# Superconductivity by Berry connection from many-body wave functions: revisit
to Andreev$-$Saint-James reflection and Josephson effect

Hiroyasu Koizumi  Division of Quantum Condensed Matter Physics, Center for
Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577,
Japan

###### Abstract

Although the standard theory of superconductivity based on the BCS theory is a
successful one, several experimental results indicate the necessity for a
fundamental revision.

We argue that the revision is on the origin of the phase variable for
superconductivity; this phase appears as a consequence of the electron-pairing
in the standard theory, but its origin is a Berry connection arising from
many-body wave functions. When this Berry connection is non-trivial, it gives
rise to a collective mode that generates supercurrent; this collective mode
creates number-changing operators for particles participating in this mode,
and these number-changing operators stabilize the superconducting state by
exploiting the Cooper instability.

In the new theory, the role of the electron-pairing is to stabilize the
nontrivial Berry connection; it is not the cause of superconductivity. In BCS
superconductors, however, the simultaneous appearance of the nontrivial Berry
connection and the electron-pairing occurs. Therefore, the electron-pairing
amplitude can be used as an order parameter for the superconducting state.

We revisit the Andreev$-$Saint-James reflection and the Josephson effect. They
are explained as consequence of the presence of the Berry connection.
Bogoliubov quasiparticles are replaced by the particle-number conserving
Bogoliubov excitations that describe the transfer of electrons between the
collective mod and single particle mode. There are two distinct cases for the
Josephson effect; one of them has the common Bogoliubov excitations for the
two superconductors in the junction, and the other does different Bogoliubov
excitations for different superconductors. The latter case is the one
considered in the standard theory; in this case, the Cooper pairs tunnel
through without Bogoliubov excitations, creating an impression that the
supercurrent is a flow of Cooper pairs; however, it does not explain the
observed ac Josephson effect under the experimental boundary condition. On the
other hand, the former case explains the ac Josephson effect under the
experimental boundary condition. In this case, it is clearly shown that the
supercurrent is a flow of electrons brought about by the non-trivial Berry
connection which provides an additional $U(1)$ gauge field.

##  I Introduction

The current standard theory of superconductivity is the one based on the BCS
theory [1]. In this theory, the order parameter of superconducting state is
the electron-pairing amplitude or the pair potential. This gives rise to an
energy gap for single-particle excitations and provides rigidity of the
superconducting state against perturbations. The standard theory has been
successfully applied to many superconducting materials; a notable point is
that it successfully explains the superconducting transition temperatures for
superconductors whose normal states are simple metals.

In 1986, high temperature superconductivity was found in ceramics [2]. Since
then, extensive efforts have been performed for the elucidation of the
mechanism of it. In spite of all the efforts, the widely-accepted theory has
not been obtained, yet. A notable point of the cuprate superconductivity is
that the superconducting transition temperature for the optimally-doped sample
is not the pairing energy gap formation temperature, but the stabilization
temperature for loop currents of the superconducting coherence length size
[3].

Efforts to elucidate the cuprate superconductivity have lead some people to
reexamine the theory of superconductivity from fundamental levels [4, 5].
Then, it is noticed that there are two solid experimental facts that point to
the need for fundamental revisions in the standard theory [4, 5].

One of them is the reversible phase transitions between normal and
superconducting phases in the $H$-$T$ plane (for Type I superconductors)[6, 7,
8, 9]. A series of work [10, 11, 12, 13] indicate that the superconducting-
normal state transition in the presence of a magnetic field occurs without
energy dissipation, and the state of the art calorimetry indicates that 99.99%
of the supercurrent stops without current carriers undergoing irreversible
collisions (see Appendix B of Ref. [6]).

However, such a transition is impossible in the standard theory; according to
the standard theory, paired electrons flow without dissipation but single
electrons flow with dissipation, thus, the supercurrent generated by the flow
of electron pairs in the magnetic field inevitably produces the Joule heat
during the superconducting to normal phase transition due to the existence of
a significant number of broken pairs that flow with dissipation.

The other is the mass of the electron in the London moment [14, 15]. Inside a
rotating superconductor, a magnetic field is created by the supercurrent
produced in the surface region. The London moment is the magnetic moment
produced by this supercurrent [16]. The London moment has been measured many
times using different materials, ranging from the conventional superconductor
[17, 18, 19, 20, 21] to the high Tc cuprates [22, 23] and heavy fermion
superconductors [24]. The results always indicate that the mass $m$ is the
free electron mass $m_{e}$ if the electron charge $q=-e$ is employed. However,
the standard theory predicts it to be an effective mass $m^{\ast}$,
contradicting the experimental results.

The resolution for the above two discrepancies is provided using a new theory
of superfluid that attributes the superfluidity to the appearance of the
nontrivial Berry connection from many-body wave functions [25, 15]. In this
theory, the supercurrent is explained as a topologically protected current
generated by the collective mode created by the nontrivial Berry connection.

A salient feature of the new theory is that it is formulated in the particle
number conserving way. In the standard theory, the particle number non-
conserving state vector is used, and the use of it gives rise to the phase
variable that explains the Meissner effect and supercurrent generation;
Bogoliubov quasiparticles are superpositions of electrons and holes, which can
only be meaningful in the particle number non-conserving formalism. On the
other hand, the particle number is conserved in the new theory; the Bogoliubov
quasiparticles are replaced by excitations that describe the transfer of
electrons between the collective mode and single-particle mode with keeping
the particle number fixed.

Since the Andreev$-$Saint-James (ASJ) reflection [26, 27] and the Josephson
effect are caused by the superconducting phase variable, it is important to
show that how they can be handled in the new theory. A purpose of the present
work is to provide explanations for them.

In the new theory, a Berry connection arises from interactions through wave
functions. Those electrons whose coordinates are the arguments of the same
wave function, actually, interact through the gauge field created by the wave
function they share [25]. This gauge field is calculated as a Berry
connection. It is a $U(1)$ gauge field, just like the $U(1)$ gauge field of
the electromagnetism. Thus, there are two $U(1)$ gauge fields in the system.

The presence of the Berry connection modifies Maxwell’s equations, and the
Lorentz interaction between charged particles and electromagnetic field as
will be discussed in the present work. Actually, the Lorentz interaction gives
rise to Aharonov-Bohm type effects [28] that cannot be described by the
Lorentz force. This modification of the Lorentz interaction affects the
magnetic energy part of the phenomenological Ginzburg-Landau theory [29] in
such a way that Abrikosov’s vortices [30] appear, naturally. Then, the
superconducting coherence length is regard as the core size of the loop
currents generated by the Berry connection that exist even without the applied
magnetic field.

The organization of the present work is as follows; in Section II, the
modification of Maxwell’s equations in the presence of the Berry connection
from many-body wave functions is explained. In Section III, the modification
of the magnetic energy part of the Ginzburg-Landau theory and its consequences
are discussed. Supercurrent flow in the new and standard theories are compared
in Section IV. In Section V, the Andreev$-$Saint-James reflection is
revisited. In Section VI, the Josephson effect is revised. Lastly, we conclude
the present work in Section VII.

##  II Modification of Maxwell’s equations in the presence of the Berry
connection from many-body wave functions

From the view point of the Feynman path integral formalism of quantum
mechanics [31], the wave function is the sum of contributions from all paths
each contributes an exponential whose phase is the classical action divided by
$\hbar$ for the path in question.

For the system of charged particles and electromagnetic field, the classical
action $S$ are composed of the following three terms,

$\displaystyle S=S_{1}+S_{2}+S_{3}$ (1)

where

$\displaystyle S_{1}=\sum_{i}{m\over 2}\int dt\ \dot{\bf r}_{i}^{2}$ (2)

is the action for the particles,

$\displaystyle S_{2}=-\int d^{3}rdt\ \left[\rho\phi^{\rm em}({\bf
r},t)-{1\over c}{\bf j}\cdot{\bf A}^{\rm em}({\bf r},t)\right]=-q\sum_{i}\int
dt\ \left[\phi^{\rm em}({\bf r}_{i},t)-{1\over c}\dot{\bf r}_{i}\cdot{\bf
A}^{\rm em}({\bf r}_{i},t)\right]$ (3)

is the action for the interaction between the field and particles, and

$\displaystyle S_{3}={1\over{8\pi}}\int d^{3}rdt\ \left[({\bf E}^{\rm
em})^{2}-({\bf B}^{\rm em})^{2}\right]={1\over{8\pi}}\int d^{3}rdt\
\left[\left(-\nabla\phi^{\rm em}-{1\over c}{{\partial{\bf A}^{\rm
em}}\over{\partial t}}\right)^{2}-\left(\nabla\times{\bf A}^{\rm
em}\right)^{2}\right]$ (4)

is the action for the field. Here $\phi^{\rm em}$ and ${\bf A}^{\rm em}$ are
the scalar and vector potentials for the electromagnetic field, respectively;
$\rho$ and ${\bf j}$ are the electric charge and current densities,
respectively; $q$ and $m$ are the charge and mass of the particle,
respectively.

In the following we consider the case where the electric field ${\bf E}^{\rm
em}$ is absent, and only the magnetic field ${\bf B}^{\rm em}$ is present. We
will consider the case where ${\bf E}^{\rm em}$ is present, later, to deal
with the ac Josephson effect.

