# FAST polarization mapping of the SNR VRO 42.05.01

Li Xiao,11affiliation: National Astronomical Observatories, CAS, Beijing
100012, China; Email<EMAIL_ADDRESS>22affiliation: Key Laboratory of FAST,
NAOC, Chinese Academy of Science, Beijing 100012, China 33affiliation: Guizhou
Radio Astronomical Observatory, Guizhou University, Guiyang 550000, China Ming
Zhu,11affiliationmark:  22affiliationmark:  33affiliationmark:  Xiao-Hui
Sun,44affiliation: Department of Astronomy, Yunnan University, and Key
Laboratory of Astroparticle Physics of Yunnan Province, Kunming, 650091, China
Peng Jiang 11affiliationmark:  22affiliationmark:  33affiliationmark:  Chun
Sun 11affiliationmark:  22affiliationmark:  33affiliationmark:

###### Abstract

We have obtained the polarization data cube of the VRO 42.05.01 supernova
remnant at 1240 MHz using the Five-hundred-meter Aperture Spherical radio
Telescope (FAST). Three-dimensional Faraday Synthesis is applied to the FAST
data to derive the Faraday depth spectrum. The peak Faraday depth map shows a
large area of enhanced foreground RM of $\sim$60 rad m-2 extending along the
remnant’s “wing” section, which coincides with a large-scale HI shell at $-20$
km s-1. The two depolarization patches within the “wing” region with RM of 97
rad m-2 and 55 rad m-2 coincide with two HI structures in the HI shell.
Faraday screen model fitting on the Canadian Galactic Plane Survey (CGPS) 1420
MHz full-scale polarization data reveals a distance of $0.7-0.8d_{SNR}$ in
front of the SNR with enhanced regular magnetic field there. The highly piled-
up magnetic field indicates that the HI shell at $-20$ km s-1 could originate
from an old evolved SNR.

– radiation mechanisms: non-thermal – methods: observational – ISM: supernova
remnants – ISM:evolution

##  1 Introduction

A supernova remnant shock compresses the magnetic field in the interstellar
medium (ISM) with a factor of four or more in the evolved stage (Fürst & Reich
2004; Helder et al. 2012), which generates enhanced synchrotron emission and
retains the information of ambient interacted medium. Radio polarization
observations directly detect the perpendicular magnetic field configuration
across supernova remnants (SNRs), and give the line-of-sight component of the
magnetic field by measuring the Faraday rotation of the polarization angles at
different frequencies. It provides important magnetic field configuration
information for understanding the dynamics of shock, post-shock gas, and the
interaction with the ISM.

VRO 42.05.01 (G166.0$+$4.3) is a shell-type SNR with a peculiar morphology
consisting of a semicircular “shell” and a triangular “wing” structure
(Pineault et al. 1985, 1987). It has been observed in various bands to
investigate the evolution in the inhomogeneous medium with two different
densities (Summary review in Arias et al. (2019a)). Radio observations from
low to high frequencies have indicated different indices and compression
ratios in these two sections (Leahy & Tian 2005; Gao et al. 2011; Arias et al.
2019b; Xiao et al. 2022). The polarized emission has been observed in the
Canadian Galactic Plane Survey (CGPS) at 1420 MHz, with most regions
depolarized (Kothes et al. 2006). A large-scale HI shell at $-20$ km s-1 with
the corresponding footprint on the $U/Q$ map extends across the remnant,
probably acting as a foreground Faraday screen (Kothes & Landecker 2004). The
remnant has also been observed in the Sino-German Galactic plane survey at 4.8
GHz (Gao et al. 2011). The magnetic field vectors in the two “wing” shock
shells show different deviations from tangential to the shock front,
indicating different rotation measures.

In this paper, we present the polarization observation of VRO 42.05.01 at 1240
MHz with the Five-hundred-meter Aperture Spherical radio Telescope (FAST).
FAST has the advantage of a large aperture with high resolution and high
sensitivity (Jiang et al. 2020) and has produced good scientific outputs (Qian
et al. 2020). The 19-beam wide band multi-channel polarization receiver from
1050$-$1450 MHz enables it to measure the polarization of extended SNRs at
multi frequencies simultaneously (Sun et al. 2022; Gao et al. 2022). Through
the Faraday rotation synthesis technology (Brentjens & de Bruyn 2005), we can
efficiently obtain the Faraday depth spectrum and retrieve the polarization
properties towards VRO 42.05.01.

The paper is organized as follows. The observation and data reduction are
described in Sect. 2. The comparison of the FAST polarization map with the
CGPS map is shown in Sect. 3.1. The Faraday rotation synthesis and
depolarization analysis are presented in Sect. 3.2 and 3.3. In Sect. 4, we
make a discussion on the depolarization feature in the southern wing region.
Conclusions are summarized in Sect. 5.

##  2 Observations and Data reduction

The FAST polarized continuum mapping of VRO 42.05.01 was observed twice using
the multibeam on-the-fly mode (2RA + 2DEC) in 2020 April and 2021 February,
respectively. The observation setting and parameters are listed in paper I
(Xiao et al. 2022). 3C138 is observed as the calibrator. The spectral backend
with a 65536 channel covering the 1050$-$1450 MHz band recorded the four
polarization outputs ($I_{1}$, $I_{2}$, $U$, $V$) from the FAST 19-beam linear
feeds. After intensity calibration, the data were converted to antenna
temperature (K $T_{a}$), and the Stokes parameters are derived as
$I=I_{1}+I_{2}$, $Q=I_{1}-I_{2}$, $U$ and $V$. The leakage caused by the
mismatch between the amplitudes and phases of the gains of the two linear
feeds is small and corrected using the 10 K injected reference signal injected
every two seconds following the method in Sun et al. (2021).

