Proceedings of International Conference on Physics in ... - KEK

the dispersion relation p0 = Ep − µ for particles). Equation
(8) implies that eA1γ1 = eA1γ0γ 5 and thus −eA1 can
be identified as the axial (or chiral) chemical potential µ5.
Therefore, we can conclude;
J 1 V = 1
π
∫
dx µ5, (17)
which correctly recovers the (3+1)-dimensional chiral
magnetic current (4) once we multiply this by the Landau
level density, eB/(2π). That is,
jV = µ5
π
−→ jV = |eB|
2π
(in (1+1) dimensions)
· µ5
π
(in (3+1) dimensions), (18)
which coincides with Eq. (4).
Here, it is clear that the longitudinal gauge field A 1 ,
which is the Chern-Simons number in (1+1) dimensions,
plays the role <strong>of</strong> the chiral chemical potential µ5 in (3+1)
dimensions. We note, however, that there is an important
difference; usually µ5 is introduced by hand as a constant,
but in (1+1) dimensions A 1 must have t-dependence to allow
for nonzero QW . We can think <strong>of</strong> a concrete “instanton”
configuration in (1+1) dimensions simply as
A 1 (t, x) = 2πQW t
= −Et, (19)
eL T
where we limit ourselves to the spatially homogeneous case
and denote the spatial and temporal extents as L and T ,
respectively, and then we have
j 1 V (t) = J 1 V (t) eE
= t. (20)
L π
From this, again, if multiplied by the Landau-level degeneracy
we can correctly recover the current generation rate
given by Eq. (6), i.e.
d(ejV )
dt = e2E (in (1+1) dimensions)
π
−→ d(ejV ) |eB|
=
dt 2π · e2E (in (3+1) dimensions), (21)
π
which coincides with Eq. (6).
In the same way we can get a finite axial-vector current
at finite quark chemical potential µ. To see the anomalous
nature the important fact is that the relation between the
density and the chemical potential is given by the quantum
anomaly in (1+1) dimensions, i.e.
n = − eA0
, (22)
π
which results from the anomaly. One can derive this expression
directly from n = ⟨ψ † (x)ψ(x)⟩ by inserting
the gauge field as lim y 0 →x 0 ψ † (y) exp[−ie ∫ dtA 0 ]ψ(x).
From this we can immediately reach,
J 1 ∫
5 = dx n = 1
∫
dx µ, (23)
π
which represents the chiral separation effect. This is again
the anomaly relation exactly same as that in (3+1) dimensions
once multiplied by the Landau level density eB/2π.
SCHWINGER MODEL
So far the arguments and the resulting expressions are
quite general. From now on we shall go into the dynamical
properties calculating microscopic quantities in a solvable
(1+1)-dimensional model, i.e. the massless Schwinger
model. The easiest way to accomplish a calculation in the
Schwinger model is to use mapping onto a free bosonic
theory. In our case, however, the bosonization rule is a
bit more complicated than usual because we deal with not
only fermionic fields (such as the chiral condensate) but
also gauge fields (such as the electric field). So, the Lagrangian
density <strong>of</strong> the corresponding theory should be
L = 1
2 (∂µ θ)(∂µθ) − mγ(∂ µ θ)(∂µϕ) − 1
2 (∂µ ϕ)∂ 2 (∂µϕ)
(24)
with the boson mass,
m 2 γ = e2
. (25)
π
If the ϕ-field is integrated out, we get a theory only in terms
<strong>of</strong> the θ-field that is free and has a mass by mγ. Such a
scalar theory is usually used with the bosonization rule;
j µ
V = ¯ ψγ µ ψ = 1
√ π ϵ µν ∂νθ, (26)
j µ
5 = ¯ ψγ µ γ 5 ψ = − 1
√ π ∂ µ θ, (27)
¯ψψ = −c mγ : cos(2 √ πθ) : (28)
with the normal ordering : :. Now we remark that ϕ in
the Lagrangian density (24) comes from the gauge field,
A µ = −ϵ µν∂νϕ (where ϕ includes an instanton-like configuration
∼ 1
2Et2 which does not satisfy the periodic
boundary condition in the t-direction). Then the electric
field takes a form E = ∂2ϕ. Once we integrate the θ-field
out from the theory, after the Gaussian integration in the
functional formalism, Eq. (27) is replaced by
j µ
5
= − 1
√ π ∂ µ θ → − mγ
√π ∂ µ ϕ = − e
π ∂µ ϕ. (29)
The anomaly relation is then derived as
∂µj µ
5
= − e
π ∂2 ϕ = − e
π E = −2qW , (30)
which is fully consistent with the anomaly relation (11).
In the same manner we can express the vector current in
terms <strong>of</strong> ϕ, and then we find,
j µ
V
e
=
π ϵµν µν ∂ν
∂νϕ = 2ϵ
∂2 qW . (31)
It is easy to confirm that this result is fully consistent with
the previous relation. That is, after the spatial integration
for the µ = 1 component (or ϕ and qW ), the spatial derivative
∂1 drops and the right-hand side simplifies as −2/∂0,
that is just a t-integration. Therefore the right-hand side finally
becomes −2QW together with the spatial integration,
and hence we obtain J 1 V = −2QW .