In (<ref>), we adjusted the trivial phase $2\pi i$ for the convenience of the later analysis; this is related to a  gauge transformation of Wu and Yang [29].
The corresponding energy eigenvalues are

\begin{eqnarray}\label{energy eigenvalue}
E_{\pm}= w_{\pm}^{\dagger}(t)\hat{H}w_{\pm}(t) =\mp (\mu\hbar B\cos\alpha).
\end{eqnarray}

From now on, we concentrate on $\Omega_{+}$ associated with the lower energy eigenvalue $E_{+}$, i.e., the spin is pointing to the direction of the applied magnetic field; the monopole $\Omega_{-}$ associated with the energy eigenvalue $E_{-}$ is described by $-\Omega_{+}$ up to a gauge transformation of Wu and Yang.
We then have an azimuthally symmetric  monopole-like potential [98]

\begin{eqnarray}\label{new potential}
{\cal A}_{\varphi}
= \frac{\hbar }{2B\sin\theta} [1 - \cos\Theta(\theta, \eta) ]
\end{eqnarray}

and ${\cal A}_{\theta} ={\cal A}_{B} =0$, where we defined

\begin{eqnarray}\label{effective angle}
\Theta(\theta, \eta)=\theta-\alpha(\theta,\eta).
\end{eqnarray}

The standard Dirac monopole [37] is recovered when one sets $\alpha(\theta,\eta)=0$ (or in the ideal adiabatic limit $\eta=\infty$ in (<ref>)), namely, $\Theta=\theta$ in (<ref>) and when $B$ is identified with the radial coordinate $r$ in the real space.
The crucial parameter $\Theta(\theta, \eta)$ is shown in Fig.B.2.

The relation between $\theta$ and $\Theta(\theta, \eta)=\theta-\alpha(\theta,\eta)$ parameterized by $\eta$. We have the exact relations $\Theta(\theta, \infty)=\theta$, $\Theta(\theta, 1)=\theta/2$ and $\Theta(\theta, 0)=0$, respectively, for $\eta=\infty$, $\eta=1$ and $\eta=0$. Topologically, $\eta>1$ corresponds to a monopole, $\eta=1$ corresponds to a half-monopole, and $\eta<1$ corresponds to a dipole, respectively.
Note that $\cos \theta_0 =-\eta$  with $\eta<1$, for which $\partial\Theta(\theta, \eta)/\partial\theta=0$.  See  [98].

The Dirac string appears at the singularity of the potential (<ref>). There exists no singularity at $\theta=0$ since $\Theta(\theta, \eta)\rightarrow 0$ for $\theta\rightarrow 0$.
The singularity does not appear at the origin $B=0$ with any fixed $T$ since $\alpha(\theta,\eta)\rightarrow \theta$ for $B\rightarrow 0$, namely, if one uses $\Theta(\theta, \eta)=\theta -\alpha(\theta,\eta) \rightarrow 0$ for $\eta=\mu BT/\pi\rightarrow 0$ in (<ref>). In fact the potential vanishes at $B=0$ for any finite $T$; we have a useful relation in the non-adiabatic domain $\eta=\mu BT/\pi \ll 1$, namely, $\alpha(\theta)\simeq \theta -\eta \sin\theta \ {\rm for} \ 0\leq \eta \ll 1$ in (<ref>) [98], which implies

\begin{eqnarray}
{\cal A}_{\varphi}
&\simeq& \frac{\hbar}{4B}(\mu TB/\pi)^{2}\sin\theta
\end{eqnarray}

that has no singularity associated with the Dirac string at $\theta=\pi$ near $B=0$ and linearly vanishes in $B$.
Thus the Dirac string can appear only at $\theta=\pi$ and only when $\Theta(\pi, \eta)\neq 0$, namely, for $\eta\geq 1$ in Fig.B.2 or equivalently

\begin{eqnarray}
B\geq \frac{\pi}{\mu T}
\end{eqnarray}

for any fixed finite $T$; the end of the Dirac string is located at $\frac{\pi}{\mu T}$ and $\theta=\pi$.  The total magnetic flux passing through a small circle C  around the Dirac string at the point $B$ and $\theta=\pi$
is given by the potential (<ref>)

\begin{eqnarray}\label{Stokes}
\oint_{C} {\cal A}_{\varphi} B\sin\theta d\varphi =\frac{e_{M}}{2}(1-\cos\Theta(\pi, \eta))
\end{eqnarray}

with $e_{M}=2\pi\hbar$. See a schematic figure in Fig.B.3  [98].

A schematic picture with fixed $T$.  Geometrical center and the origin of the Dirac string are displaced by the distance $B=\pi/\mu T$. An infinitesimal circle $C$ surrounds the Dirac string indicated by a wavy line.

