# Neglected $U(1)$ phase in the Schrödinger representation of quantum
mechanics and particle number conserving formalisms for superconductivity

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

###### Abstract

Superconductivity is reformulated as a phenomenon in which a stable velocity
field is created by a $U(1)$ phase neglected by Dirac in the Schrödinger
representation of quantum mechanics. The neglected phase gives rise to a
$U(1)$ gauge field expressed as the Berry connection from many-body wave
functions. The inclusion of this gauge field transforms the standard particle-
number non-conserving formalism of superconductivity to a particle-number
conserving one with many results of the former unaltered. In other words, the
new formalism indicates that the current standard one is an approximation that
effectively takes into account this neglected $U(1)$ gauge field by employing
the particle-number non-conserving formalism. Since the standard and new
formalisms are physically different, conflicting results are predicted in some
cases. We reexamine the Josephson relation and show that a capacitance
contribution of the Josephson junction to the $U(1)$ phase is missing in the
standard formalism, and inclusion of it indicates that the standard theory
actually does not agree with the experiment while the new one does. It is also
shown that the dissipative quantum phase transition in Josephson junctions
predicted in the standard theory does not exit in the new one in accordance
with the recent experimental result.

##  I Introduction

After the success of the BCS theory for superconductivity [1], it is now
widely-believed that superconductivity is a phenomenon of gauge symmetry
breaking. The order parameter for the superconducting phase is a macroscopic
wave function given by the expectation value of the electron-pair field
operator [2]; in order to have non-zero expectation values for it, a particle-
number non-conserving formalism is required.

The original intention for the use of this particle-number non-conserving
formalism is to facilitate calculations involving electron pair-correlation
that yields an energy gap [1]. The particle number non-conserving formalism is
very practical for this purpose since it yields the total energy with an
estimated relative error proportional to $N^{-1/2}$, where $N$ is the total
number of the particles in the system; thus the error will be negligibly small
for a bulk system with very large $N$ [3].

The BCS theory successfully explains a number of pairing energy gap related
effects; especially, it provides a way to calculate the superconducting
transition temperature $T_{c}$ as the pairing energy gap formation
temperature. Surprisingly, a $U(1)$ phase arising in the particle-number non-
conserving formalism turned out to be the essential ingredient for
supercurrent generation. It brings about the supercurrent related effects [4],
which were predicted by Josephson using the Bogoliubov’s particle-number non-
conserving formalism of the BCS theory [5]; and the predicted effects were
confirmed by experiments [6, 7].

In spite of all those successes, the use of the particle-number non-conserving
formalism is uncomfortable since it is sensible to consider that
superconductivity also occurs in isolated systems with fixed number of
particles. Some researchers put significant efforts to restore the particle-
number conservation in the formalism for superconductivity [8]. Such efforts
are based on the idea of the Bose-Einstein condensation of electron pairs.
However, it is not applicable for the cases where the spatial overlapping of
electron-pairs is significant since it is not legitimate to treat electron-
pairs as bosons in such cases. Thus, the Bose-Einstein condensation based
refinement will not be able to explain the whole variety of superconductors.

The discovery of high transition temperature superconductors in cuprates upset
the comfortable agreement between theory and experiment for superconductivity
[9]. At present, no widely accepted theory exists for this “cuprate
superconductivity”. The electron pairing gap with $d$-wave symmetry exists;
however, $T_{c}$ is not identified as this gap formation temperature. The
experiment indicates that $T_{c}$ corresponds to the stabilization temperature
for nano-sized loop currents [10]; this view is supported by the fact that the
specific near $T_{c}$ resembles that of the three dimensional $XY$ model [11].
Besides, there are a number of phenomena that indicate the presence of loop
currents in the pseudogap and superconducting phases [12, 13, 14, 15, 16, 17,
18].

It is also notable that the normal state of the cuprate from which the
superconducting state appears is not a simple metal; the strong electron-
lattice interaction causes the small lattice polaron formation and its
relevance to superconductivity has been experimentally verified [19]; and the
strong electron repulsion in this system makes the parent compound an
antiferromagnetic insulator [20, 21], and the origin of the $d$-wave gap seems
to be this strong repulsion. Peculiar enough, in spite of all those strong
electron-lattice and electron-electron interactions expected to increase the
effective mass, the mass for the supercurrent carrier is that of free electron
one according to the London moment measurement [22]. Actually, this sharply
contradicts the BCS theory, since it predicts that the mass appears in the
London moment is the effective mass of the normal metallic phase.

Furthermore, this London moment mass discrepancy problem is not just for the
cuprate superconductivity, but for all other superconductors as well; the free
electron mass is observed universally in the London moment experiments [23].

Recently, it is argued that the $U(1)$ phase that appears in the particle-
number non-conserving formalism is actually the $U(1)$ phase neglected by
Dirac in the Schrödinger’s representation of quantum mechanics [24]. In this
new theory, the mass in the London moment is the free electron mass and the
$U(1)$ phase variable appears with keeping the number of particles [25].

Let us explain this new theory succinctly below. In the Schrödinger’s
representation, the momenta $p_{r}$’s conjugate to canonical coordinates
$q_{r}$’s are given by

$\displaystyle p_{r}=-i\hbar{{\partial}\over{\partial q_{r}}},\quad
r=1,\cdots,n$ (1)

According to the Heisenberg formulation of quantum theory [26], commutation
relations

$\displaystyle[q_{r},q_{s}]=0,\quad[p_{r},p_{s}]=0,\quad[q_{r},p_{s}]=i\hbar\delta_{rs}$
(2)

are more fundamental than the derivative representation of the momenta.
Actually, the following $p_{r}$’s are also legitimate

$\displaystyle p_{r}=-i\hbar{{\partial}\over{\partial q_{r}}}+{{\partial
F}\over{\partial q_{r}}},\quad r=1,\cdots,n$ (3)

where $F$ is a function of $q_{1},\cdots,q_{n}$, and also of time $t$.

Dirac argued that $F$ can be removed by the following transformation of the
wave function,

$\displaystyle\psi(q_{1},\cdots,q_{n})\rightarrow
e^{i\gamma}\psi(q_{1},\cdots,q_{n})$ (4)

where $\gamma$ is related to $F$ by $F=\hbar\gamma+\mbox{constant}$, thus, we
can always use $p_{r}$’s in Eq. (1). The standard calculation obtains
$e^{i\gamma}\psi(q_{1},\cdots,q_{n})$ as a whole by employing a finite number
of basis functions, thus, $e^{i\gamma}$ is not considered explicitly.

Contrary to the standard procedure, there are cases where $e^{i\gamma}$ should
be considered as an extra degree-of-freedom [25]. Actually, the condition
$[p_{r},p_{s}]=0$ in Eq. (3) requires the following condition

$\displaystyle{\partial\over{\partial q_{s}}}{{\partial F}\over{\partial
q_{r}}}-{\partial\over{\partial q_{r}}}{{\partial F}\over{\partial q_{s}}}=0$
(5)

and this condition can be violated at some points giving rise to toological
defects. In such cases, $\gamma$ is expressed as a nontrivil Berry connection
from many-body wave functions given by

$\displaystyle\gamma=\sum_{j=1}^{N}\int_{0}^{{\bf r}_{j}}{\bf A}^{\rm
MB}_{\Psi}({\bf r}^{\prime},t)\cdot d{\bf r}^{\prime}$ (6)

where ${\bf A}^{\rm MB}_{\Psi}({\bf r},t)$ is given by

$\displaystyle{\bf A}^{\rm MB}_{\Psi}({\bf r},t)={1\over{\hbar\rho({\bf
r})}}{\rm Re}\left\\{\int d\sigma_{1}d{\bf x}_{2}\cdots d{\bf
x}_{N}\Psi^{\ast}({\bf r},\sigma_{1},\cdots,{\bf
x}_{N})(-i\hbar\nabla)\Psi({\bf r},\sigma_{1},\cdots,{\bf x}_{N})\right\\}$
(7)

‘$\rm{Re}$’ denotes the real part, $\Psi$ is the total wave function, ${\bf
x}_{i}$ collectively stands for the coordinate ${\bf r}_{i}$ and the spin
$\sigma_{i}$ of the $i$th electron (the coordinates $q_{r},\ r=1,\cdots,n$ in
Eq. (1) correspond to ${\bf r}_{i},\ i=1,\cdots,N$ with $n=3N$ [27, 28]),
$-i\hbar\nabla$ is the Schrödinger’s momentum operator for the coordinate
vector ${\bf r}$, and $\rho({\bf r},t)$ is the number density calculated from
$\Psi$. This the Berry connection is obtained by regarding ${\bf r}$ as the
“adiabatic parameter”[29].