For the $Q$, $U$, and $V$ data along the R.A. or decl. direction, a baseline
correction was first made by linearly fitting the two ends of each scan and
subsequently subtracted to remove unrelated large-scale emissions. Radio
frequency interference (RFI) caused by communication satellites and navigation
satellites (1160-1280 MHz) and other interference were flagged the same as
that in $I$ in the frequency zone. Then we converted the antenna temperature
to the main-beam brightness temperature (K $T_{B}$) by dividing the main beam
efficiency ($\eta_{b}$) calculated from the calibrator (Sun et al. 2021), and
smoothed each channel map to a common 4$\arcmin$ angular resolution.

Finally, all four $Q$ maps and $U$ maps were binned every 20 channels to
improve the noise level and weaved together to destripe the scanning effects
in the Fourier domain (Emerson & Graeve 1988). We averaged the data over the
full FAST band by taking medians of all the frequency channels to form the
combined $Q$ and $U$ maps. The polarized intensity $PI$ corrected from the
positive noise bias was calculated as $PI=\sqrt{Q^{2}+U^{2}-(1.2\sigma)^{2}}$
(Wardle & Kronberg 1974), where the rms noise $\sigma$ measured from the
average combined $Q$, $U$ maps are both about 4 mK Tb. The polarization angle
$\psi$ is derived as $\psi=\frac{1}{2}\arctan 2(U,Q)$.

Figure 1:  Left panel: The FAST full bandwidth polarization intensity map of
VRO 42.05.01 at a central frequency of 1240 MHz. The angular resolution is
4$\arcmin$. Contours show the total intensities of VRO 42.05.01 start at 350
mK and increase by a step of 300 mK. Right panel: The CGPS 1420 MHz
polarization intensity map of VRO 42.05.01 with the large-scale polarized
emission restored from Effelsberg and the DRAO 26-m single antenna telescopes.
The angular resolution is 1$\arcmin$. The contours are the same as the left
panel. The bars show the orientation of the magnetic field B, with lengths
proportional to the CGPS polarization intensity.

##  3 Results

###  3.1 Comparison with the CGPS 1420 MHz polarization map

The FAST full-bandwidth 1240 MHz polarization intensity map of VRO 42.05.01
derived from the averaged $Q$ and $U$ data is presented in Fig. 1. There is a
weak polarization emission counterpart for the shock “shell”. The polarized
patches in the center of VRO 42.05.01 with a comparable 10$-12$ mK Tb
intensity with that in the outer region, might originate from the foreground
diffuse polarization emission. A zero-polarized intensity canal structure with
a length of 27$\arcmin$ passes across the wing region. It has a width of about
$4\arcmin$, similar to the beam size, and might be caused by different
polarization angles of the polarized patches on each side.

For comparison, the combined Canadian Galactic Plane Survey (CGPS) 1420 MHz
polarization intensity map of VRO 42.05.01 (Landecker et al. 2010) is shown in
the right panel of Fig. 1. It has an angular resolution of $\sim 1\arcmin$,
and has restored the large-scale polarized emission by combing data from
Effelsberg and the Dominion Radio Astrophysical Observatory (DRAO) 26-m single
antenna telescopes. Polarized emission is present only in the shell peak and
the tip edge of the triangle wing region in the CGPS synthesis map (Fig.13 in
Kothes et al. (2006)), which turns into voids after baselevel restoration. The
depolarization feature in the southern wing region (two depolarization patches
separated with a filament) is also identical in a way of opposite values to
the FAST 1240 MHz polarization intensity map. The canal structure becomes
polarized filaments in the CGPS 1420 MHz map, while the polarization patches
on both sides turned into depolarized regions.

The large-scale emission component is important for the interpretation of
polarized structures caused by Faraday rotation in the interstellar medium
(Reich 2006). The FAST 1240 MHz polarization data lacks the large-scale
polarized emission due to the baselevel subtraction. After adding the large-
scale polarization vectors in the CGPS map, the polarized canals become
polarized filament emission. The polarized patches in the southern wing region
become depolarized regions, indicating there exists Faraday rotation effect.
The polarization angle of the background polarization emission has been
modulated and canceled the foreground polarization emission to cause a
depression in the polarization intensity (detail discussion in Sect. 4.1).

Figure 2: The Faraday depth spectra and fitted Faraday depth $\phi$ components
(green) towards polarization patches in the region of VRO 42.05.01 as marked
in the maxPI map of Fig. 4.

###  3.2 the Faraday Synthesis analysis

We used the widely used RM synthesis to reconstruct the Faraday depth spectrum
from the frequency cubes of $Q$ and $U$ (Brentjens & de Bruyn 2005). The
method is based on that the observed polarization intensity $P(\lambda^{2})$
and the Faraday dispersion function $F(\phi)$ are Fourier transform pairs
(Burn 1966)

$P(\lambda^{2})=Q(\lambda^{2})+iU(\lambda^{2})=\int
F(\phi)e^{2i\phi\lambda^{2}}d\phi,$ (1)

where $\lambda$ is the wavelength and the $\phi$ is the Faraday depth defined
as $\phi=0.812\int n_{e}B_{||}dr$, integral of the thermal electron density
with the magnetic field along the line-of-sight from the source to the
observer. By fitting the peak of the Faraday depth spectrum, we can obtain the
Faraday depth integrated towards the source, which is equivalent to the
rotation measure (RM).

We applied the RM synthesis to the FAST data of VRO 42.05.01 for each pixel
using the RM-Tools package (Purcell et al. 2020). The algorithm returns the
peak Faraday depth $\phi_{peak}$, the peak polarized intensity
$F(\phi_{peak})$, and the polarized intensity cube along Faraday depth
$F(\phi)$. For the FAST frequency range and sampling, the width of the RM
spread function is about 90 rad m-2, and the Faraday depth has a step interval
of 14 rad m-2. We displayed Faraday spectra and fitted peak components towards
six polarized patches as marked in the maxPI map in Fig. 2. Most of the
Faraday spectra show one peak above 3$\sigma_{PI}$ level. The spectra towards
the polarized patches in the northern wing and in the southeast outside peak
at 0 rad m-2 indicate, that they come from the foreground and the polarized
emission of the remnant is fully depolarized. The spectrum in the shell shows
a peak of 14 rad m-2. The spectra towards the polarized patches in the
southern wing (4, 5) and in the southwest outside (6) show large RM peaks of
55, 97, and 97 rad m-2, respectively. The spectrum 6 also shows another weak
negative RM component, which might come from a foreground local feature. Here
we just discuss the positive RM.