In our previous work [25, 32, 15], it is shown that the Berry connection
arising from many-body wave functions modifies the momentum operator
$-i\hbar\nabla$ in the Schrödinger representation of quantum mechanics as
follows

$\displaystyle-i\hbar\nabla\longrightarrow-i\hbar\nabla+\hbar{\bf
A}_{\Phi}^{\rm MB}$ (5)

where ${\bf A}_{\Phi}^{\rm MB}$ is the Berry connection defined by

$\displaystyle{\bf A}^{\rm MB}_{\Phi}({\bf r},t)=-i\langle n_{\Phi}({\bf
r},t)|\nabla|n_{\Phi}({\bf r},t)\rangle$ (6)

and $|n_{\Phi}({\bf r})\rangle$ is the parameterized wave function with the
parameter ${\bf r}$ and integration coordinates ${\bf r}_{2},\cdots{\bf
r}_{N}$ given by

$\displaystyle\langle{\bf r}_{2},\cdots,{\bf r}_{N}|n_{\Phi}({\bf
r},t)\rangle={{\Phi({\bf r},{\bf r}_{2},\cdots,{\bf
r}_{N},t)}\over{|C_{\Phi}({\bf r},t)|^{{1\over 2}}}}$ (7)

with $|C_{\Phi}({\bf r},t)|$ being the normalization constant given by

$\displaystyle|C_{\Phi}({\bf r},t)|=\int d{\bf r}_{2}\cdots d{\bf
r}_{N}\Phi({\bf r},{\bf r}_{2},\cdots)\Phi^{\ast}({\bf r},{\bf r}_{2},\cdots)$
(8)

Inclusion of $\hbar{\bf A}_{\Phi}^{\rm MB}$ means the inclusion of the gauge
field that describes the interaction between particles through the wave
function they share.

As a consequence, the effective vector potential in the system becomes

$\displaystyle{\bf A}^{\rm eff}={\bf A}^{\rm em}+{\bf A}^{\rm fic},\quad{\bf
A}^{\rm fic}=\hbar{\bf A}_{\Phi}^{\rm MB}$ (9)

due to the presence of the “fictitious” vector potential ${\bf A}^{\rm fic}$.

Actually, ${\bf A}^{\rm fic}$ is given by

$\displaystyle{\bf A}^{\rm fic}={{\hbar c}\over{2e}}\nabla\chi$ (10)

where $\chi$ is an angular variable with period $2\pi$ [33]. This appears
through the spin-twisting itinerant motion of electrons, and the spin-twisting
is caused by the Rashba spin-orbit interaction. Although the energy gain by
the spin-twisting is very small, it gives rise to the non-trivial Berry
connection; the Berry connection creates the number changing operators that
make it possible to gain energy by exploiting the Cooper instability.

By including the Berry connection, $S_{2}$ becomes,

$\displaystyle S_{2}^{\prime}={1\over c}\int d^{3}rdt\ {\bf j}\cdot{\bf
A}^{\rm eff}({\bf r},t)$ (11)

where we only retained the term with vector potential assuming that electric
field is absent. This gives rise to “Lorentz force”. The Lorenz force from
${\bf A}^{\rm fic}$ is zero; however, ${\bf A}^{\rm fic}$ may affect the
dynamics of charged particles through the Aharonov-Bohm effect [28]. Thus, we
call this term, the Lorentz interaction term, instead of the Lorentz force
term.

Since the electromagnetic field energy is the energy stored in the space
through the Lorentz interaction derived from $S_{2}^{\prime}$, $S_{3}$ should
be modified as

$\displaystyle S_{3}^{\prime}=-{1\over{8\pi}}\int d^{3}rdt\ ({\bf B}^{\rm
eff})^{2}$ (12)

where

$\displaystyle{\bf B}^{\rm eff}={\bf B}^{\rm em}+{\bf B}^{\rm fic},\quad{\bf
B}^{\rm fic}=\nabla\times{\bf A}^{\rm fic}={{\hbar
c}\over{2e}}\nabla\times\nabla\chi$ (13)

${\bf B}^{\rm fic}$ may not be zero due to the fact that $\chi$ may be multi-
valued.

Using $S_{2}^{\prime}$ and $S_{3}^{\prime}$, two of the Maxwell’s equations
are modified as

$\displaystyle\nabla\cdot{\bf B}^{\rm eff}$ $\displaystyle=$ $\displaystyle 0$
(14) $\displaystyle\nabla\times{\bf B}^{\rm eff}$ $\displaystyle=$
$\displaystyle{{4\pi}\over c}{\bf j}$ (15)

The first one gives rise to a Dirac monopole as shown below. It is written as

$\displaystyle\nabla\cdot{\bf B}^{\rm em}=-\nabla\cdot(\nabla\times{\bf
A}^{\rm fic})$ (16)

When the both sides of the above equation are integrated for a closed region
with surface ${\rm Sf}$, we have

$\displaystyle\int_{\rm Sf}d{\bf S}\cdot{\bf B}^{\rm em}=-\int_{\rm Sf}d{\bf
S}\cdot(\nabla\times{\bf A}^{\rm fic})$ (17)

We split ${\rm Sf}$ into two surfaces ${\rm Sf}_{1}$ and ${\rm Sf}_{2}$ with
common boundary loop, $C=\partial({\rm Sf}_{1})=-\partial({\rm Sf}_{2})$.
Then, we have

$\displaystyle\int_{{\rm Sf}_{1}}d{\bf S}\cdot(\nabla\times{\bf A}^{\rm
fic})+\int_{{\rm Sf}_{2}}d{\bf S}\cdot(\nabla\times{\bf A}^{\rm
fic})=\int_{\partial({\rm Sf}_{1})}d{\bf r}\cdot{\bf A}^{\rm
fic}+\int_{\partial({\rm Sf}_{1})}d{\bf r}\cdot{\bf A}^{\rm fic}$ (18)

We examine the case in which singularities exist in ${\bf A}^{\rm fic}$. Let
us consider a closed surface $S$ with boundary $C=\partial S$, and ${\bf
A}^{\rm fic}$ has a singularity in $S$. Then, we have

$\displaystyle\int_{C}d{\bf r}\cdot{{\hbar
c}\over{2e}}\nabla\chi={{hc}\over{2e}}n$ (19)

where $n$ is an integer.

If we have $n=0$ for $\partial({\rm Sf}_{1})$ term, and $n=1$ for
$\partial({\rm Sf}_{2})$ term, we have

$\displaystyle\int_{\rm Sf}d{\bf S}\cdot{\bf B}^{\rm em}={{hc}\over{2e}}$ (20)

This shows that a monopole with magnetic charge ${{hc}\over{2e}}$ exists in
the region enclosed by ${\rm Sf}$. This corresponds to the monopole considered
by Dirac [34].

The second one in Eq. (15) is equal to

$\displaystyle\nabla\times{\bf B}^{\rm em}$ $\displaystyle=$
$\displaystyle{{4\pi}\over c}{\bf j}$ (21)

since $\nabla\times{\bf B}^{\rm fic}=0$ is satisfied as shown below.

It is well-known that $\nabla\chi$ in ${\bf A}^{\rm fic}$ can be decomposed as

$\displaystyle\nabla\chi=\nabla\chi_{0}+\nabla f,\quad\nabla^{2}\chi_{0}=0$
(22)

where $f$ is a single-valued, and $\chi_{0}$ may be multi-valued. Thus, we
have

$\displaystyle\nabla\times{\bf B}^{\rm
fic}=\nabla\times(\nabla\times\nabla\chi_{0})=\nabla(\nabla^{2}\chi_{0})-\nabla^{2}\nabla\chi_{0}=0$
(23)

As a consequence, Eq. (14) is reduced to the original one in Eq. (21)

##  III Modification of the magnetic field energy in the Ginzburg-Landau
theory due to the presence of the Berry connection from many-body wave
functions

The Ginzburg-Landau theory [29] is based on the London theory [16]. In the
London theory, the velocity field for electrons in superconductors is given by

$\displaystyle{\bf v}=-{q\over{mc}}\left({\bf A}^{\rm em}-{{c\hbar}\over
q}\nabla\chi^{\rm super}\right)$ (24)

where $\chi^{\rm super}$ is the superpotential assumed to exist in
superconductors.

The Ginzburg-Landau theory uses a free energy consists of the material part
and the magnetic field part. It assumes the presence of the effective wave
function of superconducting electrons $\Psi_{\rm GL}$ in the superconducting
phase.

Using $\Psi_{\rm GL}$, the material part of the free energy for a
superconductor is given by

$\displaystyle F_{\rm mat}=F_{\rm normal}+\int
d^{3}r{1\over{2m}}\left|\left({\hbar\over i}\nabla-{q\over c}{\bf A}^{\rm
em}\right)\Psi_{\rm GL}\right|^{2}+\int d^{3}r\left(\alpha|\Psi_{\rm
GL}|^{2}+{\beta\over 2}|\Psi_{\rm GL}|^{4}\right)$ (25)

where $\alpha<0$, $\beta>0$ are parameters.