This flux agrees with the integrated flux outgoing from a sphere with a radius $B$ covering the monopole
due to Stokes' theorem, since no singularity appears except for the Dirac string.  For $\eta>1$, one sees from Fig.B.2 that the above flux is given by $e_{M}=2\pi\hbar$ and thus Dirac's quantization condition is satisfied in the sense $\exp[-ie_{M}/\hbar]=1$. On the other hand, the flux vanishes for $\eta<1 (i.e., B<\frac{\pi}{\mu T})$ and $\Theta(\pi, \eta)=0$, and thus the object changes to a dipole [98].

The topology change is seen in the change of $\Theta(\pi)=\pi$ for $\eta>1$ to $\Theta(\pi)=0$ for $\eta<1$ in Fig.B.2.  But we have a well-defined potential (“half-monopole”) at the boundary $\eta=1$

\begin{eqnarray}\label{half sphere}
{\cal A}_{\varphi}=\frac{\hbar}{2B\sin\theta}\left(1-\cos\frac{1}{2}\theta \right)
\end{eqnarray}

for $\theta\neq \pi$.

§.§ Fixed $T$ configurations

We analyze the behavior of the magnetic monopole-like object (<ref>) in more detail for fixed $T$ and varying B;
this is close to the description of a monopole in the real space if one identifies $B$ with the radial variable $r$ of the real space.
The topology and topology change of Berry's phase when regarded as a magnetic monopole defined in the parameter space of $\vec{B}$ is specified by the parameter $\eta$, as is suggested  by a discrete jump of the end point $\lim_{\theta\rightarrow\pi}\Theta(\theta, \eta)$ in Fig.B.2.

Using the exact potential (<ref>)
we have an analogue of the magnetic flux in the parameter space $\vec{B}=B(\sin\theta\cos\varphi,
\sin\theta\sin\varphi,\cos\theta )$,

\begin{eqnarray}\label{magnetic flux}
{\cal B}\equiv \nabla\times {\cal A}
=\frac{\hbar}{2}\frac{\frac{\partial\Theta(\theta,\eta)}{\partial\theta}\sin\Theta(\theta,\eta)}{\sin\theta}\frac{1}{B^{2}}{\bf e}_{B} - \frac{\hbar}{2}\frac{\frac{\partial\Theta(\theta,\eta)}{\partial B}\sin\Theta(\theta,\eta)}{B\sin\theta}{\bf e}_{\theta}
\end{eqnarray}

for $\theta\neq \pi$ and $B\neq 0$ with ${\bf e}_{B}=\frac{\vec{B}}{B}$, and  ${\bf e}_{\theta}$ is a unit vector in the direction $\theta$ in the spherical coordinates.
We have

\begin{eqnarray}
\frac{\partial\Theta(\theta,\eta)}{\partial\theta} = \frac{\eta(\eta+\cos\theta)}{1+\eta^{2}+2\eta\cos\theta},
\end{eqnarray}

by noting $
\frac{\partial\alpha(\theta,\eta)}{\partial\theta}=\frac{1+\eta\cos\theta}{(\eta+\cos\theta)^{2}+\sin^{2}\theta}$ in (<ref>),
and thus
\frac{\partial\Theta(\theta,\eta)}{\partial\theta}=0$
at $\cos\theta_{0}=-\eta$ for $\eta<1$.
The factor in the  second term in (<ref>) is given by recalling $\eta=\mu TB/\pi$,

\begin{eqnarray}
\frac{\partial\Theta(\theta,\eta)}{\partial B}&=&\frac{\mu T}{\pi}\frac{\partial\Theta(\theta,\eta)}{\partial \eta}\nonumber\\
\end{eqnarray}

using (<ref>) and (<ref>). Thus we have (by setting $e_{M}=2\pi\hbar$)

\begin{eqnarray}\label{magnetic flux2}
{\cal B}&\equiv& \nabla\times {\cal A}\nonumber\\
&=&\frac{e_{M}}{4\pi}\frac{\sin\Theta(\theta,\eta)}{\sin\theta}\frac{\eta}{B^{2}}\frac{1}{1+\eta^{2}+2\eta\cos\theta}[(\eta+\cos\theta){\bf e}_{B} -\sin\theta{\bf e}_{\theta}]
\end{eqnarray}

We also have from (<ref>),

\begin{eqnarray}\label{alpha parameter}
\cos\alpha=\frac{\eta+\cos\theta}{\sqrt{1+\eta^{2}+2\eta\cos\theta}}, \ \ \  \sin\alpha=\frac{\sin\theta} {\sqrt{1+\eta^{2}+2\eta\cos\theta}},
\end{eqnarray}

and thus

\begin{eqnarray}
\sin\Theta(\theta,\eta)&=&\sin(\theta-\alpha) \nonumber\\
&=&\sin\theta\cos\alpha -\cos\theta\sin\alpha\nonumber\\
&=&\frac{\eta\sin\theta}{\sqrt{1+\eta^{2}+2\eta\cos\theta}} ,\end{eqnarray}

and similarly $\cos\Theta(\theta,\eta)

