# Berry connection from many-body wave functions and superconductivity:
Calculations by the particle number conserving Bogoliubov-de Gennes equations

Hiroyasu Koizumi  Center for computational sciences, University of Tsukuba,
Tsukuba, Ibaraki, Japan<EMAIL_ADDRESS>Alto Ishikawa
Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba,
Ibaraki, Japan

###### Abstract

A fundamentally revised version of superconductivity theory has been put
forward by the present authors since the standard theory of superconductivity
based on the BCS theory cannot explain superconductivity in cuprates
discovered in 1986, and reexaminations on several experimental results on the
conventional superconductors indicate the necessity for a fundamental
revision.

The revision is made on the origin of the superconducting phase variable,
which is attributed to a Berry connection arising from many-body wave
functions. With this revision, the theory can be cast into a particle number
conserving formalism.

We have developed a method to calculate superconducting states with the Berry
connection using the particle number conserving version of the Bogoliubov-de
Gennes equations. An example calculation is made for a model originally built
for cuprate superconductors.

##  I Introduction

Since the present work deals with a fundamental revision of the standard
theory of superconductivity, we start with historical evolution of
superconductivity theory.

After studying the gauge theories in quantum mechanics [1], London developed a
phenomenological theory of superconductivity from the view point of the wave
mechanics. He put forward the equation

$\displaystyle\hbar\nabla\chi^{\rm super}=m{\bf v}+{q\over c}{\bf A}^{\rm em}$
(1)

which explains the Meissner effect [2], where $\hbar$ is Planck’s constant
divided by $2\pi$, $\chi^{\rm super}$ is the superpotential, $q$ and $m$ are
the charge and mass of the particle, $c$ is the speed of light in vacuum, and
${\bf A}^{\rm em}$ is the electromagnetic vector potential, respectively.

London argued that the wave function for the superconducting state is rigid in
the sense that when a magnetic field is applied, and the wave function is
modified only by the change of the phase factor

$\displaystyle\Psi_{0}\rightarrow\Psi=\Psi_{0}e^{{i}\sum_{j}\chi^{\rm
super}({\bf r}_{j})}$ (2)

where $\Psi_{0}$ is the wave function when magnetic field is absent, and
$\Psi$ is that when it is present [2].

The supercurrent density ${\bf j}_{s}$ can be obtained using the rigid wave
function in Eq. (2) as

$\displaystyle{\bf j}_{s}=qn_{s}{\bf v}=-{{q^{2}n_{s}}\over{mc}}\left({\bf
A}^{\rm em}-{{\hbar c}\over q}\nabla\chi^{\rm super}\right)$ (3)

where $n_{s}$ is the number density of the charge carriers. This explains the
diamagnetic current that flows when a magnetic field is applied.

When Eq. (3) is combined with one of Maxwell’s equation

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

the equation that explains the exclusion of magnetic field from inside
superconductors (Meissner effect),

$\displaystyle\nabla^{2}{\bf B}^{\rm em}=-{1\over{\lambda_{L}^{2}}}{\bf
B}^{\rm em}$ (5)

is obtained, where the London penetration depth

$\displaystyle\lambda_{L}=\sqrt{{mc^{2}}\over{4\pi q^{2}n_{s}}}$ (6)

was introduced. This is the length scale where a magnetic field penetrates
into a superconductor.

The superpotential gives rise to fluxoid,

$\displaystyle\Phi_{\rm fluxoid}={{c\hbar}\over q}\oint\nabla\chi^{\rm
super}\cdot d{\bf r}$ (7)

In a ring-shaped superconductor where the magnetic field is excluded from the
bulk, the flux through the ring is given by

$\displaystyle\oint_{C}{\bf A}^{\rm em}\cdot d{\bf r}=\Phi_{\rm fluxoid}$ (8)

where $C$ is a loop along the ring inside the superconductor where ${\bf
j}_{s}=0$.

Experiments confirmed the presence of the fluxoid given by

$\displaystyle\Phi_{\rm fluxoid}={{hc}\over{2e}}n$ (9)

where $n$ is an integer [3].

Note that

$\displaystyle{\bf A}^{\rm eff}={\bf A}^{\rm em}-{{\hbar c}\over
q}\nabla\chi^{\rm super}$ (10)

must be gauge invariant since ${\bf j}_{s}$ in Eq. (3) is gauge invariant.
Thus, the angular variable $\chi^{\rm super}$ that makes ${\bf A}^{\rm eff}$
gauge invariant exists in superconductors.

The theory based on the wave mechanics formalism did not succeed in explaining
superconductivity. The first successful microscopic theory for
superconductivity, the BCS theory, was built by the formalism using creation
and annihilation operators [4].

Let $c^{\dagger}_{{\bf k}\sigma}$ be the creation operator for the electron in
the metal with wave vector ${\bf k}$ and spin $\sigma$, the BCS theory uses
the ground state composed of different particle number states

$\displaystyle|{\rm BCS}(\theta)\rangle=\prod_{\bf k}\left(u_{\bf k}+v_{\bf
k}c^{\dagger}_{{\bf k}\uparrow}c^{\dagger}_{-{\bf
k}\downarrow}e^{{i}{\theta}}\right)|{\rm vac}\rangle$ (11)

where $u_{\bf k}$ and $v_{\bf k}$ are real parameters that satisfy $u_{\bf
k}^{2}+v_{\bf k}^{2}=1$, and $|{\rm vac}\rangle$ is the vacuum that satisfy

$\displaystyle c_{{\bf k}\sigma}|{\rm vac}\rangle=0$ (12)

The ground state $|{\rm BCS}(\theta)\rangle$ is a linear combination of
different particle number states, thus, breaks the conservation of the
particle number. It has a degeneracy with respect to the choice of $\theta$,
and breaking the global $U(1)$ gauge invariance, the invariance of the
physical state by the change $c_{{\bf k}\sigma}\rightarrow e^{-{i\over
2}\theta}c_{{\bf k}\sigma},c^{\dagger}_{{\bf k}\sigma}\rightarrow e^{{i\over
2}\theta}c^{\dagger}_{{\bf k}\sigma}$, where $\theta$ is a constant.

Using the particle number non-conserving formalism, the BCS theory provides a
way to calculate the superconducting transition temperature as the energy gap
formation temperature by the electron-pairing.

A salient feature of the BCS theory is the presence of the Bogoliubov
excitation with an energy gap formed by the electrton-pairing [5]. This
excitation is most clearly seen if we use the Bogoliubov operators given by

$\displaystyle\gamma_{{\bf k}\uparrow}$ $\displaystyle=$ $\displaystyle
u_{k}e^{-{i\over 2}\theta}c_{{\bf k}\uparrow}-v_{k}e^{{i\over
2}\theta}c^{\dagger}_{-{\bf k}\downarrow}$ $\displaystyle\gamma_{-{\bf
k}\downarrow}$ $\displaystyle=$ $\displaystyle u_{k}e^{-{i\over
2}\theta}c_{-{\bf k}\downarrow}+v_{k}e^{{i\over 2}\theta}c^{\dagger}_{{\bf
k}\uparrow}$ (13)

where they satisfy

$\displaystyle\gamma_{{\bf k}\uparrow}|{\rm
BCS}(\theta)\rangle=0,\quad\gamma_{-{\bf k}\downarrow}|{\rm
BCS}(\theta)\rangle=0$ (14)

This indicates that the superconducting state is the “vacuum of the Bogoliubov
quasiparticlres”, which replaces Eq. (12).

Using the Bogoliubov operators, the BCS Hamiltonian can be cast into the
following,

$\displaystyle H_{\rm BCS}=\sum_{{\bf k},\sigma}E_{\bf
k}\gamma^{\dagger}_{{\bf k}\sigma}\gamma_{{\bf k}\sigma}+E_{\rm const}$ (15)

where

$\displaystyle E_{\bf k}=\sqrt{({\cal E}({\bf k}))^{2}+\Delta_{\bf k}^{2}}$
(16)

with ${\cal E}({\bf k})$ being the energy of Bloch electrons measured from the
Fermi energy; $\Delta_{\bf k}$ is the energy gap given by

$\displaystyle\Delta_{\bf k}=2E_{\bf k}u_{k}v_{k},\quad u_{k}^{2}={1\over
2}\left(1+{{{\cal E}({\bf k})}\over{E_{\bf k}}}\right),\quad v_{k}^{2}={1\over
2}\left(1-{{{\cal E}({\bf k})}\over{E_{\bf k}}}\right)$ (17)

$E_{\bf k}$ explains the energy gap spectrum observed in superconductors.

In the BCS theory, the induced current by a magnetic field is calculated as a
linear response to ${\bf A}^{\rm em}$, which gives the formula very similar to
Eq. (3) without the term with $\nabla\chi^{\rm super}$. The obtained current
is not gauge invariant, and the gauge invariance of the induced current became
a big issue. After seminal works by Anderson [6, 7], a final resolution was
provided by Nambu[8], culminating to the idea “spontaneous gauge symmetry
breaking”. Nambu showed the the Nambu-Goldstone mode appears due to the
degeneracy of the ground state with respect to the choice of $\theta$, and
this mode retrieves the gauge invariance.

Independent of the BCS theory, a phenomenological theory was developed by
Ginzburg and Landau [9] based in the London theory of superconductivity. The
gauge invariant supercurrent is most easily calculated by this Ginzburg-Landau
theory [9]. In this theory a free energy for the superconducting state is
given as a functional of an order parameter

$\displaystyle\Psi_{\rm GL}=n_{s}^{1/2}e^{i\chi^{\rm super}}$ (18)

The free energy is composed of a material part

$\displaystyle F_{\rm mat}=F_{\rm normal}+\int
d^{3}r{1\over{2m}}\left|\left({\hbar\over i}\nabla-{q\over c}{\bf A}^{\rm
em}\right)\Psi_{\rm GL}\right|^{2}+\int d^{3}r\left(\alpha|\Psi_{\rm
GL}|^{2}+{\beta\over 2}|\Psi_{\rm GL}|^{4}\right)$ (19)

where $\alpha<0$, $\beta>0$ are parameters and $F_{\rm normal}$ is the free
energy of the normal phase, and a magnetic field part

$\displaystyle F_{\rm mag}=\int d^{3}r{1\over{8\pi}}\left({\bf B}^{\rm
em}\right)^{2}$ (20)

In the Ginzburg-Landau theory, the supercurrent is calculated as

$\displaystyle{\bf j}_{s}=-c{{\delta F_{\rm mat}}\over{\delta{\bf A}^{\rm
em}({\bf r})}}=-{{q^{2}n_{s}}\over{mc}}\left({\bf A}^{\rm em}-{{\hbar c}\over
q}\nabla\chi^{\rm super}\right)$ (21)

in accordance with Eq. (3). The gauge invariance is retrieved by $\chi^{\rm
super}$ [10].

The connection between the BCS and Ginzburg-Landau theories was provided by
Gor’kov [11]. The Gor’kov’s derivation yields,

$\displaystyle m=2m^{\ast},\quad q=-2e$ (22)

where $m^{\ast}$ is the effective mass of electron in the metal.

Although the Gor’kov derivation has been considered as a valid one, it
disagrees with the experimental value $m=2m_{e}$ from the London moment
measreuemen where $m_{e}$ is the free electron mass [12, 13].

A notable point of the Ginzburg-Landau theory is that it introduces the
coherence length

$\displaystyle\xi_{\rm GL}=\sqrt{\hbar^{2}\over{2m|\alpha|}}$ (23)

The BCS theory also introduces a similar quantity, the BCS coherence length,

$\displaystyle\xi_{\rm BCS}={{\hbar v_{\rm Fermi}}\over{\pi\Delta}}$ (24)

Currently, it is interpreted that it is the size of the electron-pair, where
$\Delta$ is the energy gap and $v_{\rm Fermi}$ is the velocity of the electron
at the Fermi energy [4].

Based on the Ginzburg-Landau theory, Abrikosov showed that there are two types
of superconductors depending on the ratio of $\lambda_{L}$ and $\xi_{\rm GL}$,

$\displaystyle\mbox{type I }\quad{\lambda_{L}\over{\xi_{\rm
GL}}}<{1\over\sqrt{2}};\quad\quad\mbox{type II
}\quad{\lambda_{L}\over{\xi_{\rm GL}}}>{1\over\sqrt{2}},$ (25)

In type II superconductors, applied magnetic field can penetrate the sample as
vortices, and $\xi_{\rm GL}$ is the core size of such vortices [14].

Remarkable effects caused by $\chi^{\rm super}$ were predicted by Josephson
[15]. They are called “dc Josephson effect” and “ac Josephson effect”, and
occur in the junction composed of two superconductors separated by a thin
barrier (or insulator region) depicted in Fig. 1.

Figure 1: Josephson junction SL-I-SR (superconductor-insulator-superconductor
junction) considered by Josephson. $\mu_{R}$ and $\mu_{L}$ are chemical
potentials of SL and SR, respectively. When voltage $V$ exists the relation
$eV=\mu_{R}-\mu_{L}$ holds.

The current through the junction is given by

$\displaystyle J=J_{c}\sin\phi$ (26)

where $J_{c}$ is a parameter for the junction, and $\phi$ is given using the
gauge invariant ${\bf A}^{\rm eff}$ as

$\displaystyle\phi=-{q\over{\hbar c}}\int_{L}^{R}\left({\bf A}^{\rm
em}-{{\hbar c}\over q}\nabla\chi^{\rm super}\right)+\phi_{0}$ (27)

where $\phi_{0}$ is a constant and the integration is taken along the line
connecting the two superconductors.

The ac Josephson effect occurs when there is a voltage across the junction.
When the voltage is $V$, time variation of $\phi$ given by

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

occurs.

This effect is most clearly seen if a radiation field with frequency $f$
exists; the resonance condition $\dot{\phi}=2\pi fn$, where $n$ is an integer,
gives

$\displaystyle V={{hf}\over{2e}}n$ (29)

Then, steps are observed in the $I$-$V$ curve (current-voltage curve) [16].

Josephson obtained Eq. (28) based on the BCS theory by assuming $q=-2e$ [15].
His derivation had been long considered to be right one. However, it has been
argued that it misses a contribution; Eq. (28) can be obtained by $q=-e$ [17,
18, 19, 20, 21].

Before the discovery of high temperature superconductivity in 1986 [22], the
BCS theory and its extensions have been very successful and was believed that
superconductivity was a well-understood solved problem. However, the cuprate
superconductivity is markedly different from the BCS superconductivity in a
number of ways; for example, the magnetism is harmful in the BCS
superconductivity since it causes the breaking of electron-pairing, however
magnetism coexists with superconductivity in the cuprate [23]; the
superconducting transition temperature is not the energy gap formation
temperature, but corresponds to the stabilizing temperature for loop currents
of $\xi_{\rm GL}$ size in optimally doped sample of the cuprates [24]; the
charge carriers in the room temperature become small polarons at low
temperatures [25].

Since the cuprate superconductivity is so different from the BCS
superconductivity, a theory much departs from the standard one is expected to
be needed [17]. The superconducting state transition temperature indicates the
importance of $\xi_{\rm GL}$-sized loop currents, and it has been argued that
$\xi_{\rm GL}$-sized spin-vortices and loop currents appear around the small
polarons in the bulk, explaining the magnetic excitation spectrum observed by
inelastic neutron scattering [26].

Efforts toward the elucidation of the cuprate superconductivity lead some
researchers to reexamine superconductivity from very fundamental levels. Those
efforts revealed that there are some experimental facts disagreeing with the
standard theory even in the conventional superconductors [27]. We would like
to point out four major problems below:

1) Supercurrent in the standard theory contradicts the reversible
superconducting-normal phase transition in a magnetic field observed in type I
superconductors.

Magnetic field is excluded from the bulk of superconductors, thereby the
supercurrent is generated in the surface region of width $\lambda_{L}$ in Eq.
(6) as the screening current for the magnetic field.

However, the Meissner effect is not just the exclusion of the magnetic field.
It also indicates that the superconducting state is a thermodynamically stable
phase in the $T-H$ plane, where $T$ is the temperature and $H$ is the external
magnetic field. Thus, the surface current stops without generating
irreversible Joule heat during the superconducting to currentless normal
metallic phase transition. Note that the thermodynamically stable phase of the
normal metal in a magnetic field is currentless.

Actually, after the discovery of the Meissner effect, the reversibility of the
superconducting-normal state transition in the presence of a magnetic field
was confirmed [28, 29, 30, 31, 32]; and the state of the art calorimetry
indicates that 99.99% of the supercurrent stops without current carriers
undergoes irreversible collisions (see Appendix B of Ref. [33]). Hirsch argued
that the standard theory of superconductivity cannot explain how this
transition is possible since the supercurrent generated by the flow of
electron pairs inevitably produces the Joule heat due to the existence of a
significant number of broken pairs that flow with dissipation during the
transition [33, 34, 35].

It is also note worthy that the use of the linear response theory to calculate
the supercurrent may be problematic since it is tied to the fluctuation-
dissipation relation [36], and the current calculated by it usually causes
dissipation. This point has drawn renewed attention in relation to the absence
of a dissipative quantum phase transition in Josephson junctions [37, 38, 39].

One may say that the Ginzburg-Landau theory is derivable from the BCS theory
[11], the supercurrent generation in the latter theory can be used as the
evidence that the BCS theory explains the Meissner effect. However, the
Gor’kov’s derivation from BCS to Ginzburg-Landau theories disagrees in the
mass in the London moment as explained below.

2) Experiments indicate that the mass in the London moment is the free
electron mass, however, the standard theory predicts it to be an effective
mass.

The London moment is the magnetic moment generated by a rotating
superconductor in a magnetic field [40, 2]. Inside the superconductor magnetic
field, the “London field”, is produced by the supercurrent flowing in the
surface region.