We can express $\Psi_{\rm GL}$ using the supercurrent carrier density $n_{s}$
and the superpotential $\chi^{\rm super}$ as

$\displaystyle\Psi_{\rm GL}=n_{s}^{1/2}e^{i\chi^{\rm super}}$ (26)

Then, the kinetic term becomes

$\displaystyle\int d^{3}r{1\over{2m}}\left|\left({\hbar\over i}\nabla-{q\over
c}{\bf A}^{\rm em}\right)\Psi_{\rm GL}\right|^{2}=F_{k}+\int
d^{3}r{{\hbar^{2}(\nabla n_{s})^{2}}\over{8m\ n_{s}}}$ (27)

where the supercurrent kinetic energy is given by

$\displaystyle F_{k}=\int d^{3}r{m\over{2}}n_{s}{\bf v}^{2}=\int
d^{3}r{{q^{2}n_{s}}\over{2mc^{2}}}\left({\bf A}^{\rm eff}\right)^{2}$ (28)

Here, ${\bf A}^{\rm eff}$ in Eq. (9) is used by identifying

$\displaystyle{\bf A}^{\rm fic}=-{{c\hbar}\over q}\nabla\chi^{\rm
super}={{c\hbar}\over{2e}}\nabla\chi,$ (29)

assuming that $\chi^{\rm super}$ arises from the Berry connection.

In the standard theory, the Ginzburg-Landau theory is derived from the BCS
theory, yielding $q=-2e$ [35]. In this case the mass of the charge carriers
becomes $m=2m^{\ast}$, thus, ${m\over q}$ in the London moment becomes
$-{m^{\ast}\over e}$, disagrees with the experimental value $-{m_{e}\over e}$.
Thus, the accepted derivation of the Ginzburg-Landau theory from the standard
theory is incorrect.

We will use $q=-e$ here, and identify

$\displaystyle\nabla\chi^{\rm super}=\nabla\chi$ (30)

with $m=m_{e}$. This $m=m_{e}$ can be explained in the new theory [15].

Using $F_{k}$, the current density ${\bf j}$ is calculated as

$\displaystyle{\bf j}=-c{{\delta F_{k}}\over{\delta{{\bf A}^{\rm
em}}}}=-{{e^{2}n_{s}}\over{m_{e}c}}{\bf A}^{\rm eff}$ (31)

Since

$\displaystyle{\bf j}=-en_{s}{\bf v}$ (32)

Eq. (31) is equivalent to the London equation in Eq. (24). Actually, the above
relation should be the definition of $n_{s}$ through ${\bf j}$ and ${\bf v}$.

Using $q=-e$ and $m=m_{e}$, the velocity field ${\bf v}$ is given by

$\displaystyle{\bf v}={e\over{m_{e}c}}\left({\bf A}^{\rm em}+{{c\hbar}\over
e}\nabla\chi^{\rm super}\right)={e\over{m_{e}c}}\left({\bf A}^{\rm
em}+{{c\hbar}\over e}\nabla\chi\right)={e\over{m_{e}c}}{\bf A}^{\rm eff}$ (33)

Now, let su consider the magnetic field part. It is given by

$\displaystyle F_{m}=\int d^{3}r{1\over{8\pi}}\left({\bf B}^{\rm
eff}\right)^{2}$ (34)

using ${\bf B}^{\rm eff}$ in place of ${\bf B}^{\rm em}$. This is different
from the one employed by the original GL work.

The stationary condition of $F_{k}+F_{m}$ with respect to the variation of
${\bf A}^{\rm fic}$ yields,

$\displaystyle-{1\over c}{\bf j}+{1\over{4\pi}}\nabla\times{\bf B}^{\rm
eff}=-{1\over c}{\bf j}+{1\over{4\pi}}\nabla\times{\bf B}^{\rm em}=0$ (35)

where Eq. (23) is used. This is one of the Maxwell’s equations.

Using Eq. (31) and neglecting the spatial variation of $n_{s}$, we have

$\displaystyle\nabla\times{\bf
j}=-{{n_{s}e^{2}}\over{m_{e}}}\left[\nabla\times{\bf A}^{\rm
em}+\nabla\times{\bf A}^{\rm fic}\right]=-{{n_{s}e^{2}}\over{m_{e}}}\left[{\bf
B}^{\rm em}+\nabla\times{\bf A}^{\rm fic}\right]$ (36)

From Eq. (35), the following relation is obtained,

$\displaystyle\nabla\times{\bf
j}={{c}\over{4\pi}}\nabla\times(\nabla\times{\bf B}^{\rm
em})=-{{c}\over{4\pi}}\nabla^{2}{\bf B}^{\rm em}$ (37)

Here, $\nabla\cdot{\bf B}^{\rm em}=0$ is assumed.

Combining Eqs. (36) and (37), the following is obtained,

$\displaystyle\nabla^{2}{\bf B}^{\rm em}-{{1}\over{\lambda^{2}}}{\bf B}^{\rm
em}={{1}\over{\lambda^{2}}}\nabla\times{\bf A}^{\rm
fic}={{1}\over{\lambda^{2}}}{\bf B}^{\rm fic}$ (38)

where $\lambda$ is the London penetration depth

$\displaystyle\lambda=\sqrt{{m_{e}c}\over{4\pi n_{s}e^{2}}}$ (39)

Now we consider the loop current formation by following Abrikosov [30]. The
characteristic length scale for the spatial variation of $n_{s}$ in the
Ginzburg-Landau theory is

$\displaystyle\xi_{\rm GL}=\sqrt{\hbar^{2}\over{2m_{e}|\alpha|}}$ (40)

Abrikosov argued that if $\lambda\gg\xi_{\rm GL}$ is satisfied and the
singularity of $\nabla\chi$ is along the $z$-axis, Eq. (38) can be
approximated as

$\displaystyle\nabla^{2}{\bf B}^{\rm em}-{{1}\over{\lambda^{2}}}{\bf B}^{\rm
em}={{1}\over{\lambda^{2}}}\Phi^{\rm fic}{\bf e}_{z}\delta^{(2)}({\bf r})$
(41)

in the region away from the core, where $\Phi^{\rm fic}$ is given by

$\displaystyle\Phi^{\rm fic}=\int_{S}{\bf B}^{\rm fic}\cdot d{\bf
S}=\int_{C}{\bf A}^{\rm fic}\cdot d{\bf
r}={{c\hbar}\over{2e}}\int_{C}\nabla\chi\cdot d{\bf r}$ (42)

and $\delta^{(2)}({\bf r})$ is the delta function in two-dimension with
singularities along the $z$-axis.

The solution is known to be ${\bf B}^{\rm em}={B}^{\rm em}(\rho){\bf e}_{z}$,
where ${B}^{\rm em}(\rho)$ is given by

$\displaystyle{B}^{\rm em}(\rho)=-2\pi\Phi^{\rm
fic}\lambda^{-2}K_{0}(\rho/\lambda)$ (43)

Here $K_{0}$ is the modified Bessel function of the 2nd kind, and $\rho$ is
the distance from the $z$-axis.

In the BCS theory, a different coherence length

$\displaystyle\xi_{\rm BCS}={{\hbar v_{\rm Fermi}}\over{\pi\Delta}}.$ (44)

is defined, where $v_{\rm Fermi}$ is the velocity of the electron at the Fermi
energy.

It is known that $\xi_{\rm GL}$ and $\xi_{\rm BCS}$ are similar in size at
very low temperatures for BCS superconductors. However, $\xi_{\rm BCS}$ is
regarded as the size of the Cooper pair in contrast to $\xi_{\rm GL}$, which
is the size of the core of the loop current.

In our previous work, it has been argued that the Cooper pair formation is
accompanied by the loop current formation that encircles a section of the
Fermi surface. This loop current is stabilized by a Rashba spin-orbit
interaction, and gives rise to ${\bf A}^{\rm fic}$ given in Eq. (10) [33, 32].
We can relate $\xi_{\rm BCS}$ to the core size of this loop current.

First, we associate $\xi_{\rm BCS}$ to the wave number $q_{c}$ that has an
excitation equal to the gap energy $\Delta$,

$\displaystyle\Delta=\hbar q_{c}v_{\rm BCS}$ (45)

If we identify

$\displaystyle\xi_{\rm BCS}={1\over{\pi q_{c}}}$ (46)

we obtain Eq. (44). Thus, $\xi_{\rm BCS}$ is an estimate of the size of the
loop current whose excitation energy is equal to the gap energy.

The presence of the loop current by ${\bf A}^{\rm fic}$ is plausible from
another experimental fact. It explains the reversible phase transitions in a
magnetic field [9]; the reversible energy transfer between the kinetic energy
of the supercurrent and the magnetic field energy is explained as due to the
abrupt change of loop currents associated with the change of the winding
numbers for $\chi$ in ${\bf A}^{\rm fic}$. Let us see this point, below.