We finally have the azimuthally symmetric magnetic field from (<ref>) [99]

\begin{eqnarray}\label{magnetic flux3}
{\cal B}
&=&\frac{e_{M}}{4\pi}\frac{\eta^{2}}{B^{2}}\frac{1}{(1+\eta^{2}+2\eta\cos\theta)^{3/2}}[(\eta+\cos\theta){\bf e}_{B} -\sin\theta{\bf e}_{\theta}].
\end{eqnarray}

We note that $B/\eta=\pi/\mu T$ and, $\theta=\pi$ define the end  point of the Dirac string in the fixed $T$ picture. The magnetic field
${\cal B}$  is not singular at $\theta=\pi$ for $\eta>1$ which shows that the Dirac string is not observable if it satisfies the Dirac quantization condition, which is the case of the present model. In the adiabatic limit $\eta\rightarrow\infty$ ($\pi/\mu T\rightarrow 0$ with fixed $B$) in (<ref>), the outgoing magnetic flux agrees with that of the Dirac monopole

\begin{eqnarray}
{\cal B}
=\frac{e_{M}}{4\pi}\frac{1}{B^{2}}{\bf e}_{B}
\end{eqnarray}

located at the origin (level crossing point)  in the parameter space. This is the common magnetic monopole field associated with Berry's phase in the precise adiabatic approximation. At the origin $B=0$ with fixed finite $T$, which corresponds to the nonadiabatic limit $\eta=\mu BT/\pi\rightarrow 0$, the magnetic field (<ref>) approaches a constant field parallel to the z-axis

\begin{eqnarray}
{\cal B}
=\frac{e_{M}}{4\pi}\left(\frac{\mu T}{\pi}\right)^{2}[\cos\theta{\bf e}_{B} -\sin\theta{\bf e}_{\theta}].
\end{eqnarray}

A precise view of the magnetic flux generated by the monopole-like object (<ref>) is shown in Fig.B.4.  Note that $ {\cal B}\rightarrow 0$ in the nonadiabatic limit $T\rightarrow 0$ with any fixed $B$.

In passing, we comment on the notational conventions in this section: $\vec{B}(t)$ stands for the externally applied magnetic field to define the original Hamiltonian in (<ref>) and $\vec{B}$ is used to specify the parameter space to define Berry's phase, and ${\cal B}$ stands for the “magnetic field” generated by Berry's phase in the parameter space. The calligraphic symbols ${\cal A},\ {\cal B}$, ${\cal \nabla}$ and the bold ${\bf e}$ stand for vectors without writing arrows.

Arrows indicating the direction and magnitude of the magnetic flux from the azimuthally symmetric monopole-like object associated with Berry's phase (<ref>) in the fixed  $T$ picture. Two spheres with radii $B>\pi/\mu T$ (i.e., $\eta>1$) and $B<\pi/\mu T$ (i.e., $ \eta<1$) are shown. The wavy line stands for the Dirac string with the end located at $B=\pi/\mu T$ and $\theta=\pi$ from which the magnetic flux is imported. Only in the ideal adiabatic limit $T\rightarrow\infty$, the end of the Dirac string and the geometrical center of Berry's phase which is located at the origin agree.

§.§ Lensing of Dirac monopole in Berry's phase

It is  shown that the monopole associated with Berry's phase is mathematically regarded as a Dirac monopole moving away from the level crossing point of the parameter space driven by the force generated by the nonadiabatic rotating external field with finite period $T=2\pi/\omega<\infty$ in Berry's model [99].  This picture corresponds to a description of the monopole-like object by changing $T$ with fixed $B$.  We consider the configuration in Fig.B.5.

A geometric picture in the 3-dimensional parameter space $\vec{B}$ with a sphere centered at $O$ and the radius $B$ by assuming azimuthal symmetry. We suppose that  a genuine azimuthally symmetric Dirac monopole is located at the point $O^{\prime}$ in the parameter space.  The distance between $O$ and $O^{\prime}$ is chosen at
$\overline{OO^{\prime}} = B/\eta$. The three angles $\theta$, $\alpha$ and $\Theta=\theta-\alpha$ are shown. The observer is located at the point $P$. The wavy line indicates the Dirac string.