Let us consider a superconducting sphere rotating about its symmetry axis with
constant angular velocity ${\bm{\omega}}$. Then, the velocity at the position
${\bf r}$ measured from the center of the sphere is given by

$\displaystyle{\bf v}={\bm{\omega}}\times{\bf r}$ (30)

Electrons inside the superconductor move with this velocity to shield the
background positive charge. Substituting this in Eq. (1) and using the
relation ${\bf B}^{\rm em}=\nabla\times{\bf A}^{\rm em}$, we have

$\displaystyle{\bf B}^{\rm em}=-{{2mc}\over q}{\bm{\omega}}$ (31)

This is the London field, where $m$ is mass, $q$ is charge, and $c$ is the
speed of light.

The London moment and field have been measured many times using different
materials, ranging from the conventional superconductor [41, 42, 43, 44, 45]
to the high Tc cuprates [46, 47] and heavy fermion superconductors [48]. The
results always indicate that the mass $m$ is the free electron mass $m_{e}$ if
$q=-e$ is used, not the effective mass $m^{\ast}$ predicted by the standard
theory including the Gorkov’s derivation.

3) In the standard theory, the breakdown of the global $U(1)$ gauge invariance
or the non-conservation of the particle number is essential.

It is sensible to consider that the particle number is conserved in an
isolated superconductor. However, the standard theory requires to use the
particle number non-conserving formalism as manifested in the use of Eq. (11)
[49, 50]. It has been also claimed that the use of $\theta$ as a physically
meaningful parameter is against the superselection rule for charge [51]. Thus,
it has been argued that current formalism is an approximation of the true
formalism in which the particle number is conserved [52, 50].

4) The derivation of the ac Josephson effect takes into account only the half
of the contributions.

We shall revisit the ac Josephson effect: let us calculate $\dot{\phi}$ using
Eq. (27),

$\displaystyle\dot{\phi}=-{q\over{\hbar c}}\int_{R}^{L}d{\bf
r}\cdot\left(\partial_{t}{\bf A}^{\rm em}-{{\hbar
c}\over{q}}\nabla\partial_{t}{\chi}^{\rm
super}\right)={q\over\hbar}\int_{R}^{L}d{\bf r}\cdot{\bf E}^{\rm
em}-\left.{q\over\hbar}\left(-\varphi^{\rm
em}-{\hbar\over{q}}\partial_{t}{\chi}^{\rm super}\right)\right|^{L}_{R}$ (32)

where the relation

$\displaystyle{\bf E}^{\rm em}=-{1\over c}\partial_{t}{\bf A}^{\rm
em}-\nabla\varphi^{\rm em}$ (33)

is used.

Figure 2: Josephson junction in a real experimental situation, connected to
leads, denoted as LeadL and LeadR. The ac Josephson effect is observed as the
quantized voltage given in Eq. (29) under the application a microwave of
frequency $f$ and a dc current through the junction.

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

$\displaystyle{q\over\hbar}\int_{R}^{L}d{\bf r}\cdot{\bf E}^{\rm
em}=-{{qV}\over\hbar}$ (34)

where $V$ is the voltage across the junction.

The second one is

$\displaystyle-\left.{q\over\hbar}\left(-\varphi^{\rm
em}-{\hbar\over{q}}\partial_{t}{\chi}^{\rm
super}\right)\right|^{L}_{R}={q\over\hbar}{{\mu_{L}-\mu_{R}}\over e}$ (35)

Here, the time-component partner of Eq. (10),

$\displaystyle\varphi^{\rm eff}=\varphi^{\rm em}+{\hbar\over{q}}\partial^{\rm
super}_{t}{\chi}$ (36)

is identified as

$\displaystyle\varphi^{\rm eff}={\mu\over e}$ (37)

where $\mu$ is the chemical potential.

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

$\displaystyle eV=\mu_{R}-\mu_{L}$ (38)

Thus, the sum of the two contributions yields

$\displaystyle\dot{\phi}=-{{2qV}\over\hbar}$ (39)

In order to obtain the result in Eq. (28), $q$ needs to be $q=-e$, this
contradicts the value used by Josephson $q=-2e$.

The above discrepancy comes from the fact that only one of the two
contributions in Eq. (32) is included in the Josephson’s derivation [15, 53].

The first term with ${\bf E}^{\rm em}$ in Eq. (32) arises when charged
particles go through the electric field in the insulator region. The second
term with $\varphi^{\rm em}$ and $\chi^{\rm super}$ arises due to the fact
that the chemical potentials are maintained by the contact with the leads; in
order to keep the chemical potential constant, electrons enter from one of the
leads and exit to the other lead as electrons tunnel through “I” region in
Fig. 2. Since experiments are performed in the presence of a dc current in
addition to a microwave field, it is sensible to consider that the two
contributions both exit.

If $q=-2e$ is used as in the standard theory, the Josephson relation becomes
$\dot{\phi}={{4eV}\over\hbar}$. In this case, half-integers will appear in Eq.
(29). Indeed, such effects have been observed [54].

As Eq. (3) indicates, the most important ingredient for supercurrent
generation is $\chi^{\rm super}$. If an angular variable$\chi$ with period
$2\pi$ that satisfy

$\displaystyle\nabla\chi^{\rm super}=-{1\over 2}\nabla\chi$ (40)

exits, the experimental values of fluxoid are obtained by $q=-e$. $q=-2e$
contributions may exist since the electron pairing occurs in the system;
however, the view that the supercurrent is the flow of paired-electrons will
not be valid according to the problem given above in the item 1). It is also
noteworthy to recognize that careful treatments of the spatial and temporal
fluctuations of ${\bf E}^{\rm em}$ are needed to understand Josephson effect
related phenomena, correctly [55, 21].

The new theory is the one that can be formulated in a particle number
conserving manner [56, 19, 57, 18, 13, 21]. We will present a method to
calculate properties of superconducting states, such as diamagnetic response
current and persistent current, using the particle number conserving
formalism.

The organization of the present work is as follows: In Section II particle
number conserving version of the BCS theory is explained. This formalism
becomes possible due to the presence of non-trivial Berry connection. In
Section III model calculations are performed using the particle number
conserving Bogoliubov-de Gennes equations derived from the particle number
conserving version of the BCS theory. A model constructed for the cuprate
superconductivity is used to demonstrate calculations. We conclude this work
with some remarks in Section IV.

Four appendices are included: Berry connection from many-body wave functions
and the modification of Maxwell’s equations are explained in Appendix A;
appearance of ${\bf A}^{\rm fic}$ from spin-twisting itinerant motion of
electrons is explained in Appendix B; reversible superconducting-normal metal
phase transition in a magnetic field is discussed in Appendix C; and
collective mode $\chi$ and associated number changing operators are derived in
Appendix D.

##  II Particle number conserving version of the BCS theory using the
superconducting phase from the Berry connection

In the original BCS theory, electrons in the normal metallic state are assumed
to be well-described by the free electrons with the effective mass $m^{\ast}$.
Then, the electron field operators are given by

$\displaystyle\hat{\Psi}_{\sigma}({\bf r})={1\over\sqrt{\cal V}}\sum_{\bf
k}e^{i{\bf k}\cdot{\bf r}}c_{{\bf k}\sigma}$ (41)

where ${\cal V}$ is the volume of the system.

Using the Bogoliubov operators [58, 59] they are expressed as

$\displaystyle\hat{\Psi}_{\uparrow}({\bf r})$ $\displaystyle=$
$\displaystyle{1\over\sqrt{\cal V}}\sum_{\bf k}e^{{i\over 2}\theta}e^{i{\bf
k}\cdot{\bf r}}\left(\gamma_{{\bf k}\uparrow}u_{k}-\gamma^{\dagger}_{-{\bf
k}\downarrow}v_{k}\right)$ $\displaystyle\hat{\Psi}_{\downarrow}({\bf r})$
$\displaystyle=$ $\displaystyle{1\over\sqrt{\cal V}}\sum_{\bf k}e^{{i\over
2}\theta}e^{i{\bf k}\cdot{\bf r}}\left(\gamma_{{\bf
k}\downarrow}u_{k}+\gamma^{\dagger}_{-{\bf k}\uparrow}v_{k}\right)$ (42)

Now we depart from the standard theory by replacing $e^{\pm{i\over 2}\theta}$
in Eq. (42) by the number changing operators $e^{\pm{i\over 2}\hat{\chi}}$.
Here, we do not specify the coordinate in $\hat{\chi}$ (we will introduce the
coordinate dependence, later). Then, the Bogoliubov transformation in Eq. (13)
becomes

$\displaystyle\gamma_{{\bf k}\uparrow}$ $\displaystyle=$ $\displaystyle
u_{k}e^{{i\over 2}\hat{\chi}}c_{{\bf k}\uparrow}-v_{k}e^{-{i\over
2}\hat{\chi}}c^{\dagger}_{-{\bf k}\downarrow}$ $\displaystyle\gamma_{-{\bf
k}\downarrow}$ $\displaystyle=$ $\displaystyle u_{k}e^{{i\over
2}\hat{\chi}}c_{-{\bf k}\downarrow}+v_{k}e^{-{i\over
2}\hat{\chi}}c^{\dagger}_{{\bf k}\uparrow}$ (43)

Note that the above Bogoliubov operators conserve particle numbers. Terms like
$e^{{i\over 2}\hat{\chi}}c_{{\bf k}\sigma}$ can be interpreted that an
electron in the $({\bf k},\sigma)$ single-electron mode is removed and an
electron is added to the collective mode described by $\chi$; those like
$e^{-{i\over 2}\hat{\chi}}c^{\dagger}_{{\bf k}\sigma}$ create an electron in
the $({\bf k},\sigma)$ single-electron mode and subtract an electron from the
collective mode described by $\chi$. Thus, the Bogoliubov operators cause the
fluctuation of the number of electrons in the collective mode by transferring
electrons between single-particle modes and collective mode. As a consequence,
the ground state becomes a linear combination of states with different number
of electrons in the collective mode [56]. This state replaces the state with
fluctuating total number of electrons (given by Eq. (11)) in the standard
theory.

By including ${1\over\sqrt{\cal V}}e^{i{\bf k}\cdot{\bf r}}$ in $u_{k}$ and
$v_{k}$, we define

$\displaystyle u_{\bf k}({\bf r})={1\over\sqrt{\cal V}}e^{i{\bf k}\cdot{\bf
r}}u_{k},\quad v_{\bf k}({\bf r})={1\over\sqrt{\cal V}}e^{i{\bf k}\cdot{\bf
r}}v_{k}$ (44)

Then, the field operators become

$\displaystyle\hat{\Psi}_{\uparrow}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\bf k}e^{-{i\over 2}\hat{\chi}({\bf r})}\left(\gamma_{{\bf
k}\uparrow}u_{\bf k}({\bf r})-\gamma^{\dagger}_{{\bf
k}\downarrow}v^{\ast}_{\bf k}({\bf r})\right)$
$\displaystyle\hat{\Psi}_{\downarrow}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\bf k}e^{-{i\over 2}\hat{\chi}({\bf r})}\left(\gamma_{{\bf
k}\downarrow}u_{\bf k}({\bf r})+\gamma^{\dagger}_{{\bf
k}\uparrow}v^{\ast}_{\bf k}({\bf r})\right)$ (45)

where the spatial dependence is included in $\hat{\chi}$.

Now we allow the coordinate dependent functions other than plane waves. We use
the label $n$ in place of the wave number ${\bf k}$ for them. As a result, the
field operators become

$\displaystyle\hat{\Psi}_{\uparrow}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{n}e^{-{i\over 2}\hat{\chi}({\bf
r})}\left(\gamma_{{n}\uparrow}u_{n}({\bf
r})-\gamma^{\dagger}_{{n}\downarrow}v^{\ast}_{n}({\bf r})\right)$
$\displaystyle\hat{\Psi}_{\downarrow}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{n}e^{-{i\over 2}\hat{\chi}({\bf
r})}\left(\gamma_{{n}\downarrow}u_{n}({\bf
r})+\gamma^{\dagger}_{{n}\uparrow}v^{\ast}_{n}({\bf r})\right)$ (46)

The particle number conserving Bogoliubov operator $\gamma_{n\sigma}$
satisfies

$\displaystyle\gamma_{n\sigma}|{\rm Gnd}(N)\rangle=0$ (47)

where $|{\rm Gnd}(N)\rangle$ is the ground state with the total number of
particles $N$.

The ground state also satisfies

$\displaystyle e^{\pm{i\over 2}\hat{\chi}({\bf r})}|{\rm
Gnd}(N)\rangle=e^{\pm{i\over 2}{\chi}({\bf r})}|{\rm Gnd}(N\pm 1)\rangle$ (48)

The above phase factor gives

$\displaystyle\langle{\rm Gnd}(N)|e^{{i\over 2}\hat{\chi}({\bf
r}_{f})}e^{-{i\over 2}\hat{\chi}({\bf r}_{i})}|{\rm Gnd}(N)\rangle=e^{{i\over
2}\int_{{\bf r}_{i}}^{{\bf r}_{f}}\nabla{\chi}\cdot d{\bf r}}$ (49)

indicating that the phase ${1\over 2}\int_{{\bf r}_{i}}^{{\bf
r}_{f}}\nabla{\chi}\cdot d{\bf r}$ is acquired when the particle travels from
${\bf r}_{i}$ to ${\bf r}_{f}$, in accordance with the presence of ${\bf
A}^{\rm fic}$ in Eq. (200).

Let us consider the electronic Hamiltonian expressed by the field operators

$\displaystyle H=\sum_{\sigma}\int d^{3}r\hat{\Psi}^{\dagger}_{\sigma}({\bf
r})h({\bf r})\hat{\Psi}_{\sigma}({\bf r})-{1\over
2}\sum_{\sigma,\sigma^{\prime}}\int d^{3}rd^{3}r^{\prime}V_{\rm eff}({\bf
r},{\bf r}^{\prime})\hat{\Psi}^{\dagger}_{\sigma}({\bf
r})\hat{\Psi}^{\dagger}_{\sigma^{\prime}}({\bf
r}^{\prime})\hat{\Psi}_{\sigma^{\prime}}({\bf
r}^{\prime})\hat{\Psi}_{\sigma}({\bf r})$ (50)

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

$\displaystyle h({\bf r})={1\over{2m_{e}}}\left({\hbar\over i}\nabla+{e\over
c}{\bf A}^{\rm em}\right)^{2}+U({\bf r})-\mu$ (51)

and $U({\bf r})$ is the single particle potential energy, $\mu$ is the
chemical potential, and $-V_{\rm eff}$ is the effective interaction between
electrons. In the BCS theory, $-V_{\rm eff}$ describes the attractive
interaction between electrons via the virtual phonon exchange.

We perform the following mean field approximation

$\displaystyle H^{\rm MF}$ $\displaystyle=$ $\displaystyle\sum_{\sigma}\int
d^{3}r\hat{\Psi}^{\dagger}_{\sigma}({\bf r})h({\bf r})\hat{\Psi}_{\sigma}({\bf
r})+\int d^{3}rd^{3}r^{\prime}\left[\Delta({\bf r},{\bf
r}^{\prime})\hat{\Psi}^{\dagger}_{\uparrow}({\bf
r})\hat{\Psi}^{\dagger}_{\downarrow}({\bf r}^{\prime})e^{-{i\over
2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf r}^{\prime}))}+{\rm H.c.}\right]$ (52)
$\displaystyle+$ $\displaystyle\int d^{3}rd^{3}r^{\prime}{{|\Delta({\bf
r},{\bf r}^{\prime})|^{2}}\over{V_{\rm eff}({\bf r},{\bf r}^{\prime})}}$

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

$\displaystyle\Delta({\bf r},{\bf r}^{\prime})=V_{\rm eff}({\bf r},{\bf
r}^{\prime})\langle e^{{i\over 2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf
r}^{\prime}))}\hat{\Psi}_{\uparrow}({\bf r})\hat{\Psi}_{\downarrow}({\bf
r^{\prime}})\rangle$ (53)

Due to the factor $e^{{i\over 2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf
r}^{\prime})}$ the expectation value can be calculated using the particle
number fixed state.

Using commutation relations for $\hat{\Psi}^{\dagger}_{\sigma}({\bf r})$ and
$\hat{\Psi}_{\sigma}({\bf r})$,

$\displaystyle\\{\hat{\Psi}_{\sigma}({\bf
r}),\hat{\Psi}^{\dagger}_{\sigma^{\prime}}({\bf
r}^{\prime})\\}=\delta_{\sigma\sigma^{\prime}}\delta({\bf r}-{\bf
r}^{\prime})$ $\displaystyle\\{\hat{\Psi}_{\sigma}({\bf
r}),\hat{\Psi}_{\sigma^{\prime}}({\bf r}^{\prime})\\}=0$
$\displaystyle\\{\hat{\Psi}^{\dagger}_{\sigma}({\bf
r}),\hat{\Psi}^{\dagger}_{\sigma^{\prime}}({\bf r}^{\prime})\\}=0$ (54)

the following relations are obtained

$\displaystyle\left[\hat{\Psi}_{\uparrow}({\bf r}),{H}_{\rm MF}\right]$
$\displaystyle=$ $\displaystyle{h}({\bf r})\hat{\Psi}_{\uparrow}({\bf r})+\int
d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})\hat{\Psi}^{\dagger}_{\downarrow}({\bf r}^{\prime})e^{-{i\over
2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf r}^{\prime}))}$
$\displaystyle\left[\hat{\Psi}_{\downarrow}({\bf r}),{H}_{\rm MF}\right]$
$\displaystyle=$ $\displaystyle{h}({\bf r})\hat{\Psi}_{\downarrow}({\bf
r})-\int d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})\hat{\Psi}^{\dagger}_{\uparrow}({\bf r}^{\prime})e^{-{i\over
2}(\hat{\chi}({\bf r})+\hat{\chi}({\bf r}^{\prime}))}$ (55)

The particle number conserving Bogoliubov operators $\gamma_{n\sigma}$ and
$\gamma^{\dagger}_{n\sigma}$ obey fermion commutation relations. They are
chosen to satisfy

$\displaystyle\left[{H}_{\rm
MF},\gamma_{n\sigma}\right]=-\epsilon_{n}\gamma_{n\sigma},\quad\left[{H}_{\rm
MF},\gamma^{\dagger}_{n\sigma}\right]=\epsilon_{n}\gamma^{\dagger}_{n\sigma}$
(56)

yielding the diagonalized ${H}_{\rm MF}$,

$\displaystyle{H}_{\rm
MF}=E_{g}+\sum_{n,\sigma}\epsilon_{n}\gamma^{\dagger}_{n\sigma}\gamma_{n\sigma}$
(57)

where the excitation energies satisfy $\epsilon_{n}>0$, and $E_{g}$ is the
ground state energy.