The Berry phase formalism employed above should be considered as a way to
detect topological singularities of the connection (points of the violation of
Eq. (5)), and the adiabatic assumption of the parameter is disregarded. It is
noteworthy that Dirac considered such singularities in relation to the
magnetic monopole [30]. Note also that ${\bf A}^{\rm MB}_{\Psi}({\bf r},t)$ is
a many-body quantity since its value for particle labeled ‘$1$’ at ${\bf
r}_{1}={\bf r}$ arises from variables for other particles; thus, it is absent
for the hydrogen problem solved by Schrödinger [31]. The connection ${\bf
A}^{\rm MB}_{\Psi}({\bf r},t)$ generates a topologically protected velocity
field (that corresponds to ${{\partial F}\over{\partial q_{r}}}$ in Eq. (3))
and the supercurrent is generated by it [25].

This new theory quantizes the field generated by ${\bf A}^{\rm MB}_{\Psi}$
which is considered as a collective mode. The quantization of this mode
enables to take into account of the fluctuation of the number of electrons
participating in it. The new theory captures many of the important results
obtained by the standard one since the Bogoliubov excitations are identical to
the standard one, and also removes some of the shortcomings of it, such as the
mass discrepancy problem in the London moment [32, 27, 25]. The mass for the
supercurrent velocity field is the free electron mass since the velocity field
appears in the modification of the momentum operators in Eq. (3) with the free
electron mass. It is also notable that some results that are attributed to the
electron-pair supercurrent flow are explained without pairing electrons. This
indicates that the electron-pair formation is not the essential ingredient of
superconductivity; its role is to stabilize the velocity field that generates
the supercurrent, thus, $T_{c}$ agrees with the paring gap formation
temperature in some superconductors.

In the present work, we explain the new theory of superconductivity putting
emphasis on Josephson effects related phenomena. The organization of the
present work is as follows: we revisit the flux quantization and Josephson
effects in Section II. We examine the Josephson relation and the Ambegaokar-
Baratoff relation [33] in Section III. A practical way to include the Dirac’s
neglected $U(1)$ phase is explained in Section IV. The absence of the
dissipative quantum phase transition in Josephson junctions [34, 35, 36, 37]
is discussed in Section V. Lastly, we conclude the present work by mentioning
implications of the present new theory to the superfluidity of bosons in
Section VI.

##  II Flux Quantization and Josephson effects

In order to take into account the electron pairing effect, the BCS used the
following variational state vector,

$\displaystyle|{\rm BCS}\rangle=\prod_{\bf k}(u_{\bf k}+e^{i\theta}v_{\bf
k}c^{\dagger}_{{\bf k}\uparrow}c^{\dagger}_{-{\bf k}\downarrow})|{\rm
vac}\rangle$ (8)

where $|{\rm vac}\rangle$ is the vacuum state that satsifies

$\displaystyle c_{{\bf k}\sigma}|{\rm vac}\rangle=0$ (9)

where $c_{{\bf k}\sigma}$ is the annihilation operator for the conduction
electron with effective mass $m^{\ast}$, the wave vector ${\bf k}$, and spin
$\sigma$; $c^{\dagger}_{{\bf k}\sigma}$ is the creation operator for the same
electron, and $u_{\bf k}$ and $v_{\bf k}$ are variational parameters [1]. The
phase factor $e^{i\theta}$ is a number; it plays an important role in the
Josephson effects.

This BCS state, $|{\rm BCS}\rangle$, is a vacuum of the following Bogoliubov
operators [38],

$\displaystyle\gamma_{{\bf k}0}^{\rm BCS}$ $\displaystyle=$ $\displaystyle
u_{\bf k}e^{-{i\over 2}\theta}c_{{\bf k}\uparrow}-v_{\bf k}e^{{i\over
2}\theta}c_{-{\bf k}\uparrow}^{\dagger}$ $\displaystyle\gamma_{-{\bf k}1}^{\rm
BCS}$ $\displaystyle=$ $\displaystyle u_{\bf k}e^{-{i\over 2}\theta}c_{-{\bf
k}\downarrow}+v_{\bf k}e^{{i\over 2}\theta}c_{{\bf k}\uparrow}^{\dagger}$ (10)

It satisfies

$\displaystyle\gamma_{{\bf k}0}^{\rm BCS}|{\rm
BCS}\rangle=0,\quad\gamma_{-{\bf k}1}^{\rm BCS}|{\rm BCS}\rangle=0$ (11)

Excitations from the ground sate are the Bogoliubov excitations generated by
$(\gamma_{{\bf k}0}^{\rm BCS})^{\dagger}$ and $(\gamma_{-{\bf k}1}^{\rm
BCS})^{\dagger}$ [39]. The ground state $|{\rm BCS}\rangle$ is a mixed state
of different particle-numbers and the Bogoliubov operators do not conserve the
particle-number.

We do not derive but just mention here that the particle-number conserving
Bogoliubov operators are constructed in the new theory. The details will be
found in [25]. In the new theory, $|{\rm BCS}\rangle$ is replaced by $|{\rm
Gnd}\rangle$

$\displaystyle|{\rm Gnd}\rangle=\prod_{\bf k}(u_{\bf k}+v_{\bf
k}c^{\dagger}_{{\bf k}\uparrow}c^{\dagger}_{-{\bf
k}\downarrow}e^{-i\hat{X}})|{\rm Cnd}\rangle$ (12)

where the state vector $|{\rm Cnd}\rangle$ corresponds to the state for the
wave function $\Psi$ in Eq. (7). It corresponds to the state in which all the
electrons participate in the collective mode described by $\gamma$ in Eq. (6).
The phase factor $e^{-i\hat{X}}$ is an operator associated with $\gamma$. We
will explain it, later.

For convenience, we use the following $\chi$, instead of ${\bf A}_{\Psi}^{\rm
MB}$ in Eqs. (6) and (7):

$\displaystyle{{\chi({\bf r},t)}}=-2\int^{{\bf r}}_{0}{\bf A}_{\Psi}^{\rm
MB}({\bf r}^{\prime},t)\cdot d{\bf r}^{\prime}$ (13)

Then, $\gamma$ is given by

$\displaystyle\gamma=-{1\over 2}\sum_{j=1}^{N}\chi({\bf r}_{j},t)$ (14)

As explained in Refs. [27, 28, 25], the many-electron wave function $\Psi$ is
expressed as

$\displaystyle\Psi({\bf x}_{1},\cdots,{\bf x}_{N},t)=\exp\left(-{i\over
2}\sum_{j=1}^{N}\chi({\bf r}_{j},t)\right)\Psi_{0}({\bf x}_{1},\cdots,{\bf
x}_{N},t)$ (15)

with $\Psi_{0}$ being a currentless wave function. $\Psi_{0}$ is obtained by
the requirement of the minimum energy with using the momentum operators of the
standard Schrödinger representation. In this wave function, the number of
particles participating in the collective mode is $N$ as is indicated in the
phase factor $\exp\left(-{i\over 2}\sum_{j=1}^{N}\chi({\bf r}_{j},t)\right)$.

Now, we include the fluctuation of the number of particles participating in
the collective mode. Let us denote this number by $N_{\chi}$ and its eigenket
as $|N_{\chi}\rangle$. With this notation, $|{\rm Cnd}\rangle$ is expressed as
$|{\rm Cnd}\rangle=|{N}_{\chi}=N\rangle$.

By quantizing the collective mode $\chi$, the number changing operators
$e^{\pm i{1\over 2}\hat{X}}$ are constructed [25]. They satisfy the following
relations

$\displaystyle e^{\pm i{1\over
2}\hat{X}}|{N}_{\chi}\rangle\propto|{N}_{\chi}\pm 1\rangle$ (16)

Thus, $e^{-i{1\over 2}\hat{X}}$ is the number changing operator that decreases
the number of electrons participating in the collective mode by one,
$e^{i{1\over 2}\hat{X}}$ increases by one. Especially, $e^{-i\hat{X}}$ reduces
the number by two.

The comparison of Eqs. (8) and (12) indicates the following correspondence

$\displaystyle e^{i\theta}\rightarrow e^{-i\hat{X}},\quad|{\rm
vac}\rangle\rightarrow|{\rm Cnd}\rangle$ (17)

exists between the BCS theory and new one. The electron pair creation
operators $e^{i\theta}c^{\dagger}_{{\bf k}\uparrow}c^{\dagger}_{-{\bf
k}\downarrow}$ in $|{\rm BCS}\rangle$ is replaced by $c^{\dagger}_{{\bf
k}\uparrow}c^{\dagger}_{-{\bf k}\downarrow}e^{-i\hat{X}}$ in $|{\rm
Gnd}\rangle$; the latter decreases the number of electrons participating in
the collective mode by two in addition to increasing the number of electrons
in single-particle motion states. As a consequence, the particle number is
fixed in $|{\rm Gnd}\rangle$.