Figure 3: Left: The peak RM map reconstructed from the Faraday synthesis
method for VRO 42.05.01. Pixels with maxPI values below 30 mK have been set to
nan. Middle and right: the polarized intensity at 55 rad m-2 and 97 rad m-2 to
display the large RM distribution separately.  Figure 4: The FAST maximum
polarized intensity of VRO 42.05.01 at 1240 MHz derived from the Faraday
synthesis method.

The peak Faraday depth $\phi_{peak}$ map towards VRO 42.05.01 is presented in
Fig. 3. It shows the large Faraday depth area ($\phi>$ 60 rad m-2) in the
southwest extending along the wing region (also the Galactic longitude
direction). The depolarization patches within the wing region in the CGPS map
of Fig. 1 correspond to a maximum Faraday depth of 97 rad m-2 and 55 rad m-2,
respectively. The polarized intensity at these two values in the Faraday depth
cube $F(\phi)$ are shown separately in Fig. 3.

We presented the maximum polarized intensity map in Fig. 4. The canal and
polarized patches in the wing region are almost the same as in the
polarization map averaged from $Q$ and $U$. However, in the southern and
western edges, the maxPI map shows some weak polarized patches, which are
depolarized in the averaged polarization map. Considering the large foreground
RM of 60 rad m-2, averaging over all the frequency channels might have caused
some bandwidth depolarization towards these regions.

###  3.3 Depolarization in the shell

In most regions of VRO 42.05.01 the polarization has been smeared out. Only
weak polarized emission is detected in the shell region both in FAST 1240 and
CGPS 1420 MHz maps. For a medium where synchrotron emission and Faraday
rotation co-exist, depth depolarization is the main mechanism to cause
depolarization. Polarized emission at different depths experiences different
Faraday rotations, which reduce or even cancel the polarization intensity when
added together. Following Sokoloff et al. (1998), it is wavelength-dependent
and the depolarization $P_{\lambda}$, the ratio of the observed polarization
percentage $PC$ at $\lambda$ to intrinsic $PC$ is related to RM as

$P_{\lambda}={\rm sin}(2|RM|\lambda^{2})/2|RM|\lambda^{2}$ (2)

Figure 5:  The Urumqi 4.8 GHz polarization percentage map of VRO 42.05.01. The
image has an angular resolution of 9.5$\arcmin$. Contours show the total
intensities at 4.8 GHz start at 6 mK, and increase by a step of 5 mK. The bars
show the orientation of the magnetic field B, with lengths proportional to the
polarized intensity.

We presented the Urumqi 4.8 GHz polarization percentage map (Gao et al. 2011)
in Fig. 5. At 4.8 GHz, the depth depolarization is negligible and it traces
approximately the intrinsic polarization emission. The polarization percentage
$PC$ in the shell region is about 20%. The wing region has a different $PC$
distribution, with the southern section having a higher $PC$ of 24%, and a
lower $PC$ of 15% in the northern area. By comparing the polarization degree
in the shell at 1240 MHz ($PC\sim 3\%$) with that at 4.8 GHz, we obtained a
relative depolarization $PC_{1.24}/PC_{5}$ of 0.85. It corresponds to an
intrinsic RM of 20 rad m-2 in the shell peak. The positive sign of RMs is
adopted based on the deviation of polarization angles at 4.8 GHz. The value is
comparable with the Faraday synthesis result in Fig. 3.

Figure 6: The CGPS integrated HI map with 3 channels at $-20$ km s-1 towards
VRO 42.05.01. White contours show the CGPS total intensity of VRO 42.05.01
smoothed to 4$\arcmin$, ranging from 4.8 K in a step of 240 mK. Black contours
show the Rotation measure map at 55 rad m-2 at 26 and 60 mK level.

##  4 Discussion

###  4.1 depolarization feature caused by Faraday screens

VRO 42.05.01 was reported to overlap with a partial large-scale HI shell with
a radius of 4$\arcdeg$ at $-20$ km s-1 in the CGPS polarization survey (Kothes
& Landecker 2004). As shown in Fig. 6, the HI shell seems to go across and
have separated the “shell” and “wing” region. This is consistent with the RM
peak distribution derived from the FAST polarization data of Fig. 3. The large
foreground RM component at 55 rad m-2 in the wing section also extends along
the HI shell.

The depolarized patches in the wing region of VRO 42.05.01 seem to coincide
with two HI sheet structures in the HI shell. As revealed in the CGPS restored
polarization intensity map smoothed to 4$\arcmin$ in Fig. 7, the lower end of
the depolarized patch (B) with a width of 5.0$\arcmin$ overlaps with the HI
cloud below and extends upward. The depolarized arc (A) with a width of
2.4$\arcmin$ goes along the left periphery of the upper HI cloud and gradually
extends to coincide with the right side of the cloud. It seems that the HI
sheet structures have enhanced rotation measures and acted as Faraday screens,
which rotated the background polarized emission and caused depolarization
after adding up with the foreground emission.

Figure 7: The CGPS 1420 MHz restored polarization map of VRO 42.05.01 smoothed
to 4$\arcmin$. Black contours show the CGPS $-20$ km s-1 HI channel map
smoothed to 4$\arcmin$, ranging from 76.3 K in a step of 2.5 K. White contours
is the same as Fig. 6.