We then have

\begin{align}
\overline{O^{\prime}P}^{2}
&=B^{2}+\left(\frac{B}{\eta}\right)^{2} -2B\left(\frac{B}{\eta} \right)\cos(\pi-\theta)
\nonumber\\
&=\frac{B^{2}}{\eta^{2}}\left[ 1 + \eta^{2} + 2\eta\cos\theta \right],
\end{align}

and the unit vector ${\bf e}$ in the direction of $\vec{O^{\prime}P}$ is

\begin{align}
{\bf e}
&=\cos\alpha {\bf e}_{B} -\sin\alpha {\bf e}_{\theta}
\end{align}

with $ {\bf e}_{B}=\vec{B}/B$ and $ {\bf e}_{\theta}$ is a unit vector in the direction of $\theta$ in the spherical coordinates.
Then the magnetic flux of Dirac's monopole with a magnetic charge $e_{M}$ located at $O^{\prime}$ when observed at the point P is given by

\begin{align}\label{Dirac monopole}
{\cal B}^{\prime} &= \frac{e_{M}}{4\pi}\frac{1}{\overline{O^{\prime}P}^{2}}{\bf e}\nonumber\\
&=\frac{e_{M}}{4\pi}\frac{\eta^{2}}{B^{2}}\frac{1}{1+\eta^{2}+2\eta\cos\theta}(\cos\alpha {\bf e}_{B} -\sin\alpha {\bf e}_{\theta}).
\end{align}

Next we fix the parameter $\alpha$. We have $
(B/\eta)^{2}=B^{2} + \overline{O^{\prime}P}^{2} -2B\overline{O^{\prime}P}\cos\alpha$
which gives

\begin{align}
\cos\alpha&=\frac{1}{2B(B/\eta)\sqrt{1 + \eta^{2} + 2\eta\cos\theta}}\left[B^{2} + \left(\frac{B}{\eta} \right)^{2}(1 + \eta^{2} + 2\eta\cos\theta)-\left(\frac{B}{\eta} \right)^{2} \right]\nonumber\\
&=\frac{\eta+\cos\theta}{\sqrt{1 + \eta^{2} + 2\eta\cos\theta}}
\end{align}

and  from the geometrical relation $\frac{B\sin\alpha}{B\sin\theta}=\frac{B/\eta}{(B/\eta)\sqrt{1 + \eta^{2} + 2\eta\cos\theta}}$ ,

\begin{eqnarray}
\sin\alpha=\frac{\sin\theta}{\sqrt{1 + \eta^{2} + 2\eta\cos\theta}}.
\end{eqnarray}

The parameter $\alpha$ agrees with the parameter in (<ref>).
The azimuthally symmetric flux (<ref>) is thus given by [99]

\begin{eqnarray}\label{Dirac monopole2}
{\cal B}^{\prime}
=\frac{e_{M}}{4\pi}\frac{\eta^{2}}{B^{2}}\frac{1}{(1+\eta^{2}+2\eta\cos\theta)^{3/2}}[(\eta+\cos\theta) {\bf e}_{B} -(\sin\theta){\bf e}_{\theta}]
\end{eqnarray}

which agrees with the flux given by Berry's phase (<ref>).

This agreement of two expressions (<ref>) and (<ref>)  shows that the Dirac monopole originally at the level crossing point in the parameter space  formally appears to drift away by the distance $B/\eta=\pi/\mu T$ in the parameter space when the precise adiabaticity condition $T=\infty$ [4] is spoiled by the finite $T$.
It is interesting
that two dynamical parameters,  the strength of the external magnetic field and the period in Berry's model, are converted to very different geometrical parameters in Berry's phase, namely, the shape of the monopole and the distance of the deviation of the monopole from the level crossing point.
The observed magnetic field on the sphere with a radius $B$, which is controlled by the observer, thus changes when one changes the parameter $T$ that determines the end of the Dirac string located at $\pi/\mu T$ in the parameter space.
This geometrical picture is useful when one draws the precise magnetic flux from the monopole-like object for finite $T$ as in Fig.B.6 and it is essential when one attempts to understand the motion of a particle in the magnetic field.

Arrows indicating the direction and magnitude of the  magnetic flux observed at the point P with fixed $B$ and $\theta$ when the end of Dirac string at $\pi/\mu T$ is varied from the point $\pi/\mu T<B$ (Fig.B.6(a)) to the boundary $\pi/\mu T=B$ (Fig.B.6(b)) and then to the point $\pi/\mu T>B$ (Fig.B.6(c)), which correspond to the change of the basic parameter $\eta=\mu BT/\pi$ from the adiabatic domain $\eta=2.5>1$ to the boundary  $\eta=1$ and then to the nonadiabatic domain $\eta=0.5<1$, respectively. The wavy line stands for the Dirac string with the end at $B=\pi/\mu T$ and $\theta=\pi$. These figures after a suitable rescaling may also be interpreted as the results with the end of the Dirac string kept fixed at $\pi/\mu T$ and $\theta=\pi$ and varying the distance $B$, starting with a large $B>\pi/\mu T$  (Fig.B.6(a)) toward a small $B<\pi/\mu T$  (Fig.B.6(c)) in the parameter space, such as two spheres in Fig.B.4. See [99].

