17 Axion Electrodynamics in Magnetoelectric Media
485
A z<0 = 2q 1 ˆ
e ϕ
∞
0
λ − cos (λ − z) − (λ + + k) sin (λ − z)
1
λ
2
− + λ
2
+
+ 2 k 2 + kλ + ( 1 + 2 )
k J 1 (kρ)e
λ + z−kz
dk,
(17.74)
and, above the surface, we obtain
A z>0 = 2q 1 ˆ
e ϕ
∞
0
λ −
1
λ
2
− + λ
2
+
+ 2 k 2 + kλ + ( 1 + 2 )
k J 1 (kρ)e
−k(z+z
) dk.
(17.75)
The magnetic field is obtained as B = ∇ × A. The right panel of Fig. 17.5 shows
the magnetic field B (in units of q
2 ) induced by an electric charge in vacuum at
z
= 1// close to the WSM TaAs as a function of the dimensionless coordinates
ρ and z. We observe that in the present case, the behaviour of the field lines
is significantly different from the radially directed field lines appearing when the
charge is close to a TI. We will discuss this point later.
The physical origin of the magnetic field is the bulk Hall current circulating
around the symmetry axis, J Hall = σ xy
ˆ
e ρ · E z<0
ˆ
e ϕ , which is induced by the inplane component of the electric field produced by the charge. In this way, each cross
section (perpendicular to the symmetry axis) of the bulk Hall current resembles the
surface Hall current induced by an electric charge near to a topological insulator.
This suggests that a 3D Weyl semimetallic phase can be understood as an infinite
number of 2+1 Dirac subsystems (one for each value of z in the bulk) supporting a
surface Hall current [100].
Now we go back to the discussion of the behaviour of the magnetic field lines in
Fig. 17.5. A close inspection reveals that below the surface of the WSM, centered
at the position of the image charge, the B-field lines wind in an axisymmetric way
as if about a loop of current, similar to those of a physical magnetic dipole. This
suggests that we consider a multipole expansion of the magnetic field and determine
the dominant contribution. Still, we recall that the source of the magnetic field is not
localized (since J Hall is defined in the whole bulk), and hence the standard multipole
expansion for localized sources does not necessarily applies. In the far zone, the
integral in (17.75) can be approximated in powers of 1/r . The dominant contributions
are
A
(1)
≈
3
32
q
π
2
tan(θ/2)(13 + 3 cos θ)
r 3/2
ˆ
e ϕ ,
A
(2)
≈ −
q
1 +
2
1
sin θ
r 2 ˆ
e ϕ ,
(17.76)
where θ is the angle from the z-axis to the observation point, i.e. r cos θ = ˆ
e z · r and
r =
ρ 2 + (z + z ) 2 . We observe that the leading term is a fractional multipole, with
no analogue in standard electromagnetism. The measurement of this contribution
would be a unique signature of the anomalous Hall effect in the bulk of 3D WSMs.
The second term can be successfully compared with the vector potential produced
485
A z<0 = 2q 1 ˆ
e ϕ
∞
0
λ − cos (λ − z) − (λ + + k) sin (λ − z)
1
λ
2
− + λ
2
+
+ 2 k 2 + kλ + ( 1 + 2 )
k J 1 (kρ)e
λ + z−kz
dk,
(17.74)
and, above the surface, we obtain
A z>0 = 2q 1 ˆ
e ϕ
∞
0
λ −
1
λ
2
− + λ
2
+
+ 2 k 2 + kλ + ( 1 + 2 )
k J 1 (kρ)e
−k(z+z
) dk.
(17.75)
The magnetic field is obtained as B = ∇ × A. The right panel of Fig. 17.5 shows
the magnetic field B (in units of q
2 ) induced by an electric charge in vacuum at
z
= 1// close to the WSM TaAs as a function of the dimensionless coordinates
ρ and z. We observe that in the present case, the behaviour of the field lines
is significantly different from the radially directed field lines appearing when the
charge is close to a TI. We will discuss this point later.
The physical origin of the magnetic field is the bulk Hall current circulating
around the symmetry axis, J Hall = σ xy
ˆ
e ρ · E z<0
ˆ
e ϕ , which is induced by the inplane component of the electric field produced by the charge. In this way, each cross
section (perpendicular to the symmetry axis) of the bulk Hall current resembles the
surface Hall current induced by an electric charge near to a topological insulator.
This suggests that a 3D Weyl semimetallic phase can be understood as an infinite
number of 2+1 Dirac subsystems (one for each value of z in the bulk) supporting a
surface Hall current [100].
Now we go back to the discussion of the behaviour of the magnetic field lines in
Fig. 17.5. A close inspection reveals that below the surface of the WSM, centered
at the position of the image charge, the B-field lines wind in an axisymmetric way
as if about a loop of current, similar to those of a physical magnetic dipole. This
suggests that we consider a multipole expansion of the magnetic field and determine
the dominant contribution. Still, we recall that the source of the magnetic field is not
localized (since J Hall is defined in the whole bulk), and hence the standard multipole
expansion for localized sources does not necessarily applies. In the far zone, the
integral in (17.75) can be approximated in powers of 1/r . The dominant contributions
are
A
(1)
≈
3
32
q
π
2
tan(θ/2)(13 + 3 cos θ)
r 3/2
ˆ
e ϕ ,
A
(2)
≈ −
q
1 +
2
1
sin θ
r 2 ˆ
e ϕ ,
(17.76)
where θ is the angle from the z-axis to the observation point, i.e. r cos θ = ˆ
e z · r and
r =
ρ 2 + (z + z ) 2 . We observe that the leading term is a fractional multipole, with
no analogue in standard electromagnetism. The measurement of this contribution
would be a unique signature of the anomalous Hall effect in the bulk of 3D WSMs.
The second term can be successfully compared with the vector potential produced