We tried to apply the Faraday screen model to the CGPS 1420 MHz polarization
map. We extracted data along the slice in Fig.7. The $PI$ and $PA$
distribution in the rectangle region along the slice direction is presented in
Fig. 8. The distance of VRO 42.05.01 is 3.24 kpc from the optical extinction
of background stars (Zhao et al. 2020). The total depolarization in the wing
region indicates that all detected polarized emission at 1.24 GHz comes from
the foreground, closer to us than the remnant. It is consistent with the CGPS
polarization horizon estimation of 3.0 kpc at $l<140\arcdeg$ (Kothes &
Landecker 2004) from depolarization properties of HII regions and SNRs.
Assuming that the Galactic disk has a uniform polarized emissivity and
rotation measure distribution, we fitted the ratio of polarization intensity
and polarization angle ($PI_{on}/PI_{off}$, $PA_{on}/PA_{off}$) through the
Faraday screens with that in the off surrounding region according to the
formulas in (Sun et al. 2011), to obtain their relative distances with the
remnant and the rotated angles. The fitted distances of these two Faraday
screens are 0.7$-$0.8$d_{SNR}$. The depths of the Faraday screens assuming
equal to the mean width are then estimated about 1.8 and 3.8 pc. The fitted
maximum rotated angles shown in Fig. 8 are about $-$30$\arcdeg$ and
$-$70$\arcdeg$, with an ambiguity of $\pm n\pi$ (n=0, 1, 2, $\cdots$). Based
on the RM synthesis results of 55 and 97 rad m-2 towards these two regions,
the corresponding rotated angles are 150$\arcdeg$ and 290$\arcdeg$,
respectively.

Figure 8: The distribution of the CGPS $PI$, $PA$ along the slice in the
southern wing region of VRO 42.05.01. The fitted rotation angle $\psi$
according to the Faraday screen model are $-$30$\arcdeg$ and $-$70$\arcdeg$
with an ambiguity of n$\pi$ (n=0, 1, 2, $\cdots$).

The non-detection of radiation from the Faraday screens in the total intensity
gives a limit on the electron density. According to Rohlfs & Wilson (2004),
the contribution of warm thermal gas to the FAST 1240 MHz brightness
temperature could be estimated as $T=T_{e}\tau=2.26n_{e}^{2}L$ mK, where
$\tau$ is the opacity, $L$ is the radiation depth in pc and $T_{e}$ is the
electron temperature taken to be 8000 K. Take $5\times rms$ (12 mK) in the
total intensity as the non-detection level, the upper limits of electron
density in the Faraday screens are derived as 1.7 and 1.2 cm-3. The resulting
lower limits for the regular magnetic field strength along the line of sight
are about 22 and 26 $\mu$G. However, the value is inversely proportional to
the square root of the depth $L$ and thus largely depends on the geometry of
the depolarized screens. If the depolarized structures are sheets or a shell
periphery with ten times large depths, the magnetic field upper limit will go
down to $6-8$ $\mu$G.

The large-scale HI shell at $-20$ km s-1 was suggested to be swept up by an
old SNR evolved into the dissipation phase (Kothes & Landecker 2004). The
thermal and radio emission in the shocked shell has declined and the interior
temperature drops, while the piled-up regular magnetic field and HI shell
remain. A similar case of HI bubbles associated with old remnants has been
reported in Xiao & Zhu (2014).

###  4.2 The RM in the wing region

The large foreground RM (97 rad m-2) filament extending across the northern
wing shell explains the different polarization angles in the northern wing
shock shell in the Urumqi 4.8 GHz map of Fig. 5. Based on the polarization
angle deviation from the alignment with the shocked shell at 4.8 GHz, we can
roughly estimate the RM in the wing region. In the outer northern wing shock-
shell of VRO 42.05.01, the polarization angle deviates less than 20$\arcdeg$,
indicating a lower RM of 80 rad m-2. After subtracting the foreground
contribution, the RM in the northern shell is about 20 rad m-2. The
polarization angle deviation in the southern wing shock-shell is small. The
high polarization percentage there indicates enhanced regular magnetic field
with less random magnetic field than other regions.

##  5 Summary

We have obtained the polarization data cube of the SNR VRO 42.05.01 at 1240
MHz using the FAST radio telescope and applied RM-Synthesis to derive the
Faraday depth spectrum. Only the “shell” in the low RM region shows weak
polarization emission. The peak Faraday depth map shows a large foreground RM
region of 60 rad m-2 extending along the wing section, which coincides with a
large-scale HI shell at $-20$ km s-1. The two depolarization patches in the
wing region with RM of 97 and 55 rad m-2 coincide with two HI structures in
the HI shell. Faraday screen model fitting on the CGPS 1420 MHz full-scale
polarization data reveals that it lies at 2.2-2.6 kpc in the Perseus arm with
an enhanced regular magnetic field. The highly piled-up magnetic field
indicates that the HI shell at $-20$ km s-1 could originate from an evolved
old SNR.

We thank the journal referee for critical reading and valuable comments to
improve the paper. We acknowledge the support from the National Key R&D
Program of China (2018YFE0202900), and the help from FAST colleagues for using
the server. The research presented in this paper has used data from the
Canadian Galactic Plane Survey, a Canadian project with international
partners, supported by the Natural Sciences and Engineering Research Council.

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11institutetext: Division of Quantum Condensed Matter Physics, Center for
Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577,
Japan

Theories and models of superconducting state

# Superconductivity by Berry connection from many-body wave functions: a
generalized Hartree-Fock approximation

Hiroyasu Koizumi

###### Abstract

A fundamental revision of superconductivity theory that resolves the
supercurrent carrier mass contradiction (the standard theory predicts it to be
the effective mass but the London moment measurement indicates it to be the
free electron mass) is presented, using a generalized Hatree-Fock
approximation that takes into account a Berry connection from many-body wave
functions. The new theory explains the pairing energy gap formation
accompanying the superconductivity transition in the same manner as the
standard theory, yet, provides the free electron carrier mass in accordance
with the London moment measurement.

###### pacs:

74.20.-z

##  1 Introduction

The standard theory of superconductivity is based on the BCS theory. The
origin of superconducting states is the electron-pair formation in this theory
[1]. By using the variational state vector

$\displaystyle|{\rm BCS}(\theta)\rangle=\prod_{\bf k}(u_{\bf
k}+e^{i\theta}v_{\bf k}c^{\dagger}_{{\bf k}\uparrow}c^{\dagger}_{-{\bf
k}\downarrow})|{\rm vac}\rangle$ (1)

the BCS theory provides a way to calculate the energy gap formation due to the
electron-pairing, where $|{\rm vac}\rangle$ is the vaccuum, $c^{\dagger}_{{\bf
k}\sigma}$ is the creation operator for the conduction electron of effective
mass $m^{\ast}$ with the wave vector ${\bf k}$ and spin $\sigma$.