By taking into account only the change of ${\bf A}^{\rm fic}$ during the phase
transition in the time interval $\Delta t$, the change of the kinetic energy
is given by

$\displaystyle\Delta F_{k}$ $\displaystyle=$ $\displaystyle\int
d^{3}r{{e^{2}n_{s}}\over{m_{e}c^{2}}}{\bf A}^{\rm eff}\cdot\int_{t}^{t+\Delta
t}\partial_{t}{\bf A}^{\rm fic}dt$ (47) $\displaystyle=$
$\displaystyle-{1\over c}\int d^{3}r{\bf j}\cdot\int_{t}^{t+\Delta
t}\partial_{t}{\bf A}^{\rm fic}dt$

and the change of the magnetic field energy is given by

$\displaystyle\Delta F_{m}$ $\displaystyle=$ $\displaystyle\int
d^{3}r{1\over{4\pi}}{\bf B}^{\rm eff}\cdot\int_{t}^{t+\Delta
t}\partial_{t}{\bf B}^{\rm fic}dt$ (48) $\displaystyle=$ $\displaystyle\int
d^{3}r{1\over{4\pi}}\nabla\times{\bf B}^{\rm eff}\cdot\int_{t}^{t+\Delta
t}\partial_{t}{\bf A}^{\rm fic}dt$ $\displaystyle=$ $\displaystyle{1\over
c}\int d^{3}r{\bf j}\cdot\int_{t}^{t+\Delta t}\partial_{t}{\bf A}^{\rm fic}dt$

Thus, the energy conservation, $\Delta F_{m}+\Delta F_{k}=0$, is satisfied
[9].

The situation considered above may be smoothly connected to the ${\bf B}^{\rm
em}=0$ case. If this is the case, similar loop currents from ${\bf A}^{\rm
fic}$ with a loop current arrangement with zero macroscopic current should
exit even for the ${\bf B}^{\rm em}=0$ case. Then, the phase transition
between the superconducting and normal phases are accompanied by creation and
annihilation of such loop currents as in the ${\bf B}^{\rm em}\neq 0$ case.

Let us consider the energy balance at the transition point for the ${\bf
B}^{\rm em}=0$ case mentioned above. In this case ${\bf A}^{\rm em}$ can be
taken as a pure gauge, and we assume that it cancels ${\bf A}^{\rm fic}$
except in the core region of the size $\xi$.

For simplicity, we assume that singularities of $\chi$ form vortices along the
$z$-direction. Let us pick up one of them, and take the $z$-axis along it.
Then, we have

$\displaystyle n_{s}=n_{0}e^{-{{2\rho}\over{\xi}}}$ (49)

near the vortex, where $n_{0}$ is a constant.

The sum of the energies from the spatial variation of $n_{s}$ and the term
linear to $n_{s}$ in Eqs. (25) and (27) given by

$\displaystyle\int d^{3}r{{\hbar^{2}(\nabla n_{s})^{2}}\over{8m_{e}\
n_{s}}}+\int d^{3}r\alpha|\Psi_{\rm GL}|^{2}=\int d^{3}r{{\hbar^{2}(\nabla
n_{s})^{2}}\over{8m_{e}\ n_{s}}}+\int d^{3}r\alpha n_{s}$ (50)

becomes zero when $\xi$ satisfies the relation

$\displaystyle\xi=\xi_{\rm GL}$ (51)

If $\xi>\xi_{\rm GL}$, the above sum becomes negative, indicating that the
vortex formation may be possible if the energy gain from it is more than the
energy deficit due to the core formation. In other words, if the energy
deficit by the creation of vortex cores is compensated by the energy gain
obtained by the generation of the non-trivial Berry connection, the loop
currents generation by the Berry connection will be realized.

In the new theory, the energy gain by the electron-pair formation requires the
presence of the number changing operators that arise from the non-trivial
Berry connection ${\bf A}^{\rm fic}$. Thus, $\xi_{\rm BCS}$ given in Eq. (44)
can be regarded as the estimate for the minimum size of the vortex core for
the loop current generated by ${\bf A}^{\rm fic}$ that exit even for the ${\bf
B}^{\rm em}=0$ case, which corresponds to $\xi_{\rm GL}$.

##  IV Supercurrent flow

Let us compare the supercurrent in the new theory and that in the standard
theory.

In the standard theory, the supercurrent is a flow of Cooper pairs. In the
original BCS theory, the normal metallic state is assumed to be well-described
by the free electrons with the effective mass $m^{\ast}$. Then, the electron
field operators are given by

$\displaystyle\hat{\Psi}_{\sigma}({\bf r})={1\over\sqrt{V}}\sum_{\bf
k}e^{i{\bf k}\cdot{\bf r}}c_{{\bf k}\sigma}$ (52)

The electron-paring amplitude is expressed as

$\displaystyle\langle\hat{\Psi}_{\downarrow}({\bf
r}_{2})\hat{\Psi}_{\uparrow}({\bf r}_{1})\rangle=\sum_{\bf q}e^{i{\bf
q}\cdot{\bf R}}\Delta_{\bf q}({\bf r})$ (53)

where

$\displaystyle\Delta_{\bf q}({\bf r})=\sum_{{\bf p}}^{\prime}e^{i{\bf
p}\cdot{\bf r}}\langle c_{-{\bf p}+{{\bf q}\over 2}\downarrow}c_{{\bf p}+{{\bf
q}\over 2}\uparrow}\rangle$ (54)

is the electron-pairing amplitude with momentum ${\bf q}$, ${\bf r}={\bf
r}_{1}-{\bf r}_{2}$ is the relative position vector between the pairing
electrons, ${\bf R}={1\over 2}({\bf r}_{1}+{\bf r}_{2})$ is the center-of-mass
position vector of the pairing electron, and the sum is over ${\bf p}$ near
the Fermi level where that the attractive interaction between electrons
exists.

Using $\Delta_{\bf q}({\bf r})$ the macroscopic wave function in the standard
theory is given by

$\displaystyle\Psi_{\rm GL}({\bf R})=gC\sum_{{\bf q}}e^{i{\bf q}\cdot{\bf
R}}\Delta_{\bf q}(0)$ (55)

where $g$ is the parameter for the attractive electron-electron interaction,
and $C$ is a constant.

Considering the change of the phase by the gauge transformation for $\langle
c_{-{\bf p}+{{\bf q}\over 2}\downarrow}c_{{\bf p}+{{\bf q}\over
2}\uparrow}\rangle$, $\nabla_{\bf R}\Psi_{\rm GL}({\bf R})$ should be modified
in the presence of the magnetic field as

$\displaystyle\nabla_{\bf R}\Psi_{\rm GL}({\bf R})\rightarrow\left[\nabla_{\bf
R}+{{2ei}\over{\hbar}}{\bf A}^{\rm em}\right]\Psi_{\rm GL}({\bf R})$ (56)

This yields the kinetic energy term in Eq. (27). Since the kinetic mass for
the electrons in the Cooper pair is the effective mass $m^{\ast}$, $m$ in it
will be $m=2m^{\ast}$, where $2$ comes from the electron-pair.

As mentioned before, this $m=2m^{\ast}$ contradicts the experimental result,
which indicates $m=2m_{e}$ when $q=-2e$ is adopted, where $m_{e}$ is free
electron mass.

Now we consider the same problem using the new theory. This problem is dealt
with our recent publication [15], but for the convenience for the later
discussion, we reproduce it succinctly in the following.

We allow the coordinate dependent functions that are different from plane
waves as the basis for the electron field operators in Eq. (52). We use the
label $n$ in place of the wave number ${\bf k}$. Then, the field operators
become

$\displaystyle\hat{\Psi}_{\uparrow}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{n}e^{{i\over 2}\hat{\chi}({\bf
r})}\left(\gamma_{{n}\uparrow}u_{n}({\bf
r})-\gamma^{\dagger}_{{n}\downarrow}v^{\ast}_{n}({\bf r})\right)$
$\displaystyle\hat{\Psi}_{\downarrow}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{n}e^{{i\over 2}\hat{\chi}({\bf
r})}\left(\gamma_{{n}\downarrow}u_{n}({\bf
r})+\gamma^{\dagger}_{{n}\uparrow}v^{\ast}_{n}({\bf r})\right)$ (57)

The particle number conserving Bogoliubov operators satisfy

$\displaystyle\gamma_{n\sigma}|{\rm Gnd}(N)\rangle=0$ (58)

where $N$ is the total number of particles.

The ground state satisfies on the operation of $e^{{i\over 2}\hat{\chi}({\bf
r})}$ as

$\displaystyle e^{{i\over 2}\hat{\chi}({\bf r})}|{\rm
Gnd}(N)\rangle=e^{{i\over 2}{\chi}({\bf r})}|{\rm Gnd}(N-1)\rangle$ (59)

The operator $e^{{i\over 2}\hat{\chi}({\bf r})}$ is the number changing
operator that removes one electron from the collective modes at the position
${\bf r}$. This also means that the Bogoliubov operators in Eq. (57) conserve
the particle number. We call them “ the particle-number conserving Bogoliubov
operators”. They describe the transfer of electrons between the collective
mode described by $\chi$ and single-particle mode (this point may be seen in
Eq. (73) in the next section).

The phase factor $e^{{i\over 2}{\chi}({\bf r})}$ arises due to the presence of
the Berry connection. The connection of geometry is embedded in the
superconducting state as

$\displaystyle\langle e^{{i\over 2}\hat{\chi}({\bf r}_{2})}e^{-{i\over
2}\hat{\chi}({\bf r}_{1})}\rangle=e^{{i\over 2}\int_{{\bf r}_{1}}^{{\bf
r}_{2}}\nabla\chi({\bf r})\cdot d{\bf r}}$ (60)

which describes the phase factor appearing when an electron is added at ${\bf
r}_{1}$ in the collective mode, and removed at ${\bf r}_{2}$ from the
collective mode.