The Bogoliubov operators in the new theory are given by

$\displaystyle\gamma_{{\bf k}0}$ $\displaystyle=$ $\displaystyle u_{\bf
k}e^{{i\over 2}\hat{X}}c_{{\bf k}\uparrow}-v_{\bf k}e^{-{i\over
2}\hat{X}}c_{-{\bf k}\uparrow}^{\dagger}$ $\displaystyle\gamma_{-{\bf k}1}$
$\displaystyle=$ $\displaystyle u_{\bf k}e^{{i\over 2}\hat{X}}c_{-{\bf
k}\downarrow}+v_{\bf k}e^{-{i\over 2}\hat{X}}c_{{\bf k}\uparrow}^{\dagger}$
(18)

They conserve the particle number, and satisfy

$\displaystyle\gamma_{{\bf k}0}|{\rm Gnd}\rangle=0,\quad\gamma_{-{\bf
k}1}|{\rm Gnd}\rangle=0$ (19)

Due to these properties, exactly the same Bogoliubov excitation spectrum of
the BCS theory is obtained [28]. Thus, the new theory captures almost all
energy gap related results of it.

Now we consider the Josephson effects. Let us consider a Josephson
(superconductor-insulator-superconductor (SIS)) junction. The hopping
Hamiltonian between the two superconductors (called ‘left’ and ‘right’,
respectively) across the insulator is given by

$\displaystyle H_{LR}=-\sum_{\sigma}T_{LR}e^{i{e\over{\hbar c}}\int^{{\bf
r}_{R}}_{{\bf r}_{L}}d{\bf r}\cdot{\bf A}^{\rm
em}}\left(c^{\dagger}_{L\sigma}c_{R\sigma}+c^{\dagger}_{R\sigma}c_{L\sigma}\right)$
(20)

where quantities for the left superconductor is labeled with $L$, and those
for the right superconductor are labeled with $R$. The phase factor
$e^{i{e\over{\hbar c}}\int^{{\bf r}_{R}}_{{\bf r}_{L}}d{\bf r}\cdot{\bf
A}^{\rm em}}$ arises due to the magnetic field ${\bf B}^{\rm
em}=\nabla\times{\bf A}^{\rm em}$. We take this Hamiltonian as a perturbation.

###  II.1 BCS superconducting state case

First, we consider the BCS superconducting state case. The state vectors for
the two superconductors are given by

$\displaystyle|{\rm BCS}_{L}\rangle$ $\displaystyle=$ $\displaystyle\prod_{\bf
k}(u_{L{\bf k}}+e^{i\theta_{L}}v_{L{\bf k}}c^{\dagger}_{L{\bf
k}\uparrow}c^{\dagger}_{L-{\bf k}\downarrow})|{\rm vac}\rangle$
$\displaystyle|{\rm BCS}_{R}\rangle$ $\displaystyle=$ $\displaystyle\prod_{\bf
k}(u_{R{\bf k}}+e^{i\theta_{R}}v_{R{\bf k}}c^{\dagger}_{R{\bf
k}\uparrow}c^{\dagger}_{R-{\bf k}\downarrow})|{\rm vac}\rangle$ (21)

where subscripts $L$ and $R$ denote quantities in the left and right
superconductors, respectively.

The nonzero perturbation for tunneling arises from the second order in
$H_{LR}$. In this case, the tunneling is due to the electron-pair transfer
between the two superconductors. The supercurrent related terms contain the
phase factor $e^{i\phi}$, where the angle $\phi$ is given by

$\displaystyle\phi=\theta_{R}-\theta_{L}+{{2e}\over{\hbar c}}\int^{{\bf
r}_{R}}_{{\bf r}_{L}}{\bf A}^{\rm em}\cdot d{\bf r}$ (22)

The second order perturbation of $H_{LR}$ yields ${{2e}\over{\hbar
c}}\int^{{\bf r}_{R}}_{{\bf r}_{L}}{\bf A}^{\rm em}\cdot d{\bf r}$, and
$\theta_{R}-\theta_{L}$ arises from the phases $e^{i\theta_{L}}$ and
$e^{i\theta_{R}}$ of the superconducting states $|{\rm BCS}_{L}\rangle$ and
$|{\rm BCS}_{R}\rangle$, respectively.

The formula in Eq. (22) indicates that the following sum is the gauge
invariant combination,

$\displaystyle\nabla\theta+{{2e}\over{\hbar c}}{\bf A}^{\rm em}$ (23)

due to the fact that the gauge invariant current depends on $\phi$.

Then, the time-component partner or the above expression (spatial component)
given by

$\displaystyle\partial_{t}\theta-{{2e}\over{\hbar}}\phi^{\rm em}$ (24)

is also gauge invarient. Here, $\phi^{\rm em}$ is the scalar potential for the
electromagnetic field.

Using Eqs. (22)-(24), the time-derivative of $\phi$ is given by

$\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle\int^{{\bf
r}_{R}}_{{\bf r}_{L}}\nabla\partial_{t}{\theta}\cdot d{\bf r}+{{2e}\over{\hbar
c}}\int^{{\bf r}_{R}}_{{\bf r}_{L}}\partial_{t}{\bf A}^{\rm em}\cdot d{\bf r}$
(25) $\displaystyle=$ $\displaystyle\int^{{\bf r}_{R}}_{{\bf
r}_{L}}\nabla\left(\partial_{t}{\theta}-{{2e}\over{\hbar}}\phi^{\rm
em}\right)\cdot d{\bf r}-{{2e}\over{\hbar}}\int^{{\bf r}_{R}}_{{\bf
r}_{L}}{\bf E}^{\rm em}\cdot d{\bf r}$

where ${\bf E}^{\rm em}$ is the electric field given by

$\displaystyle{\bf E}^{\rm em}=-{1\over c}\partial_{t}{\bf A}^{\rm
em}-\nabla\phi^{\rm em}$ (26)

This shows that the time derivative $\dot{\phi}$ is composed of two gauge
invariant terms.

The first term in Eq. (25) that contains $\partial_{t}\theta$ arises from the
chemical potential difference of the two superconductors. It is expressed as

$\displaystyle\int^{{\bf r}_{R}}_{{\bf
r}_{L}}\nabla\left(\partial_{t}{\theta}-{{2e}\over{\hbar}}\phi^{\rm
em}\right)\cdot d{\bf r}={{2eV}\over{\hbar}}$ (27)

where $V$ is the voltage across the junction. From this term, Josephson
obtained the Josephson relation [4] given by

$\displaystyle\dot{\phi}={{2eV}\over{\hbar}}$ (28)

The second term in Eq. (25) that contains ${\bf E}^{\rm em}$ is due to the
presence of the electric field between two superconductors. The presence of
this term indicates that the Josepshon junction plays a role of a capacitor.
This term is absent in the Josephson’s derivation [4]; actually, this is also
absent in the metal-superconductor tunneling calculation by Cohen et al.[40].
For the metal-superconductor junction case, the capacitance effect of the
junction is negligible, thus, this omission will be justifiable. However, for
the Josephson junction case, it is not usually negligible as is manifested in
the use of the RCSJ model [41].

From the capacitance effect of the Josephson effect, we have

$\displaystyle-{{2e}\over{\hbar}}\int^{{\bf r}_{R}}_{{\bf r}_{L}}{\bf E}^{\rm
em}\cdot d{\bf r}={{2eV}\over{\hbar}}$ (29)

due to the fact that the voltage across the capacitor part of the RCSJ model
is the same as that the Josephson junction part.

Then, Eq. (25) actually becomes

$\displaystyle\dot{\phi}={{4eV}\over{\hbar}}$ (30)

due to the two gauge invariant contributions. This indicates that if the
capacitance effect is included the observed Josephson relation is not
obtained. In other words, the presence of the capacitance effect of the
Josephson junction indicates that the correct Josephson effect cannot be
obtained by the standard theory.

###  II.2 New superconducting state case

Now, consider the new superconducting state case. We only limit the
consideration of the flux quantization problem, here. The ac Josephson effect
is considered, later.

From Eqs. (3) and (6), and the relation $F=\hbar\gamma+\mbox{constant}$, the
replacement for ${\bf p}=-i\hbar\nabla$ occurs in the following manner,

$\displaystyle-i\hbar\nabla\rightarrow-i\hbar\nabla+\hbar{\bf A}^{\rm
MB}_{\Psi}$ (31)

if the contribution of the Berry connection is included in the relevant
Hamiltonian.

When a magnetic field with vector potential ${\bf A}^{\rm em}$ exists, it is
further modified as

$\displaystyle-i\hbar\nabla+\hbar{\bf A}^{\rm
MB}_{\Psi}\rightarrow-i\hbar\nabla+\hbar{\bf A}^{\rm MB}_{\Psi}+{e\over c}{\bf
A}^{\rm em}$ (32)

This is the standard replacement of the momentum operator that occurs in the
presence of the magnetic field.

