e(vF |p|−µL )/T +1

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Chiral kinetic theory I

Chiral magnetic effect

J CME = σCME B, Z d 3p σCME ≡ (v · Ω)fp (2π)3 I

I

Consider the contribution from the left-handed chiral fermions: p ˆ ( = |p|) Ω≡∇×A= , v = ∂/∂p = vF p 2|p|3 Near equilibrium state: fp = e(vF |p|−µ1 L )/T +1 Chiral magnetic conductivity: e2 e2 σCME,L = 2 (µL + E0 ), σCME,R = − 2 (µR + E0 ) 4π 4π ⇒ J CME =

e2 µ5 B : vF independent 2π 2 9 / 23

Aside: θ-term arpproach I

Dirac action with the chiral chemical potential Z ¯ Dψ / + µ5 ψ † γ 5 ψ S = d 4 x ψi

µ5 6= 0

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Aside: θ-term arpproach I

Dirac action with the chiral chemical potential Z ¯ Dψ / + µ5 ψ † γ 5 ψ S = d 4 x ψi

I

Chiral rotation ¯ −i(µ5 t)γ5 /2 ψ → e −i(µ5 t)γ5 /2 , ψ¯ → ψe

µ5 6= 0

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Aside: θ-term arpproach I

Dirac action with the chiral chemical potential Z ¯ Dψ / + µ5 ψ † γ 5 ψ S = d 4 x ψi

I

Chiral rotation ¯ −i(µ5 t)γ5 /2 ψ → e −i(µ5 t)γ5 /2 , ψ¯ → ψe

I

µ5 6= 0

Rotated action Z ¯ Dψ / S = d 4 x ψi Z e2 + d 4 x(µ5 t)µναβ Fµν Fαβ 32π 2 µ5 = 0 10 / 23

Aside: θ-term arpproach I

Dirac action with the chiral chemical potential Z ¯ Dψ / + µ5 ψ † γ 5 ψ S = d 4 x ψi

I

Chiral rotation ¯ −i(µ5 t)γ5 /2 ψ → e −i(µ5 t)γ5 /2 , ψ¯ → ψe

I

µ5 6= 0

Rotated action Z ¯ Dψ / S = d 4 x ψi Z e2 + d 4 x(µ5 t)µναβ Fµν Fαβ 32π 2 ⇒J =

δS e2 = 2 µ5 B δA 2π

µ5 = 0 10 / 23

Outline

Introduction Chiral kinetic theory Chirality imbalance without chirality source Candidate material: SrSi2 Summary

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Chirality source I

CME current J CME =

e2 µ5 B, 2π 2

µ5 = (µL − µR )/2

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Chirality source I

CME current J CME =

e2 µ5 B, 2π 2

µ5 = (µL − µR )/2 I

Chirally balanced state (equiliblium)

equilibrium: µ5 = 0

µ5 = 0 ⇒ J CME = 0

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Chirality source I

CME current J CME =

e2 µ5 B, 2π 2

µ5 = (µL − µR )/2 I

Chirally balanced state (equiliblium)

equilibrium: µ5 = 0

µ5 = 0 ⇒ J CME = 0 I

Chirally imbalanced state (nonequiliblium) µ5 6= 0 ⇒ J CME 6= 0 see e.g. [Basar, Kharzeev, Yee (2013)]

nonequilibrium: µ5 6= 0 12 / 23

Chirality source (conventional way) B E

I

Electric field parallel to magnetic field dpz = −eE dt flips the helicity (chirality)

right handed spin

ˆR p

ˆL p left handed

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Chirality source (conventional way) B E

I

Electric field parallel to magnetic field dpz = −eE dt flips the helicity (chirality)

I

right handed spin

ˆR p

ˆL p left handed

⇔ Axial anomaly ∂µ J5µ =

1 E ·B 2π 2

[Adler (1969); Bell, Jackiw(1969)]

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Chirality source (conventional way) B E

I

Electric field parallel to magnetic field dpz = −eE dt

right handed spin

ˆL p

flips the helicity (chirality) I

left handed

⇔ Axial anomaly ∂µ J5µ =

1 E ·B 2π 2

[Adler (1969); Bell, Jackiw(1969)]

I

ˆR p

Chirally imbalanced state (nonequiliblium) µ5 ∝ E · B ⇒ J CME ∝ µ5 B ∝ (E · B)B [Son, Spivak (2013)]

J

Chirally imbalanced state 13 / 23

Chirality source (conventional way) The “conventional” chiral magnetic effect has been observed in Dirac/Weyl semimetals Experimental observations Dirac semimetals Weyl semimetals I

Bi1−x Sbx

I

[H.-J. Kim et.al. 2013] I

ZrTe5

[X. Huang et.al. (IOP) 2015] I

[Q. Li et.al. (BNL & Stony Brook U.) 2014] I

Na3 Bi

Cd3 As3 [C. Li et.al. (Peking U.) 2015]

NbAs [X. Yang et.al. (Zhejiang U.) 2015]

I

[J. Xiong et.al. (Princeton U.) 2015] I

TaAs

NbP [Z. Wang et.al. (Zheijiang U.) 2015]

I

TaP [C. Shekhar et.al. (MPI-Dresden) 2015] 14 / 23

Chirality generation in aWSMs I

[Kharzeev, Y.K., Meyer (2016)]

Different Fermi velocities for left- and right-handed fermions: vL 6= vR ⇒ different densities of states

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Chirality generation in aWSMs I

I

[Kharzeev, Y.K., Meyer (2016)]

Different Fermi velocities for left- and right-handed fermions: vL 6= vR ⇒ different densities of states Nonchiral electron source ⇔ Left- and right-handed electrons are pumped at the same rate:

dρR dρL = ⇒ µL (vL ) 6= µR (vR ) dt dt

Pour water into “left” and “right” cones with different size at the same rate.