Now, let us consider the electronic Hamiltonian expressed as

$\displaystyle H=\sum_{\sigma}\int d^{3}r\hat{\Psi}^{\dagger}_{\sigma}({\bf
r})h({\bf r})\hat{\Psi}_{\sigma}({\bf r})-{1\over
2}\sum_{\sigma,\sigma^{\prime}}\int d^{3}rd^{3}r^{\prime}V_{\rm eff}({\bf
r},{\bf r}^{\prime})\hat{\Psi}^{\dagger}_{\sigma}({\bf
r})\hat{\Psi}^{\dagger}_{\sigma^{\prime}}({\bf
r}^{\prime})\hat{\Psi}_{\sigma^{\prime}}({\bf
r}^{\prime})\hat{\Psi}_{\sigma}({\bf r})$ (61)

where $-V_{\rm eff}$ is the effective interaction between electrons, and
$h({\bf r})$ is the single-particle Hamiltonian given by

$\displaystyle h({\bf r})={1\over{2m_{e}}}\left({\hbar\over i}\nabla+{e\over
c}{\bf A}^{\rm em}\right)^{2}+U({\bf r})-\mu$ (62)

with $U({\bf r})$ being a potential energy and $\mu$ chemical potential.

We perform the mean field approximation

$\displaystyle H^{\rm MF}$ $\displaystyle=$ $\displaystyle\sum_{\sigma}\int
d^{3}r\hat{\Psi}^{\dagger}_{\sigma}({\bf r})h({\bf r})\hat{\Psi}_{\sigma}({\bf
r})+\int d^{3}rd^{3}r^{\prime}\left[\Delta({\bf r},{\bf
r}^{\prime})\hat{\Psi}^{\dagger}_{\uparrow}({\bf
r})\hat{\Psi}^{\dagger}_{\downarrow}({\bf r}^{\prime})e^{{i\over
2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf r}^{\prime}))}+{\rm H.c.}\right]$
$\displaystyle+$ $\displaystyle\int d^{3}rd^{3}r^{\prime}{{|\Delta({\bf
r},{\bf r}^{\prime})|^{2}}\over{V_{\rm eff}({\bf r},{\bf r}^{\prime})}}$

where the gap function $\Delta({\bf r},{\bf r}^{\prime})$ is defined as

$\displaystyle\Delta({\bf r},{\bf r}^{\prime})=V_{\rm eff}({\bf r},{\bf
r}^{\prime})\langle e^{-{i\over 2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf
r}^{\prime}))}\hat{\Psi}_{\uparrow}({\bf r})\hat{\Psi}_{\downarrow}({\bf
r^{\prime}})\rangle$ (64)

Due to the factor $e^{-{i\over 2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf
r}^{\prime}))}$ that increase the number of electrons by two, the expectation
value is calculated using the particle number fixed state in contrast to the
standard theory.

Using the relation in Eq. (59) and requiring $H^{\rm MF}$ to become

$\displaystyle H^{\rm
MF}=\sum_{n,\sigma}\epsilon_{n}\gamma_{n\sigma}^{\dagger}\gamma_{n\sigma}+E_{\rm
const}$ (65)

where $E_{\rm const}$ is a constant, the above are cast into the following
system of equations

$\displaystyle\epsilon_{n}u_{n}({\bf r})$ $\displaystyle=$
$\displaystyle\bar{h}({\bf r})u_{n}({\bf r})+\int d^{3}r^{\prime}\Delta({\bf
r},{\bf r}^{\prime})v_{n}({\bf r}^{\prime})$
$\displaystyle\epsilon_{n}v_{n}({\bf r})$ $\displaystyle=$
$\displaystyle-\bar{h}^{\ast}({\bf r})v_{n}({\bf r})+\int
d^{3}r^{\prime}\Delta^{\ast}({\bf r},{\bf r}^{\prime})u_{n}({\bf r}^{\prime})$
(66)

where

$\displaystyle\bar{h}({\bf r})={1\over{2m_{e}}}\left({\hbar\over
i}\nabla+{e\over c}{\bf A}^{\rm em}+{\hbar\over 2}\nabla\chi\right)^{2}+U({\bf
r})-\mu$ (67)

and

$\displaystyle\Delta({\bf r},{\bf r}^{\prime})=V_{\rm eff}({\bf r},{\bf
r}^{\prime})\sum_{n}\left[u_{n}({\bf r})v^{\ast}_{n}({\bf
r}^{\prime})(1-f(\epsilon_{n}))-u_{n}({\bf r}^{\prime})v^{\ast}_{n}({\bf
r})f(\epsilon_{n})\right]$ (68)

with $f(\epsilon_{n})$ being the Fermi function at temperature T ($k_{B}$ is
the Boltzmann constant) given by

$\displaystyle f(\epsilon_{n})=(e^{{\epsilon_{n}}\over{k_{B}T}}+1)^{-1}$ (69)

They are Bogoliubov-de Gennes equations [36] using the particle number
conserving Bogoliubov operators [25].

Note that the gauge potential in the single particle Hamiltonian $\bar{h}({\bf
r})$ is the effective one,

$\displaystyle{\bf A}^{\rm eff}={\bf A}^{\rm em}+{{\hbar
c}\over{2e}}\nabla\chi$ (70)

If we solve the system of equations composed of Eqs. (66), (67), and (68),
with the condition ${\bf A}^{\rm em}+{{\hbar c}\over{2e}}\nabla\chi=0$, we
obtain the currentless solutions for $u_{n},v_{n}$, which we denote as
$\tilde{u}_{n},\tilde{v}_{n}$.

We may construct the solution using $\tilde{u}_{n},\tilde{v}_{n}$ as

$\displaystyle u_{n}({\bf r})=\tilde{u}_{n}({\bf r})e^{-{i\over 2}{\chi}({\bf
r})},\quad v_{n}({\bf r})=\tilde{v}_{n}({\bf r})e^{{i\over 2}{\chi}({\bf r})}$
(71)

with suitably chosen $\nabla\chi$. Actually, $\nabla\chi$ is obtained by the
requirement of the conservation of local charge, and single-valuedness of the
wave function with respect to electron coordinates [32].

When the solutions are obtained by the above method, we have the velocity
field given in Eq. (33) since the velocity field from
$\tilde{u}_{n},\tilde{v}_{n}$ is zero. The supercurrent is due to this
velocity field that contains non-trivial $\nabla\chi$. In this case, $m$ is
equal to $m_{e}$ when $q=-e$ is employed, which agrees with the experiment.

##  V Andreev$-$Saint-James reflection

In the Andreev$-$Saint-James reflection, the phase factor of the pair
potential plays the central role. We examine how this phase factor arises from
the Berry connection of the new theory.

Let us consider the hopping Hamiltonian between the normal state and the
superconducting state equipped with the Berry connection,

$\displaystyle H_{\rm Super-Metal}=-t\sum_{{\bf
k},\sigma}\left(c^{\dagger}_{i\sigma}M_{{\bf k}\sigma}+M^{\dagger}_{{\bf
k}\sigma}c_{i\sigma}\right)$ (72)

where $M_{{\bf k}\sigma}^{\dagger}$ and $M_{{\bf k}\sigma}$ are creation and
annihilation operators for the electrons in the normal metal part,
respectively. $c_{i\sigma}^{\dagger}$ and $c_{i\sigma}$ are creation and
annihilation operators for the electron in the superconductor part at the
interface site $i$, respectively.

Using the number-changing operators and the Bogoliubov operators, annihilation
and creation operators for the electrons in the superconducting state are
given by

$\displaystyle c_{i\sigma}$ $\displaystyle=$
$\displaystyle\sum_{n}[u^{n}_{i}\gamma_{n\sigma}-\sigma(v^{n}_{i})^{\ast}\gamma_{n-\sigma}^{\dagger}]e^{{i\over
2}\hat{\chi}_{i}}$ $\displaystyle c^{\dagger}_{i\sigma}$ $\displaystyle=$
$\displaystyle\sum_{n}[(u^{n}_{i\sigma})^{\ast}\gamma^{\dagger}_{n\sigma}-\sigma
v^{n}_{i}\gamma_{n-\sigma}]e^{-{i\over 2}\hat{\chi}_{i}}$ (73)

where $\sigma=1$ for up-spin state, and $\sigma=-1$ for down-spin state.

Cooper-pair annihilation and creation operators are given by

$\displaystyle c_{i\uparrow}c_{i\downarrow}$ $\displaystyle=$
$\displaystyle\sum_{n,n^{\prime}}[u^{n}_{i}\gamma_{n\uparrow}-(v^{n}_{i})^{\ast}\gamma_{n\downarrow}^{\dagger}]e^{{i\over
2}\hat{\chi}_{i}}[u^{n^{\prime}}_{i}\gamma_{n^{\prime}\downarrow}+(v^{n^{\prime}}_{i})^{\ast}\gamma_{n^{\prime}\uparrow}^{\dagger}]e^{{i\over
2}\hat{\chi}_{i}}$ $\displaystyle\approx$
$\displaystyle\sum_{n}u^{n}_{i}(v^{n}_{i})^{\ast}\left[\langle\gamma_{n\uparrow}\gamma_{n\uparrow}^{\dagger}\rangle-\langle\gamma_{n\downarrow}^{\dagger}\gamma_{n\downarrow}\rangle\right]e^{{i}\hat{\chi}_{i}}$
$\displaystyle=$ $\displaystyle
e^{{i}\hat{\chi}_{i}}\sum_{n}u^{n}_{i}(v^{n}_{i})^{\ast}[1-2f(\epsilon_{n})]$
$\displaystyle c^{\dagger}_{i\downarrow}c^{\dagger}_{i\uparrow}$
$\displaystyle\approx$ $\displaystyle
e^{-{i}\hat{\chi}_{i}}\sum_{n}(u^{n}_{i})^{\ast}v^{n}_{i}[1-2f(\epsilon_{n})]$
(74)

where the products of Bogoliubov operators are replaced by their expectation
values. The above shows that Cooper-pairs can move without Bogoliubov
excitations. This feature makes an impression that supercurrent is a flow of
Cooper-pairs in the standard theory.