In terms of the original physical setting of a magnetic dipole placed in a given rotating magnetic field described by the Hamiltonian (<ref>), the cone drawn by the dipole becomes  sharper compared to the cone of the given magnetic field, which subtends the solid angle $\Omega=2\pi (1-\cos\theta)$, when the rotating speed of the external magnetic field becomes larger and the dipole moment is “left behind” (actually, not left behind, but rather follow the movement by making the polar angle $\theta$ smaller), namely, [97]

\begin{align}\label{solid angle}
\psi_{+}^{\dagger}(t)\vec{\sigma}\psi_{+}(t)&=
&=\left( \sin\Theta\cos\varphi(t), \sin\Theta\sin\varphi(t), \cos\Theta \right)\nonumber\\
\end{align}

that subtends the solid angle
$\Omega=2\pi (1-\cos\Theta)$ with $\Theta=\theta-\alpha$
; this sharper cone is effectively recognized as the drifting monopole in Berry's phase by an observer located at the point $P$ in Fig.B.5.

The above agreement of the solid angle drawn by the spinor solution (<ref>) with Berry's phase is known to be generally valid for the two-component spinor [97]: The most general orthonormal spinor bases are parameterized as

\begin{eqnarray}
\cos\frac{1}{2}\theta(t) e^{-i\varphi(t)}\\
\sin\frac{1}{2}\theta(t)
\end{array}\right), \ \ \
\sin\frac{1}{2}\theta(t)e^{-i\varphi(t)}\\
\end{array}\right)
\end{eqnarray}

that give the spin vector

\begin{align}
=\left( \sin\theta(t)\cos\varphi(t), \sin\theta(t)\sin\varphi(t), \cos\theta(t) \right)=-v_{-}^{\dagger}(t)\vec{\sigma}v_{-}(t)
\end{align}

subtending the solid angle
$\tilde{\Omega}_{\pm}=\oint (1\mp\cos\theta(t))d\varphi(t)$ for a closed movement. On the other hand, the “holonomy” which is related to Berry's phase, satisfies

\begin{eqnarray}
\oint dt ~v^{\dagger}_{\pm}(t)i\partial_{t}v_{\pm}(t)=-\frac{1}{2}\oint  (1\mp \cos\theta(t))d\varphi(t) +2\pi =-\frac{1}{2}\tilde{\Omega}_{\pm} +2\pi.
\end{eqnarray}

These two quantities thus agree up to the factor $1/2$ and  up to trivial phase $2\pi$ in the case of spinor bases. The important fact is that our exact solution of the Schrödinger equation has this structure of $v_{+}(t)$ and $v_{-}(t)$ with $\theta(t)=\theta-\alpha(\theta)$.

Coming back to (<ref>), one may thus prefer to understand that Fig.B.5 implies an analogue of the effect of lensing of Dirac's monopole, since the movement of the monopole in the parameter space is a mathematical one.  In the precise adiabatic limit with $T=\infty$ [4], the monopole is located at the level crossing point $O$, but when the effect of nonadiabatic rotation with finite $T< \infty$ is turned on, the image of the monopole is displaced to the point $O^{\prime}$ located at $\pi/\mu T$ by keeping the  topology and strength of the point-like monopole intact. In this picture, it is important that the topological monopole itself is not resolved in the nonadiabatic domain but it  disappears from  observer's view located at the point P for fixed $B$ when $\pi/\mu T = B/\eta \rightarrow {\rm large} $ with fixed $B$ (i.e., $\eta\rightarrow$ small). In the middle, the formal topology change takes place when $\pi/\mu T$ touches the sphere with the fixed radius $B$ (i.e., $\eta=1$). Even in the picture of lensing, the “magnetic flux” generated by Berry's phase  measured at the point in the parameter space specified by $(B,\theta)$ is the real flux.
It will be interesting to examine the possible experimental implications of these aspects of Berry's phase, which is expressed  by the magnetic field (<ref>), in the applications of Berry's phase.

It may be appropriate to mention that our exact solution agrees with the generic solution discussed in Section 2 for two limiting cases:
At the adiabatic limit $\eta\rightarrow \infty$, the parameter $\alpha\rightarrow 0$ in Fig.B.1, and the exact solution (<ref>) naturally reproduces the exact Dirac monopole solution. On the other hand, the phase in (<ref>) approaches

\begin{eqnarray}
\exp\left\{-\frac{i}{\hbar}\Omega_{\pm} \right\}=\exp\left\{-i\oint\frac{1\mp\cos(\theta-\alpha(\theta,\eta))}{2}d\varphi \right\} \rightarrow 1
\end{eqnarray}

in the non-adiabatic limit $\eta\rightarrow 0$ for which
$\alpha\rightarrow \theta$ in (<ref>). This agrees with the trivial phase in (<ref>)  in Section 2.

