Kinetic and magnetoresistance/

Kinetic and magnetoresistance/

Weyl metal/ chiral anomaly

Sungkit Yip

Institute of Physics

Institute of Atomic and Molecular Sciences Academia Sinica

National Center for Theoretical Sciences

arXiv:1508.01010

Attempt to formulate magnetoresistance in quasiclassical regime

Kinetic equation (no interference , localization / antilocalization)

Nielsen and Ninomiya (1983) -- quantum regime

Son and Spivak (2013) -- assumed local equilibrium at each Weyl node each specified by chemical potential

chemical potential difference related to chiral anomaly deduced magneto-resistance

Kim et al (Pohang, 2014) – Green’s function, then kinetic equation Other work: diffusive limit …

Starting point:

[left out m(k)  B]

(Q Niu, M-C Chang +collaborators )

Eqm: both sides vanish

k-space volume

(k, ..) *

* one to one correspondence to Kubo formula in linear response (conserved)

dimensionless

J = 0 in equilibrium (when E = 0) [Appendix]



I

(r,k,t) = 0

conservation of number of particles under collision

anomalous Hall To first order in E:

steady state:

only then we have continuity equation

(k)

ordinary Hall effect, ignored from here on

1. No Weyl / no chiral anomaly / single Fermi surface 2. Weyl / with chiral anomaly / multiple Fermi surfaces

(k)

~

1. Single Fermi surface

relaxation time approx:

relax to equilibrium I _col = 0 at equilibrium

[can be checked posteriori  satisfied if  is the equilibrium chemical potential

density not modified by E]

available explicitly, since on LHS, n(k) in can be replaced by f() in linear response

(k)

[ ]

Generally  _k’ n(k’) /  _k’k

advantage:

from which conductance (tensor) can be read off possible off-diagonal components

Term linear in B vanishes if time-reversal obeyed k even positive magneto-conductance

B // x:

1 T 1.6%

10³ /Bohr ² 3 T 0.016%

recent claim:

with modifications, including m(k)  B

~0.1- 0.5 % in 3 T

2. Weyl, with multiple Fermi surfaces:

sink and source if ( E  B )  0;

expect chemical potential differences between pockets

_k above and below opposite

Particles disappear / appear from the lower band, total conserved

e.g. H=

[not nec separated in k]

Below: formulation on the same basis as I gave before:

Remark: can apply kinetic equation if  not too close to Weyl points

k-space volume

diverges at Weyl point but not a concern since total particles conserved

unknown at this stage Collision integrals:

intra pocket:

conservation of particles

 _

 _

interpockets:

1 to 2:

and similarly for 2 to 1 from pocket 2 to 1

(k)

conservation:

 _

 _

if k independent:

used and similarly for 2

determines ’s after n(k) solved

Define:

similarly for 2; not necessarily equal

Restriction: !

 ^  ^1 field dependent if field dependent density of states not equal (density of states)

while if define:

can be field independent

(B = 0 values)

 ^  ^1 in general not equal

(k)

Solve kinetic equation for

1^st term: part 1 2^nd term:

x B

monopole charge

~

~ Son and Spivak close to Weyl point:

contribution from 1^st valley, similar expression for 2^nd , total = sum:

2^nd term, continued:

Enhancement in  ~  ^X  e B / c ² / D

c.f.  (B=0) ~  ₀ D v ²

Ratio:  e B / c ² / ( v D) ²  e B / c ² / k _F ⁴

 e B / c ²  ²  ^X /  ₀ 

~  ^X /  ₀ 

~  ^X /  ₀ 

seen before

! ?

(2015)

Quasiclassical kinetic equation Particle number conservation