Now, let us derive the term for the Andreev reflection by applying the second
order perturbation theory using $H_{\rm Super-Metal}$,

$\displaystyle\langle H_{\rm Super-Metal}{1\over{E_{0}-H_{0}}}H_{\rm Super-
Metal}\rangle_{\rm Bog}$ (75) $\displaystyle\approx$ $\displaystyle
t^{2}\left\langle\sum_{{\bf k},{\bf
k}^{\prime},\sigma,\sigma^{\prime}}\left[c^{\dagger}_{i\sigma}M_{{\bf
k}\sigma}{1\over{E_{0}-H_{0}}}c^{\dagger}_{i\sigma^{\prime}}M_{{\bf
k}^{\prime}\sigma^{\prime}}+M^{\dagger}_{{\bf
k}\sigma}c_{i\sigma}{1\over{E_{0}-H_{0}}}M^{\dagger}_{{\bf
k}^{\prime},\sigma^{\prime}}c_{i\sigma^{\prime}}\right]\right\rangle_{\rm
Bog}$ $\displaystyle\approx$ $\displaystyle-\sum_{{\bf k},{\bf
k}^{\prime},n}{{2t^{2}}\over{\epsilon_{n}}}\left[(u^{n}_{i})^{\ast}v^{n}_{i}M_{{\bf
k}\uparrow}M_{{\bf
k}^{\prime}\downarrow}e^{-{i}\hat{\chi}_{i}}+(v^{n}_{i})^{\ast}u^{n}_{i}M^{\dagger}_{{\bf
k}^{\prime}\downarrow}M^{\dagger}_{{\bf
k}\uparrow}e^{{i}\hat{\chi}_{i}}\right]$

where $\langle\cdots\rangle_{\rm Bog}$ denotes that the expectation value is
calculated for the products of Bogoliubov operators.

The above effective Hamiltonian indicates that when an electron in the normal
state is reflected back as a hole, the phase factor $e^{-{i}{\chi}_{i}}$ is
attached, and a hole in the normal state is reflected back as an electron, the
phase factor $e^{{i}{\chi}_{i}}$ is attached; they are salient features of the
Andreev reflection. In the following, we outline how the Andreev equations are
derived from the particle-number conserving Bogoliubov-de Gennes method by
following Ref. [37].

From Eq. (66) using (71), the particle-number conserving Bogolibov-de Gennes
equations are expressed as

$\displaystyle\epsilon_{n}\tilde{u}_{n}({\bf r})$ $\displaystyle=$
$\displaystyle{h}({\bf r})\tilde{u}_{n}({\bf r})+\int
d^{3}r^{\prime}e^{{i\over 2}(\chi({\bf r})+\chi({\bf r}^{\prime}))}\Delta({\bf
r},{\bf r}^{\prime})\tilde{v}_{n}({\bf r}^{\prime})$
$\displaystyle\epsilon_{n}\tilde{v}_{n}({\bf r})$ $\displaystyle=$
$\displaystyle-{h}^{\ast}({\bf r})\tilde{v}_{n}({\bf r})+\int
d^{3}r^{\prime}e^{-{i\over 2}(\chi({\bf r})+\chi({\bf
r}^{\prime}))}\Delta^{\ast}({\bf r},{\bf r}^{\prime})\tilde{u}_{n}({\bf
r}^{\prime})$ (76)

with $h({\bf r})$ as the single-particle Hamiltonian instead of $\bar{h}({\bf
r})$. This single-particle Hamiltonian can be used throughout the system. It
is important to have the same single-particle Hamiltonian in both the normal
metallic and superconducting parts.

Then, we may regard

$\displaystyle\bar{\Delta}\left({{{\bf r}+{\bf r}^{\prime}}\over 2},{\bf
r}-{\bf r}^{\prime}\right)=e^{{i\over 2}(\chi({\bf r})+\chi({\bf
r}^{\prime}))}\Delta({\bf r},{\bf r}^{\prime})$ (77)

as the pair potential for the Andreev reflection. Due to the phase factor
$e^{{i\over 2}(\chi({\bf r})+\chi({\bf r}^{\prime}))}$ in $\bar{\Delta}$, it
describes the Andreev reflection in which an electron is reflected back as a
hole with the phase factor $e^{-{i}{\chi}_{i}}$, and a hole is reflected back
as an electron with the phase factor $e^{{i}{\chi}_{i}}$, where $i$ is the
position of the reflection.

By taking the Fourier transformation with respect to the relative coordinate
${\bf s}={\bf r}-{\bf r}^{\prime}$, and denoting the center-of-mass coordinate
as ${\bf r}$, we have

$\displaystyle\bar{\Delta}({\bf r},{\bf k})=\int d^{3}s\bar{\Delta}({\bf
r},{\bf s})e^{-i{\bf k}\cdot{\bf s}}$ (78)

In the weak coupling case, only the wave vectors very close to the Fermi
surface are important. Then, ${\bf k}$ dependence of $\bar{\Delta}({\bf
r},{\bf k})$ comes only from the direction of the Fermi wave vector ${\bf
k}_{F}$, or the unit vector on the Fermi surface ${\bf e}_{{\bf k}_{F}}$.
Thus, we may write

$\displaystyle\bar{\Delta}_{{\bf e}_{{\bf k}_{F}}}({\bf
r})\approx\bar{\Delta}({\bf r},{\bf k}_{F})$ (79)

We also separate the fast Fermi wave vector oscillation from
$\tilde{u}_{n}({\bf r}),\tilde{v}_{n}({\bf r})$ as

$\displaystyle\bar{u}_{n}({\bf r})=e^{-i{\bf k}_{F}\cdot{\bf
r}}\tilde{u}_{n}({\bf r}),\quad\bar{v}_{n}({\bf r})=e^{-i{\bf k}_{F}\cdot{\bf
r}}\tilde{v}_{n}({\bf r})$ (80)

for convenience, since we consider the cas where $\xi_{\rm BCS}\gg k_{F}^{-1}$
is satisfied.

As a result, Eq. (76) becomes the following Andreev equations

$\displaystyle\epsilon_{n}\bar{u}_{n}({\bf r})$ $\displaystyle=$
$\displaystyle-i{\hbar^{2}\over m_{e}}{\bf k}_{F}\cdot\nabla\bar{u}_{n}({\bf
r})+\bar{\Delta}_{{\bf e}_{{\bf k}_{F}}}({\bf r})\bar{v}_{n}({\bf r})$
$\displaystyle\epsilon_{n}\bar{v}_{n}({\bf r})$ $\displaystyle=$
$\displaystyle i{\hbar^{2}\over m_{e}}{\bf k}_{F}\cdot\nabla\bar{v}_{n}({\bf
r})+\bar{\Delta}^{\ast}_{{\bf e}_{{\bf k}_{F}}}({\bf r})\bar{u}_{n}({\bf r})$
(81)

The Andreev reflection is calculated using the above equations. The effect of
the phase factor $e^{{i}{\chi}_{i}}$ is included in the phase of
$\bar{\Delta}_{{\bf e}_{{\bf k}_{F}}}({\bf r})$ at the reflection point.

##  VI Josephson effect

The Josephson effect is also the effect which arises from the phase of the
pair-potential in the standard theory [38]. We have derived the Josephson
relation by the new theory, previously [5, 39, 33, 32]; however, we reproduce
it here for the completeness of the discussion.

Note that there are two distinct cases, one is the case where the Bogoliubov
excitations are common in the two superconductors, and the other where two
separate Bogoliubov excitations exist. In the standard theory, only the latter
case is considered; however, for the explanation of the ac Josephson effect,
the former case is relevant as is indicated as is shown below. We would like
to emphasize that in the new theory, supercurrent is a flow of electrons by a
collective mode arising from the Berry connection, not a flow of Cooper-pairs.