We consider the situation where the current generation is due to the velocity
potential arising from ${\bf A}^{\rm MB}_{\Psi}$ and ${\bf A}^{\rm em}$. The
Meissner effect leads to the exclusion the magnetic field from the interior of
the superconductor, which yields

$\displaystyle 0=\hbar{\bf A}^{\rm MB}_{\Psi}+{e\over c}{\bf A}^{\rm
em}=-{\hbar\over 2}\nabla\chi+{e\over c}{\bf A}^{\rm em}$ (33)

The flux quantization arises from the above condition.

Let us consider a ring-shaped superconductor, and take a loop $C$ that goes
around the hole part of the ring through deep inside of it. Then, we have the
following relation

$\displaystyle\oint_{C}{\bf A}^{\rm em}\cdot d{\bf
r}={{c\hbar}\over{2e}}\oint_{C}\nabla\chi\cdot d{\bf r}$ (34)

This yields the flux quantization with a unit flux ${{ch}\over{2e}}$ if $\chi$
is an angular variable with period $2\pi$. The appearance of such $\chi$
occurs if electrons perform spin-twisting itinerant loop motion [25]. The
center of the spin-twisting loop is a topological singularity of the
connection where the condition in Eq. (5) is violated. However, the amplitude
of the wave function is zero at this point, thus, the wave function does not
show any anomalies. The topological singularity here is characterized by the
winding number ${1\over{2\pi}}\oint_{C}\nabla\chi\cdot d{\bf r}$.

In the new theory, the observed flux quantization is attributed to this
singularity in the Berry connection; it is characterized by the topological
integer, ‘winding number’; ‘$2e$’ in ${{ch}\over{2e}}$ arises from the spinor
character of the spin function for the spin-twisting itinerant loop motion of
electrons.

##  III The Josephson relation and the Ambegaokar-Baratoff relation

In the new theory, the superconductivity is due to the presence of $\chi$. The
paired electrons are not the essential ingredient. In the standard theory, the
Josephson relation and the Ambegaokar-Baratoff relation are considered to be
the manifestation that electron-pairs are supercurrent carries [33]; however,
the new theory explains them, differently, as will be explained below.

###  III.1 The Josephson relation

Let us now consider the current flow though the Josephson junction in the new
theory. The tunneling effect can be obtained by the first order perturbation
of $H_{LR}$.

To deal with the Josephson junction problem, we use the coordinate dependent
basis functions $u_{n}({\bf r}),v_{n}({\bf r})$ instead of $u_{\bf k}$ and
$v_{\bf k}$ [39, 42]. Then, the electron field operators become

$\displaystyle\hat{\Psi}_{\uparrow}({\bf r},t)$ $\displaystyle=$
$\displaystyle\sum_{n}e^{-{i\over 2}\hat{\chi}({\bf
r},t)}\left(\gamma_{{n}\uparrow}u_{n}({\bf
r})-\gamma^{\dagger}_{{n}\downarrow}v^{\ast}_{n}({\bf r})\right)$
$\displaystyle\hat{\Psi}_{\downarrow}({\bf r},t)$ $\displaystyle=$
$\displaystyle\sum_{n}e^{-{i\over 2}\hat{\chi}({\bf
r},t)}\left(\gamma_{{n}\downarrow}u_{n}({\bf
r})+\gamma^{\dagger}_{{n}\uparrow}v^{\ast}_{n}({\bf r})\right)$ (35)

where the particle number conserving Bogoliubov operators satisfy

$\displaystyle\gamma_{n\sigma}|{\rm Gnd}(N)\rangle=0$ (36)

with $N$ being the total number of particles. The operator $e^{-{i\over
2}\hat{\chi}({\bf r},t)}$ is the coordinate and time dependent version of
$e^{-{i\over 2}\hat{X}}$; the operation of $e^{-{i\over 2}\hat{\chi}({\bf
r},t)}$ on the ground state yields

$\displaystyle e^{-{i\over 2}\hat{\chi}({\bf r},t)}|{\rm
Gnd}(N)\rangle=e^{-{i\over 2}{\chi}({\bf r},t)}|{\rm Gnd}(N-1)\rangle$ (37)

We consider a lattice system where the site $i$ corresponds to the coordinate
${\bf r}_{i}$. Using the number-changing operators and the Bogoliubov
operators defined above, annihilation and creation operators for the electrons
are given by

$\displaystyle c_{i\sigma}$ $\displaystyle=$
$\displaystyle\sum_{n}[u^{n}_{i}\gamma_{n\sigma}-\sigma(v^{n}_{i})^{\ast}\gamma_{n-\sigma}^{\dagger}]e^{-{i\over
2}\hat{\chi}_{i}}$ $\displaystyle c^{\dagger}_{i\sigma}$ $\displaystyle=$
$\displaystyle\sum_{n}[(u^{n}_{i\sigma})^{\ast}\gamma^{\dagger}_{n\sigma}-\sigma
v^{n}_{i}\gamma_{n-\sigma}]e^{{i\over 2}\hat{\chi}_{i}}$ (38)

where $u_{i}^{n}$, $v_{i}^{n}$ , and $\hat{\chi}_{i}$ represent $u_{n}({\bf
r}_{i})$, $v_{n}({\bf r}_{i})$, and $\hat{\chi}({\bf r}_{i})$, respectively;
values of $\sigma$ are $\sigma=1$ for up-spin state, and $\sigma=-1$ for down-
spin state.

Using the particle-number conserving Bogoliubov operators, $H_{LR}$ is
expressed as

$\displaystyle H_{LR}$ $\displaystyle=$ $\displaystyle-T_{LR}e^{{i\over
2}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot{\bf A}^{\rm em}}$ (39)
$\displaystyle\times\sum_{n,m}\Big{[}((u^{n}_{L})^{\ast}\gamma^{\dagger}_{n\downarrow}+v^{n}_{L}\gamma_{n\uparrow})(u^{m}_{R}\gamma_{m\downarrow}+(v^{m}_{R})^{\ast}\gamma_{m\uparrow}^{\dagger})$
$\displaystyle+((u^{n}_{L})^{\ast}\gamma^{\dagger}_{n\uparrow}-v^{n}_{L}\gamma_{n\downarrow})(u^{m}_{R}\gamma_{m\uparrow}-(v^{m}_{R})^{\ast}\gamma_{m\downarrow}^{\dagger})\Big{]}+\mbox{h.c.}$

The phase corresponding to $\phi$ in Eq. (22) is now given by

$\displaystyle\phi_{new}=-{1\over 2}(\chi_{R}-\chi_{L})+{{e}\over{\hbar
c}}\int^{{\bf r}_{R}}_{{\bf r}_{L}}{\bf A}^{\rm em}\cdot d{\bf r}$ (40)

which arise from the factor $e^{{i\over
2}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot{\bf A}^{\rm em}}$ in Eq. (39).

Its time-derivative is given by

$\displaystyle\dot{\phi}_{new}$ $\displaystyle=$ $\displaystyle-{1\over
2}\int^{{\bf r}_{R}}_{{\bf r}_{L}}\nabla\partial_{t}{\chi}\cdot d{\bf
r}+{{e}\over{\hbar c}}\int^{{\bf r}_{R}}_{{\bf r}_{L}}\partial_{t}{\bf A}^{\rm
em}\cdot d{\bf r}$ (41) $\displaystyle=$ $\displaystyle-{1\over 2}\int^{{\bf
r}_{R}}_{{\bf
r}_{L}}\nabla\left(\partial_{t}{\chi}+{{2e}\over{\hbar}}\phi^{\rm
em}\right)\cdot d{\bf r}-{{e}\over{\hbar}}\int^{{\bf r}_{R}}_{{\bf r}_{L}}{\bf
E}^{\rm em}\cdot d{\bf r}$ $\displaystyle=$ $\displaystyle{{2eV}\over\hbar}$

By including the capacitance effect of the junction, the correct Josephson
relation is obtained in the first order perturbation. In other words, the new
theory obtains the correct Josephson relation with including the capacitance
effect of the Josephson junction.

###  III.2 The Ambegaokar-Baratoff relation

Let us briefly explain how the Ambegaokar-Baratoff relation is obtained in the
new theory. The detail will be found in Ref. [43].

In order to obtain the Ambegaokar-Baratoff relation, we need to consider the
case where the Bogoliubov excitations in the two superconductors are
different. Then, the tunneling occurs as the second order perturbation. Let us
denote the two different Bogoliubov excitations as
$\gamma^{\dagger}_{Ln\sigma},\gamma_{Ln\sigma}$ for the left superconductor,
and $\gamma^{\dagger}_{Rn\sigma},\gamma_{Rn\sigma}$ for the right
superconductor, respectively.

The Ambegaokar-Baratoff relation is relevant near the normal-superconducting
transition temperature, where the superconducting coherence length is very
small. Therefore, it is reasonable to assume that the he Bogoliubov
excitations in the two superconductors are different.