From Eqs. (55), (56), and (57), the following system of equations are obtained

$\displaystyle\epsilon_{n}e^{-{i\over 2}\hat{\chi}({\bf r})}u_{n}({\bf r})$
$\displaystyle=$ $\displaystyle h({\bf r})e^{-{i\over 2}\hat{\chi}({\bf
r})}u_{n}({\bf r})+\int d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})e^{-{i\over 2}\hat{\chi}({\bf r})}v_{n}({\bf r}^{\prime})$
$\displaystyle\epsilon_{n}e^{-{i\over 2}\hat{\chi}({\bf r})}v^{\ast}_{n}({\bf
r})$ $\displaystyle=$ $\displaystyle-h({\bf r})e^{-{i\over 2}\hat{\chi}({\bf
r})}v^{\ast}_{n}({\bf r})+\int d^{3}r^{\prime}\Delta({\bf r},{\bf
r}^{\prime})e^{-{i\over 2}\hat{\chi}({\bf r})}u^{\ast}_{n}({\bf r}^{\prime})$
(58)

Using the relation in Eq. (48), the above are cast into the following,

$\displaystyle\epsilon_{n}u_{n}({\bf r})$ $\displaystyle=$
$\displaystyle\bar{h}({\bf r})u_{n}({\bf r})+\int d^{3}r^{\prime}\Delta({\bf
r},{\bf r}^{\prime})v_{n}({\bf r}^{\prime})$
$\displaystyle\epsilon_{n}v_{n}({\bf r})$ $\displaystyle=$
$\displaystyle-\bar{h}^{\ast}({\bf r})v_{n}({\bf r})+\int
d^{3}r^{\prime}\Delta^{\ast}({\bf r},{\bf r}^{\prime})u_{n}({\bf r}^{\prime})$
(59)

where

$\displaystyle\bar{h}({\bf r})={1\over{2m_{e}}}\left({\hbar\over
i}\nabla+{e\over c}{\bf A}^{\rm em}-{\hbar\over 2}\nabla\chi\right)^{2}+U({\bf
r})-\mu$ (60)

and

$\displaystyle\Delta({\bf r},{\bf r}^{\prime})=V_{\rm eff}({\bf r},{\bf
r}^{\prime})\sum_{n}\left[u_{n}({\bf r})v^{\ast}_{n}({\bf
r}^{\prime})(1-f(\epsilon_{n}))-u_{n}({\bf r}^{\prime})v^{\ast}_{n}({\bf
r})f(\epsilon_{n})\right]$ (61)

$f(\epsilon_{n})$ is the Fermi function given by

$\displaystyle f(\epsilon_{n})={1\over{e^{\epsilon_{n}\over{k_{\rm B}T}}+1}}$
(62)

where $k_{\rm B}$ is the Boltzmann constant.

The above system of equations are Bogoliubov-de Gennes equations [58] using
the particle number conserving Bogoliubov operators [56].

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

$\displaystyle{\bf A}^{\rm eff}={\bf A}^{\rm em}-{{\hbar
c}\over{2e}}\nabla\chi$ (63)

In the BCS superconductor, the energy gain by the electron-pair formation
exceeds the energy loss by performing the spin-twisting itinerant motion that
is necessary to have the number changing operators; thus, even if the spin-
orbit interaction is negligibly small, the spin-twisting itinerant motion will
occur. This will be the reason for the simultaneous occurrence of
superconductivity and energy gap formation.

By solving the system of equations composed of Eqs. (59), (60), and (61), with
the condition ${\bf A}^{\rm eff}=0$, we obtain the currentless solutions for
$u_{n},v_{n}$, which we denote as $\acute{u}_{n},\acute{v}_{n}$.

Using $\acute{u}_{n},\acute{v}_{n}$, we express ${u}_{n},{v}_{n}$ as

$\displaystyle u_{n}({\bf r})=\acute{u}_{n}({\bf r})e^{{i\over 2}{\chi}({\bf
r})},\quad v_{n}({\bf r})=\acute{v}_{n}({\bf r})e^{-{i\over 2}{\chi}({\bf
r})}$ (64)

We obtain $\chi$ in the following manner. First, we note that $\chi$ has to
satisfy constraints,

$\displaystyle\oint_{C_{\ell}}\nabla\chi\cdot d{\bf r}=2\pi
w_{C_{\ell}}[\chi]$ (65)

where $w_{C_{\ell}}[\chi]$ is the integer called the winding number. The value
$w_{C_{\ell}}[\chi]$ needs to satisfy the condition similar to the one in Eq.
(229) to make the wave function single-valued.

We obtain $\chi$ for the ground state by minimizing the total energy under the
above constraint. For this purpose, we use the following functional,

$\displaystyle F[\nabla\chi]=E[\nabla\chi]+\sum_{\ell=1}^{N_{\rm
loop}}{{\lambda_{\ell}}}\left(\oint_{C_{\ell}}\nabla\chi\cdot d{\bf r}-2\pi
w_{C_{\ell}}[\chi]\right),$ (66)

where

$\displaystyle E[\nabla\chi]=\langle{\rm Gnd}|H_{\rm eff}|{\rm Gnd}\rangle$
(67)

is obtained using Eq. (64); $\lambda_{\ell}$’s are Lagrange multipliers, and
$\\{C_{1},\cdots,C_{N_{\rm loop}}\\}$ are boundaries of plaques of the lattice
(we assume we have the lattice version of the Hamiltonian); and $N_{\rm loop}$
is the total number of plaques of the lattice.

The minimum condition of $F[\nabla\chi]$ with respect to the variation of
$\nabla\chi$ yields,

$\displaystyle{{\delta
E[\nabla\chi]}\over{\delta\nabla\chi}}+\sum_{\ell=1}^{N_{\rm
loop}}{{\lambda_{\ell}}}{{\delta}\over{\delta\nabla\chi}}\oint_{C_{\ell}}\nabla\chi\cdot
d{\bf r}=0$ (68)

Then, the current density is given by

$\displaystyle{\bf J}={{2e}\over\hbar}{{\delta
E}\over{\delta\nabla\chi}}=-{{2e}\over\hbar}\sum_{\ell=1}^{N_{\rm
loop}}{{\lambda_{\ell}}}{{\delta}\over{\delta\nabla\chi}}\oint_{C_{\ell}}\nabla\chi\cdot
d{\bf r}$ (69)

This indicates that the current flowing in superconducting states is a
collection of loop currents.

Expressing Eqs. (65) and (68) using the discrete lattice Hamiltonian, a system
of equations for $\nabla\chi$ is obtained,

$\displaystyle\sum_{k\leftarrow j}L_{k\leftarrow j}^{\ell}\tau_{k\leftarrow
j}$ $\displaystyle=$ $\displaystyle 2\pi w_{C_{\ell}}[\chi]$ (70)
$\displaystyle{{\partial E(\\{\tau_{k\leftarrow
j}\\})}\over{\partial\tau_{k\leftarrow j}}}$ $\displaystyle+$
$\displaystyle\sum_{\ell=1}^{N_{\rm
loop}}{{\lambda_{\ell}}}{{\partial}\over{\partial\tau_{k\leftarrow
j}}}\sum_{k\leftarrow j}L_{k\leftarrow j}^{\ell}\tau_{k\leftarrow j}=0$ (71)

where ${k\leftarrow j}$ indicates the bond that starts from site $j$ and ends
at site $k$, $\tau_{k\leftarrow j}$ is the difference of $\chi$ for the bond
$k\leftarrow j$

$\displaystyle\tau_{k\leftarrow j}=\chi_{k}-\chi_{j}$ (72)

and $L_{k\leftarrow j}^{\ell}$ is defined as

$\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.$ (76)

Note that a set of parameters $\\{w_{C_{\ell}}[\chi]\\}$ must be supplied as
part of boundary conditions.

We take the branch of $\chi_{j}$ that satisfies the difference of value from
the nearest neighbor site $k$ is in the range,

$\displaystyle-\pi\leq\chi_{j}-\chi_{k}<\pi$ (77)

Figure 3: An example of a simply-connected lattice constructed from square
lattice. $5\times 5$ square lattice in ${\bf a}$ becomes a simply-connected
lattice in ${\bf b}$ by removing 16 bonds.

From ($\chi_{j}-\chi_{k}$)’s, we rebuild $\chi$. The rebuilding process is as
follows: first, we pick a value for the initial $\chi_{1}$ (say $\chi_{1}=0$).
After fixing the value of $\chi_{1}$, we calculate $\chi_{2}$ by
$\chi_{2}=\chi_{1}+(\chi_{2}-\chi_{1})$, where the site $2$ is connected to
the site $1$ by a nearest neighbor bond. The step where value $\chi_{j}$ is
derived from the already evaluated value of $\chi_{k}$ is given by

$\displaystyle\chi_{j}=\chi_{k}+(\chi_{j}-\chi_{k})$ (78)

where the sites ${j}$ and $k$ are connected by a bond in the path for the
rebuilding of $\chi$. This process is continued until values at all accessible
sites are evaluated once and only once.

By this rebuilding process, a single path is constructed from the site $1$ to
other sites $k\neq 1$, which is achieved by making the region simply-connected
by removing some bonds (see Fig. 3). We denote the path from the site $1$ to
other sites $k\neq 1$ by $C_{1\rightarrow k}$. Then, the value $\chi_{k}$ is
given by

$\displaystyle\chi_{k}\approx\chi_{1}+\int_{C_{1\rightarrow k}}\nabla\chi\cdot
d{\bf r}$ (79)

When a magnetic field ${\bf B}^{\rm em}=\nabla\times{\bf A}^{\rm em}$ is
applied, the energy functional in Eq. (67) is modified as

$\displaystyle E[\nabla\chi]\ \rightarrow\ E\left[\nabla\chi-{{2e}\over{\hbar
c}}{\bf A}^{\rm em}\right]$ (80)

This leads to replace $\tau_{j\leftarrow i}$ in $E$ by

$\displaystyle\tau_{j\leftarrow i}\ \rightarrow\ \tau_{j\leftarrow
i}-{{2e}\over{\hbar c}}\int^{j}_{i}{\bf A}^{\rm em}\cdot d{\bf r}$ (81)

By this modification, a magnetic field can be taken into account.

Lastly in this section, we would like to consider the mass in the London
moment. In the present formalism, the velocity field ${\bf v}$ is given by

$\displaystyle{\bf v}={e\over{m_{e}c}}{\bf A}^{\rm
em}-{\hbar\over{2m_{e}}}\nabla\chi$ (82)

since the velocity field from $\tilde{u}_{n},\tilde{v}_{n}$ is zero. This
indicates that $m$ in Eq. (1) is $m_{e}$ in agreement with the value observed
in the London moment.

##  III Model calculations

###  III.1 Model Hamiltonian

In this section, a model is solved using the particle number conserving
version of the Bogoliubov-de Gennes equations. This model is originally built
to study the cuprate superconductivity. Since the purpose of the present work
is to explain the new theory, the model calculation below should be consider
as an example exercise. The validity of the model as a model for the cuprate
superconductivity will be dealt elsewhere.

The model is composed of two layers, a surface layer and a bulk layer (see
Fig. 4). The superconductivity occurs in the bulk, however, many experimental
results show contributions from the surface region, thus, the surface layer
effects must be included to interpret experiments. The experimental results
seem to indicate that the small polaron formation and coexisting magnetic
moment effects are strong in the bulk, however, they are weak in the surface
region.

Thus, we developed a model containing two layers to reproduce experimental
results. It is also notable that although the electron-pairing does not occur
in the bulk layer, supercurrent flows there in this model; thus, it
demonstrates that the primary importance for superconductivity is the
appearance of the non-trivial Berry connection.

Figure 4: Simply-connected lattice constructed by removing some bonds for the
two-layer model. We consider the model with two 7$\times$7 square lattices
stacked in the $z$ direction. There are four small polarons in the bulk layer
indicated by “M” and “A”. Each “M” indicates a center of a spin-vortex with
winding number $+1$, and “A” a center of a spin-vortex with winding number
$-1$.

###  III.2 Bulk layer

The model for the bulk layer is constructed based on the fact that the parent
compound is a Mott insulator [60], which is well-described by the two
dimensional Hubbard model with large on-site repulsion

$\displaystyle U\gg t_{1}$ (83)

where $U$ is the on-site repulsion parameter and $t_{1}$ is the nearest
neighbor hopping parameter. The bulk CuO2 plane of the cuprate is taken as a
two-dimensional square lattice, where a copper atom resides at each site.
Oxygens between copper atoms are not explicitly taken into account in the
present model.

When holes are doped, they form small polarons in the bulk [25, 61]. At low
temperatures, the mobility of the small polaron is very small; thus, the
supercurrent is not due to the hopping of small polarons.

The inelastic neutron scattering experiments indicate spin-wave excitations
with spin lying in CuO2 plane [23]. Thus, the magnetic moment coexisting with
superconductivity in the bulk. The magnetic excitation spectrum has a
hourglass-shaped dispersion, suggesting the stripes of spins or spin-vortices
[62, 26].

By taking into account the above observation, the following bulk Hamiltonian
is constructed,

$\displaystyle H_{\rm{EHFS}}$ $\displaystyle=$ $\displaystyle-
t_{1}\sum_{\langle
i,j\rangle_{1},\sigma}[c^{\dagger}_{i\sigma}c_{j\sigma}+{\rm
H.c.}]+U\sum_{j}c^{\dagger}_{j\uparrow}c_{j\uparrow}c^{\dagger}_{j\downarrow}c_{j\downarrow}+J_{h}\sum_{\left<i,j\right>_{h}}{\bf
S}_{i}\cdot{\bf S}_{j}$ (84) $\displaystyle+$
$\displaystyle\lambda\sum_{h}[c^{\dagger}_{h+y\downarrow}c_{h-x\uparrow}-c^{\dagger}_{h+y\uparrow}c_{h-x\downarrow}+c^{\dagger}_{h+x\downarrow}c_{h-y\uparrow}-c^{\dagger}_{h+x\uparrow}c_{h-y\downarrow}$
$\displaystyle+$
$\displaystyle\text{i}(c^{\dagger}_{h-x\downarrow}c_{h-y\uparrow}+c^{\dagger}_{h-x\uparrow}c_{h-y\downarrow})+\text{i}(c^{\dagger}_{h+y\downarrow}c_{h+x\uparrow}+c^{\dagger}_{h+y\uparrow}c_{h+x\downarrow})+\text{h.c.}])$
$\displaystyle-$ $\displaystyle\mu_{\rm
bulk}\sum_{j,\sigma}c_{j\sigma}^{\dagger}c_{j\sigma}$

where $c^{\dagger}_{j\sigma}$ and $c_{j\sigma}$ are creation and annihilation
operators for the electron at $j$th site with spin $\sigma$, respectively.

In this Hamiltonian, the doped holes are assumed to form small polarons and
immobile, thus, the effectively-half-filled situation (EHFS) is realized,
where the number of electrons and the number of accessible sites for the
electron are equal. We take the hole-occupied sites as inaccessible sites for
electrons, thus, not included in the sum.

The first term describes the electron hopping with transfer integral $t_{1}$;
$\langle i,j\rangle_{1}$ indicates that the sum is taken over 1st nearest
neighbor hoppings.

The second term describes the on-site Coulomb repulsion; $U$ is significantly
larger that $t_{1}$ (actually, we adopt $U=8t_{1}$ in the present work).

The third term describes the antiferromagnetic exchange interaction between
electron spins around each doped hole; $\langle i,j\rangle_{h}$ indicates that
the sum is taken over pairs around holes, i.e., each hole (denoted by $h$)
accompanies four sites around it (denoted by $h-x,h+x,h-y$ and $h+y$; they are
the nearest neighbor sites of $h$ in $-x$, $+x$, $-y$, and $+y$ directions,
respectively), and there are six site-pairs that contribute to $\langle
i,j\rangle_{h}$ for each doped hole. The parameter $J_{h}$ is taken to be
$0.5J_{\rm AF}$ where $J_{\rm AF}={{4t_{1}^{2}}\over U}$ is the
antiferromagnetic exchange parameter for the parent compound [63].

The fourth term is the Rashba interaction term [64], which we only include
around the holes [65]. The fifth term is the chemical potential term.

Components of the spin operator ${\bf
S}_{j}=({S}^{x}_{j},S^{y}_{j},S^{z}_{j})$ are given by

$\displaystyle{S}^{x}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(c^{\dagger}_{j\uparrow}c_{j\downarrow}+c^{\dagger}_{j\downarrow}c_{j\uparrow})$
$\displaystyle{S}^{y}_{j}$ $\displaystyle=$
$\displaystyle\frac{i}{2}(-c^{\dagger}_{j\uparrow}\
c_{j\downarrow}+c^{\dagger}_{j\downarrow}c_{j\uparrow})$
$\displaystyle{S}^{z}_{j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(c^{\dagger}_{j\uparrow}c_{j\uparrow}-c^{\dagger}_{j\downarrow}c_{j\downarrow})$
(85)

The chemical potential $\mu_{\rm bulk}$ is chosen to control the number of
small polarons in EHFS.