The supercurrent is calculated as the linear response to the vector potential,
yielding the carrier mass to have the effective mass $m^{\ast}$. The carrier
mass has been measured through the London moment for many materials [2]. The
results always contradict the BCS theory, indicating that it is the free
electron mass $m_{e}$ instead of the effective mass $m^{\ast}$.

In the present theory, we will argue that the contradiction is resolved if
$\theta$ in Eq. (1) is defined differently by using the Berry phase from many-
body wave functions [3, 4]. The phase $\theta$ arises in the standard theory
the global $U(1)$ gauge symmetry breaking [5, 6], the number of particles in
the superconductor is not definite; however, something corresponding to
$\theta$ arises from spin-twisting itinerant motion of electrons, and
inclusion of it resolves the carrier mass contradiction with keeping the
particle number in the superconductor constant.

##  2 Berry Connection for Many-Body Wave Functions

Let us first explain the Berry phase from many-body wave functions. We
consider the ground state wave function of a system of $N_{e}$ electrons,
$\Psi({\bf x}_{1},\cdots,{\bf x}_{N_{e}},t)$, where ${\bf x}_{j}=({\bf
r}_{j},\sigma_{j})$ denotes the coordinate of the $j$th particle ${\bf r}_{j}$
and its spin $\sigma_{j}$.

A Berry connection associated with this wave function can be defined as

$\displaystyle{\bf A}^{\rm MB}_{\Psi}({\bf r},t)=-i\langle n_{\Psi}({\bf
r},t)|\nabla_{\bf r}|n_{\Psi}({\bf r},t)\rangle$ (2)

where $|n_{\Psi}({\bf r},t)\rangle$ is defined as

$\displaystyle\langle\sigma_{1},{\bf x}_{2},\cdots,{\bf
x}_{N_{e}}|n_{\Phi}({\bf r},t)\rangle={{\Psi({\bf r},\sigma_{1},{\bf
x}_{2},\cdots,{\bf x}_{N_{e}},t)}\over{|C_{\Psi}({\bf r},t)|^{{1\over 2}}}}$
(3)

where $|C_{\Psi}({\bf r},t)|$ is the normalization constant given by

$\displaystyle|C_{\Psi}({\bf r},t)|=\int d\sigma_{1}d{\bf x}_{2}\cdots d{\bf
x}_{N_{e}}\Psi({\bf r},\sigma_{1},{\bf
x}_{2},\cdots)\Psi^{\ast}(\sigma_{1},{\bf x},{\bf x}_{2},\cdots)$ (4)

A more explicit expression for ${\bf A}^{\rm MB}_{\Psi}({\bf r},t)$ is given
by

$\displaystyle{\bf A}^{\rm MB}_{\Psi}({\bf r},t)$ $\displaystyle=$
$\displaystyle-i\int d\sigma_{1}d{\bf x}_{2}\cdots d{\bf
x}_{N_{e}}{{\Psi^{\ast}({\bf r},\sigma_{1},{\bf x}_{2},\cdots,{\bf
x}_{N_{e}},t)}\over{|C_{\Psi}({\bf r},t)|^{{1\over 2}}}}\nabla_{\bf
r}{{\Psi({\bf r},\sigma_{1},{\bf x}_{2},\cdots,{\bf
x}_{N_{e}},t)}\over{|C_{\Psi}({\bf r},t)|^{{1\over 2}}}}$ $\displaystyle=$
$\displaystyle{1\over{\hbar\rho({\bf r},t)}}{\rm Re}\left\\{\int
d\sigma_{1}\cdots d{\bf x}_{N_{e}}\Psi^{\ast}({\bf r},\sigma_{1},\cdots,{\bf
x}_{N_{e}},t){\bf p}_{\bf r}\Psi({\bf r},\sigma_{1},\cdots,{\bf
x}_{N_{e}},t)\right\\}$

Thus, ${\bf A}^{\rm MB}_{\Psi}({\bf r},t)$ can be identified as a velocity
field generated by the wave function $\Psi$ multiplied by $m_{e}/\hbar$; it
describes the effect on the electron with ${\bf r}_{1}$ by the interactions
with other particles with ${\bf x}_{2},\cdots,{\bf x}_{N_{e}}$ through the
wave function they share. The same effect exists for other particles; thus,
${\bf A}^{\rm MB}_{\Psi}({\bf r},t)$ is the common velocity field to all the
particles forming the wave field $\Psi$, which can be taken as a Schrödinger
field.

Using $\Psi$ and ${\bf A}_{\Psi}^{\rm MB}$, we define another function
$\Psi_{0}$

$\displaystyle\Psi_{0}({\bf x}_{1},\cdots,{\bf x}_{N_{e}},t)=\Psi({\bf
x}_{1},\cdots,{\bf x}_{N_{e}},t)\exp\left(-i\sum_{j=1}^{N_{e}}\int_{0}^{{\bf
r}_{j}}{\bf A}_{\Psi}^{\rm MB}({\bf r}^{\prime},t)\cdot d{\bf
r}^{\prime}\right)$ (6)

The Berry connection for this state is calculated to be zero

$\displaystyle{\bf A}^{\rm MB}_{\Psi_{0}}({\bf r},t)=-i\langle
n_{\Psi_{0}}({\bf r},t)|\nabla_{\bf r}|n_{\Psi_{0}}({\bf r},t)\rangle=0$ (7)

Thus, $\Psi$ can be decomposed into the velocity field zero part $\Psi_{0}$,
and the nonzero velocity part $e^{i\sum_{j}\int_{0}^{{\bf r}_{j}}{\bf
A}_{\Psi}^{\rm MB}({\bf r}^{\prime},t)\cdot d{\bf r}^{\prime}}$.