§.§ Smooth topology change

As for the smooth transition from a monopole to a dipole, it corresponds to the process of the shrinking of the sphere with a radius $B$ covering the end of the Dirac string located at $\pi/\mu T$ to a smaller sphere for which $B<\pi/\mu T$ as in Fig.B.4. When the sphere touches the end of the Dirac string (at $\eta=1$) in the middle, one encounters a  “half monopole” (<ref>) with the outgoing flux which is half of the full monopole $e_{M}/2=\pi\hbar$. See Stokes' theorem (<ref>) with $\Theta(\pi, \eta=1)=\pi/2$ in Fig.B.2. At this specific point, the Dirac string becomes observable [98], corresponding to the Aharonov-Bohm effect [27] of the electron in the magnetic flux generated by the superconducting Cooper pair [100].  As for more details, see the discussion below.
It is then natural to attach the end of the Dirac string to an infinitesimally small opening on the sphere (see Fig.B. 6b) forming a closed sphere and thus leading to the vanishing net outgoing flux, which corresponds to a dipole. The idea of the half monopole at $\eta=1$ is interesting, but it is natural to incorporate it as a part of a dipole. The monopole-like object (<ref>) is always  a dipole if one counts the Dirac string as in Fig.B.4  and Stokes' theorem (<ref>) always holds.  In this sense, no real topology change takes place for the movement of $B$, from large $B$ to small $B$ with fixed $T$, except for the fact that the unobservable Dirac string becomes observable at $B=\pi/\mu T$ and triggers the topology change from a monopole to a dipole.

A more detailed analysis of this smooth topology change goes as follows:
From a point of view of the net outgoing flux, we see the full flux with $e_{M}=2\pi\hbar$ in Fig.B.6a and the half flux with $e_{M}/2$ in Fig.B.6b and then no net flux in Fig.B.6c, corresponding to $\Theta(\pi)$ with $\pi$, $\pi/2$ and $0$, respectively, in (<ref>).
Thus these configurations are very distinct.
On the other hand, Stokes' theorem (<ref>) formally shows a smooth transition among distinct topologies specified by $\Theta(\pi)$ with $\pi$, $\pi/2$ and $0$.  Our smoothness argument of topology change in Berry's phase is based on the Stokes theorem.  But the arguments of Dirac [37] and Wu and Yang [29] to distinguish different configurations are important. Namely, if the Dirac string is not observable, then we ignore it physically and identify a monopole.  This unobservability critically depends on the magnetic charge of the monopole-like object and leads to the quantization of the charge in the case of the genuine Dirac monopole [37, 29]. In the present case, the magnetic charge is fixed by the formula of Berry's phase at $e_{M}=2\pi\hbar$.  Thus if the magnetic flux carried by the Dirac string satisfies the unobservability condition, we regard the monopole-like object as a physical monopole, and otherwise no physical monopole, namely, we regard only a combination of the monopole-like object and the accompanied string as a physical entity [98].

By keeping this criterion in mind, we start with an analysis of the adiabatic configuration with $\eta=\mu TB/\pi >1$ such as in Fig.B.6a.  The argument of Wu and Yang is to consider the singularity-free potentials in the upper and lower hemispheres

\begin{align}
{\cal A}_{\varphi +}
&=  \frac{e_{M}}{4\pi B\sin\theta}(1 - \cos\Theta(\theta)),\nonumber\\
{\cal A}_{\varphi -}
&=  \frac{e_{M}}{4\pi B\sin\theta}(- 1 - \cos\Theta(\theta)),
\end{align}

using the potential in (<ref>) with $e_{M}=2\pi\hbar$ by denoting the potential there as ${\cal A}_{\varphi +}$.  Note that ${\cal A}_{\varphi -}$ here is different from $\Omega_{-}$ in (<ref>) which is associated with the energy eigenvalue $w_{-}Hw_{-}$.
These two potentials are related by a gauge transformation

\begin{eqnarray}
{\cal A}_{\varphi -}={\cal A}_{\varphi +} - \frac{\partial\Lambda}{B\sin\theta\partial \varphi}
\end{eqnarray}

