532
E. Kamenetskii
asymmetry. In this case, the membrane functions ˜
η cannot be considered as singlevalued functions, and the question arises of the validity of the energy orthogonality
relation for MS-wave modes.
To restore the ND boundary conditions and thus the completeness of eigenfunctions ˜
η, we need introducing a certain surface magnetic current
j
(m)
s
top
circulating
on a lateral surface of the disk. This is a topological current, which compensates
the term
iμ a
1
r
∂ ˜
η
∂θ
r =R −
in the EM boundary conditions [55, 56, 60]. Evidently,
for a given direction of a bias magnetic field (that is, for a given sign of μ a ),
there can be two, clockwise and counterclockwise, quantities of the circulating
magnetic current. The topological current
j
(m)
s
top
is defined by the velocity of
an irrotational border flow. This flow is observable via the circulation integral of
the gradient
∇ θ δ =
1
R
∂δ ±
∂θ
r =R
e θ , where δ ± is a double-valued edge wave function on contour L = 2π R. On a lateral surface of a quasi-2D ferrite disk, one can
distinguish two different functions δ ± , which are the counterclockwise and clockwise rotating-wave edge functions with respect to a membrane function ˜
η. The
spin-half wave-function δ ± changes its sign when the regular-coordinate angle θ
is rotated by 2π . As a result, one has the eigenstate spectrum of MDM oscillations
with topological phases accumulated by the edge wave function δ. A circulation of
gradient
∇ θ δ along contour L = 2π R gives a non-zero quantity when an azimuth
number is a quantity divisible by
1
2
. A line integral around a singular contour L:
1
L
i
∂δ ±
∂θ
(δ ± )
∗ dL =
2π
0
i
∂δ ±
∂θ
(δ ± )
∗
r ==
dθ is an observable quantity. Because
of the existing the geometrical phase factor on a lateral boundary of a ferrite disk,
MDM oscillations are characterized by a pseudo-electric field (the gauge field) .
The pseudo-electric field can be found as
. The field is the Berry
curvature. The corresponding flux of the gauge field through a circle of radius is
obtained as:
, where
Ξ
(e)
±
are
quantized fluxes of pseudo-electric fields, K is the normalization coefficient. Each
MDM is quantized to a quantum of an emergent electric flux. There are the positive and negative eigenfluxes. These different-sign fluxes should be nonequivalent
to avoid the cancellation. It is evident that while integration of the Berry curvature
over the regular-coordinate angle θ is quantized in units of 2π , integration over the
spin-coordinate angle θ
θ
=
1
2
θ
is quantized in units of π . The physical meaning
of coefficient K concerns the property of a flux of a pseudo-electric field. The Berry
mechanism provides a microscopic basis for the surface magnetic current at the interface between gyrotropic and nongyrotropic media. Following the spectrum analysis
of MDMs in a quasi-2D ferrite disk one obtains pseudo-scalar axion-like fields and
edge chiral magnetic currents. The anapole moment for every mode n is calculated
as [55, 56, 60].
E. Kamenetskii
asymmetry. In this case, the membrane functions ˜
η cannot be considered as singlevalued functions, and the question arises of the validity of the energy orthogonality
relation for MS-wave modes.
To restore the ND boundary conditions and thus the completeness of eigenfunctions ˜
η, we need introducing a certain surface magnetic current
j
(m)
s
top
circulating
on a lateral surface of the disk. This is a topological current, which compensates
the term
iμ a
1
r
∂ ˜
η
∂θ
r =R −
in the EM boundary conditions [55, 56, 60]. Evidently,
for a given direction of a bias magnetic field (that is, for a given sign of μ a ),
there can be two, clockwise and counterclockwise, quantities of the circulating
magnetic current. The topological current
j
(m)
s
top
is defined by the velocity of
an irrotational border flow. This flow is observable via the circulation integral of
the gradient
∇ θ δ =
1
R
∂δ ±
∂θ
r =R
e θ , where δ ± is a double-valued edge wave function on contour L = 2π R. On a lateral surface of a quasi-2D ferrite disk, one can
distinguish two different functions δ ± , which are the counterclockwise and clockwise rotating-wave edge functions with respect to a membrane function ˜
η. The
spin-half wave-function δ ± changes its sign when the regular-coordinate angle θ
is rotated by 2π . As a result, one has the eigenstate spectrum of MDM oscillations
with topological phases accumulated by the edge wave function δ. A circulation of
gradient
∇ θ δ along contour L = 2π R gives a non-zero quantity when an azimuth
number is a quantity divisible by
1
2
. A line integral around a singular contour L:
1
L
i
∂δ ±
∂θ
(δ ± )
∗ dL =
2π
0
i
∂δ ±
∂θ
(δ ± )
∗
r ==
dθ is an observable quantity. Because
of the existing the geometrical phase factor on a lateral boundary of a ferrite disk,
MDM oscillations are characterized by a pseudo-electric field (the gauge field) .
The pseudo-electric field can be found as
. The field is the Berry
curvature. The corresponding flux of the gauge field through a circle of radius is
obtained as:
, where
Ξ
(e)
±
are
quantized fluxes of pseudo-electric fields, K is the normalization coefficient. Each
MDM is quantized to a quantum of an emergent electric flux. There are the positive and negative eigenfluxes. These different-sign fluxes should be nonequivalent
to avoid the cancellation. It is evident that while integration of the Berry curvature
over the regular-coordinate angle θ is quantized in units of 2π , integration over the
spin-coordinate angle θ
θ
=
1
2
θ
is quantized in units of π . The physical meaning
of coefficient K concerns the property of a flux of a pseudo-electric field. The Berry
mechanism provides a microscopic basis for the surface magnetic current at the interface between gyrotropic and nongyrotropic media. Following the spectrum analysis
of MDMs in a quasi-2D ferrite disk one obtains pseudo-scalar axion-like fields and
edge chiral magnetic currents. The anapole moment for every mode n is calculated
as [55, 56, 60].