Since $H_{\rm{EHFS}}$ is too difficult to handle as it is, we use a mean field
version of it

$\displaystyle H^{\rm{HF}}_{\rm{EHFS}}$ $\displaystyle=$ $\displaystyle-
t_{1}\sum_{\langle
i,j\rangle_{1},\sigma}[c^{\dagger}_{i\sigma}c_{j\sigma}+{\rm H.c.}]$ (86)
$\displaystyle+$ $\displaystyle
U\sum_{j}[(-\frac{2}{3}\langle{S^{z}_{j}}\rangle+\frac{1}{2})c^{\dagger}_{j\uparrow}c_{j\uparrow}+(\frac{2}{3}\langle{S^{z}_{j}}\rangle+\frac{1}{2})c^{\dagger}_{j\downarrow}c_{j\downarrow}$
$\displaystyle-$
$\displaystyle\frac{2}{3}(\langle{S^{x}_{j}}\rangle-\rm{i}\langle{S^{y}_{j}}\rangle)c^{\dagger}_{j\uparrow}c_{j\downarrow}-\frac{2}{3}(\langle{S^{x}_{j}}\rangle+\rm{i}\langle{S^{y}_{j}}\rangle)c^{\dagger}_{j\downarrow}c_{j\uparrow}-\frac{2}{3}\langle{\bf
S}_{j}\rangle^{2}]$ $\displaystyle+$ $\displaystyle
J_{h}\sum_{\left<i,j\right>_{h}}[\frac{1}{2}(\langle{S^{z}_{i}}\rangle
c^{\dagger}_{j\uparrow}c_{j\uparrow}+\langle{S^{z}_{j}}\rangle
c^{\dagger}_{i\uparrow}c_{i\uparrow})-\frac{1}{2}(\langle{S^{z}_{i}}\rangle
c^{\dagger}_{j\downarrow}c_{j\downarrow}+\langle{S^{z}_{j}}\rangle
c^{\dagger}_{i\downarrow}c_{i\downarrow})$ $\displaystyle+$
$\displaystyle\frac{1}{2}\\{(\langle{S^{x}_{i}}\rangle-\rm{i}\langle{S^{y}_{i}}\rangle)c^{\dagger}_{j\uparrow}c_{j\downarrow}+(\langle{S^{x}_{j}}\rangle-\rm{i}\langle{S^{y}_{j}}\rangle)c^{\dagger}_{i\uparrow}c_{i\downarrow}\\}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\\{(\langle{S^{x}_{i}}\rangle+\rm{i}\langle{S^{y}_{i}}\rangle)c^{\dagger}_{j\downarrow}c_{j\uparrow}+(\langle{S^{x}_{j}}\rangle+\rm{i}\langle{S^{y}_{j}}\rangle)c^{\dagger}_{i\downarrow}c_{i\uparrow}\\}-\langle{\bf
S}_{i}\rangle\cdot\langle{\bf S}_{j}\rangle]$ $\displaystyle+$
$\displaystyle\lambda\sum_{h}[c^{\dagger}_{h+y\downarrow}c_{h-x\uparrow}-c^{\dagger}_{h+y\uparrow}c_{h-x\downarrow}+c^{\dagger}_{h+x\downarrow}c_{h-y\uparrow}-c^{\dagger}_{h+x\uparrow}c_{h-y\downarrow}$
$\displaystyle+$
$\displaystyle\text{i}(c^{\dagger}_{h-x\downarrow}c_{h-y\uparrow}+c^{\dagger}_{h-x\uparrow}c_{h-y\downarrow})+\text{i}(c^{\dagger}_{h+y\downarrow}c_{h+x\uparrow}+c^{\dagger}_{h+y\uparrow}c_{h+x\downarrow})+\text{h.c.}]$
$\displaystyle-$ $\displaystyle\mu_{\rm
bulk}\sum_{j,\sigma}c_{j\sigma}^{\dagger}c_{j\sigma}$

where $\langle\hat{O}\rangle$ denotes the expectation value of the operator
$\hat{O}$. This Hamiltonian yields states with spin-vortices around small
polarons (see Fig.5). These spin-vortices induce loop currents called the
“spin-vortex-induced loop currents”, due to the appearance of a non-trivial
Berry connection.

Figure 5: Spin texture obtained. a: The spin-texture of the two layer. The
spins in the surface layer are very small. b: The normalized spin-textures in
the bulk (top), and that in the surface (bottom). Spin-vortices are created in
the surface layer due to the influence of those in the bulk layer.

###  III.3 Surface layer

The Hamiltonian for the surface layer is following

$\displaystyle H_{\rm surf}$ $\displaystyle=$ $\displaystyle-
t_{1}\sum_{\langle
i,j\rangle_{1},\sigma}[c^{\dagger}_{i\sigma}c_{j\sigma}+{\rm
H.c.}]-t_{2}\sum_{\langle
i,j\rangle_{2},\sigma}[c^{\dagger}_{i\sigma}c_{j\sigma}+{\rm
H.c.}]+U\sum_{j}c^{\dagger}_{j\uparrow}c_{j\uparrow}c^{\dagger}_{j\downarrow}c_{j\downarrow}-\mu_{\rm
surf}\sum_{j,\sigma}c_{j\sigma}^{\dagger}c_{j\sigma}$

This is the Hubbard model including the second nearest neighbor hopping. The
second nearest neighbor hopping with the parameter $t_{2}$ is included to have
a curved Fermi surface in the angle-resolved photoemission spectrum. We use
$t_{2}=-0.12t_{1}$ in this work.

The most important difference from the bulk Hamiltonian is the absence of the
small polaron effect. The spin-orbit interaction term is also absent. The
chemical potential $\mu_{\rm surf}$ is so chosen that the bulk and surface
hole densities are equal.

Since $H_{\rm surf}$ is too difficult to handle as it is, we construct a mean-
field version via the $t-J$ model [66] for the Hubbard model,

$\displaystyle H^{\rm HF}_{\rm surf}$ $\displaystyle=$ $\displaystyle-
t_{1}\sum_{\langle i,j\rangle_{1},\sigma}[(1-\langle
n_{i,-\sigma}\rangle)c^{\dagger}_{i\sigma}c_{j\sigma}(1-\langle
n_{j,-\sigma}\rangle)+{\rm H.c.}]$ (88) $\displaystyle-$ $\displaystyle
t_{2}\sum_{\langle i,j\rangle_{2},\sigma}[(1-\langle
n_{i,-\sigma}\rangle)c^{\dagger}_{i\sigma}c_{j\sigma}(1-\langle
n_{j,-\sigma}\rangle)+{\rm H.c.}]$ $\displaystyle+$
$\displaystyle\sum_{\langle i,j\rangle_{1}}(\Delta_{ij}e^{-{i\over
2}\hat{\chi}_{j}}e^{-{i\over
2}\hat{\chi}_{i}}c_{i\uparrow}^{\dagger}c_{j\downarrow}^{\dagger}+{\rm
H.c.})-\mu_{\rm surf}\sum_{j,\sigma}c_{j\sigma}^{\dagger}c_{j\sigma}$

where the pairing amplitude or pair potential $\Delta_{ij}$ is defined as

$\displaystyle\Delta_{ij}=-{{4t_{1}^{2}}\over U}\langle e^{{i\over
2}\hat{\chi}_{j}}e^{{i\over
2}\hat{\chi}_{i}}(c_{j\downarrow}c_{i\uparrow}-c_{j\uparrow}c_{i\downarrow})\rangle=\Delta_{ji}$
(89)

Due to the presence of the number changing operators, $e^{{i\over
2}\hat{\chi}_{j}}$ and $e^{{i\over 2}\hat{\chi}_{i}}$, $\Delta_{ij}$ is
obtained in the particle number conserving formalism. The appearance of the
number changing operator is due to the spin-vortex formation in the bulk;
thus, the electron pair formation is a secondary effect. To have the
superconducting state, stable loop currents that can generate a macroscopic
persistent current should be realized.

The pair potential $\Delta_{ij}$ gives rise to $d$-wave pairing gap, and
explains the Fermi arc [67]. We would like to emphasized that the appearance
of the $d$-wave pairing gap is secondary effect enabled by the appearance of
the non-trivial Berry connection in the bulk; this generates the collective
mode for supercurrent, and particle number changing operators. The particle
number changing operators make it possible to gain energy by the $d$-wave
pairing gap formation.

###  III.4 Interlayer hopping

The bulk and surface layers are connected by the inter-layer hopping. The
following is the inter-layer hopping Hamiltonian,

$\displaystyle H_{\rm inter-layer}$ $\displaystyle=$ $\displaystyle-
t_{3}\sum_{\langle
i_{s},j_{b}\rangle_{z},\sigma}[(1-n_{i_{s},-\sigma})c^{\dagger}_{i_{s}\sigma}c_{j_{b}\sigma}(1-n_{j_{b},-\sigma})+{\rm
H.c.}+{{4t_{3}^{2}}\over U}\sum_{\langle i_{s},j_{b}\rangle_{z}}\left({\bf
S}_{i_{s}}\cdot{\bf S}_{j_{b}}-{1\over 4}n_{i_{s}}n_{j_{b}}\right)$

where an antiferromagnetic exchange interaction between spins in the bulk-
layer and those in the surface-layer is included using a $t-J$ model
approximation.

Since $H_{\rm inter-layer}$ is still too difficult to handle, we use the
following mean field version,

$\displaystyle H^{\rm HF}_{\rm inter-layer}=-t_{3}\sum_{\langle
i_{s},j_{b}\rangle_{z},\sigma}[(1-\langle
n_{i_{s},-\sigma}\rangle)c^{\dagger}_{i_{s}\sigma}c_{j_{b}\sigma}(1-\langle
n_{j_{b},-\sigma}\rangle)+{\rm H.c.}]$ (91) $\displaystyle+$
$\displaystyle{{2t_{3}^{2}}\over U}\sum_{\langle
i_{s},j_{b}\rangle_{z}}\Big{[}(\langle{S}^{x}_{i_{s}}\rangle-i\langle{S}^{y}_{i_{s}}\rangle)c^{\dagger}_{j_{b}\uparrow}c_{j_{b}\downarrow}+(\langle{S}^{x}_{i_{s}}\rangle+i\langle{S}^{y}_{i_{s}}\rangle)c^{\dagger}_{j_{b}\uparrow}c_{j_{b}\downarrow}$
$\displaystyle+$ $\displaystyle(\langle{S}^{z}_{i_{s}}\rangle-{1\over
2}\langle
n_{i_{s}}\rangle)c^{\dagger}_{j_{b}\uparrow}c_{j_{b}\uparrow}-(\langle{S}^{z}_{i_{s}}\rangle+{1\over
2}\langle
n_{i_{s}}\rangle)c^{\dagger}_{j_{b}\downarrow}c_{j_{b}\downarrow})\Big{]}$
$\displaystyle+$ $\displaystyle{{2t_{3}^{2}}\over U}\sum_{\langle
i_{s},j_{b}\rangle_{z}}\Big{[}(\langle{S}^{x}_{j_{b}}\rangle-i\langle{S}^{y}_{j_{b}}\rangle)c^{\dagger}_{i_{s}\uparrow}c_{i_{s}\downarrow}+(\langle{S}^{x}_{j_{b}}\rangle+i\langle{S}^{y}_{j_{b}}\rangle)c^{\dagger}_{i_{s}\uparrow}c_{i_{s}\downarrow}$
$\displaystyle+$ $\displaystyle(\langle{S}^{z}_{j_{b}}\rangle-{1\over
2}\langle
n_{j_{b}}\rangle)c^{\dagger}_{i_{s}\uparrow}c_{i_{s}\uparrow}-(\langle{S}^{z}_{j_{b}}\rangle+{1\over
2}\langle
n_{j_{b}}\rangle)c^{\dagger}_{i_{s}\downarrow}c_{i_{s}\downarrow})\Big{]}$
$\displaystyle-$ $\displaystyle{{4t_{3}^{2}}\over U}\sum_{\langle
i_{s},j_{b}\rangle_{z}}\left(\langle{\bf S}_{i_{s}}\rangle\cdot\langle{\bf
S}_{j_{b}}\rangle-{1\over 4}\langle n_{i_{s}}\rangle\langle
n_{j_{b}}\rangle\right)$

We adopt $t_{3}=0.01t_{1}$ in the following calculations. Through this
Hamiltonian, the non-trivial Berry connection generated in the bulk propagates
in the surface.

###  III.5 The total Hamiltonian for the two-layer model

Over all, our model Hamiltonian is given by

$\displaystyle H_{\rm eff}$ $\displaystyle=$ $\displaystyle H^{\rm HF}_{\rm
EHFS}+H^{\rm HF}_{\rm surf}+H^{\rm HF}_{\rm inter-layer}$ $\displaystyle=$
$\displaystyle\sum_{i,j,\sigma,\sigma^{\prime}}h_{i\sigma,j\sigma^{\prime}}c_{i\sigma}^{\dagger}c_{j\sigma^{\prime}}+\sum_{i,j}[\Delta_{ij}e^{-{i\over
2}\hat{\chi}_{j}}e^{-{i\over
2}\hat{\chi}_{i}}c_{i\uparrow}^{\dagger}c_{j\downarrow}^{\dagger}+\Delta^{\ast}_{ij}e^{{i\over
2}\hat{\chi}_{j}}e^{{i\over
2}\hat{\chi}_{i}}c_{j\downarrow}c_{i\uparrow}]+E_{\rm const}$

###  III.6 Particle number conserving Bogoliubov-de Gennes equations

In order to obtain the particle number conserving Bogoliubov-de Gennes
equations from $H_{\rm eff}$ in Eq. (LABEL:Heff), we perform the following
Bogoliubov transformation

$\displaystyle c_{i\uparrow}$ $\displaystyle=$
$\displaystyle\sum^{\prime}_{n}[u^{n}_{i\uparrow}\gamma_{n}-(v^{n}_{i\uparrow})^{\ast}\gamma_{n}^{\dagger}]e^{-{i\over
2}\hat{\chi}_{i}}$ $\displaystyle c_{i\downarrow}$ $\displaystyle=$
$\displaystyle\sum^{\prime}_{n}[u^{n}_{i\downarrow}\gamma_{n}+(v^{n}_{i\downarrow})^{\ast}\gamma_{n}^{\dagger}]e^{-{i\over
2}\hat{\chi}_{i}}$ (93)

Using the Bogoliubov operators $\gamma_{n}$ and $\gamma^{\dagger}_{n}$,
$H_{\rm eff}$ is expressed as

$\displaystyle H_{\rm
eff}=\sum_{n}^{\prime}E_{n}\gamma^{\dagger}_{n}\gamma_{n}+E^{\prime}_{\rm
const.}$ (94)

where $E^{\prime}_{\rm const.}$ is a constant and “$\sum_{n}^{\prime}$”
denotes that the sum is take over $E_{n}>0$, and the ground state $|{\rm
Gnd}\rangle$ satisfies

$\displaystyle\gamma_{n}|{\rm Gnd}\rangle=0,\quad E_{n}>0$ (95)

The Bogoliubov operators are fermion operators that satisfy

$\displaystyle\\{\gamma_{n},\gamma^{\dagger}_{m}\\}=\delta_{nm},\quad\\{\gamma^{\dagger}_{n},\gamma^{\dagger}_{m}\\}=0,\quad\\{\gamma_{n},\gamma_{m}\\}=0$
(96)

thus, the commutation relations

$\displaystyle[\gamma^{\dagger}_{n},H_{\rm
eff}]=-E_{n}\gamma^{\dagger}_{n},\quad[\gamma_{n},H_{\rm
eff}]=E_{n}\gamma_{n}$ (97)

are obtained.

The following commutation relations are obtained between $H_{\rm eff}$ and
$c_{i\sigma}$, $c^{\dagger}_{i\sigma}$,

$\displaystyle[c_{i\uparrow},H_{\rm
eff}]=\sum_{j,\sigma^{\prime}}h_{i\uparrow,j\sigma^{\prime}}c_{j\sigma^{\prime}}+\sum_{j}\Delta_{ij}e^{-{i\over
2}\hat{\chi}_{j}}e^{-{i\over 2}\hat{\chi}_{i}}c_{j\downarrow}^{\dagger}$
$\displaystyle[c_{i\uparrow}^{\dagger},H_{\rm
eff}]=-\sum_{j,\sigma^{\prime}}h_{j\sigma^{\prime},i\uparrow}c^{\dagger}_{j\sigma^{\prime}}-\sum_{j}\Delta^{\ast}_{ij}e^{{i\over
2}\hat{\chi}_{j}}e^{{i\over 2}\hat{\chi}_{i}}c_{j\downarrow}$
$\displaystyle[c_{i\downarrow},H_{\rm
eff}]=\sum_{j,\sigma^{\prime}}h_{i\downarrow,j\sigma^{\prime}}c_{j\sigma^{\prime}}-\sum_{j}\Delta_{ji}e^{-{i\over
2}\hat{\chi}_{j}}e^{-{i\over 2}\hat{\chi}_{i}}c_{j\uparrow}^{\dagger}$
$\displaystyle[c_{i\downarrow}^{\dagger},H_{\rm
eff}]=-\sum_{j,\sigma^{\prime}}h_{j\sigma^{\prime},i\downarrow}c^{\dagger}_{j\sigma^{\prime}}+\sum_{j}\Delta^{\ast}_{ji}e^{{i\over
2}\hat{\chi}_{j}}e^{{i\over 2}\hat{\chi}_{i}}c_{j\uparrow}$ (98)

Using Eqs. (98), (93), and (97), the Bogoliubov-de Gennes equations are
obtained,