\begin{eqnarray}
\Lambda=\frac{e_{M}}{2\pi} \varphi.
\end{eqnarray}

The physical condition is

\begin{align}
\exp\left[ -\frac{i}{\hbar}\oint {\cal A}_{\varphi -}B\sin\theta d\varphi \right]&=\exp \left[-\frac{i}{\hbar}\oint{\cal A}_{\varphi +}B\sin\theta d\varphi +\frac{i}{\hbar}\oint \frac{\partial\Lambda}{B\sin\theta\partial \varphi}B\sin\theta d\varphi \right]\nonumber\\
&=\exp \left[-\frac{i}{\hbar}\oint{\cal A}_{\varphi +}B\sin\theta d\varphi \right]
\end{align}

which is in fact satisfied since the gauge term gives
$\exp[i e_{M}/\hbar]=\exp[2\pi i]=1$ and thus defines a monopole.
Note that the physical condition in the present context is that the Schrödinger wave function (<ref>) is single valued under the gauge transformation. It is confirmed that the present argument of gauge transformation is equivalent to the evaluation of the phase change induced by the Dirac string [29].
The fact that the physical condition is satisfied
shows that the magnetic charge

\begin{eqnarray}\label{magnetic charge}
e_{M}=2\pi \hbar
\end{eqnarray}

is properly quantized satisfying the Dirac quantization condition, although we have no analogue of an electric coupling in the present case unlike the original Dirac monopole [37].

In contrast, for the transitional domain $\eta=\mu TB/\pi =1$
such as in Fig.B.6b
we have two potentials from (<ref>)

\begin{align}
{\cal A}_{\varphi +}&=\frac{e_{M}}{4B\sin\theta}\left(1-\cos\frac{\theta}{2}\right)\nonumber\\
{\cal A}_{\varphi -}&=\frac{e_{M}}{4B\sin\theta}\left(-\cos\frac{\theta}{2}\right)
\end{align}

which are well-defined in the upper and lower hemispheres, respectively, and are related by the gauge transformation

\begin{eqnarray}
{\cal A}_{\varphi -}={\cal A}_{\varphi +} - \frac{\partial\Lambda}{B\sin\theta\partial \varphi}
\end{eqnarray}

\begin{eqnarray}
\Lambda=\frac{e_{M}}{4\pi} \varphi.
\end{eqnarray}

The physical condition

\begin{align}
\exp\left[-\frac{i}{\hbar}\oint {\cal A}_{\varphi -}B\sin\theta d\varphi \right]&=\exp\left[-\frac{i}{\hbar}\oint{\cal A}_{\varphi +}B\sin\theta d\varphi +\frac{i}{\hbar}\oint \frac{\partial\Lambda}{B\sin\theta\partial \varphi}B\sin\theta d\varphi \right]\nonumber\\
&=\exp\left[-\frac{i}{\hbar}\oint{\cal A}_{\varphi +}B\sin\theta d\varphi \right]
\end{align}

is not satisfied since the gauge transformation gives

\begin{eqnarray}\label{half monopole phase}
\exp[ie_{M}/2\hbar]=\exp[i\pi ]=-1.
\end{eqnarray}

We thus conclude that the half-monopole at the transitional value of the parameter $\eta=1$ with the magnetic charge $e_{M}/2$ cannot describe a physical monopole; it is physical  as a combination of  the monopole-like object, which generates the outgoing flux, and the accompanied Dirac string  [A half-monopole with a magnetic charge $e_{M}/2$ gives a non-trivial phase (<ref>) and thus the Dirac string is not unobservable. In fact this phase of $\exp[i\pi]$ is the same as the Aharonov-Bohm phase of an electron in the magnetic field generated by the superconducting current of the Cooper pair in the experiment by Tonomura [100].  In our criterion following the analysis of Wu and Yang [29], the Dirac string thus becomes a physical
observable just as the outgoing flux from the monopole-like object. ], although the Dirac string is actually defined only at $B=\pi/\mu T$. Topologically, it is thus the same as the dipole for $\eta<1$ in Fig.B.6c.

§.§  Longuet-Higgins phase change rule in an exactly solvable model

It is instructive to see the implications of  an exactly solvable model of Berry's phase discussed in this appendix  on the analysis of Longuet-Higgins phase change rule, which is the manifestation of topological properties of Berry's phase, namely, the phase change  when the particle circles the level crossing point.
The exact solution, (<ref>) with (<ref>),  gives (after adjusting the notation including $B\rightarrow r$)

\begin{eqnarray}
= \frac{\hbar }{2r\sin\frac{\pi}{2}} \left[1 - \cos\left( \frac{\pi}{2}-\alpha(\frac{\pi}{2}, \eta ) \right) \right] =\frac{\hbar }{2r} \left[1 - \sin \alpha( \frac{\pi}{2}, \eta ) \right]
\end{eqnarray}

if one sets $\theta=\frac{\pi}{2}$ to project the monopole to the two-dimensional subspace. The phase change factor is then given by

\begin{eqnarray}\label{phase change factor}
\exp \left\{ -\frac{i}{\hbar}\oint \hbar A^{(+)}_{\varphi} rd\varphi \right\}
=\exp\left\{-i\pi \left[1 - \sin \alpha(\frac{\pi}{2}, \eta) \right] \right\}.
\end{eqnarray}

It is important that the quantity $\alpha(\frac{\pi}{2}, \eta)$ controls the phase change factor instead of the naively expected $\theta$. The parameter $\alpha(\frac{\pi}{2}, \eta)$ is given in Fig.B.1, and one finds $\alpha(\frac{\pi}{2}, \infty) = 0$,   $\alpha(\frac{\pi}{2}, 1) = \pi/4$,  $\alpha(\frac{\pi}{2}, 0) = \pi/2$, respectively.