Let us consider a superconductor-insulator-superconductor (SIS) junction. The
hopping Hamiltonian between the two superconductors (we use labels $L$ for
left and $R$ for right superconductors) across the insulator is given by

$\displaystyle
H_{LR}=-\sum_{\sigma}T_{LR}\left(c^{\dagger}_{L\sigma}c_{R\sigma}+c^{\dagger}_{R\sigma}c_{L\sigma}\right)$
(82)

Let us first consider the case where the insulator part is very thin and the
Bogoliubov excitations in the two superconductors are the same, denoted by
$\gamma^{\dagger}_{n\sigma},\gamma_{n\sigma}$. In this case $H_{LR}$ is re-
expressed using the number-changing operators and the Bogoliubov operators as

$\displaystyle H_{LR}$ $\displaystyle=$ $\displaystyle-T_{LR}e^{-{i\over
2}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot{\bf A}^{\rm em}}$ (83)
$\displaystyle\times\sum_{n,m}\Big{[}((u^{n}_{L})^{\ast}\gamma^{\dagger}_{n\downarrow}+v^{n}_{L}\gamma_{n\uparrow})(u^{m}_{R}\gamma_{m\downarrow}+(v^{m}_{R})^{\ast}\gamma_{m\uparrow}^{\dagger})$
$\displaystyle+((u^{n}_{L})^{\ast}\gamma^{\dagger}_{n\uparrow}-v^{n}_{L}\gamma_{n\downarrow})(u^{m}_{R}\gamma_{m\uparrow}-(v^{m}_{R})^{\ast}\gamma_{m\downarrow}^{\dagger})\Big{]}+\mbox{h.c.}$

Then, the effective hopping Hamiltonian that does not cause Bogoliubov
excitations is given by

$\displaystyle H_{J}^{e}=C\cos\left[{{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot\left({\bf A}^{\rm em}+{{\hbar
c}\over{2e}}\nabla\chi\right)}+\alpha\right]$ (84)

where $C$ and $\alpha$ are parameters given through the following relations

$\displaystyle{1\over
2}Ce^{i\alpha}=-2T_{LR}\sum_{n}v^{n}_{L}(v^{n}_{R})^{\ast}+\mbox{h.c.}$ (85)

Now we consider the second case where the Bogoliubov excitations in the two
superconductors are different. We denote them,
$\gamma^{\dagger}_{Ln\sigma},\gamma_{Ln\sigma}$ for the left superconductor,
and $\gamma^{\dagger}_{Rn\sigma},\gamma_{Rn\sigma}$ for the right
superconductor.

In this case $H_{LR}$ is re-expressed as

$\displaystyle H_{LR}$ $\displaystyle=$ $\displaystyle-T_{LR}e^{-{i\over
2}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot{\bf A}^{\rm em}}$ (86)
$\displaystyle\times\sum_{n,m}\Big{[}((u^{n}_{L})^{\ast}\gamma^{\dagger}_{Ln\downarrow}+v^{n}_{L}\gamma_{Ln\uparrow})(u^{m}_{R}\gamma_{Rm\downarrow}+(v^{m}_{R})^{\ast}\gamma_{Rm\uparrow}^{\dagger})$
$\displaystyle+((u^{n}_{L})^{\ast}\gamma^{\dagger}_{Ln\uparrow}-v^{n}_{L}\gamma_{Ln\downarrow})(u^{m}_{R}\gamma_{Rm\uparrow}-(v^{m}_{R})^{\ast}\gamma_{Rm\downarrow}^{\dagger})\Big{]}+\mbox{h.c.}$

In this case, the current flow without Bogoliubov excitations require the
second order perturbation. As a consequence, the current flow is the flow of
electron pairs. The standard theory only considers this case; this is one of
the reasons that the supercurrent is identified as the flow of electron pairs
in the standard theory.

The second order effective Hamiltonian with taking average over the Bogoliubov
excitations is given by

$\displaystyle\left\langle H_{LR}{1\over{E_{0}-H_{0}}}H_{LR}\right\rangle_{\rm
Bog}\approx-\Big{\langle}\sum_{m,n,m^{\prime},n^{\prime}}T_{LR}^{2}\left[e^{-{i\over
2}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot{\bf A}^{\rm
em}}v_{L}^{n}u_{R}^{m}(\gamma_{Ln\uparrow}\gamma_{Rm\downarrow}-\gamma_{Ln\downarrow}\gamma_{Rm\uparrow})+(L\leftrightarrow
R)\right]$ $\displaystyle\times$
$\displaystyle{1\over{\epsilon_{m}^{R}+\epsilon_{n}^{L}}}\left[e^{-{i\over
2}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot{\bf A}^{\rm
em}}(u_{L}^{n^{\prime}}v_{R}^{m^{\prime}})^{\ast}(\gamma^{\dagger}_{Ln^{\prime}\downarrow}\gamma^{\dagger}_{Rm^{\prime}\uparrow}-\gamma^{\dagger}_{Ln^{\prime}\uparrow}\gamma^{\dagger}_{Rm^{\prime}\downarrow})+(L\leftrightarrow
R)\right]\Big{\rangle}_{\rm Bog}$ $\displaystyle\approx$
$\displaystyle-\sum_{m,n}{{2T_{LR}^{2}}\over{\epsilon_{m}^{R}+\epsilon_{n}^{L}}}\left[v_{L}^{n}u_{R}^{m}(u_{L}^{n}v_{R}^{m})^{\ast}e^{-{i}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{{2e}\over{\hbar
c}}\int_{R}^{L}d{\bf r}\cdot{\bf A}^{\rm
em}}+(v_{L}^{n}u_{R}^{m})^{\ast}u_{L}^{n}v_{R}^{m}e^{{i}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{i{{2e}\over{\hbar
c}}\int_{R}^{L}d{\bf r}\cdot{\bf A}^{\rm
em}})+|u_{L}^{n}v_{R}^{m}|^{2}+|v_{L}^{n}u_{R}^{m}|^{2}\right]$

Thus, the effective hopping Hamiltonian for this case is

$\displaystyle H_{J}^{2e}=C^{\prime}\cos\left({{2e}\over{\hbar
c}}\int_{R}^{L}d{\bf r}\cdot\left[{\bf A}^{\rm em}+{{\hbar
c}\over{2e}}\nabla\chi\right]+\alpha^{\prime}\right)$ (88)

where $C^{\prime}$ and $\alpha^{\prime}$ are parameters given through the
following relations,

$\displaystyle{1\over
2}C^{\prime}e^{i\alpha^{\prime}}=-\sum_{m,n}{{2T_{LR}^{2}}\over{\epsilon_{m}^{R}+\epsilon_{n}^{L}}}(v_{L}^{n}u_{R}^{m})^{\ast}u_{L}^{n}v_{R}^{m}$
(89)

This is the well-known result in the standard theory. It gives rise to the
Ambegaokar-Baratoff relation[40] for the dc Josephson effect[38].

$H_{J}^{2e}$ is used to explain the ac Josephson effect in the standard
theory; however, the new theory indicates that $H_{J}^{e}$ is the right one to
use as explained below.

Let us write the current through the junction as

$\displaystyle J=J_{c}\sin\phi$ (90)

where

$\displaystyle J_{c}=C{e^{2}\over{\hbar}}$ (91)

and

$\displaystyle\phi={{e\over{\hbar c}}\int_{R}^{L}d{\bf r}\cdot\left({\bf
A}^{\rm em}+{{\hbar c}\over{2e}}\nabla\chi\right)}+\alpha$ (92)

Then, the time-derivative of $\phi$ is calculated as

$\displaystyle\dot{\phi}={e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot\left(\partial_{t}{\bf A}^{\rm em}+{{\hbar
c}\over{2e}}\nabla\partial_{t}{\chi}\right)=-{e\over\hbar}\int_{R}^{L}d{\bf
r}\cdot{\bf E}^{\rm em}+\left.{e\over\hbar}\left(-\varphi^{\rm
em}+{\hbar\over{2e}}\partial_{t}{\chi}\right)\right|^{L}_{R}$ (93)

where

$\displaystyle{\bf E}^{\rm em}=-{1\over c}\partial_{t}{\bf A}^{\rm
em}-\nabla\varphi^{\rm em}$ (94)

is used.

There are two contributions for $\dot{\phi}$. The first one is

$\displaystyle-{e\over\hbar}\int_{R}^{L}d{\bf r}\cdot{\bf E}^{\rm
em}={{eV}\over\hbar}$ (95)

where

$\displaystyle V=-\int_{R}^{L}d{\bf r}\cdot{\bf E}^{\rm em}$ (96)

is the voltage across the junction.

The second contribution includes the following

$\displaystyle\varphi^{\rm eff}=\varphi^{\rm
em}-{\hbar\over{2e}}\partial_{t}{\chi}$ (97)

It is actually the time-component of the four vector whose spatial components
are

$\displaystyle{\bf A}^{\rm eff}={\bf A}^{\rm em}+{{\hbar
c}\over{2e}}\nabla{\chi}$ (98)

It is gauge invariant since ${\bf A}^{\rm eff}$ is gauge invariant. Actually,

$\displaystyle\int d^{3}r\varphi^{\rm eff}\rho$ (99)

appears in the Hamiltonian, thus, $\varphi^{\rm eff}$ should be related to the
chemical potential $\mu$ as

$\displaystyle\mu=e\varphi^{\rm eff}$ (100)

Then, Eq. (93) becomes

$\displaystyle\dot{\phi}={e\over\hbar}V+{1\over\hbar}\left(\mu_{R}-\mu_{L}\right)$
(101)

The balance between the voltage and the chemical potential difference yields,

$\displaystyle eV=\mu_{R}-\mu_{L}$ (102)

Thus, the following Josephson relation is obtained

$\displaystyle\dot{\phi}={{2eV}\over\hbar}$ (103)

In the standard theory, $H_{J}^{2e}$ is used to obtain the Josephson relation
without including the contribution from the chemical potential difference.
However, the chemical potential difference term exists in the real
experimental situation since two superconductors in the junction are connected
to difference leads with different chemical potentials. Thus, the omission of
this term in the standard theory is not valid. If this term is included, the
derivation using $H_{J}^{2e}$ gives $\dot{\phi}={{4eV}\over\hbar}$, instead of
Eq. (103), which disagrees with experiments.