Usually, the case is considered where the common velocity field is zero in the
ground state. However, we consider the case where it is not zero in the
following.

##  3 The Hartree-Fock approximation with non-trivial ${\bf A}_{\Psi}^{\rm
MB}$

Let us revisit the Hartree-Fock approximation, where an approximate wave
function $\Psi$ is given by a Slater determinant of single-particle wave
functions $\psi_{k_{j}}({\bf x}_{\ell})$,

$\displaystyle\Psi({\bf x}_{1},\cdots,{\bf
x}_{N_{e}})={1\over\sqrt{N_{e}!}}\left|\begin{array}[]{ccc}\psi_{k_{1}}({\bf
x}_{1})&\cdots&\psi_{k_{N_{e}}}({\bf x}_{1})\\\ \vdots&\vdots&\vdots\\\
\psi_{k_{1}}({\bf x}_{N_{e}})&\cdots&\psi_{k_{N_{e}}}({\bf x}_{N_{e}})\\\
\end{array}\right|$ (11)

Usually, the basis functions are taken to be

$\displaystyle\phi_{k_{i}}({\bf x}_{1})=\varphi_{\lambda_{i}}({\bf
r}_{1})\langle\sigma_{1}|\uparrow\rangle\mbox{ or }\varphi_{\lambda_{i}}({\bf
r}_{1})\langle\sigma_{1}|\downarrow\rangle$ (12)

where $\varphi_{\lambda_{i}}({\bf r}_{1})$ is a single-valued function of the
coordinate ${\bf r}_{1}$, and $\langle\sigma_{1}|\uparrow\rangle$ and
$\langle\sigma_{1}|\downarrow\rangle$ are up and down spin functions of the
spin coordinate $\sigma_{1}$, respectively.

Now, we consider the case where single-particle states are those of spin-
twisting itinerant motion. In this case, the spin function is coordinate-
dependent, and the requirement that $\phi_{k_{i}}({\bf x}_{1})$ is a single-
valued with respect to the coordinate ${\bf r}_{1}$ must be imposed.

The Hartree-Fock approximation has been developed by tacitly assuming ${\bf
A}_{\Psi}^{\rm MB}=0$; in this case, the obtained approximate wave function is
that for $\Psi_{0}$. We express it as

$\displaystyle\Psi_{0}({\bf x}_{1},\cdots,{\bf
x}_{N_{e}})={1\over\sqrt{N_{e}!}}\left|\begin{array}[]{ccc}\tilde{\psi}_{k_{1}}({\bf
x}_{1})&\cdots&\tilde{\psi}_{k_{N_{e}}}({\bf x}_{1})\\\
\vdots&\vdots&\vdots\\\ \tilde{\psi}_{k_{1}}({\bf
x}_{N_{e}})&\cdots&\tilde{\psi}_{k_{N_{e}}}({\bf x}_{N_{e}})\\\
\end{array}\right|$ (16)

If the electrons perform spin-twisting itinerant motion, $\Psi_{0}$ may be a
multi-valued function of coordinates. In this cate, $\Psi_{0}$ is not a
legitimate wave function since it is not single-valued with respect to the
coordinates.

When $\Psi_{0}$ is multi-valued, we construct a single-valued $\Psi$ as

$\displaystyle\Psi({\bf x}_{1},\cdots,{\bf
x}_{N_{e}})=\exp\left(i\sum_{j=1}^{N_{e}}\int_{0}^{{\bf r}_{j}}{\bf
A}_{\Psi}^{\rm MB}({\bf r}^{\prime},t)\cdot d{\bf r}^{\prime}\right)\Psi_{0}$
(17)

This is the legitimate wave function; and this is the wave function we employ
for the extended Hartree-Fock procedure presented in this work. It is not a
Slater determinant of single-particle wave functions; it contains many-body
effects through the gauge field ${\bf A}_{\Psi}^{\rm MB}$.

The form of the wave function in Eq. (17) and the fact that $\Psi_{0}$ is a
currentless state read that the kinetic energy is a sum of the contribution
from $e^{-i\sum_{j}\int_{0}^{{\bf r}_{j}}{\bf A}_{\Psi}^{\rm MB}({\bf
r}^{\prime},t)\cdot d{\bf r}^{\prime}}$ and $\Psi_{0}$ [7]; the former is
given by

$\displaystyle\int{\hbar^{2}\over{2m_{e}}}{\bf A}_{\Psi}^{2}\rho({\bf
r})d^{3}r$ (18)

where $\rho({\bf r})$ is the electron number density. Supercurrent is
generated from this term. It is not the linear response current assumed in the
standard theory [1].

For convenience, let us define an angular variable $\chi$ by

$\displaystyle{{\chi({\bf r})}\over 2}=\int^{{\bf r}}_{0}{\bf A}_{\Psi}^{\rm
MB}({\bf r}^{\prime},t)\cdot d{\bf r}^{\prime}$ (19)

It is a multi-valued function of ${\bf r}$ when the Berry connection is
nontrivial.

We denote the total energy as $E[\chi]$, indicating that it is a functional of
$\chi$. We can construct $\chi$ using $\tilde{\psi}_{k_{i}}$ obtained from the
standard Hartree-Fock procedure by imposing the single-valued condition on
$\tilde{\psi}_{k_{j}}({\bf x})\exp\left(i{\chi\over 2}\right)$ and the
conservation of the local charge [8]. We will demonstrate it using an example,
later.