In the adiabatic limit $\eta\rightarrow \infty$, we have $\alpha(\frac{\pi}{2}, \infty) = 0$ and thus the conventional phase change rule with $(-1)$ is given by (<ref>). For $\eta=1$, which approximately corresponds to the topology change from a monopole to a dipole in the present model,
we have the Longuet-Higgins phase change factor (<ref>)

\begin{eqnarray}
\exp\{-i\pi[1 - 1/\sqrt{2}] \}.
\end{eqnarray}

In the nonadiabatic limit $\eta\rightarrow 0$, $\sin \alpha(\frac{\pi}{2}, \eta)\rightarrow 1$ and we have a unit phase change factor; thus the phase change rule is gone. This variation of the phase change factor, when one goes to off the precise adiabatic limit, shows how the topology is modified and eventually lost.

A physical picture of this topology change is seen in terms of the motion of the spin placed in the rotating magnetic field in the present section. This is illustrated in Fig.B.7 for the case with fixed  $T=2\pi/\omega$ but changing $B (=r)$.

Horizontal views of the magnetic field $\vec{B}$ rotating around the origin in the horizontal plane (i.e., $x-y$-plane, and thus the vertical direction is $z$-axis) with a fixed angular velocity $\omega$ and a free spin (denoted by a bold arrow)  placed at the origin. For a slow adiabatic rotation and strong magnetic field, the spin stays on the horizontal plane and follows the rotating magnetic field, as in Fig.B.7(a) and gives the Longuet-Higgins phase change factor (<ref>) at $(-1)$. When the magnetic field approaches the non-adiabatic configurations, which implies the weak field $\vec{B}$ in the present case (or rapid rotation), the spin starts to stand up effectively from the horizontal plane to the $z$-direction still following the rotating magnetic field with the angular velocity $\omega$ as in Fig.B.7(b); the effective polar angle becomes $\pi/2 - \alpha$.  For the angle $\alpha=\pi/2$, for which the spin becomes effectively perpendicular to the magnetic field, the Longuet-Higgins phase change factor (<ref>) becomes unity, and no phase change rule appears.

Figure B.7 shows that the spin is aligned to the direction of the magnetic field confined in the $x-y$ plane in the adiabatic limit of strong magnetic field $B$. But the spin   gradually “stands up” effectively to the direction of $z$ axis when the magnetic field in the $x-y$ plane becomes weaker with the fixed angular velocity $\dot{\varphi}=\omega$, although the spin follows the angular velocity $\omega$ of the magnetic field. The formula (<ref>) shows that the Longuet-Higgins phase change factor gradually deviates from the ideal value $(-1)$. This is a picture how the phase change rule specified by Berry's phase at $\theta=\pi/2$ is gradually lost  in the non-adiabatic domain in the present model.

§.§ Some remarks

To understand the magnetic flux generated by the Dirac monopole in Berry's phase explicitly, we used a mathematical description of the topology change in Berry's phase in terms of a geometrical movement of Dirac's monopole caused by the nonadiabatic variations of parameters in Berry's model. It is remarkable that the monopole itself formally remains in tact without being resolved even in the nonadiabatic domain, but moves away from the level crossing point as in Fig. B.5.
Traditionally, we are accustomed to understanding the topology change in terms of the winding and unwinding of some topological obstruction. The present geometrical description of topology change in terms of the moving monopole is a hitherto unknown mechanism. This new mechanism  partly arises from the fact that Berry's phase is not a simple monopole but rather a complex of the monopole and the level crossing point located at the origin of coordinates.  If one instead understands Berry's phase as a simple monopole, one would find a novel class of monopoles [101, 102].

We analyzed the monopole-like object and its topology appearing in Berry's phase by treating $\vec{B}$ as a given classical parameter.  If one adds other physical considerations, there appear some conditions on the parameters of the exactly solvable model (<ref>). For example, the two energy levels in (<ref>) cross at $\alpha(\theta;\eta)=\pi/2$, which is related to the topology change from a monopole to a dipole in an intricate way, as is seen in Fig.B.1.

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