By using the two sets of Bogoliubov operatores, $H_{LR}$ is given by

$\displaystyle H_{LR}$ $\displaystyle=$ $\displaystyle-T_{LR}e^{{i\over
2}(\hat{\chi}_{L}-\hat{\chi}_{R})}e^{-i{e\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot{\bf A}^{\rm em}}$ (42)
$\displaystyle\times\sum_{n,m}\Big{[}((u^{n}_{L})^{\ast}\gamma^{\dagger}_{Ln\downarrow}+v^{n}_{L}\gamma_{Ln\uparrow})(u^{m}_{R}\gamma_{Rm\downarrow}+(v^{m}_{R})^{\ast}\gamma_{Rm\uparrow}^{\dagger})$
$\displaystyle+((u^{n}_{L})^{\ast}\gamma^{\dagger}_{Ln\uparrow}-v^{n}_{L}\gamma_{Ln\downarrow})(u^{m}_{R}\gamma_{Rm\uparrow}-(v^{m}_{R})^{\ast}\gamma_{Rm\downarrow}^{\dagger})\Big{]}+\mbox{h.c.}$

The current flow across the junction is obtained by the second order
perturbation of the above. The results is the same as the Ambegaokar-Baratoff
relation obtained for the standard theory.

##  IV A practical way to obtain ${\bf A}^{\rm MB}_{\Psi}$ in lattice systems

${\bf A}^{\rm MB}_{\Psi}$ cannot be obtained form the Schrödinger equation. We
explain how to obtain ${\bf A}^{\rm MB}_{\Psi}$ in real calculations.

###  IV.1 Particle-number conserving Bogoliubov-de Gennes equations

First, we assume that the electronic Hamiltonian is given by

$\displaystyle H=\sum_{\sigma}\int d^{3}r\hat{\Psi}^{\dagger}_{\sigma}({\bf
r})h({\bf r})\hat{\Psi}_{\sigma}({\bf r})-{1\over
2}\sum_{\sigma,\sigma^{\prime}}\int d^{3}rd^{3}r^{\prime}V_{\rm eff}({\bf
r},{\bf r}^{\prime})\hat{\Psi}^{\dagger}_{\sigma}({\bf
r})\hat{\Psi}^{\dagger}_{\sigma^{\prime}}({\bf
r}^{\prime})\hat{\Psi}_{\sigma^{\prime}}({\bf
r}^{\prime})\hat{\Psi}_{\sigma}({\bf r})$ (43)

where $h({\bf r})$ is the single-particle Hamiltonian given by

$\displaystyle h({\bf r})={1\over{2m_{e}}}\left({\hbar\over i}\nabla+{e\over
c}{\bf A}^{\rm em}\right)^{2}+U({\bf r})-\mu$ (44)

and $-V_{\rm eff}$ is the effective interaction between electrons.

We consider the mean field solution of the above by neglecting the time-
dependence of $\chi$. The mean-field version of the above Hamiltonian is given
by

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

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

$\displaystyle\Delta({\bf r},{\bf r}^{\prime})=V_{\rm eff}({\bf r},{\bf
r}^{\prime})\langle e^{{i\over 2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf
r}^{\prime}))}\hat{\Psi}_{\uparrow}({\bf r})\hat{\Psi}_{\downarrow}({\bf
r^{\prime}})\rangle$ (46)

The following commutation relations are obtained from Eqs. (35), (LABEL:eq43),
and the fermion commutation relations for the Bogoliubov operators

$\displaystyle\left[\hat{\Psi}_{\uparrow}({\bf r}),{H}_{\rm MF}\right]$
$\displaystyle=$ $\displaystyle{h}({\bf r})\hat{\Psi}_{\uparrow}({\bf r})+\int
d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})\hat{\Psi}^{\dagger}_{\downarrow}({\bf r}^{\prime})e^{-{i\over
2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf r}^{\prime}))}$
$\displaystyle\left[\hat{\Psi}_{\downarrow}({\bf r}),{H}_{\rm MF}\right]$
$\displaystyle=$ $\displaystyle{h}({\bf r})\hat{\Psi}_{\downarrow}({\bf
r})-\int d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})\hat{\Psi}^{\dagger}_{\uparrow}({\bf r}^{\prime})e^{-{i\over
2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf r}^{\prime}))}$ (47)

The particle number conserving Bogoliubov operators $\gamma_{n\sigma}$ and
$\gamma^{\dagger}_{n\sigma}$ are so constructed that they satisfy

$\displaystyle\left[{H}_{\rm
MF},\gamma_{n\sigma}\right]=-\epsilon_{n}\gamma_{n\sigma},\quad\left[{H}_{\rm
MF},\gamma^{\dagger}_{n\sigma}\right]=\epsilon_{n}\gamma^{\dagger}_{n\sigma}$
(48)

with $\epsilon_{n}\geq 0$. By this construction, ${H}_{\rm MF}$ is
diagonalized as

$\displaystyle{H}_{\rm
MF}=E_{g}+\sum_{n,\sigma}\epsilon_{n}\gamma^{\dagger}_{n\sigma}\gamma_{n\sigma}$
(49)

where $E_{g}$ is the ground state energy.

From Eqs. (47), (48), and (49), the equations for the required Bogoliubov
operators (i.e, equations to obtained parameters $u_{n}$’s, $v_{n}$’s, and
$\epsilon_{n}$’s for the Bogoliubov operators) are

$\displaystyle\epsilon_{n}e^{-{i\over 2}\hat{\chi}({\bf r})}u_{n}({\bf r})$
$\displaystyle=$ $\displaystyle h({\bf r})e^{-{i\over 2}\hat{\chi}({\bf
r})}u_{n}({\bf r})+\int d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})e^{-{i\over 2}\hat{\chi}({\bf r})}v_{n}({\bf r}^{\prime})$
$\displaystyle\epsilon_{n}e^{-{i\over 2}\hat{\chi}({\bf r})}v^{\ast}_{n}({\bf
r})$ $\displaystyle=$ $\displaystyle-h({\bf r})e^{-{i\over 2}\hat{\chi}({\bf
r})}v^{\ast}_{n}({\bf r})+\int d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})e^{-{i\over 2}\hat{\chi}({\bf r})}u^{\ast}_{n}({\bf r}^{\prime})$
(50)

The above are cast into the following

$\displaystyle\epsilon_{n}u_{n}({\bf r})$ $\displaystyle=$
$\displaystyle\bar{h}({\bf r})u_{n}({\bf r})+\int d^{3}r^{\prime}\Delta({\bf
r},{\bf r}^{\prime})v_{n}({\bf r}^{\prime})$
$\displaystyle\epsilon_{n}v_{n}({\bf r})$ $\displaystyle=$
$\displaystyle-\bar{h}^{\ast}({\bf r})v_{n}({\bf r})+\int
d^{3}r^{\prime}\Delta^{\ast}({\bf r},{\bf r}^{\prime})u_{n}({\bf r}^{\prime})$
(51)

using the relation in Eq. (37), where the single-particle Hamiltonian
$\bar{h}$ is given by

$\displaystyle\bar{h}({\bf r})={1\over{2m_{e}}}\left({\hbar\over
i}\nabla+{e\over c}{\bf A}^{\rm em}-{\hbar\over 2}\nabla\chi\right)^{2}+U({\bf
r})-\mu$ (52)

and the energy gap function $\Delta({\bf r},{\bf r}^{\prime})$ is given by

$\displaystyle\Delta({\bf r},{\bf r}^{\prime})=V_{\rm eff}({\bf r},{\bf
r}^{\prime})\sum_{n}\left[u_{n}({\bf r})v^{\ast}_{n}({\bf
r}^{\prime})(1-f(\epsilon_{n}))-u_{n}({\bf r}^{\prime})v^{\ast}_{n}({\bf
r})f(\epsilon_{n})\right]$ (53)

Here, $f(\epsilon_{n})$ is the Fermi function.

###  IV.2 Calculation of $\Psi_{0}$

By solving the system of equations Eqs. (51), (52), and (53), self-
consistently, $\Psi_{0}$ is obtained, first. This is a currentless state in
accordance with the so-called “Bloch’s theorem” [44]. Next, $\chi$ is obtained
using $\Psi_{0}$, and requiring the two conditions, i.e., the single-
valuedness requirement of $\Psi$ as a function of coordinates, and the
requirement of the local charge conservation.

###  IV.3 Calculation of $\chi$

We consider lattice systems and explain a way to obtain values of $\chi$ at
lattice sites. The value of $\chi$ at the $j$th lattice site is denoted by
$\chi_{j}$. For this purpose, a three dimensional lattice is flattened to a
two dimensional one. Examples are shown in Figs. 1 and 2.

Figure 1: Flattening a 3D lattice to 2D lattice. The shaded face in the 3D
lattice is removed and converted to a 2D lattice. Figure 2: Flattening a 3D
lattice to 2D lattice. The two shaded faces in the 3D lattice are removed and
converted to a 2D lattice.