The appearance of $\chi$ generates a collective mode. Let us quantize it by
treating $\Psi$ as a Schrödinger field, and employing the canonical
quantization procedure. For this purpose, we consider the following Lagrangian

$\displaystyle{\cal
L}=\langle\Psi|i\hbar\partial_{t}-H|\Psi\rangle=i\hbar\langle\Psi_{0}|\partial_{t}|\Psi_{0}\rangle-\hbar\int
rdrd\phi\rho{\dot{\chi}\over 2}-\langle\Psi|H|\Psi\rangle$ (20)

The canonical conjugate momentum of $\chi$ is calculated as

$\displaystyle\pi_{\chi}={{\delta{\cal
L}}\over{\delta\dot{\chi}}}=-{\hbar\over 2}\rho$ (21)

The canonical quantization condition is given by $[\pi_{\chi}({\bf
r},t),\chi({\bf r}^{\prime},t)]=-i\hbar\delta({\bf r}-{\bf r}^{\prime})$,
yielding

$\displaystyle[\rho({\bf r},t),{1\over 2}\chi({\bf
r}^{\prime},t)]=i\delta({\bf r}-{\bf r}^{\prime})$ (22)

Next, we introduce the following boson field operators,

$\displaystyle\psi_{\chi}^{\dagger}({\bf r})=\sqrt{\rho({\bf r})}e^{-{i\over
2}\chi({\bf r})},\quad\psi_{\chi}({\bf r})=e^{{i\over 2}\chi({\bf
r})}\sqrt{\rho({\bf r})}$ (23)

They satisfy the following commutation relation

$\displaystyle[\psi_{\chi}({\bf r}),\psi_{\chi}^{\dagger}({\bf
r}^{\prime})]=\delta({\bf r}-{\bf r}^{\prime})$ (24)

Then, we construct the following boson creation and annihilation operators

$\displaystyle B^{\dagger}_{\chi}=\int rdrd\phi\psi_{\chi}^{\dagger}({\bf
r}),\quad B_{\chi}=\int rdrd\phi\psi_{\chi}({\bf r})$ (25)

that satisfy the following commutation relation

$\displaystyle[B_{\chi},B_{\chi}^{\dagger}]=1$ (26)

Finally, we obtain the following number operator for the collective mode,

$\displaystyle\hat{N}_{\chi}=B_{\chi}^{\dagger}B_{\chi},\quad\hat{N}_{\chi}|{N}_{\chi}\rangle={N}_{\chi}|{N}_{\chi}\rangle$
(27)

The number ${N}_{\chi}$ is the number of electrons participating in the
collective mode described by $\chi$, and $|{N}_{\chi}\rangle$ denotes its
eigenstate.

We can also construct a phase operator $\hat{X}$ that is conjugate with
$\hat{N}_{\chi}$; it is defined through
$B_{\chi}^{\dagger}=\sqrt{\hat{N}_{\chi}}e^{-i\hat{X}}$ and
$B_{\chi}=e^{i\hat{X}}\sqrt{\hat{N}_{\chi}}$.

The phase and number operators have the following commutation relation

$\displaystyle[e^{i\hat{X}},\hat{N}_{\chi}]=e^{i\hat{X}}$ (28)

thus, $e^{\pm i\hat{X}}$ are number changing operators that satisfy $e^{\pm
i\hat{X}}|{N}_{\chi}\rangle=|{N}_{\chi}\mp 2\rangle$. These number changing
operators are crucial for the generation of the pairing gap with keeping the
total number of number electrons constant.

Note that since $\chi$ and $\rho$ are now operators, $\chi$ and $\rho$ appear
in Eq. (18) should be regarded as expectation values of the corresponding
operators.

##  4 An example for a two-dimensional system of electrons with a pairing
interaction

To investigate the situation where ${\bf A}_{\Psi}^{\rm MB}\neq 0$, let us
consider a two dimensional system with the following single particle
Hamiltonian

$\displaystyle
h=-{{\hbar^{2}}\over{2m_{e}}}(\partial_{x}^{2}+\partial_{y}^{2})+U(r)$ (29)

For simplicity we assume that the potential $U$ depends only on $r$.

The coordinate part of the wave function is given as the product of an angular
function and a radial function given by

$\displaystyle\varphi_{nm}(r,\phi)={1\over\sqrt{2\pi}}e^{im\phi}R_{n|m|}(r),\quad
h\varphi_{nm}(r,\phi)=E_{nm}\varphi_{nm}(r,\phi)$ (30)

where $x=r\cos\phi,y=r\sin\phi$, $m$ is an integer, $n$ is a natural number
that denotes the number of nodes of the radial wave function, $R_{n|m|}(r)$,
which depends on $m$ through $|m|$. The energy is given by $E_{nm}$.

Usually, the wave functions

$\displaystyle\psi_{nm\uparrow}=\varphi_{nm}(r,\phi)\langle\sigma|\uparrow\rangle,\quad\psi_{nm\downarrow}=\varphi_{nm}(r,\phi)\langle\sigma|\downarrow\rangle$
(31)

are used by adopting the coordinate independent spin functions
$|\uparrow\rangle$ and $|\downarrow\rangle$. However, we consider the
following spin functions

$\displaystyle|\Sigma_{a}(\phi)\rangle={1\over\sqrt{2}}(e^{-{i\over
2}\phi}\sin\zeta|\uparrow\rangle+e^{{i\over
2}\phi}\cos\zeta|\downarrow\rangle),\quad|\Sigma_{b}(\phi)\rangle={1\over\sqrt{2}}(-e^{-{i\over
2}\phi}\sin\zeta|\uparrow\rangle+e^{{i\over
2}\phi}\cos\zeta|\downarrow\rangle)$ (32)

and use the following wave functions,

$\displaystyle\tilde{\psi}_{nma}=\varphi_{nm}(r,\phi)\langle\sigma|\Sigma_{a}(\phi)\rangle,\quad\tilde{\psi}_{nmb}=\varphi_{nm}(r,\phi)\langle\sigma|\Sigma_{b}(\phi)\rangle$
(33)

Expectation values of the components of spin for $|\Sigma_{a}(\phi)\rangle$
are given by

$\displaystyle\langle\Sigma_{a}(\phi)|s_{x}|\Sigma_{a}(\phi)\rangle={\hbar\over
2}\cos\phi\sin\zeta,\langle\Sigma_{a}(\phi)|s_{y}|\Sigma_{a}(\phi)\rangle={\hbar\over
2}\sin\phi\sin\zeta,\langle\Sigma_{a}(\phi)|s_{z}|\Sigma_{a}(\phi)\rangle={\hbar\over
2}\cos\zeta$ (34)

It describing spin-twisting around the $z$-axis. For simplicity, we consider
the case where $\zeta$ is constant in the following. Expectation values of the
components of spin for $|\Sigma_{b}(\phi)\rangle$ are given by
$\langle\Sigma_{b}(\phi)|{\bf
s}|\Sigma_{b}(\phi)\rangle=-\langle\Sigma_{a}(\phi)|{\bf
s}|\Sigma_{a}(\phi)\rangle$, thus, it describes the spin-twisting motion with
the spin direction opposite to $|\Sigma_{a}(\phi)\rangle$.