##  VII Concluding remarks

In the standard theory of superconductivity, the origin of superconductivity
is the electron-pairing. In this theory, the induced current by a magnetic
field is calculated by the linear response to the vector potential, and the
supercurrent is identified as the dissipationless flow of the paired-
electrons, while single electrons flow with dissipation.

The above supercurrent description suffers from the following serious
problems: 1) it contradicts the reversible superconducting-normal phase
transition in a magnetic field observed in type I superconductors; 2) the
gauge invariance of the supercurrent induced by a magnetic field requires the
breakdown of the global $U(1)$ gauge invariance, or the non-conservation of
the particle number; 3) the explanation of the ac Josephson effect is based on
the boundary condition that is different from the real experimental one; 4)
the measured London moment indicates the mass for the superconducting carrier
is the free electron mass $m_{e}$ if the electron charge $q=-e$ is used
although the standard theory predicts it to be the effective mass $m^{\ast}$.

The standard theory relies on the non-zero value of Eq. (53), and actually,
the cause of the above problems is partly due to the belief that the non-zero
value of Eq. (53) is a physical consequence of the Cooper instability.

In the new theory, Eq. (53) that is calculated using the particle number non-
conserving state is replaced by Eq. (64) that is calculated using the particle
number conserving state. This is possible due to the presence of the Berry
connection that generates the number-changing operators. It appears that the
standard theory takes into account the presence of the number-changing
operators brought about by the Berry connection, by using the particle-number
non-conserving approximation [41]. This approximation works well for some
purposes. However, it also causes serious contradictions.

The new theory is a significant departure from the standard one. We hope that
the elucidation of the cuprate superconductivity is achieved with it.

## References

  * Bardeen  _et al._ [1957] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108, 1175 (1957).
  * Bednorz and Müller [1986] J. G. Bednorz and K. A. Müller, Possible high tc superconductivity in the ba-la-cu-o system, Z. Phys. B 64, 189 (1986).
  * Emery and Kivelson [1995] V. J. Emery and S. A. Kivelson, Importance of phase fluctuation in superconductors with small superfluid density, Nature 374, 434 (1995).
  * Hirsch [2009] J. Hirsch, Bcs theory of superconductivity: it is time to question its validity, Physica Scripta 80, 035702 (2009).
  * Koizumi [2011] H. Koizumi, Spin-vortex superconductivity, J. Supercond. Nov. Magn. 24, 1997 (2011).
  * Hirsch [2017] J. E. Hirsch, Momentum of superconducting electrons and the explanation of the Meissner effect, Phys Rev. B 95, 014503 (2017).
  * Hirsch [2018] J. E. Hirsch, Entropy generation and momentum transfer in the superconductor-normal and normal-superconductor phase transitions and the consistency of the conventional theory of superconductivity, International Journal of Modern Physics B 32, 1850158 (2018).
  * Hirsch [2020] J. E. Hirsch, Inconsistency of the conventional theory of superconductivity, EPL 130, 17006 (2020).
  * Koizumi [2020a] H. Koizumi, Reversible superconducting-normal phase transition in a magnetic field and the existence of topologically-protected loop currents that appear and disappear without Joule heating, EPL 131, 37001 (2020a).
  * Keesom and Kok [1934] W. Keesom and J. Kok, Measurements of the latent heat of thallium connected with the transition, in a constant external magnetic field, from the supraconductive to the non-supraconductive state, Physica 1, 503 (1934).
  * Keesom and Van Laer [1936] W. Keesom and P. Van Laer, Measurements of the latent heat of tin in passing from the supraconductive to the non-supraconductive state, Physica 3, 371 (1936).
  * Keesom and van Laer [1937] W. Keesom and P. van Laer, Measurements of the latent heat of tin while passing from the superconductive to the non-superconductive state at constant temperature, Physica 4, 487 (1937).
  * van Laer and Keesom [1938] P. H. van Laer and W. H. Keesom, On the reversibility of the transition processs between the superconductive and the normal state, Physica 5, 993 (1938).
  * Hirsch [2013] J. E. Hirsch, The london moment: what a rotating superconductor reveals about superconductivity, Physica Scripta 89, 015806 (2013).
  * Koizumi [2020b] H. Koizumi, London moment, london’s superpotential, nambu-goldstone mode, and berry connection from many-body wave functions, DOI: 10.1007/s10948-021-05827-9 J. Supercond. Nov. Magn. (2020b), arXiv:2011.10701 [cond-mat.supr-con] .
  * London [1950] F. London,  _Superfluids_ , Vol. 1 (Wiley, New York, 1950).
  * Hildebrandt [1964] A. F. Hildebrandt, Magnetic field of a rotating superconductor, Phys. Rev. Lett. 12, 190 (1964).
  * Zimmerman and Mercereau [1965] J. E. Zimmerman and J. E. Mercereau, Compton wavelength of superconducting electrons, Phys. Rev. Lett. 14, 887 (1965).
  * Brickman [1969] N. F. Brickman, Rotating superconductors, Phys. Rev. 184, 460 (1969).
  * Tate  _et al._ [1989] J. Tate, B. Cabrera, S. B. Felch, and J. T. Anderson, Precise determination of the Cooper-pair mass, Phys. Rev. Lett. 62, 845 (1989).
  * Tate  _et al._ [1990] J. Tate, S. B. Felch, and B. Cabrera, Determination of the Cooper-pair mass in niobium, Phys. Rev. B 42, 7885 (1990).
  * Verheijen  _et al._ [1990a] A. Verheijen, J. van Ruitenbeek, R. de Bruyn Ouboter, and L. de Jongh, The London moment for high temperature superconductors, Physica B: Condensed Matter 165-166, 1181 (1990a), lT-19.
  * Verheijen  _et al._ [1990b] A. A. Verheijen, J. M. van Ruitenbeek, R. de Bruyn Ouboter, and L. J. de Jongh, Measurement of the London moment in two high-temperature superconductors, Nature 345, 418 (1990b).
  * Sanzari  _et al._ [1996] M. A. Sanzari, H. L. Cui, and F. Karwacki, London moment for heavy-fermion superconductors, Applied Physics Letters 68, 3802 (1996).
  * Koizumi [2020c] H. Koizumi, Explanation of superfluidity using the Berry connection for many-body wave functions, J. Supercond. Nov. Magn. 33, 1697 (2020c).
  * Andreev [1964] A. F. Andreev, Thermal conductivity of the intermediate state of superconductors, Sov. Phys. JETP. 19 19, 1228 (1964).
  * Saint-James [1964] D. Saint-James, Excitations élémentaires au voisinage de la surface de séparation d’un métal normal et d’un métal superconducteur, J. Phys. France 25, 899 (1964).
  * Aharonov and Bohm [1959] Y. Aharonov and D. Bohm, Significance of elecromagnetic potentials in the quantum theory, Phys. Rev. 115, 167 (1959).
  * Ginzburg and Landau [1950] V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh. Exsp. Teor. Fiz. 20, 1064 (1950).
  * Abrikosov [1957] A. A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5, 1174 (1957).
  * Feynman [1965] R. P. Feynman,  _Quantum mechanics and path integrals_ (McGraw-Hill companies, Inc., 1965).
  * Koizumi and Ishikawa [2020] H. Koizumi and A. Ishikawa, Theory of supercurrent in superconductors, International Journal of Modern Physics B 34, 2030001 (2020), https://doi.org/10.1142/S0217979220300017 .
  * Koizumi [2020d] H. Koizumi, Possible occurrence of superconductivity by the $\pi$-flux Dirac string formation due to spin-twisting itinerant motion of electrons, Symmetry 12, 776 (2020d).
  * Dirac [1931] P. Dirac, Quantised singularities in the electromagnetic field, Proc. Roy. Soc. London 133, 60 (1931).
  * Gor’kov [1959] L. P. Gor’kov, Microscopic derivation of the Ginzburg-Landau equations in the theory of superconductivity, Sov. Phys. JETP 36, 1364 (1959).
  * de Gennes [1966] P. G. de Gennes,  _Superconductivity of Metals and Alloys_ (W. A. Benjamin, Inc., 1966).
  * Zhu [2016] J.-X. Zhu,  _Bogoliubov-de Gennes Method and Its Applications_ (Springer, 2016).
  * Josephson [1962] B. D. Josephson, Possible new effects in superconductive tunneling, Phys. Lett. 1, 251 (1962).
  * Koizumi and Tachiki [2015] H. Koizumi and M. Tachiki, Supercurrent generation by spin-twisting itinerant motion of electrons: Re-derivation of the ac josephson effect including the current flow through the leads connected to josephson junction, J. Supercond. Nov. Magn. 28, 61 (2015).
  * Ambegaokar and Baratoff [1963] V. Ambegaokar and A. Baratoff, Tunneling between superconductors, Phys. Rev. Lett. 10, 486 (1963).
  * Peierls [1991] R. Peierls, Spontaneously broken symmetries, J. Phys. A 24, 5273 (1991).