We take the branch of $\chi_{j}$ that satisfies the difference of value
between sites connected by bonds (say site $k$) is in the range,

$\displaystyle-\pi\leq\chi_{j}-\chi_{k}<\pi$ (54)

We obtain $(\chi_{k}-\chi_{j})$’s by minimizing the following functional

$\displaystyle F[\nabla\chi]=E[\nabla\chi]+\sum_{\ell=1}^{N_{\rm
loop}}{{\lambda_{\ell}}}\left(\oint_{C_{\ell}}\nabla\chi\cdot d{\bf r}-2\pi
w_{C_{\ell}}[\chi]\right),$ (55)

where

$\displaystyle E[\nabla\chi]=\langle{\Psi}|H_{MF}|{\Psi}\rangle$ (56)

Here, Lagrange multipliers $\lambda_{\ell}$’s are introduced to impose the
winding number condtions for loops; $w_{C_{\ell}}[\chi]$ is the winding number
around the loop $C_{\ell}$ given by

$\displaystyle
w_{C_{\ell}}[\chi]={1\over{2\pi}}\oint_{C_{\ell}}\nabla\chi\cdot d{\bf r}$
(57)

where $\\{C_{1},\cdots,C_{N_{\rm loop}}\\}$ are boundaries of plaques of the
lattice, where $N_{\rm loop}$ is equal to the number of plaques of the
lattice. In Fig. 1, a $N_{\rm loop}=5$ case is depicted; in Fig. 2 a $N_{\rm
loop}=9$ case is depicted. The winding numbers are so chosen that the obtained
wave function becomes a single-valued function of coordinates.

The values of $(\chi_{k}-\chi_{j})$’s are obtained as solutions of the
following system of equations;

$\displaystyle{{\delta E[\nabla\chi]}\over{\delta\nabla\chi}}$
$\displaystyle+$ $\displaystyle\sum_{\ell=1}^{N_{\rm
loop}}{{\lambda_{\ell}}}{{\delta}\over{\delta\nabla\chi}}\oint_{C_{\ell}}\nabla\chi\cdot
d{\bf r}=0$ (58) $\displaystyle\oint_{C_{\ell}}\nabla\chi\cdot d{\bf r}$
$\displaystyle=$ $\displaystyle 2\pi w_{C_{\ell}}[\chi]$ (59)

which become

$\displaystyle{{\partial E(\\{\tau_{k\leftarrow
j}\\})}\over{\partial\tau_{k\leftarrow j}}}$ $\displaystyle+$
$\displaystyle\sum_{\ell=1}^{N_{\rm
loop}}{{\lambda_{\ell}}}{{\partial}\over{\partial\tau_{k\leftarrow
j}}}\sum_{k\leftarrow j}L_{k\leftarrow j}^{\ell}\tau_{k\leftarrow j}=0$ (60)
$\displaystyle\sum_{k\leftarrow j}L_{k\leftarrow j}^{\ell}\tau_{k\leftarrow
j}$ $\displaystyle=$ $\displaystyle 2\pi w_{C_{\ell}}[\chi]$ (61)

in the lattice system; here, the sum is taken over the bonds ${k\leftarrow
j}$, and $\tau_{k\leftarrow j}$ is the difference of $\chi$ for the bond
$\\{k\leftarrow j\\}$ given by

$\displaystyle\tau_{k\leftarrow j}=\chi_{k}-\chi_{j}$ (62)

$L_{k\leftarrow j}^{\ell}$ denotes quantities defined below

$\displaystyle L_{k\leftarrow j}^{\ell}=\left\\{\begin{array}[]{cl}-1&\mbox{
if $k\leftarrow j$ exists in $C_{\ell}$ in the clockwise direction}\\\
1&\mbox{ if $k\leftarrow j$ exists in $C_{\ell}$ in the counterclockwise
direction}\\\ 0&\mbox{ if $k\leftarrow j$ does not exist in
$C_{\ell}$}\end{array}\right.$ (66)

Figure 3: Construction of a simply-connected lattice by removing five bonds.
Figure 4: Construction of a simply-connected lattice by removing nine bonds.

After $(\chi_{k}-\chi_{j})$’s are obtained, we rebuild $\chi$ from them. For
this purpose, we remove some of the bonds from the flattened two-dimensional
lattice so that the connection of the bonds becomes a simply-connected one,
i.e., so that there is only one path to reach a lattice point from the first
lattice point denoted by ‘$1$’. Then, $\chi_{k}$ is obtained as

$\displaystyle\chi_{k}\approx\chi_{1}+\int_{C_{1\rightarrow k}}\nabla\chi\cdot
d{\bf r}$ (67)

where $C_{1\rightarrow k}$ is the unique path that connects from the site $1$
to the site $k$. Note that the integration here means actually a sum.

In real calculations, we can obtain $\tau_{j\leftarrow i}$’s without obtaining
${{\lambda_{\ell}}}$’s. Actually, we use the conservation of the local charge.
Let us explain this method below.

First, we note that the current density is given by

$\displaystyle J_{j\leftarrow i}={{2e}\over\hbar}{{\partial
E}\over{\partial\tau_{j\leftarrow i}}}$ (68)

where $J_{j\leftarrow i}$ is the current through the bond between sites $i$
and $j$ in the direction $j\leftarrow i$. Then, the conservation of charge at
site $j$ is given by

$\displaystyle 0=J^{\rm EX}_{j}+\sum_{i}{{2e}\over\hbar}{{\partial
E}\over{\partial\tau_{j\leftarrow i}}}$ (69)

where $J^{\rm EX}_{j}$ is the external current that enters through site $j$.
We use this in place of Eq. (60).

In order to impose conditions in Eq. (61), $\tau_{j\leftarrow i}$ is split
into a multi-valued part $\tau_{j\leftarrow i}^{0}$ and single-valued part
$f_{j\leftarrow i}$ as

$\displaystyle\tau_{j\leftarrow i}=\tau_{j\leftarrow i}^{0}+f_{j\leftarrow i}$
(70)

where $\tau_{j\leftarrow i}^{0}$ satisfies the constraint in Eq. (61)

$\displaystyle
w_{C_{\ell}}[\chi]={1\over{2\pi}}\sum_{i=1}^{N_{\ell}}\tau^{0}_{{C_{\ell}(i+1)}\leftarrow{C_{\ell}(i)}}$
(71)

and $f_{j\leftarrow i}$ satisfies

$\displaystyle
0={1\over{2\pi}}\sum_{i=1}^{N_{\ell}}f_{{C_{\ell}(i+1)}\leftarrow{C_{\ell}(i)}}$
(72)

The number of $\tau_{j\leftarrow i}$ to be evaluated is equal to the number of
the bonds. The number of equations in Eq. (61) is equal to the number of the
plaques. The number of equations from Eq. (69) is equal to the number of
sites$-1$, due to the fact that the total charge is fixed in the calculation.

The number of unknowns to be obtained and the number of equations have the
following relation

$\displaystyle[\mbox{\\# bonds}]=[\mbox{\\# plaques}]+[\mbox{\\# sites}-1]$
(73)

where ‘# A’ means ‘the number of $A$’. This is the equality of the number of
unknowns and that of equations.

This actually agrees with the Euler’s theorem for the two-dimensional lattice
given by

$\displaystyle[\mbox{\\# edges}]=[\mbox{\\# faces}]+[\mbox{\\# vertices}-1]$
(74)

Example calculations will be found in Ref. [45].

##  V Absence of dissipative quantum phase transition in Josephson junction

Due to the advent of superconducting qubits, a theory for quantization of the
circuit Hamiltonian for superconducting qubit systems has been developed [46].
In this theory, Josephson junctions are important elements. The new theory
gives different description of them, thus, how the modification of the theory
occurs is very important. Actually, the theory is unaltered practically.
However, the dissipative quantum phase transition in Josephson junction
predicted by the standard theory [34, 35] is absent in the new theory. The
absence of it, however, in accordance with the recent experimental result [36,
37]. We will explain it in this section.

###  V.1 Lagrangians for circuit elements

We first, consider the quantization of circuits containing Josephson junctions
[46, 47]. As usual, the node flux at the $n$th node is given by

$\displaystyle\Phi_{n}=c\int^{t}_{-\infty}V_{n}(t^{\prime})dt^{\prime}$ (75)

where $V_{n}$ is node voltage. It is used as the dynamical variable.