$\displaystyle E_{n}u^{n}_{i\uparrow}$ $\displaystyle=$
$\displaystyle\sum_{j\sigma^{\prime}}e^{{i\over
2}\hat{\chi}_{i}}h_{i\uparrow,j\sigma^{\prime}}e^{-{i\over
2}\hat{\chi}_{j}}u^{n}_{j\sigma^{\prime}}+\sum_{j}\Delta_{ij}v^{n}_{j\downarrow}$
$\displaystyle E_{n}u^{n}_{i\downarrow}$ $\displaystyle=$
$\displaystyle\sum_{j\sigma^{\prime}}e^{{i\over
2}\hat{\chi}_{i}}h_{i\downarrow,j\sigma^{\prime}}e^{-{i\over
2}\hat{\chi}_{j}}u^{n}_{j\sigma^{\prime}}+\sum_{j}\Delta_{ji}v^{n}_{j\uparrow}$
$\displaystyle E_{n}v^{n}_{i\uparrow}$ $\displaystyle=$
$\displaystyle-\sum_{j}e^{-{i\over
2}\hat{\chi}_{i}}h^{\ast}_{i\uparrow,j\uparrow}e^{{i\over
2}\hat{\chi}_{j}}v^{n}_{j\uparrow}+\sum_{j}e^{-{i\over
2}\hat{\chi}_{i}}h^{\ast}_{i\uparrow,j\downarrow}e^{{i\over
2}\hat{\chi}_{j}}v^{n}_{j\downarrow}+\sum_{j}\Delta_{ij}^{\ast}u^{n}_{j\downarrow}$
$\displaystyle E_{n}v^{n}_{i\downarrow}$ $\displaystyle=$
$\displaystyle\sum_{j}e^{-{i\over
2}\hat{\chi}_{i}}h^{\ast}_{i\downarrow,j\uparrow}e^{{i\over
2}\hat{\chi}_{j}}v^{n}_{j\uparrow}-\sum_{j}e^{-{i\over
2}\hat{\chi}_{i}}h^{\ast}_{i\downarrow,j\downarrow}e^{{i\over
2}\hat{\chi}_{j}}v^{n}_{j\downarrow}+\sum_{j}\Delta_{ji}^{\ast}u^{n}_{j\uparrow}$
(99)

In order to solve the above system of equations, we replace the operator
$\hat{\chi}_{j}$ by its associated scalar value ${\chi}_{j}$. Then, the
Bogoliubov-de Gennes equations is put into the following matrix form,

$\displaystyle\sum_{j}M_{ij}\phi^{n}_{j}=E_{n}\phi^{n}_{i}$ (100)

where

$\displaystyle
M_{ij}=\left(\begin{array}[]{cccc}h_{i\uparrow,j\uparrow}e^{{i\over
2}({\chi}_{i}-{\chi}_{j})}&h_{i\uparrow,j\downarrow}e^{{i\over
2}({\chi}_{i}-{\chi}_{j})}&0&\Delta_{ij}\\\
h_{i\downarrow,j\uparrow}e^{{i\over
2}({\chi}_{i}-{\chi}_{j})}&h_{i\downarrow,j\downarrow}e^{{i\over
2}({\chi}_{i}-{\chi}_{j})}&\Delta_{ji}&0\\\
0&\Delta_{ij}^{\ast}&-h^{\ast}_{i\uparrow,j\uparrow}e^{-{i\over
2}({\chi}_{i}-{\chi}_{j})}&h^{\ast}_{i\uparrow,j\downarrow}e^{-{i\over
2}({\chi}_{i}-{\chi}_{j})}\\\
\Delta_{ji}^{\ast}&0&h^{\ast}_{i\downarrow,j\uparrow}e^{-{i\over
2}({\chi}_{i}-\chi_{j})}&-h^{\ast}_{i\downarrow,j\downarrow}e^{-{i\over
2}({\chi}_{i}-{\chi}_{j})}\end{array}\right)$ (105)

and

$\displaystyle\phi^{n}_{i}=\left(\begin{array}[]{cccc}u^{n}_{i\uparrow}\\\
u^{n}_{i\downarrow}\\\ v^{n}_{i\uparrow}\\\
v^{n}_{i\downarrow}\end{array}\right)$ (110)

Note that solutions for $E_{n}$ and $-E_{n}$ are connected; the solution for
$-E_{n}$ is given by

$\displaystyle\left(\begin{array}[]{cccc}-(v^{n}_{i\uparrow})^{\ast}\\\
(v^{n}_{i\downarrow})^{\ast}\\\ -(u^{n}_{i\uparrow})^{\ast}\\\
(u^{n}_{i\downarrow})^{\ast}\end{array}\right)$ (115)

Therefore, we only need to obtain half of the solutions to Eq. (100).

Self-consistent solutions are obtained by solving the above equations using
the self-consistent fields,

$\displaystyle\langle n_{i\uparrow}\rangle$ $\displaystyle=$
$\displaystyle\langle
c^{\dagger}_{i\uparrow}c_{i\uparrow}\rangle=\sum_{n_{1}}^{\prime}\sum_{n_{2}}^{\prime}\langle
e^{{i\over
2}\hat{\chi}_{i}}[u^{n_{1}}_{i\uparrow}\gamma_{n_{1}}-(v^{n_{1}}_{i\uparrow})^{\ast}\gamma_{n_{1}}^{\dagger}]^{\dagger}[u^{n_{2}}_{i\uparrow}\gamma_{n_{2}}-(v^{n_{2}}_{i\uparrow})^{\ast}\gamma_{n_{2}}^{\dagger}]e^{-{i\over
2}\hat{\chi}_{i}}\rangle$ $\displaystyle=$
$\displaystyle\sum_{n_{1}}^{\prime}\sum_{n_{2}}^{\prime}\langle e^{{i\over
2}{\chi}_{i}}[(u^{n_{1}}_{i\uparrow})^{\ast}\gamma^{\dagger}_{n_{1}}-(v^{n_{1}}_{i\uparrow})\gamma_{n_{1}}][u^{n_{2}}_{i\uparrow}\gamma_{n_{2}}-(v^{n_{2}}_{i\uparrow})^{\ast}\gamma_{n_{2}}^{\dagger}]e^{-{i\over
2}{\chi}_{i}}\rangle$ $\displaystyle=$
$\displaystyle\sum_{n_{1}}^{\prime}\sum_{n_{2}}^{\prime}[(u^{n_{1}}_{i\uparrow})^{\ast}u^{n_{2}}_{i\uparrow}\langle\gamma^{\dagger}_{n_{1}}\gamma_{n_{2}}\rangle-(u^{n_{1}}_{i\uparrow})^{\ast}(v^{n_{2}}_{i\uparrow})^{\ast}\langle\gamma^{\dagger}_{n_{1}}\gamma_{n_{2}}^{\dagger}\rangle+(v^{n_{1}}_{i\uparrow})u^{n_{2}}_{i\uparrow}\langle\gamma_{n_{1}}\gamma_{n_{2}}\rangle+(v^{n_{1}}_{i\uparrow})(v^{n_{2}}_{i\uparrow})^{\ast}\langle\gamma_{n_{1}}\gamma_{n_{2}}^{\dagger}\rangle]$
$\displaystyle=$
$\displaystyle\sum_{n}^{\prime}[|u^{n}_{i\uparrow}|^{2}f(E_{n})+|v^{n}_{i\uparrow}|^{2}f(-E_{n})]=\sum_{n}|u^{n}_{i\uparrow}|^{2}f(E_{n})=\sum_{n}|v^{n}_{i\uparrow}|^{2}f(-E_{n})$
$\displaystyle\langle n_{i\downarrow}\rangle$ $\displaystyle=$
$\displaystyle\sum_{n}^{\prime}[|u^{n}_{i\downarrow}|^{2}f(E_{n})+|v^{n}_{i\downarrow}|^{2}f(-E_{n})]=\sum_{n}|u^{n}_{i\downarrow}|^{2}f(E_{n})=\sum_{n}|v^{n}_{i\downarrow}|^{2}f(-E_{n})$
(116) $\displaystyle\langle{S}^{x}_{i}\rangle$ $\displaystyle=$
$\displaystyle-{1\over
2}\sum_{n}^{\prime}[v^{n}_{i\uparrow}(v^{n}_{i\downarrow})^{\ast}+{\rm
c.c}]f(-E_{n})+{1\over
2}\sum_{n}^{\prime}[u^{n}_{i\uparrow}(u^{n}_{i\downarrow})^{\ast}+{\rm
c.c}]f(E_{n})$ $\displaystyle\langle{S}^{y}_{i}\rangle$ $\displaystyle=$
$\displaystyle{i\over
2}\sum_{n}^{\prime}[v^{n}_{i\uparrow}(v^{n}_{i\downarrow})^{\ast}-{\rm
c.c}]f(-E_{n})-{i\over
2}\sum_{n}^{\prime}[u^{n}_{i\uparrow}(u^{n}_{i\downarrow})^{\ast}-{\rm
c.c}]f(E_{n})$ $\displaystyle\langle{S}^{z}_{i}\rangle$ $\displaystyle=$
$\displaystyle{1\over
2}\sum_{n}^{\prime}[v^{n}_{i\uparrow}(v^{n}_{i\uparrow})^{\ast}-v^{n}_{i\downarrow}(v^{n}_{i\downarrow})^{\ast}]f(-E_{n})+{1\over
2}\sum_{n}^{\prime}[u^{n}_{i\uparrow}(u^{n}_{i\uparrow})^{\ast}-u^{n}_{i\downarrow}(u^{n}_{i\downarrow})^{\ast}]f(E_{n})$
(117)

and

$\displaystyle\Delta_{ij}={{2t_{1}^{2}}\over
U}\sum^{\prime}_{n}[u^{n}_{i\uparrow}(v^{n}_{j\downarrow})^{\ast}+u^{n}_{j\downarrow}(v^{n}_{i\uparrow})^{\ast}+u^{n}_{i\downarrow}(v^{n}_{j\uparrow})^{\ast}+u^{n}_{j\uparrow}(v^{n}_{i\downarrow})^{\ast}]\tanh{{E_{n}}\over{2k_{B}T}}$
(118)

where following relations are used

$\displaystyle\langle\gamma^{\dagger}_{n}\gamma_{n}\rangle=f(E_{n}),\quad\langle\gamma_{n}\gamma^{\dagger}_{n}\rangle=1-f(E_{n})=f(-E_{n}),$
(119)

We solve the system of equations in Eq. (100) by the Car-Parrinello method
[68], taking $\phi^{n}_{i}$ as “time-dependent” variable $\phi^{n}(t)$ at the
$i$th site that follows the Newtonian dynamics

$\displaystyle m_{\rm
CP}\ddot{\phi}^{n}_{i}(t)=-\sum_{j}M_{ij}\phi^{n}_{j}(t)+E_{n}\phi^{n}_{i}(t)-\eta_{\rm
CP}\dot{\phi}^{n}_{i}(t)$ (120)

where $m_{\rm CP}$ and $\eta_{\rm CP}$ are mass and friction coefficient for
the variable, respectively; here, the wave functions $\\{\phi^{n}(t)\\}$ are
orthonormalized during the calculation.

###  III.7 Spin-vortices and multi-valued wave functions

The self-consistent wave functions obtained in the previous section, actually,
become multi-valued with respect to electron coordinates due to the presence
of spin-vortices. The spin-texture obtained for a model system composed of the
two-layers shown in Fig. 4 is depicted in Fig. 5.

If $u^{n}_{i\sigma}$ and $v^{n}_{i\sigma}$ are obtained just by minimizing
energy, the resulting state is a currentless state that satisfies the so-
called “Bloch’s theorem” [69]. This corresponds to the solution with constant
$\chi$.

Let us examine this currentless state, more closely. Replacing the operator
$\hat{\chi}_{i}$ in Eq. (93) by the scalar value of the underlying Berry
connection, we obtain

$\displaystyle\left(\begin{array}[]{c}c_{i\uparrow}\\\
c_{i\downarrow}\end{array}\right)=e^{-{i\over
2}{\chi}_{i}}\sum_{n}^{\prime}\left(\begin{array}[]{cc}u^{n}_{i\uparrow}&-(v^{n}_{i\uparrow})^{\ast}\\\
u^{n}_{i\downarrow}&(v^{n}_{i\downarrow})^{\ast}\end{array}\right)\left(\begin{array}[]{c}\gamma_{n}\\\
\gamma^{\dagger}_{n}\end{array}\right)$ (127)

and the currentless result corresponds to the solution with constant
$\chi_{j}$. Since the single-valued solution corresponds to multi-valued
$\chi_{j}$, the currentless $u^{n}_{i\sigma}$ and $v^{n}_{i\sigma}$ are
actually multi-valued.

When the bulk Hamiltonian yields spin-vortices lying in the $xy$ plane (in the
CuO2 plane), the on-site $U$ term in $H^{\rm HF}_{\rm EHFS}$ in Eq. (86)
becomes

$\displaystyle
U\sum_{j}[\frac{1}{2}(c^{\dagger}_{j\uparrow}c_{j\uparrow}+c^{\dagger}_{j\downarrow}c_{j\downarrow})-\frac{2}{3}\left|\langle{\bf
S}_{j}\rangle\right|(e^{-i\xi_{j}}c^{\dagger}_{j\uparrow}c_{j\downarrow}+e^{i\xi_{j}}c^{\dagger}_{j\downarrow}c_{j\uparrow})-\frac{2}{3}\langle{\bf
S}_{j}\rangle^{2}]$ (128)

We can abosrbe the spin-twisting factor $e^{\pm i\xi_{j}}$ using
$\tilde{c}^{\dagger}_{j\uparrow},\tilde{c}_{j\downarrow}$ as

$\displaystyle c^{\dagger}_{j\uparrow}=e^{{i\over
2}\xi_{j}}\tilde{c}^{\dagger}_{j\uparrow},\quad c_{j\downarrow}=e^{{i\over
2}\xi_{j}}\tilde{c}_{j\downarrow}$ (129)

Then, Eq. (127) becomes

$\displaystyle\left(\begin{array}[]{c}\tilde{c}_{i\uparrow}\\\
\tilde{c}_{i\downarrow}\end{array}\right)=\sum_{n}^{\prime}\left(\begin{array}[]{cc}\tilde{u}^{n}_{i\uparrow}&-(\tilde{v}^{n}_{i\uparrow})^{\ast}\\\
\tilde{u}^{n}_{i\downarrow}&(\tilde{v}^{n}_{i\downarrow})^{\ast}\end{array}\right)\left(\begin{array}[]{c}\gamma_{n}\\\
\gamma^{\dagger}_{n}\end{array}\right)$ (136)

This is essentially equivalent to the solution with constant $\chi$ in Eq.
(127). This means that $\tilde{u}^{n}_{i\sigma}$ and $\tilde{v}^{n}_{i\sigma}$
are obtained from the energy minimization only calculation if
${c}^{\dagger}_{j\uparrow},{c}_{j\downarrow}$ are replaced by
$\tilde{c}^{\dagger}_{j\uparrow},\tilde{c}_{j\downarrow}$. We do this in the
following.

We obtain $\tilde{u}^{n}_{i\sigma}$ and $\tilde{v}^{n}_{i\sigma}$ from the
following Hamiltonian, in which ${c}^{\dagger}_{j\sigma}$ and ${c}_{j\sigma}$
are replaced by $\tilde{c}^{\dagger}_{j\sigma},\tilde{c}_{j\sigma}$,

$\displaystyle\sum_{j}\tilde{M}_{ij}\tilde{\phi}^{n}_{j}=\tilde{E}_{n}\tilde{\phi}^{n}_{i}$
(137)

where $\tilde{M}_{ij}$ and $\tilde{\phi}^{n}_{i}$ are given by

$\displaystyle\tilde{M}_{ij}=\left(\begin{array}[]{cccc}h_{i\uparrow,j\uparrow}e^{{i\over
2}({\xi}_{i}-{\xi}_{j})}&h_{i\uparrow,j\downarrow}e^{-{i\over
2}({\xi}_{i}+{\xi}_{j})}&0&\Delta_{ij}e^{{i\over 2}({\xi}_{i}-{\xi}_{j})}\\\
h_{i\downarrow,j\uparrow}e^{-{i\over
2}({\xi}_{i}+{\xi}_{j})}&h_{i\downarrow,j\downarrow}e^{-{i\over
2}({\xi}_{i}-{\xi}_{j})}&\Delta_{ji}e^{-{i\over 2}({\xi}_{i}-{\xi}_{j})}&0\\\
0&\Delta_{ij}^{\ast}e^{-{i\over
2}({\xi}_{i}-{\xi}_{j})}&-h^{\ast}_{i\uparrow,j\uparrow}e^{-{i\over
2}({\xi}_{i}-{\xi}_{j})}&h^{\ast}_{i\uparrow,j\downarrow}e^{-{i\over
2}({\xi}_{i}+{\xi}_{j})}\\\ \Delta_{ji}^{\ast}e^{{i\over
2}({\xi}_{i}-{\xi}_{j})}&0&h^{\ast}_{i\downarrow,j\uparrow}e^{{i\over
2}({\xi}_{i}+{\xi}_{j})}&-h^{\ast}_{i\downarrow,j\downarrow}e^{{i\over
2}({\xi}_{i}-{\xi}_{j})}\end{array}\right)$ (142)

and

$\displaystyle\tilde{\phi}^{n}_{i}=\left(\begin{array}[]{cccc}\tilde{u}^{n}_{i\uparrow}\\\
\tilde{u}^{n}_{i\downarrow}\\\ \tilde{v}^{n}_{i\uparrow}\\\
\tilde{v}^{n}_{i\downarrow}\end{array}\right)$ (147)

respectively. Note that $\tilde{u}^{n}_{i\sigma}$, $\tilde{v}^{n}_{i\sigma}$
are single-valued since the multi-valued part $e^{\pm{i\over 2}\xi}$ are
removed.

###  III.8 Evaluation of Berry connection

Using $\tilde{u}^{n}_{i\sigma}$, $\tilde{v}^{n}_{i\sigma}$, Eq. (127) is
rewritten as

$\displaystyle\left(\begin{array}[]{c}c_{i\uparrow}\\\
c_{i\downarrow}\end{array}\right)=\sum_{n}^{\prime}\left(\begin{array}[]{cc}e^{-{i\over
2}(\chi_{i}+\xi_{i})}\tilde{u}^{n}_{i\uparrow}&-e^{-{i\over
2}(\chi_{i}+\xi_{i})}(\tilde{v}^{n}_{i\uparrow})^{\ast}\\\ e^{-{i\over
2}(\chi_{i}-\xi_{i})}\tilde{u}^{n}_{i\downarrow}&e^{-{i\over
2}(\chi_{i}-\xi_{i})}(\tilde{v}^{n}_{i\downarrow})^{\ast}\end{array}\right)\left(\begin{array}[]{c}\gamma_{n}\\\
\gamma^{\dagger}_{n}\end{array}\right)$ (154)

Now, we need to obtain the multi-valued $\chi$. The angular variable $\chi$ is
obtained from conditions,

$\displaystyle w_{C_{\ell}}[\xi]+w_{C_{\ell}}[\chi]=\mbox{even number}\mbox{
for any loop $C_{\ell}$}$ (155)

If the above conditions are satisfied, $e^{-{i\over 2}(\chi_{i}\pm\xi_{i})}$
appeared in Eq. (154) become single-valued.