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Chirality generation in aWSMs I

I

[Kharzeev, Y.K., Meyer (2016)]

Different Fermi velocities for left- and right-handed fermions: vL 6= vR ⇒ different densities of states Nonchiral electron source ⇔ Left- and right-handed electrons are pumped at the same rate:

dρR dρL = ⇒ µL (vL ) 6= µR (vR ) dt dt I

Chiral chemical potential 3 ∆v dµ µ5 ' τ5 2 V dt ∆v ≡ vL −vR , V ≡ (vL +vR )/2, µ ≡ (µL +µR )/2

Pour water into “left” and “right” cones with different size at the same rate.

τ5 : chirality relaxation time 15 / 23

Experimental setup I

Apply a time dependent AC voltage to the gate, which pumps the system with nonchiral electrons.

+

B

A

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Experimental setup I

Apply a time dependent AC voltage to the gate, which pumps the system with nonchiral electrons.

I

The current is then measured through two gates applied in the direction of the external magnetic field.

+

B

A

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Experimental setup I

Apply a time dependent AC voltage to the gate, which pumps the system with nonchiral electrons.

I

The current is then measured through two gates applied in the direction of the external magnetic field.

I

No electric field in the direction of the current ⇒ No Ohmic current. +

B

A

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Experimental setup

Current density response for ∆v /V = 0.1, |B| = 1T, and frequencies: 1 GHz (blue solid), 2 GHz (yellow dashed) and 3 GHz (green dotted). c.f. 2 |jCME | ' 0.4[mA/mm ] for conventional CME. 17 / 23

Outline

Introduction Chiral kinetic theory Chirality imbalance without chirality source Candidate material: SrSi2 Summary

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Inversion and reflection symmetry (Two dimensional) Brillouin zone

Discrete symmetries restrict I

Positions of the Weyl cones

I

Shapes (dispersion relations) of Weyl cones

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Inversion and reflection symmetry (Two dimensional) Brillouin zone

Discrete symmetries restrict I

Positions of the Weyl cones

I

Shapes (dispersion relations) of Weyl cones

Inversion symmetric

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Inversion and reflection symmetry (Two dimensional) Brillouin zone

Inversion symmetric

Discrete symmetries restrict I

Positions of the Weyl cones

I

Shapes (dispersion relations) of Weyl cones

Reflection symmetric

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Inversion and reflection symmetry (Two dimensional) Brillouin zone

Inversion symmetric

Discrete symmetries restrict I

Positions of the Weyl cones

I

Shapes (dispersion relations) of Weyl cones

Reflection symmetric

Asymmetric

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SrSi2 The band structure calculation shows “SrSi2 lacks both mirror and inversion symmetries.” [S.-M. Huang et.al. (2015)]

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SrSi2 The band structure calculation shows “SrSi2 lacks both mirror and inversion symmetries.” [S.-M. Huang et.al. (2015)]

I

The Weyl nodes with opposite chiralities are located at different energies.

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SrSi2 The band structure calculation shows “SrSi2 lacks both mirror and inversion symmetries.” [S.-M. Huang et.al. (2015)]

I I

The Weyl nodes with opposite chiralities are located at different energies. Density of states differ for the left- and right-handed Weyl cones.

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SrSi2 The band structure calculation shows “SrSi2 lacks both mirror and inversion symmetries.” [S.-M. Huang et.al. (2015)]

I I I

The Weyl nodes with opposite chiralities are located at different energies. Density of states differ for the left- and right-handed Weyl cones. Electron pumping generates the chiral chemical potential: dµ µ5 ∝ b0 dt 20 / 23

Current response

[Kharzeev, Y.K., Meyer (2016)]

Current density response for frequency 1GHz, |B| = 1T, T = 20K, and each curve corresponds to energy shift: 1 meV (blue solid), 5 meV (yellow dashed) and 10 meV (green dotted). 21 / 23

Outline

Introduction Chiral kinetic theory Chirality imbalance without chirality source Candidate material: SrSi2 Summary

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Summary I

Chirality imbalance is built into the band structure of aWSMs ⇒ aCME does not rely on an external source of chirality imbalance.

I

Pump a time-dependent nonchiral current ⇒ Change the chemical potential in the left and right chiral Weyl cones at a different rate ⇒ The chirality imbalance is generated.

I

aCME current is the same order of magnitude, and possibly even stronger, as the conventional CME current.

I

SrSi2 is predicted to be a Weyl semimetal with broken reflection and inversion symmetries.

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Crystal and electronic structure of SrSi2

1/2

Current response

[Kharzeev, Y.K., Meyer (2016)]

Temperature dependence of the amplitudes of current density response for frequency 1 GHz, |B| = 1T, and each curve corresponds to energy shift: 1 meV (blue solid), 5 meV (yellow dashed), 10 meV (green dotted) and 20 meV (red dot-dashed). 2/2

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