The voltage between nodes $n$ and $m$ is given by

$\displaystyle\dot{\Phi}_{n}-\dot{\Phi}_{m}$ (76)

The Lagrangian for a capacitor with capacitance $C$ existing between node $n$
and node $m$ is given by

$\displaystyle{\cal
L}_{C}={1\over{2c^{2}}}C\left(\dot{\Phi}_{n}-\dot{\Phi}_{m}\right)^{2}$ (77)

When a superconductor exists between nodes $n$ and $m$, we employ the
following formula

$\displaystyle\Phi_{n}-\Phi_{m}=\int^{{\bf r}_{n}}_{{\bf r}_{n}}{1\over
2}\left[{1\over c}{\bf A}^{\rm em}({\bf
r},t)-{{\hbar}\over{2e}}\nabla\chi({\bf r},t)\right]\cdot d{\bf r}$ (78)

This is different form the standard one since the factor ${1\over 2}$ is
multiplied. The reason for the use of it is the following: according to Eq.
(40), the time-derivative of the flux difference is calculated as

$\displaystyle\dot{\Phi}_{n}-\dot{\Phi}_{m}$ $\displaystyle=$
$\displaystyle\int^{{\bf r}_{n}}_{{\bf r}_{m}}{1\over 2}\left[{1\over
c}\dot{\bf A}^{\rm em}({\bf
r},t)-{{\hbar}\over{2e}}\partial_{t}\nabla\chi({\bf r},t)\right]\cdot d{\bf
r}=V$ (79)

where $V$ is the voltage across the nodes. Thus, Eq. (78) yields the correct
voltage.

The current through a Josephson junction that exists between nodes $n$ and $m$
is given by

$\displaystyle J=J_{c}\sin\phi$ (80)

where $\phi$ is the gauge invariant phase difference between the two
superconductors in the junction, and $\phi$ should satisfy the Josephson
relation $\dot{\phi}={{2eV}\over{\hbar c}}$.

According to Eq. (40), such a $\phi$ is given by

$\displaystyle\phi={e\over{\hbar}}\int_{{\bf r}_{n}}^{{\bf
r}_{m}}\left[{1\over c}{\bf A}^{\rm
em}-{{\hbar}\over{2e}}\nabla\chi\right]={{2e}\over{\hbar
c}}\left[{\Phi}_{m}-{\Phi}_{n}\right]$ (81)

Then, the Lagrangian for the Josephson junction ${\cal L}_{J}$ including the
capacitance effect is given by

$\displaystyle{\cal L}_{J}=E_{J}\cos{{2e}\over{\hbar
c}}\left({\Phi}_{m}-{\Phi}_{n}\right)+{C_{J}\over{2c^{2}}}(\dot{\Phi}_{n}-\dot{\Phi}_{m})^{2}$
(82)

with $E_{J}={{\hbar J_{c}}\over{2e}}$.

The Lagrangian for an inductor with inductance $L$ exiting between nodes $n$
and $m$ is given by

$\displaystyle{\cal
L}_{L}=-{1\over{2L}}\left({\Phi}_{n}-{\Phi}_{m}\right)^{2}$ (83)

The minus sign here is due to the fact that it is a potential energy term with
respect to the dynamical variable ${\Phi}_{n}$.

###  V.2 Circuit quantization for a Cooper-pair box

$1$$E_{J}$$3$$C_{J}$$C_{g}$$2$$-$$+$$V_{g}$ Figure 5: Cooper pair box. The
Josephson junction is composed of components denoted as $E_{J}$ and $C_{J}$.
$C_{g}$ denotes the gate capacitor, and $V_{g}$ supplies the gate voltage.

Let us consider the Cooper-pair box depicted in Fig 5 The Lagrangian for it is
given by

$\displaystyle{\cal
L}_{CPB}={1\over{2c^{2}}}C_{g}(\dot{\Phi}_{3}-\dot{\Phi}_{2})^{2}+{1\over{2c^{2}}}C_{J}(\dot{\Phi}_{3}-\dot{\Phi}_{1})^{2}+E_{J}\cos{{2e}\over{\hbar
c}}\left({\Phi}_{3}-{\Phi}_{1}\right)$ (84)

The conjugate momentum for $\Phi_{3}$ is

$\displaystyle{{\partial{\cal
L}_{CPB}}\over{\partial\dot{\Phi}_{3}}}={1\over{c^{2}}}C_{g}(\dot{\Phi}_{3}-\dot{\Phi}_{2})+{1\over{c^{2}}}C_{J}(\dot{\Phi}_{3}-\dot{\Phi}_{1})={1\over
c}Q_{3}$ (85)

where $Q_{3}$ is charge at node $3$ given by

$\displaystyle Q_{3}={1\over c}C_{g}(\dot{\Phi}_{3}-\dot{\Phi}_{2})+{1\over
c}C_{J}(\dot{\Phi}_{3}-\dot{\Phi}_{1})$ (86)

The Lagrange equation for $\Phi_{3}$ and $\dot{\Phi}_{3}$ yields the equation
of the conservation for the charge at node $3$,

$\displaystyle\partial_{t}{Q_{3}}+J_{c}\sin{{2e}\over{\hbar
c}}\left({\Phi}_{3}-{\Phi}_{1}\right)=0$ (87)

We take $\dot{\Phi}_{1}=0$ since it is the ground We also have

$\displaystyle(\dot{\Phi}_{1}-\dot{\Phi}_{2})=cV_{g}$ (88)

since we have a voltage generator between nodes $1$ and $2$

Overall, the Lagrangian is given by

$\displaystyle{\cal
L}_{CPB}={1\over{2c^{2}}}(C_{g}+C_{J})\dot{\Phi}_{3}^{2}+{1\over
c}C_{g}V_{g}\dot{\Phi}_{3}+{1\over{2c^{2}}}C_{g}\dot{\Phi}_{3}^{2}+E_{J}\cos{{2e}\over{\hbar
c}}{\Phi}_{3}$ (89)

with $Q_{3}={1\over c}(C_{g}+C_{J})\dot{\Phi}_{3}+V_{g}C_{g}$

Neglecting a constant term, the Hamiltonian is obtained from ${\cal L}_{CPB}$
as

$\displaystyle{\cal
H}_{CPB}={1\over{2(C_{g}+C_{J})}}(Q_{3}-C_{g}V_{g})^{2}-E_{J}\cos{{2e}\over{\hbar
c}}{\Phi_{3}}$ (90)

We introduce $\hat{n}$ and $\hat{\phi}$ by

$\displaystyle\hat{n}=-{Q_{3}\over{2e}},\quad\hat{\phi}={{2e}\over{\hbar
c}}\Phi_{3}$ (91)

Then, the Hamiltonian becomes

$\displaystyle{\cal H}_{CPB}=E_{C}(\hat{n}-n_{g})^{2}-E_{J}\cos\hat{\phi}$
(92)

where

$\displaystyle E_{C}={(2e)^{2}\over{2(C_{g}+C_{J})}},\quad
n_{g}={{C_{g}V_{g}}\over{2e}}$ (93)

From the canonical quantization condition

$\displaystyle[Q_{3},\Phi_{3}]=-i\hbar$ (94)

we have the following relation,

$\displaystyle[\hat{n},\hat{\phi}]=i$ (95)

The above leads to the commutation relation,

$\displaystyle[e^{i\hat{\phi}},\hat{n}]=e^{i\hat{\phi}}$ (96)

Note that the above $e^{i\hat{\phi}}$ essentially corresponds to the number-
changing operator in Eq (16) or Eq (37)

Strictly speaking, $\hat{\phi}$ is not a hermitian operator [48, 49] due to
the fact that $\hat{\phi}$ is an operator for an angular variable If we
consider the eigenstates of $\hat{n}$ given by

$\displaystyle\hat{n}|n\rangle=n|n\rangle$ (97)

where $n$ is an integer, the angular variable nature of $\phi$ is manifested
by the relation

$\displaystyle e^{\pm i\hat{\phi}}|n\rangle=|n\pm 1\rangle$ (98)

An adequate quantization of ${\cal H}_{CPB}$ is obtained using the basis
$\\{|n\rangle\\}$

$\displaystyle{\cal H}_{CPB}=\sum_{n}\left[E_{C}(n-n_{g})^{2}|n\rangle\langle
n|-{1\over 2}E_{J}(|n+1\rangle\langle n|+|n-1\rangle\langle n|)\right]$ (99)

Transmon qubit states are obtained as the eigenstates of the above
Hamiltonian.

###  V.3 Current feeding to Josephson junction and absence of dissipative
quantum phase transition

Now, we consider the absence of the dissipative quantum phase transition in
Josephson junctions.

Let us consider the following Hamiltonian

$\displaystyle{\cal H}=E_{C}\hat{n}^{2}-E_{J}\cos\hat{\phi}-\hat{\phi}I$ (100)

used for the prediction of the dissipative quantum phase in the standard
theory. This is obtained from Eq. (92) by putting $V_{g}=0$ and introducing
the term $-\hat{\phi}I$ to include the current bias. This type of current bias
was introduce by Anderson [50]. It should be noted that this Hamiltonian is
not self-adjoint (or hermitian) due to the fact that $\hat{\phi}$ is not self-
adjoint [48, 49].