Due to the presence of spin-vortices $\xi$ is multi-valued, and contains jump-
of-value points of an integral multiple of $2\pi$. To establish the condition
in Eq. (155), $\xi$ and $\chi$ must have the same jump-of-value points.

Figure 6: Loops used to impose boundary conditions in Eq. (155). a: loops in
the bulk and surface layers. a: loops in the bulk-surface interface region.
Some of the loops in the system are removed since they are not independent; as
a consequence, all the cells are accessible from outside without breaking any
walls framed by loops.

To have the same jump-of-value points for $\xi$ and $\chi$, we do the
following: first, we rebuild $\xi$ from differences of their values between
bonds; $\xi_{j}$’s are obtained from the expectation values of the spin
components $S^{x}_{j}$ and $S^{y}_{j}$; we choose a particular branch for
$\xi_{j}$. The system we consider below has the antiferromagnetic background;
this gives rise to the background value of $\xi$,
$\xi^{0}_{j}=\pi(j_{x}+j_{y}+j_{z})$ in the units the lattice constant, where
$(j_{x},j_{y},j_{z})$ is the $xyz$ coordinates of the $j$th site.

In order to separate the antiferromagnetic contribution from $\xi$, we
introduce angular variable $\eta$,

$\displaystyle\eta_{j}=\xi_{j}-\pi(j_{x}+j_{y}+j_{z})$ (156)

where $\eta_{j}$ is $\eta$ at the $j$th site. We take the branch of $\eta_{j}$
that satisfies the following condition,

$\displaystyle-\pi\leq\eta_{j}-\eta_{k}<\pi$ (157)

where $k$ is a nearest neighbor site of $j$.

From ($\eta_{j}-\eta_{k}$)’s, we construct ($\xi_{j}-\xi_{k}$)’s. After
($\xi_{j}-\xi_{k}$)’s are obtained, we rebuild $\xi$ from them. The process is
as follows: first, we pick a value for the initial $\xi_{1}$. After fixing the
value of $\xi_{1}$, we calculate $\xi_{2}$ by
$\xi_{2}=\xi_{1}+(\xi_{2}-\xi_{1})$, where the site $2$ is connected to the
site $1$ by a nearest neighbor bond. The step where value $\xi_{j}$ is derived
from the already evaluated value of $\xi_{k}$ is given by

$\displaystyle\xi_{j}=\xi_{k}+(\xi_{j}-\xi_{k})$ (158)

where the sites ${j}$ and $k$ are connected by a bond in the path for the
rebuilding of $\xi$. This process is continued until values at all accessible
sites are evaluated once and only once.

By this rebuilding process, a single path is constructed from the site $1$ to
other sites $k\neq 1$, which is achieved by making the region singly-connected
by removing some bonds (see Fig. 4). We denote the path from the site $1$ to
other sites $k\neq 1$ by $C_{1\rightarrow k}$. Then, the value $\xi_{k}$ is
given as

$\displaystyle\xi_{k}\approx\xi_{1}+\int_{C_{1\rightarrow k}}\nabla\xi\cdot
d{\bf r}$ (159)

Let us summarize the calculation so far.

  1. 1.

Solve Eq. (100) self-consistently using the Car-Parrinello method in Eq.
(120). Thereby, we obtain Eqs. (116), (117), and (118).

  2. 2.

From Eq. (117), $\xi_{k}$’s are obtained. Next, $\xi_{k}$’s are rebuilt using
Eq. (159). Thereby, we have $\xi_{k}$’s for which we know the positions of the
jump-of-values.

  3. 3.

From the rebuilt $\xi_{k}$’s, construct
$\tilde{c}^{\dagger}_{j\sigma},\tilde{c}_{j\sigma}$ in Eq. (129). Using Eqs.
(116), (117), and (118) obtained in Step 1), solve Eq. (137). Thereby,
$\tilde{u}^{n}_{i\sigma}$, $\tilde{v}^{n}_{i\sigma}$ are obtained.

  4. 4.

$e^{\pm{i\over 2}\xi_{i}}$, $\tilde{u}^{n}_{i\sigma}$,
$\tilde{v}^{n}_{i\sigma}$, ${u}^{n}_{i\sigma}$ and ${v}^{n}_{i\sigma}$ are
obtained; ${u}^{n}_{i\uparrow}=e^{-{i\over
2}\xi_{i}}\tilde{u}^{n}_{i\uparrow}$, ${u}^{n}_{i\downarrow}=e^{{i\over
2}\xi_{i}}\tilde{u}^{n}_{i\downarrow}$, ${v}^{n}_{i\uparrow}=e^{{i\over
2}\xi_{i}}\tilde{v}^{n}_{i\uparrow}$, ${v}^{n}_{i\downarrow}=e^{-{i\over
2}\xi_{i}}\tilde{v}^{n}_{i\downarrow}$.

Next, we obtain $\chi_{k}$’s that satisfy Eq. (155). The requirement in Eq.
(155) is given by

$\displaystyle
w_{C_{\ell}}[\chi]={1\over{2\pi}}\sum_{i=1}^{N_{\ell}}\tau_{{C_{\ell}(i+1)}\leftarrow{C_{\ell}(i)}}$
(160)

where $w_{C_{\ell}}[\chi]$ is the number supplied as a boundary condition.
$C_{\ell}(i)$ is the $i$th site in the loop $C_{\ell}$, where
$C_{N_{\ell}+1}=C_{1}$ with $N_{\ell}$ is the number of sites in $C_{\ell}$.

In order to impose conditions in Eq. (163), $\tau_{j\leftarrow i}$ is split
into a multi-valued part $\tau_{j\leftarrow i}^{0}$ and single-valued part
$f_{j\leftarrow i}$ as

$\displaystyle\tau_{j\leftarrow i}=\tau_{j\leftarrow i}^{0}+f_{j\leftarrow i}$
(161)

where $\tau_{j\leftarrow i}^{0}$ is given to satisfy

$\displaystyle
w_{C_{\ell}}[\chi]={1\over{2\pi}}\sum_{i=1}^{N_{\ell}}\tau^{0}_{{C_{\ell}(i+1)}\leftarrow{C_{\ell}(i)}}$
(162)

and $f_{j\leftarrow i}$ is solved to satisfy

$\displaystyle
0={1\over{2\pi}}\sum_{i=1}^{N_{\ell}}f_{{C_{\ell}(i+1)}\leftarrow{C_{\ell}(i)}}$
(163)

The above is the equation corresponding to Eq. (70). They are imposed on
independent loops in the system, as shown in Fig. 6.

For Eq. (71), we use the conservation of the local charge at sites instead. As
will be explained later, it gives rise to a sufficient number of equations to
obtain unknowns.

The conservation of the local charge at the $k$th site is

$\displaystyle 0=\sum_{\ell\in b_{k}}J_{k\leftarrow\ell}$ (164)

where $b_{k}$ is a set of sites that are connected to the $k$th site.
$J_{k\leftarrow\ell}$ is the current through the bond between sites $k$ and
$\ell$ in the direction $k\leftarrow\ell$,

$\displaystyle J_{k\leftarrow\ell}$ $\displaystyle=$
$\displaystyle{{2e}\over\hbar}{{\partial
E}\over{\partial\tau_{k\leftarrow\ell}}}={{2e}\over\hbar}{{\partial}\over{\partial\tau_{k\leftarrow\ell}}}\langle{\rm
Gnd}|\sum_{i,j,\sigma,\sigma^{\prime}}h_{i\sigma,j\sigma^{\prime}}c_{i\sigma}^{\dagger}c_{j\sigma^{\prime}}|{\rm
Gnd}\rangle$ (165) $\displaystyle=$
$\displaystyle{{2e}\over\hbar}{{\partial}\over{\partial\tau_{k\leftarrow\ell}}}\sum_{i,j,\sigma,\sigma^{\prime}}h_{i\sigma,j\sigma^{\prime}}\sum_{n_{1}}^{\prime}\sum_{n_{2}}^{\prime}\langle
e^{{i\over
2}\hat{\chi}_{i}}[(u^{n_{1}}_{i\sigma})^{\ast}\gamma_{n_{1}}^{\dagger}-\sigma(v^{n_{1}}_{i\sigma})\gamma_{n_{1}}][u^{n_{2}}_{j\sigma^{\prime}}\gamma_{n_{2}}-\sigma^{\prime}(v^{n_{2}}_{j\sigma^{\prime}})^{\ast}\gamma_{n_{2}}^{\dagger}]e^{-{i\over
2}\hat{\chi}_{j}}\rangle$ $\displaystyle=$
$\displaystyle{{2e}\over\hbar}\sum_{i,j,\sigma,\sigma^{\prime}}h_{i\sigma,j\sigma^{\prime}}\sum_{n}^{\prime}{{\partial
e^{{i\over 2}\tau_{i\leftarrow
j}}}\over{\partial\tau_{k\leftarrow\ell}}}\left[(u^{n}_{i\sigma})^{\ast}(u^{n}_{j\sigma^{\prime}})f(E_{n})+\sigma\sigma^{\prime}(v^{n}_{i\sigma})(v^{n}_{j\sigma^{\prime}})^{\ast}f(-E_{n})\right]$
$\displaystyle=$
$\displaystyle{{2e}\over\hbar}\sum_{n,i,j,\sigma}^{\prime}{{\partial
e^{{i\over 2}\tau_{i\leftarrow
j}}}\over{\partial\tau_{k\leftarrow\ell}}}\Big{\\{}h_{i\sigma,j\sigma}[(u^{n}_{i\sigma})^{\ast}u^{n}_{j\sigma}f(E_{n})+(v^{n}_{j\sigma})^{\ast}v^{n}_{i\sigma}f(-E_{n})]$
$\displaystyle+$ $\displaystyle
h_{i\sigma,j-\sigma}[(u^{n}_{i\sigma})^{\ast}u^{n}_{j-\sigma}f(E_{n})-(v^{n}_{j-\sigma})^{\ast}v^{n}_{i\sigma}f(-E_{n})]\Big{\\}}$

We employ an iterative improvement of the approximate solutions by using the
linearized version of Eq. (164) given by

$\displaystyle 0\approx{{2e}\over\hbar}\sum_{i}{{\partial
E(\\{\tau_{j\leftarrow i}^{0}\\})}\over{\partial\tau_{j\leftarrow
i}}}+{{2e}\over\hbar}\sum_{i}{{\partial^{2}E(\\{\tau_{j\leftarrow
i}^{0}\\})}\over{\partial(\tau_{j\leftarrow i})^{2}}}f_{j\leftarrow i}$ (166)

A system of equations for $f_{j\leftarrow i}$’s composed of Eqs. (163) and
(166) are solve for given $\tau_{j\leftarrow i}^{0}$’s by updating them
iteratively

$\displaystyle\tau_{j\leftarrow i}^{0\ New}=\tau_{j\leftarrow i}^{0\
Old}+f_{j\leftarrow i}$ (167)

where $\tau_{j\leftarrow i}^{0\ Old}$ is $\tau_{j\leftarrow i}^{0}$ value that
is used to obtain the current value of $f_{j\leftarrow i}$; $\tau_{j\leftarrow
i}^{0\ New}$ will be used to obtain the next $f_{j\leftarrow i}$ value.

The numerical convergence is checked by the condition

$\displaystyle\left|{{2e}\over\hbar}\sum_{i}{{\partial E(\\{\tau_{j\leftarrow
i}^{0}\\})}\over{\partial\tau_{j\leftarrow i}}}\right|<\epsilon$ (168)

where $\epsilon$ is a small number.

For the initial $\tau_{j\leftarrow i}^{0}$, we adopt the following,

$\displaystyle\tau_{j\leftarrow i}^{0\
init}=\sum_{h}w_{h}\tan^{-1}{{j_{y}-h_{y}}\over{j_{x}-h_{x}}}-\sum_{h}w_{h}\tan^{-1}{{i_{y}-h_{y}}\over{i_{x}-h_{x}}}$
(169)

where $(j_{x},j_{y},j_{z})$ and $(i_{x},i_{y},i_{z})$ are coordinates of the
sites $j$ and $i$, respectively, $h=(h_{x},h_{y},h_{z})$ is the coordinate of
the hole occupied site, and $w_{h}$ is the winding number of $\chi$ around the
hole at $h$.

The number of $\tau_{j\leftarrow i}$ to be evaluated is equal to the number of
the bonds. The number of equations in Eq. (70) is equal to the number of the
plaques. The number of equations from Eq. (164) for the conservation of charge
is equal to the number of sites$-1$, due to the fact that the total charge is
fixed in the calculation.

The equality of the number of unknowns and the number of equations gives

$\displaystyle[\mbox{\\# bonds}]=[\mbox{\\# plaques}]+[\mbox{\\# sites}-1]$
(170)

It is interesting to note that this agrees with the Euler’s theorem for the
two-dimensional lattice

$\displaystyle[\mbox{\\# edges}]=[\mbox{\\# faces}]+[\mbox{\\# vertices}-1]$
(171)

When the spin-orbit interaction exists the total energy depends on $\xi_{1}$
as shown in Fig. 7. In Fig. 8, obtained current using $\xi_{1}=0$ is depicted.

Figure 7: Dependence of the total energy on the value of $\xi_{1}$. The zero
of energy is taken to be the value for $\lambda=0$ calculation. The units of
the energy is $t_{1}$. Figure 8: Loop current obtained by the calculation. a:
Loop current in the system. Each “m” indicates a center of a loop current with
winding number $+1$, and “a” a center of a loop-current with winding number
$-1$. b: Loop current in the bulk layer viewed from above. c: Loop current in
the surface layer viewed from above.

###  III.9 Diamagnetic current produced by an application of a magnetic field

Figure 9: Supercurrent under the application of an external magnetic field.
$B=0.01$ in the units of $t_{1}=1,\hbar=1,e=1$, and lattice constant in the
plane $=1$. a: The result from the calculation. b: The linear approximation
given in Eq. (179). c: The difference between the calculated result and its
linear approximation.

One of the hallmarks of superconducting states is the diamagnetic response to
an external magnetic field. Let us consider the situation where a magnetic
field ${\bf B}^{\rm em}=\nabla\times{\bf A}^{\rm em}$ is applied. Then, the
modified energy functional in Eq. (80) is used.

This leads to replace $\tau_{j\leftarrow i}$ in $E$ by

$\displaystyle u_{j\leftarrow i}=\tau_{j\leftarrow i}-{{2e}\over{\hbar
c}}\int^{j}_{i}{\bf A}^{\rm em}\cdot d{\bf r}$ (172)

where integration is performed along the bond $j\leftarrow i$.

The calculation can be done similarly to the case for no magnetic field,
starting from the initial value

$\displaystyle u_{j\leftarrow i}^{0\ init}=\tau_{j\leftarrow i}^{0\
init}-{{2e}\over{\hbar c}}\int_{j\leftarrow i}{\bf A}^{\rm em}\cdot d{\bf r}$
(173)

Note that during the evaluation process of $\nabla\chi$, the ambiguity in the
gauge of ${\bf A}^{\rm em}$ is compensated, thus, the effective vector
potential

$\displaystyle{\bf A}^{\rm eff}={\bf A}^{\rm em}-{{\hbar
c}\over{2e}}\nabla\chi$ (174)

is invariant with respect to the choice of the gauge in ${\bf A}^{\rm em}$.

Now we apply a uniform magnetic field perpendicular to the lattice. In the
actual numerical calculations we have adopted

$\displaystyle{\bf A}^{\rm em}=\left(\begin{array}[]{c}-By\\\ 0\\\
0\end{array}\right)$ (178)

but we checked that the calculated current distribution is identical even
other gauge is employed.

In Fig. 9a, current distributions with applying the magnetic field are
depicted.

Numerical calculations indicate the following relation holds

$\displaystyle J_{j\leftarrow
i}\approx-{{4e^{2}}\over\hbar^{2}}{{\partial^{2}E[\\{0\\}]}\over{\partial(u_{j\leftarrow
i})^{2}}}\int_{i}^{j}{\bf A}^{\rm eff}\cdot d{\bf r}$ (179)

where $\\{0\\}$ means all $u_{j\leftarrow i}$’s are zero [70]. The result
using the above approximation is shown in Fig. 9b, and the difference between
the exact result and approximate one is shown in Fig. 9c. The approximate one
is almost identical to the exact one, except the current just around the small
polarons; it originates directly from the Rashba interaction.

Figure 10: Supercurrent indued by the application of an external magnetic
field. a: The induced current in the two layers. The scale for the current
arrow is enlarged from Fig. 9. b: The induced current in the bulk layer. c:
the induced current in the surface layer.

The induced current is diamagnetic as seen in Fig. 10. This will become a
screening current of the magnetic field in a large system size.

###  III.10 Zero voltage current production by an external current feeding

Another hallmark of superconducting states is the zero voltage current flow
through the sample.

Let us consider the external current feeding. When external currents are fed,
the conservation of the local charge at site $k$ in Eq. (164) is modified as

$\displaystyle 0=\sum_{\ell\in b_{k}}J_{k\leftarrow\ell}+J^{\rm EX}_{k}$ (180)

where $J^{\rm EX}_{k}$ is the external current fed at $k$.

Figure 11: Supercurrent obtained by feeding external current. a: Supercurrent
with external current. Two sets of three arrows indicate sites where external
current enters and exits. Each arrow indicates $J^{\rm EX}$. The units of the
current is $t_{1}=1,\hbar=1$, and $e=1$. b: The difference of currents with
and without external current. c: The difference of currents with and without
external current in the bulk layer viewed from above. Arrows are magnified
from those in b. Figure 12: Total energy vs $J^{\rm EX}$. The units of $J^{\rm
EX}$ is $t_{1}=1,\hbar=1$, and $e=1$. Three different Rashba parameters
$\lambda$ are employed. The zero of the energy is that of $\lambda=0$ and
$J^{\rm EX}=0$.

The results are shown in Fig. 12. The energy minimum with nonzero $J^{\rm EX}$
occurs for $\lambda\neq 0$ cases. This situation is similar to the one
envisaged by Bloch for superconducting states [71]. When the system is at the
energy minimum with nonzero $J^{\rm EX}$, zero voltage current production is
realized. In other words, persistent current flows through the sample.

The energy minimum occurs at zero $J^{\rm EX}$ for the $\lambda=0$ case. This
indicates the if the Rashba interaction is absent, the zero voltage current
production does not occur in the present model.

##  IV Concluding remarks

We have presented a fundamentally revised theory for superconductivity, and
the method to calculate properties of superconducting states based on it.

Since the early work of London, it has been repeatedly argued that theory of
superconductivity needs to have a $U(1)$ gauge field different from the
electromagnetic one to have the gauge invariant supercurrent. London called it
“superpotential”, and the current standard theory calls it “Nambu-Goldston
mode”. In the new theory, the Berry connection ${\bf A}^{\rm fic}$ provides
the necessary gauge field.

We would like to emphasize that the vector potential provided by the Berry
connection ${\bf A}^{\rm fic}$ plays a crucial role in realizing reversible
superconducting-normal metal phase transition in a magnetic field [57, 18, 13,
21]. This phenomenon cannot be explained by the standard theory [35, 57].

Besides the reversible superconducting-normal metal phase transition in a
magnetic field, the new theory solves all four problems described in
Introduction. It also provides a calculation framework for the two of the
hallmarks of superconducting states, the diamagnetic response current to an
external magnetic field, and the zero voltage current produced by an external
current feeding, microscopically. This theory may be able to elucidate the
cuprate superconductivity, where nanoscale inhomogeneity of electronic states
is important.

Lastly, we would like to note that a similar formalism using the Berry
connection is possible for superfluidity of bosons [56].

##  Appendix A Berry connection from many-body wave functions and the
modification of Maxwell’s equations

Berry’s derivation for the Berry phase uses the adiabatic assumption [72]. The
adiabatic assumption here does not necessary means a slow change. We will
explain this point below.

By following Berry, let us consider the state vector where given by

$\displaystyle|n({\bf R}(t))\rangle$ (181)

where ${\bf R}$ is the adiabatic parameter. This indicates the parametric
time-dependence occrs through time-dependent adiabatic parameter ${\bf R}(t)$.

Then, the time-dependent wave function becomes

$\displaystyle|\Psi(t)\rangle$ $\displaystyle=$ $\displaystyle e^{i\int_{{\bf
R}(0)}^{{\bf R}(t)}\langle n({\bf R}(t))|\nabla_{\bf R}|n({\bf
R}(t))\rangle\cdot d{\bf R}}|n({\bf R}(t))\rangle
e^{-{i\over\hbar}\int^{t}_{0}dt^{\prime}E_{n}(t^{\prime})}$ (182)
$\displaystyle=$ $\displaystyle|n({\bf R}(t))\rangle
e^{i\gamma_{n}(t)}e^{-{i\over\hbar}\int^{t}_{0}dt^{\prime}E_{n}(t^{\prime})}$

The phase

$\displaystyle\gamma_{n}(t)=i\int_{{\bf R}(0)}^{{\bf R}(t)}d{\bf R}\cdot{\bf
A}_{n}({\bf R})$ (183)

is the Berry phase, and

$\displaystyle{\bf A}_{n}({\bf R})=-i\langle n({\bf R})|\nabla_{\bf R}|n({\bf
R})\rangle$ (184)

is the Berry connection.

The Berry connection gives rise to a vector potential on the dynamics of ${\bf
R}$. Let us see this below, taking the Born-Oppenheimer approximation as an
example.

In the Born-Oppenheimer approximation, ${\bf R}$ is the nuclear coordinates.
The momentum conjugate to ${\bf R}$ is $-i\hbar\nabla_{\bf R}$. The electron
coordinates is denoted by ${\bf r}$. The Born-Oppenheimer total wave function
we consider is

$\displaystyle f({\bf R})\langle{\bf r}|n({\bf R})\rangle$ (185)

where $f({\bf R})$ is the nuclear wave function and $\langle{\bf r}|n({\bf
R})\rangle$ is the electronic wave function with the nuclear coordinate as the
adiabatic parameter.

Application of the momentum operator conjugate to ${\bf R}$ on the total wave
function yields

$\displaystyle-i\hbar\nabla_{\bf R}\left[f({\bf R})\langle{\bf r}|n({\bf
R})\rangle\right]=\langle{\bf r}|n({\bf R})\rangle[-i\hbar\nabla_{\bf R}f({\bf
R})]-f({\bf R})[i\hbar\nabla_{\bf R}\langle{\bf r}|n({\bf R})\rangle]$ (186)

The integration over ${\bf r}$ after applying $\langle n({\bf R})|{\bf
r}\rangle$ from the left yields

$\displaystyle[-i\hbar\nabla_{\bf R}+\hbar{\bf A}_{n}({\bf R})]f({\bf R})$
(187)

This indicates that the dynamics of ${\bf R}$ is that in the gauge field with
the vector potential $\hbar{\bf A}_{n}$. In other words, the change of the
momentum operator

$\displaystyle-i\hbar\nabla_{\bf R}\rightarrow-i\hbar\nabla_{\bf R}+\hbar{\bf
A}_{n}$ (188)

occurs due to the presence of other particles.

Now, we shall consider a wave function for $N$ electron system,

$\displaystyle\Phi({\bf r}_{1},\cdots,{\bf r}_{N},t)$ (189)

where ${\bf r}_{j}$ is the coordinate of the $j$th particle.

The Berry connection ${\bf A}_{n}$ takes into account the effect of the
electron dynamics on the nuclear dynamics through the wave function they share
(i.e., $f({\bf R})\langle{\bf r}|n({\bf R})\rangle$).

Let us calculate the Berry connection arising from $\Phi({\bf
r}_{1},\cdots,{\bf r}_{N},t)$ by regarding ${\bf r}_{1}$ as an adiabatic
parameter. In other words, we calculate the gauge field that affects the
dynamics on the particle $1$ generated by other particles $2,\cdots,{N}$
through the wave function they share (i.e., $\Phi({\bf r}_{1},\cdots,{\bf
r}_{N},t)$).

We cannot consider ${\bf r}_{1}$ as a slow variable compared with other ${\bf
r}_{2},\cdots,{\bf r}_{N}$; however, the whole system stays in the same state
described by the wave function $\Phi({\bf r}_{1},\cdots,{\bf r}_{N},t)$.

We first construct a normalized wave function $|n_{\Phi}({\bf r}_{1})\rangle$
that contains ${\bf r}_{1}$ as an adiabatic parameter,

$\displaystyle\langle{\bf r}_{2},\cdots,{\bf r}_{N}|n_{\Phi}({\bf
r}_{1},t)\rangle={{\Phi({\bf r}_{1},{\bf r}_{2},\cdots,{\bf
r}_{N},t)}\over{|C_{\Phi}({\bf r}_{1},t)|^{{1\over 2}}}}$ (190)

Here, $|C_{\Phi}({\bf r}_{1},t)|$ is the normalization constant given by

$\displaystyle|C_{\Phi}({\bf r}_{1},t)|=\int d^{3}r_{2}\cdots
d^{3}r_{N}\Phi({\bf r}_{1},{\bf r}_{2},\cdots)\Phi^{\ast}({\bf r}_{1},{\bf
r}_{2},\cdots)$ (191)

The wave function $\langle{\bf r}_{2},\cdots,{\bf r}_{N}|n_{\Phi}({\bf
r}_{1})\rangle$ is normalized to $1$ with respect to integrals over ${\bf
r}_{2},\cdots,{\bf r}_{N}$, and single-valued with respect to ${\bf r}_{1}$.
Then, the Berry connection for many-body wave functions is calculated as

$\displaystyle{\bf A}^{\rm MB}_{\Phi}({\bf r}_{1},t)=-i\langle n_{\Phi}({\bf
r}_{1},t)|\nabla_{{\bf r}_{1}}|n_{\Phi}({\bf r}_{1},t)\rangle$ (192)

As in the Born-Oppenheimer case, ${\bf A}^{\rm MB}_{\Phi}({\bf r}_{1},t)$ will
modify the momentum operator as

$\displaystyle-i\hbar\nabla_{{\bf r}_{1}}\rightarrow-i\hbar\nabla_{{\bf
r}_{1}}+\hbar{\bf A}^{\rm MB}_{\Phi}({\bf r}_{1})$ (193)

The same modification also occurs for ${\bf r}_{i},i=2\cdots,{N}$.

Let us consider the effect ${\bf A}^{\rm MB}_{\Phi}({\bf r}_{1},t)$ using the
path integral formalism of quantum mechanics [21]. According to the path
integral formalism, a wave function is a sum of contributions from all paths
each contributes an exponential whose phase is the classical action divided by
$\hbar$ for the path in question [73]. For the system of charged particles and
electromagnetic field, the classical action $S$ are composed of the following
three terms,

$\displaystyle S=S_{1}+S_{2}+S_{3}$ (194)

where

$\displaystyle S_{1}=\sum_{i}{m\over 2}\int dt\ \dot{\bf r}_{i}^{2}$ (195)

is the action for the particles,

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

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

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

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.

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.

The Berry connection ${\bf A}_{\Phi}^{\rm MB}$ modifies the momentum operator
by providing a $U(1)$ field. As a consequence, the effective vector potential
for the $U(1)$ field in the system becomes

$\displaystyle{\bf A}^{\rm eff}={\bf A}^{\rm em}+{\bf A}^{\rm fic},\quad{\bf
A}^{\rm fic}=-{{\hbar c}\over q}{\bf A}_{\Phi}^{\rm MB}$ (198)

Here, we have used the “fictitious” vector potential ${\bf A}^{\rm fic}$ in
place of ${\bf A}_{\Phi}^{\rm MB}$.

As will be shown later, ${\bf A}^{\rm fic}$ is given by

$\displaystyle{\bf A}^{\rm fic}=-{{\hbar c}\over{2e}}\nabla\chi$ (199)

where $\chi$ is an angular variable with period $2\pi$ [19].

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

$\displaystyle S_{2}^{\prime}={1\over c}\int d^{3}rdt\ {\bf j}\cdot{\bf
A}^{\rm eff}({\bf r},t)$ (200)

This gives rise to “Lorentz force”. The Lorenz force from ${\bf A}^{\rm fic}$
is zero in classical mechanics; however, ${\bf A}^{\rm fic}$ may affect the
dynamics of charged particles through the Aharonov-Bohm effect [74] in quantum
mechanics. Actually, this effect is the main concern of the present work. We
call this term, the Lorentz interaction term.

The electromagnetic field energy is the energy stored in the space through the
Lorentz interaction term. If we use $S_{2}^{\prime}$ as the Lorentz
interaction term, $S_{3}$ should be modified as

$\displaystyle S_{3}^{\prime}=-{1\over{8\pi}}\int d^{3}rdt\ ({\bf B}^{\rm
eff})^{2}$ (201)

where ${\bf B}^{\rm eff}=\nabla\times{\bf A}^{\rm eff}$.

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

The use of $S_{2}^{\prime}$ and $S_{3}^{\prime}$, modify two of the Maxwell’s
equations,

$\displaystyle\nabla\cdot{\bf B}^{\rm eff}$ $\displaystyle=$ $\displaystyle 0$
(202) $\displaystyle\nabla\times{\bf B}^{\rm eff}$ $\displaystyle=$
$\displaystyle{{4\pi}\over c}{\bf j}$ (203)

The first one gives rise to a Dirac monopole as shown below: we rewrite it as

$\displaystyle\nabla\cdot{\bf B}^{\rm em}=-\nabla\cdot(\nabla\times{\bf
A}^{\rm fic})$ (204)

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

$\displaystyle\int_{\rm Sf}d{\bf S}\cdot{\bf B}^{\rm em}=-\int_{\rm Sf}d{\bf
S}\cdot(\nabla\times{\bf A}^{\rm fic})$ (205)

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 the following relation

$\displaystyle\int_{{\rm Sf}_{1}}d{\bf S}\cdot(\nabla\times{\bf A}^{\rm
fic})+\int_{{\rm Sf}_{2}}d{\bf S}\cdot(\nabla\times{\bf A}^{\rm
fic})=\int_{\partial({\rm Sf}_{1})}d{\bf r}\cdot{\bf A}^{\rm
fic}+\int_{\partial({\rm Sf}_{1})}d{\bf r}\cdot{\bf A}^{\rm fic}$ (206)

We consider the case in which singularities exist in ${\bf A}^{\rm fic}$: let
us take a closed surface $S$ with boundary $C=\partial S$, and ${\bf A}^{\rm
fic}$ has a singularity in $S$. Then, the following relation is obtained

$\displaystyle\int_{C}d{\bf r}\cdot{{\hbar
c}\over{2e}}\nabla\chi={{hc}\over{2e}}n$ (207)

where $n$ is an integer.

If we have the case where $n=0$ for $\partial({\rm Sf}_{1})$ term and $n=1$
for $\partial({\rm Sf}_{2})$ term in Eq. (206), respectively, Eq. (205) yields

$\displaystyle\int_{\rm Sf}d{\bf S}\cdot{\bf B}^{\rm em}={{hc}\over{2e}}$
(208)

This shows that a monopole with magnetic charge ${{hc}\over{2e}}$ exists in
the region enclosed by ${\rm Sf}$ [75].

Let us show $\nabla\times{\bf B}^{\rm fic}=0$: it is well-known that
$\nabla\chi$ in ${\bf A}^{\rm fic}$ can be decomposed as

$\displaystyle\nabla\chi=\nabla\chi_{0}+\nabla f,\quad\nabla^{2}\chi_{0}=0$
(209)

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

$\displaystyle\nabla\times{\bf B}^{\rm
fic}=\nabla\times(\nabla\times\nabla\chi_{0})=\nabla(\nabla^{2}\chi_{0})-\nabla^{2}\nabla\chi_{0}=0$
(210)

As a consequence Eq. (203) is equal to

$\displaystyle\nabla\times{\bf B}^{\rm em}$ $\displaystyle=$
$\displaystyle{{4\pi}\over c}{\bf j}$ (211)

##  Appendix B Appearance of ${\bf A}^{\rm fic}$ from spin-twisting itinerant
motion of electrons.

The electrons in the normal state of the BCS superconductor are assumed to be
Bloch electrons. We consider the Bloch electron under the influence of the
Rashba interaction, and show that the spin-twisting itinerant motion is
realized [19].

We use the following Rashba interaction term

$\displaystyle H_{so}={\bm{\lambda}}({\bf r})\cdot{{\hbar{\bm{\sigma}}}\over
2}\times\left(\hat{\bf p}-{q}{\bf A}^{\rm em}({\bf r})\right),$ (212)

where ${\bm{\lambda}}({\bf r})$ is the spin-orbit coupling vector (its
direction is the internal electric field direction), $\hat{\bf
p}=-i\hbar\nabla$ is the momentum operator, and $q=-e$ is electron charge
[76].

We take into account the coordinate dependence of the spin function by
adopting the following spin function

$\displaystyle\Sigma_{1}({\bf r})=e^{-{i\over 2}\chi({\bf
r})}\left(\begin{array}[]{c}e^{-i{1\over 2}\xi({\bf r})}\sin{{\zeta({\bf
r})}\over 2}\\\ e^{i{1\over 2}\xi{\bf r})}\cos{{\zeta({\bf r})}\over
2}\end{array}\right)$ (215)

for the Bloch electron, where $\zeta({\bf r})$ and $\xi({\bf r})$ are the
polar and azimuthal angles of the spin-direction, respectively. The
expectation value of spin ${\bf s}({\bf r})=(s_{x}({\bf r}),s_{y}({\bf
r}),s_{z}({\bf r}))$ is given by

$\displaystyle s_{x}({\bf r})={\hbar\over 2}\cos\xi({\bf r})\sin\zeta({\bf
r}),\ s_{y}({\bf r})={\hbar\over 2}\sin\xi({\bf r})\sin\zeta({\bf r}),\
s_{z}={\hbar\over 2}\cos\zeta({\bf r})$ (216)

The Berry connection arising from the spin function $\Sigma_{1}({\bf r})$ is

$\displaystyle{\bf A}^{\rm fic}_{1}({\bf r})=-i{{\hbar c}\over
e}\Sigma_{1}^{\dagger}\nabla\Sigma_{1}=-{{\hbar c}\over{2e}}\nabla\chi({\bf
r})+{{\hbar c}\over{2e}}\nabla\xi({\bf r})\cos\zeta({\bf r})$ (217)

and the effective vector potential is

$\displaystyle{\bf A}_{1}^{\rm eff}={\bf A}^{\rm em}+{\bf A}_{1}^{\rm fic}$
(218)

We introduce the following wave vector ${\bf k}_{c}$ for convenience,

$\displaystyle{\bf k}_{c}={\bf q}_{c}+{e\over{\hbar c}}{\bf A}^{\rm
eff}_{1}({\bf r}_{c})$ (219)

By following the standard method for the wave packet formalism[77, 19], the
following equations of motion are obtained,

$\displaystyle\dot{\bf r}_{c}$ $\displaystyle=$
$\displaystyle{1\over\hbar}{{\partial{\cal E}}\over{\partial{\bf
k}_{c}}}+{\bm{\lambda}}({\bf r}_{c})\times{\bf s}({\bf r}_{c})$ (220)
$\displaystyle\dot{\bf k}_{c}$ $\displaystyle=$ $\displaystyle-{e\over{\hbar
c}}\dot{\bf r}_{c}\times{\bf B}^{\rm eff},$ (221)

where ${\bf B}^{\rm eff}$ is the effective magnetic field,

$\displaystyle{\bf B}^{\rm eff}=\nabla\times{\bf A}_{1}^{\rm eff}$ (222)

Eqs. (220) and (221) indicate that the cyclotron motion in the band with
energy

$\displaystyle{\cal E}({\bf k})+\hbar{\bm{\lambda}}({\bf r})\times{\bf s}({\bf
r})\cdot{\bf k}$ (223)

occurs.