In the standard theory, this Hamiltonian is expressed using the representation

$\displaystyle\hat{n}=i{\partial\over{\partial\phi}},\quad\hat{\phi}=\phi$
(101)

Then, it becomes a Hamiltonian for a particle in a potential energy
$-E_{J}\cos{\phi}-{\phi}I$ regarding ${\phi}$ as a coordinate. This potential
is often called a “washboard potential tilted by $I$”.

The above description implicitly assumes the existence of eigenket
$|\phi\rangle$ that satisfies,

$\displaystyle\hat{\phi}|\phi\rangle=\phi|\phi\rangle$ (102)

However, the eigenvalue $\phi$ is ill-defined due to the fact that
$\hat{\phi}$ is not self-adjoint and $\phi$ is multi-valued.

Using the above representation, it has been argued that the quantum mechanical
state will extend over different branches of $\phi$ (the extended-$\phi$) due
to the tunneling between wells of the washboard potential. If the shunt
resistance $R_{s}$ is added to the current equation, the time of the charge
relaxation $\tau$ is given by $\tau=R_{s}C_{J}$. Then, by denoting the quantum
resistance as $R_{q}=h/4e^{2}$, it is predicted that superconductor-insulator
transition should occur around $R_{q}/R_{s}=1$ [34, 35]. Especially,
$R_{s}\rightarrow\infty$ leads to $I\rightarrow 0$, thus, the $I=0$ insulator
state should be observed.

However, ${\cal H}$ with $I=0$ is essentially equivalent to ${\cal H}_{CPB}$.
Then, the system should be superconducting that gives rise to qubit states.
The experiment indicates that the description by the compact-$\phi$ in a
single branch is appropriate, and the dissipative quantum phase transition
does not happen. Actually, its absence has been shown in the state-of-art
experiment [36, 37].

The above discrepancy is resolved if we realize that Eq. (101) is not the only
representation that satisfies Eq. (95), and the current bias is introduced as
a condition supplied to the Hamiltonian.

The condition is given by

$\displaystyle[\hat{n},{\tilde{\phi}}]=i,\quad\hat{\phi}={\tilde{\phi}}+\phi_{0}$
(103)

where $\phi_{0}$ is the expectation value $\langle\hat{\phi}\rangle$, and
${\tilde{\phi}}$ is the fluctuation from the average that satisfies

$\displaystyle\langle{\tilde{\phi}}\rangle=0$ (104)

The value of $\phi_{0}$ is chosen to satisfy the local charge conservation,
and the current bias is given by

$\displaystyle I=J_{c}\sin{\phi}_{0}$ (105)

Thus, the same problem is solved using the modified version of ${\cal
H}_{CPB}$ given by

$\displaystyle{\cal
H}^{m}_{CPB}=E_{C}\hat{n}^{2}-E_{J}\cos(\phi_{0}+\tilde{\phi})$ (106)

The term $-I\hat{\phi}$ is absent, making this Hamiltonian self-adjoint. When
the system is approached to zero current bias condition $I\rightarrow 0$, the
phase $\phi_{0}$ shifts to ${\phi}_{0}\rightarrow 0$. In this way, the system
is superconducting all the way without quantum phase transition.

Using the condition in Eq. (104), ${\cal H}^{m}_{CPB}$ becomes

$\displaystyle{\cal H}^{m}_{CPB}$ $\displaystyle=$ $\displaystyle
E_{C}\hat{n}^{2}-E_{J}[\cos\phi_{0}\cos\tilde{\phi}-\sin\phi_{0}\sin\tilde{\phi}]$
(107) $\displaystyle\approx$ $\displaystyle
E_{C}\hat{n}^{2}-E_{J}\cos\phi_{0}\cos\tilde{\phi}$

where the following approximation is used

$\displaystyle\sin\tilde{\phi}\approx\sin\langle\tilde{\phi}\rangle=0$ (108)

The current bias reduces the barrier height from $E_{J}$ to
$E_{J}\cos\phi_{0}$ in a similar manner as the washboard potential does. This
will explain the experimental results that are explained using the non-
hermitian Hamiltonian in Eq. (100) [51, 52].

##  VI Concluding remarks

In this work, theory of superconductivity is reformulated by including the
$U(1)$ phase neglected by Dirac. It gives rise to a $U(1)$ gauge field. The
quantization of it enables to include the fluctuation of the number of
electrons participating in the collective mode of this $U(1)$ field.

We can apply a similar formalism to other superfluid phenomena. For superfluid
phase of boson systems, the order parameter is often defined using the Bose-
Einstein condensation criterion by Penrose and Onsager [53]. This criterion
uses the single-particle density matrix given by

$\displaystyle\rho_{1}({\bf r},{\bf
r}^{\prime},t)=\langle\hat{\Psi}_{B}^{\dagger}({\bf r},t)\hat{\Psi}_{B}({\bf
r}^{\prime},t)\rangle$ (109)

where $\hat{\Psi}_{B}$ is the boson field operator.

It is reexpressed using the natural orbital basis $\\{f_{i}\\}$ and the
occupation number $n_{i}$ for the state $f_{i}$

$\displaystyle\rho_{1}({\bf r},{\bf r}^{\prime},t)$ $\displaystyle=$
$\displaystyle\sum_{i}n_{i}(t)f_{i}^{\ast}({\bf r},t)f_{i}({\bf
r}^{\prime},t)$ (110) $\displaystyle=$ $\displaystyle\Psi^{\ast}({\bf
r},t)\Psi({\bf r}^{\prime},t)+\sum_{i\neq 0}n_{i}(t)f_{i}^{\ast}({\bf
r},t)f_{i}({\bf r}^{\prime},t)$

where $f_{0}$ is the condensate wave function with $n_{0}$ in the order of the
total number of particles. The order parameter $\Psi({\bf r},t)$ is identified
as $f_{0}({\bf r},t)$ [8].

There is another definition of the order parameter given by

$\displaystyle\Psi({\bf r},t)=\langle\hat{\Psi}_{B}({\bf
r}^{\prime},t)\rangle$ (111)

that requires the particle-number non-conserving formalism [54]. We may regard
this formalism as an approximation to the one that includes the Dirac’s
neglected phase.

When the Dirac’s neglected phase is included, it gives rise to a collective
mode. We may take it as the superfluid current mode, and define a new single-
particle density matrix given by

$\displaystyle\tilde{\rho}_{1}({\bf r},{\bf
r}^{\prime},t)=\langle\hat{\rho}^{1/2}({\bf r},t)e^{i\hat{\varphi}({\bf
r},t)}e^{-i\hat{\varphi}({\bf r}^{\prime},t)}\hat{\rho}^{1/2}({\bf
r}^{\prime},t)\rangle$ (112)

where $\hat{\rho}$ is the number-density of the particle participating in the
collective mode, and $e^{\pm i\hat{\varphi}}$ are the number-changing
operators for particles participating in the collective mode.

If we approximate it as

$\displaystyle\tilde{\rho}_{1}({\bf r},{\bf r}^{\prime},t)$
$\displaystyle\approx$ $\displaystyle\langle\hat{\rho}^{1/2}({\bf
r},t)e^{i{\varphi}({\bf r},t)}e^{-i{\varphi}({\bf
r}^{\prime},t)}\rangle\langle\hat{\rho}^{1/2}({\bf r}^{\prime},t)\rangle$
(113) $\displaystyle\approx$ $\displaystyle\langle\hat{\rho}^{1/2}({\bf
r},t)\rangle e^{i{\varphi}({\bf r},t)}e^{-i{\varphi}({\bf
r}^{\prime},t)}\langle\hat{\rho}^{1/2}({\bf r}^{\prime},t)\rangle$
$\displaystyle=$ $\displaystyle\rho^{1/2}({\bf r},t)e^{i{\varphi}({\bf
r},t)}e^{-i{\varphi}({\bf r}^{\prime},t)}{\rho}^{1/2}({\bf r}^{\prime},t)$

we will have the following order parameter

$\displaystyle\Psi({\bf r},t)=e^{-i{\varphi}({\bf r},t)}{\rho}^{1/2}({\bf
r},t)$ (114)

In this way, the order parameter looks similar to Eq. (111) is obtained. At
present, we do not know the exact relation between the appearance of the order
parameter in Eq. (114) and the Bose-Einstein condensation. However, a close
relation is plausible, in which the particles in the condensate state will
play a similar role as the electrons near the Fermi level play. The reduction
of the total energy is expected to occur by exploiting the interaction energy
and the number fluctuation of the particles participating in the collective
mode.

In conclusion, we present a new theory of superconductivity that includes the
neglected $U(1)$ phase by Dirac. This theory captures major experimental
results explained by the BCS theory, and lifts shortcomings of it. It also
explains the absence of the dissipative quantum phase transition in Josephson
junctions.

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