By following the Onsager’s argument[78], let us quantize the cyclotron orbit.
From Eq. (219), the Bohr-Sommerfeld relation becomes

$\displaystyle\oint_{C}\hbar{\bf q}_{c}\cdot d{\bf r}_{c}=\oint_{C}(\hbar{\bf
k}_{c}-{e\over c}{\bf A}_{1}^{\rm eff})\cdot d{\bf
r}_{c}=2\pi\hbar\left(n+{1\over 2}\right)$ (224)

where $n$ is an integer and $C$ is the closed loop that corresponds to the
section of Fermi surface enclosed by the cyclotron orbit.

Using Eq. (221), we have

$\displaystyle\oint_{C}\hbar{\bf k}_{c}\cdot d{\bf r}_{c}$ $\displaystyle=$
$\displaystyle-{e\over c}\oint_{C}d{\bf r}_{c}\cdot{\bf r}_{c}\times{\bf
B}^{\rm eff}$ (225) $\displaystyle=$ $\displaystyle{e\over c}\oint_{C}{\bf
B}^{\rm eff}\cdot{\bf r}_{c}\times d{\bf r}_{c}$ $\displaystyle=$
$\displaystyle{{2e}\over c}\oint_{C}{\bf A}_{1}^{\rm eff}\cdot d{\bf r}_{c}$

Thus, Eq. (224) becomes

$\displaystyle{e\over c}\oint_{C}{\bf A}^{\rm em}\cdot d{\bf r}_{c}+{e\over
c}\oint_{C}{\bf A}_{1}^{\rm fic}\cdot d{\bf r}_{c}=2\pi\hbar\left(n+{1\over
2}\right)$ (226)

This is the well-known quantized condition for the cyclotron motion if ${\bf
A}_{1}^{\rm fic}$ is absent.

If ${\bf A}_{1}^{\rm fic}$ is present, the above quantization condition is
satisfied even if the magnetic field is absent. In this case, Eq. (226)
yields,

$\displaystyle-\oint_{C}{1\over 2}\nabla\chi({\bf r})\cdot d{\bf
r}_{c}+\oint_{C}{1\over 2}\nabla\xi({\bf r})\cos\zeta({\bf r})\cdot d{\bf
r}_{c}=2\pi\left(n+{1\over 2}\right)$ (227)

which can be satisfied by the following two sets of conditions; one is
$\zeta={\pi\over 2}$, $w_{C}[\chi]=-1$, and $n=0$; and the other is
$\zeta={\pi\over 2}$, $w_{C}[\chi]=1$, and $n=-1$, where

$\displaystyle w_{C}[\chi]={1\over{2\pi}}\oint_{C}\nabla\chi({\bf r})\cdot
d{\bf r}$ (228)

is the winding number of $\chi$ along loop $C$.

When $\zeta={\pi\over 2}$, the single-valued condition for $\Sigma_{1}({\bf
r})$ in Eq. (215) leads to the single-valuedness of $e^{-{i\over 2}\chi({\bf
r})}e^{\pm i{1\over 2}}$. This is satisfied if the following requirement

$\displaystyle w_{C}[\chi]+w_{C}[\xi]=\mbox{even number}$ (229)

is satisfied.

The condition $w_{C}[\chi]=\pm 1$ requires that $w_{C}[\xi]$ must be odd,
thus, $w_{C}[\xi]$ is not zero. The nonzero value of $w_{C}[\xi]$ means that
electrons perform spin-twisting itinerant motion. This indicates that the
quantized cyclotron motion occurs without an external magnetic field by
performing the spin-twisting itinerant motion.

When the quantized cyclotron motion without magnetic field occurs, ${\bf
A}_{1}^{\rm fic}$ is given by

$\displaystyle{\bf A}^{\rm fic}_{1}({\bf r})=-{{\hbar
c}\over{2e}}\nabla\chi({\bf r})$ (230)

In addition to $\Sigma_{1}$ in Eq. (215), we consider the spin function
$\Sigma_{2}$ that is orthogonal to $\Sigma_{1}$,

$\displaystyle\Sigma_{2}({\bf r})=e^{-{i\over 2}\chi({\bf
r})}\left(\begin{array}[]{c}ie^{-i{1\over 2}\xi({\bf r})}\cos{{\zeta({\bf
r})}\over 2}\\\ -ie^{i{1\over 2}\xi{\bf r})}\sin{{\zeta({\bf r})}\over
2}\end{array}\right)$ (233)

The fictitious vector potential from $\Sigma_{2}$ is

$\displaystyle{\bf A}_{2}^{\rm fic}({\bf r})=-i{{\hbar c}\over
e}\Sigma_{2}^{\dagger}\nabla\Sigma_{2}=-{{\hbar c}\over{2e}}\nabla\chi({\bf
r})-{{\hbar c}\over{2e}}\nabla\xi({\bf r})\cos\zeta({\bf r})$ (234)

and the effective vector potential is

$\displaystyle{\bf A}_{2}^{\rm eff}={\bf A}^{\rm em}+{\bf A}_{2}^{\rm fic}$
(235)

When the quantized cyclotron motion without magnetic field occurs, ${\bf
A}_{2}^{\rm fic}$ is given by

$\displaystyle{\bf A}^{\rm fic}_{2}({\bf r})=-{{\hbar
c}\over{2e}}\nabla\chi({\bf r})$ (236)

just like ${\bf A}_{1}^{\rm fic}$

Let us consider the case where the many-electron wave function $\Phi$ for $N$
electrons is given as a Slater determinant of spin-orbitals $\phi_{1,1}({\bf
r})\Sigma_{1}({\bf r}),\phi_{1,2}({\bf r})\Sigma_{2}({\bf r})$, $\dots$,
$\phi_{{N\over 2},1}({\bf r})\Sigma_{1}({\bf r})$, and $\phi_{{N\over
2},2}({\bf r})\Sigma_{2}({\bf r})$, where $\phi_{j,1}({\bf r})$ and
$\phi_{j,2}({\bf r})$ are time-reversal partners and $N$ is assumed to be
even.

Then, ${\bf A}^{\rm MB}_{\Phi}$ in Eq. (192) is calculated as

$\displaystyle{\bf A}^{\rm MB}_{\Phi}$ $\displaystyle=$ $\displaystyle\
\rm{Im}{{\sum_{j=1}^{N\over 2}\left[\phi^{\ast}_{j,1}({\bf
r})\Sigma^{\dagger}_{1}({\bf r})\nabla\phi_{j,1}({\bf r})\Sigma_{1}({\bf
r})+\phi^{\ast}_{j,2}({\bf r})\Sigma^{\dagger}_{2}({\bf
r})\nabla\phi_{j,2}({\bf r})\Sigma_{2}({\bf
r})\right]}\over{\sum_{j=1}^{N\over 2}\left[\phi^{\ast}_{j,1}({\bf
r})\phi_{j,1}({\bf r})+\phi^{\ast}_{j,2}({\bf r})\phi_{j,2}({\bf r})\right]}}$
(237) $\displaystyle=$ $\displaystyle\ {e\over\hbar}{{{\bf A}^{\rm
fic}_{1}\sum_{j=1}^{N\over 2}\phi^{\ast}_{j,1}({\bf r})\phi_{j,1}({\bf
r})+{\bf A}^{\rm fic}_{2}\sum_{j=1}^{N\over 2}\phi^{\ast}_{j,2}({\bf
r})\phi_{j,2}({\bf r})}\over{\sum_{j=1}^{N\over 2}\left[\phi^{\ast}_{j,1}({\bf
r})\phi_{j,1}({\bf r})+\phi^{\ast}_{j,2}({\bf r})\phi_{j,2}({\bf r})\right]}}$

where “Im” indicates the imaginary part. Note that

$\displaystyle\sum_{j=1}^{N\over 2}\left[\phi^{\ast}_{j,1}({\bf
r})\nabla\phi_{j,1}({\bf r})+\phi^{\ast}_{j,2}({\bf r})\nabla\phi_{j,2}({\bf
r})\right]\mbox{ is real }$ (238)

since $\phi_{j,1}({\bf r})$ and $\phi_{j,2}({\bf r})$ are time-reversal
partners.

When the quantized motion of spin-twisting itinerant motion occurs, ${\bf
A}^{\rm fic}_{1}$ and ${\bf A}^{\rm fic}_{2}$ are given by ${\bf A}^{\rm
fic}_{1}={\bf A}^{\rm fic}_{2}=-{{\hbar c}\over{2e}}\nabla\chi$. Thus, we have

$\displaystyle{\bf A}^{\rm MB}_{\Phi}=-{1\over 2}\nabla\chi$ (239)

It is shown that when the pairing energy gap is formed, the smallest possible
size of the loop current from the quantized motion is estimated to be
$\xi_{\rm BCS}$ in Eq. (24) [21]. The loop currents can coexist with
displacing their centers. This suggests that $\xi_{\rm BCS}$-sized loop
currents exist in the BCS superconductor with the current directions arranged
in such a way that the macroscopic current is zero.

##  Appendix C Reversible superconducting-normal metal phase transition in a
magnetic field.

We will show that how the existence of ${\bf A}^{\rm fic}$ explains the
reversible superconducting-normal metal phase transition in a magnetic field.

Let us consider the magnetic field part of the free energy of the Ginzburg-
Landau theory is given by

$\displaystyle F_{\rm mag}=\int d^{3}r{1\over{8\pi}}\left({\bf B}^{\rm
eff}\right)^{2}$ (240)

During the phase transition, the superconducting region shrinks and the
magnetic field penetrating region increases. As a consequence
$\partial_{t}{\bf B}^{\rm em}\neq 0$ occurs. This will induce an electric
field, and if normal current exists, the Joule heating should occur. Thus, for
the reversible transition to occur, it should occur without generating normal
current.

The change of $F_{\rm mag}$ due to $\partial_{t}{\bf B}^{\rm eff}\neq 0$ is
given by

$\displaystyle\Delta F_{\rm mag}=\int d^{3}r\int_{t}^{t+\Delta
t}dt{1\over{4\pi}}{\bf B}^{\rm eff}\cdot\partial_{t}{\bf B}^{\rm eff}$ (241)

where $\Delta t$ is the time interval for this change

If the above change of ${\bf B}^{\rm eff}$ is only due to the change of ${\bf
A}^{\rm fic}$, $\Delta F_{\rm mag}$ is given by

$\displaystyle\Delta F_{\rm mag}={1\over c}\int d^{3}r{\bf
j}_{s}\cdot\int_{t}^{t+\Delta t}dt\ \partial_{t}{\bf A}^{\rm fic}$ (242)

where

$\displaystyle{\bf B}^{\rm eff}=\nabla\times({\bf A}^{\rm em}+{\bf A}^{\rm
fic}),\quad{\bf j}_{s}={c\over{4\pi}}\nabla\times{\bf B}^{\rm eff}$ (243)

are used.

The supercurrent kinetic energy part of the free energy is given by

$\displaystyle F_{\rm kin}=\int d^{3}r{m_{e}\over{2}}n_{s}{\bf v}^{2}=\int
d^{3}r{{e^{2}n_{s}}\over{2m_{e}c^{2}}}\left({\bf A}^{\rm eff}\right)^{2}$
(244)

where the supercurrent density

$\displaystyle{\bf j}_{s}=-en_{s}{\bf v}=-{{e^{2}n_{s}}\over{m_{e}c}}{\bf
A}^{\rm eff}$ (245)

is used.

The change of the supercurrent kinetic energy $\Delta F_{\rm kin}$ caused by
$\partial_{t}{\bf A}^{\rm fic}$ is

$\displaystyle\Delta F_{\rm kin}=-{1\over c}\int d^{3}r{\bf
j}_{s}\cdot\int_{t}^{t+\Delta t}dt\ \partial_{t}{\bf A}^{\rm fic}$ (246)

Therefore, according to Eqs. (242) and (246), the transition with $\Delta
F_{\rm mag}+\Delta F_{\rm kin}=0$ is realized via

$\displaystyle\int_{t}^{t+\Delta t}dt\ \partial_{t}{\bf A}^{\rm fic}$ (247)

This occurs in a quantized manner by changing the winding numbers for $\chi$,
without Joule heating. This is the key step to realize the reversible
superconducting-normal metal phase transition in a magnetic field. Other
changes occur among ${\bf j}_{s}$, ${\bf B}^{\rm em}$, $n_{s}$, to satisfy
$\nabla{\bf B}^{\rm em}={{4\pi}\over c}{\bf j}_{s}$ and
$\partial_{t}n_{s}+\nabla\cdot{\bf j}_{s}=0$. However, those changes can
proceed without Joule heating [57].

##  Appendix D Collective mode $\chi$ and associated number changing operators

We consider the quantization of the collective mode arising from $\chi$. For
this purpose, we use the Heisenberg operator formalism.

In order to have the phase arising from ${\bf A}^{\rm fic}$ in Eq. (200), we
express $\Phi$ as

$\displaystyle\Phi=\Phi_{0}e^{-{i\over 2}\sum_{j=1}^{N}\chi({\bf r}_{j})}$
(248)

by explicitly extracting the Berry phase part $e^{-{i\over
2}\sum_{j=1}^{N}\chi({\bf r}_{j})}$.

This is actually the definition for $\Phi_{0}$ through $\Phi$ and the Berry
connection,

$\displaystyle\Phi_{0}=\Phi e^{{i\over 2}\sum_{j=1}^{N}\chi({\bf r}_{j})}$
(249)

The kinetic energy part of the Hamiltonian is given by

$\displaystyle K_{0}={1\over{2m_{e}}}\sum_{j=1}^{N}\left({\hbar\over
i}\nabla_{j}\right)^{2}$ (250)

where $m_{e}$ is the electron mass. The current density calculated using the
above kinetic energy

$\displaystyle{\bf j}({\bf r})=-q\sum_{j=1}^{N}{\hbar\over{2im_{e}}}\int
d^{3}r_{1}\cdots d^{3}r_{N}\
\left[\Phi_{0}^{\ast}\nabla_{j}\Phi_{0}-\Phi_{0}\nabla_{j}\Phi_{0}^{\ast}\right]\delta({\bf
r}-{\bf r}_{j})$ (251)

is zero; i.e., $\Phi_{0}$ is a currentless state.

When the ground state does not have the non-trivial Berry connection, it is a
currentless state. In this case, we may identify $\Phi$ as $\Phi_{0}$, and the
collective coordinate $\chi$ disappears.

However, when the Berry connection is non-trivial, the collective coordinate
$\chi$ is present. We treat $\chi$ as a collective dynamical variable, and
quantize it with the help of the following Lagrangian,

$\displaystyle{\cal
L}\\!=\langle\Phi|i\hbar\partial_{t}\\!-\\!H|\Phi\rangle\\!=\\!i\hbar\langle\Phi_{0}|\partial_{t}|\Phi_{0}\rangle+{\hbar\over
2}\int\\!d{\bf r}\ {{n_{p}\dot{\chi}}}-\langle\Phi|H|\Phi\rangle$ (252)

where $n_{p}$ is the number density of the particles [79].

From the above Lagrangian, the conjugate momentum of $\chi$ is obtained as

$\displaystyle p_{\chi}={{\delta{\cal L}}\over{\delta\dot{\chi}}}={\hbar\over
2}n_{p}$ (253)

thus, $\chi$ and $n_{p}$ are canonical conjugate variables.

By following the canonical quantization condition

$\displaystyle[\hat{p}_{\chi}({\bf r},t),\hat{\chi}({\bf
r}^{\prime},t)]=-i\hbar\delta({\bf r}-{\bf r}^{\prime})$ (254)

where $\hat{p}_{\chi}$ and $\hat{\chi}$ are operators corresponding to
${p}_{\chi}$ and ${\chi}$ respectively, we obtain the following relation

$\displaystyle\left[{{\hat{n}({\bf r},t)}},\hat{\chi}({\bf
r}^{\prime},t)\right]=-2i\delta({\bf r}-{\bf r}^{\prime})$ (255)

where $\hat{n}$ is the operator corresponding to $n_{p}$.

Strictly speaking, $\hat{\chi}$ is not a hermitian operator; however, it is
known that when it is used as $e^{\pm{i\over 2}{\hat{\chi}}}$, the problem is
avoided, practically [80].

We construct the following boson field operators from $\hat{\chi}$ and
$\hat{n}$,

$\displaystyle\hat{\psi}^{\dagger}({\bf r})=\left(\hat{n}({\bf
r})\right)^{1/2}e^{{i\over 2}{\hat{\chi}({\bf r})}},\quad\hat{\psi}({\bf
r})=e^{-{i\over 2}{\hat{\chi}({\bf r})}}\left(\hat{n}({\bf r})\right)^{1/2}$
(256)

Using Eq. (255), the following relations are obtained,

$\displaystyle[\hat{\psi}({\bf r}),\hat{\psi}^{\dagger}({\bf
r}^{\prime})]=\delta({\bf r}-{\bf r}^{\prime}),\quad[\hat{\psi}({\bf
r}),\hat{\psi}({\bf r}^{\prime})]=0,\quad[\hat{\psi}^{\dagger}({\bf
r}),\hat{\psi}^{\dagger}({\bf r}^{\prime})]=0$ (257)

From Eq. (257), the following relations are obtained

$\displaystyle[e^{\pm{i\over 2}\hat{\chi}({\bf r})},\hat{\psi}^{\dagger}({\bf
r}^{\prime})\hat{\psi}({\bf r}^{\prime})]=\mp e^{\pm{i\over 2}\hat{\chi}({\bf
r})}\delta({\bf r}-{\bf r}^{\prime})$ (258)

which indicate that $e^{\pm{i\over 2}\hat{\chi}({\bf r})}$ are number changing
operators for the number density operator

$\displaystyle\hat{n}({\bf r})=\hat{\psi}^{\dagger}({\bf r})\hat{\psi}({\bf
r})$ (259)

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