To have the single-valued total wave function, we require that

$\displaystyle{\psi}_{nma}=\tilde{\psi}_{nma}e^{i{\chi\over
2}},\quad{\psi}_{nmb}=\tilde{\psi}_{nmb}e^{i{\chi\over 2}}$ (35)

are single-valued functions of the coordinate. A satisfactory $\chi$ should
only depends on $\phi$ since the multi-valuedness of $\tilde{\psi}_{nma}$ and
$\tilde{\psi}_{nmb}$ arise from their $\phi$ dependence.

The total energy $E[\chi]$ depends on $\chi$ through ${{d\chi}\over{d\phi}}$;
thus, the condition for an optimal $\chi$ that minimize the total energy is
given by

$\displaystyle 0={{\delta E[\chi]}\over{\delta\chi}}=-\nabla\cdot{{\delta
E[\chi]}\over{\delta\nabla\chi}}=-{\partial\over{\partial\phi}}{{\delta
E[\chi]}\over{\delta{{d\chi}\over{d\phi}}}}$ (36)

This it the equation for the conservation of the local charge, and yields,
$0={{d^{2}\chi}\over{d\phi^{2}}}$. Its solution is $\chi=A\phi+B$, where $A$
and $B$ are constants. The constant $B$ merely generates a constant phase
factor on the wave function. The constant $A$ must be so chosen that
$e^{{i\over 2}(A\pm 1)\phi}$ is a single-valued function of the coordinate.
This condition yields that $A$ is an odd integer.

Using the obtained $\chi$, the total energy is given by

$\displaystyle E[\chi]=\int
drd\phi{{\rho(r)\hbar^{2}}\over{2m_{e}}}\left({1\over
2}{{d\chi}\over{d\phi}}\right)^{2}+\sum_{\tilde{E}_{nm}\leq 0}2\tilde{E}_{nm}$
(37)

where $\tilde{E}_{nm}={E}_{nm}-E_{F}$ and $E_{F}$ is the Fermi energy; the
factor $2$ appears due to the fact that both $\psi_{nma}$ and $\psi_{nmb}$ are
occupied. This current carrying state is energetically higher than the
currentless state [7]. However, $e^{\pm i\hat{X}}$ make it possible to
generate a lower energy state if the electron-pairing interaction exists.

Now, introduce the pairing interaction given by

$\displaystyle H_{\rm pair}$ $\displaystyle=$
$\displaystyle\sum_{n,m,n^{\prime},m^{\prime}}V_{nm;n^{\prime}m^{\prime}}c^{\dagger}_{nm\uparrow}c^{\dagger}_{n-m\downarrow}c_{n^{\prime}-m^{\prime}\downarrow}c_{n^{\prime}m^{\prime}\uparrow}$
(38) $\displaystyle=$
$\displaystyle\sum_{n,m,n^{\prime},m^{\prime}}V_{nm;n^{\prime}m^{\prime}}c^{\dagger}_{nm\uparrow}c^{\dagger}_{n-m\downarrow}e^{i\hat{X}}e^{-i\hat{X}}c_{n^{\prime}-m^{\prime}\downarrow}c_{n^{\prime}m^{\prime}\uparrow}$

where $c^{\dagger}_{nm\sigma}$ and $c_{nm\sigma}$ are creation and
annihilation operators for $\psi_{nm\sigma},\ \sigma=\uparrow,\downarrow$,
respectively, and $1=e^{i\hat{X}}e^{-i\hat{X}}$ is inserted in the second
line.

The single-particle part of the Hamiltonian is given by

$\displaystyle
H_{0}=\sum_{n,m}\tilde{E}_{nm}(c^{\dagger}_{nm\uparrow}c_{nm\uparrow}+c^{\dagger}_{nm\downarrow}c_{nm\downarrow})$
(39)

We employ a BCS type variational ground state

$\displaystyle|{\rm
Gnd}\rangle=\prod_{n,m}(u_{nm}+v_{nm}c^{\dagger}_{nm\uparrow}c^{\dagger}_{n-m\downarrow}e^{i\hat{X}})|{\rm
Cnd}\rangle$ (40)

where $|{\rm Cnd}\rangle$ corresponds to $\Psi_{0}$, i.e., it is given by

$\displaystyle|{\rm Cnd}\rangle=\prod_{\tilde{E}_{nm}\leq
0}a^{\dagger}_{nm}b^{\dagger}_{n-m}|{\rm vac}\rangle$ (41)

where $a^{\dagger}_{nm}$ and $b^{\dagger}_{nm}$ denote creation operators for
single-particle states $\tilde{\varphi}_{nma}$ and $\tilde{\varphi}_{nmb}$,
respectively; $u_{nm}$ and $v_{nm}$ are variational parameters that satisfy
$u_{nm}^{2}+v_{nm}^{2}=1$.

The operator $c^{\dagger}_{nm\uparrow}c^{\dagger}_{n-m\downarrow}e^{i\hat{X}}$
acting on $|{\rm Cnd}\rangle$ annihilates two electrons in the occupied states
of $\tilde{\varphi}_{nma}$ and $\tilde{\varphi}_{nmb}$, and creates electrons
in $\psi_{nm\uparrow}$ and $\psi_{n-m\downarrow}$ states; thereby the number
of electrons in the collective mode is reduced by two.

The pairing energy gap is defined as

$\displaystyle\Delta_{nm}$ $\displaystyle=$
$\displaystyle-\sum_{n,m,n^{\prime},m^{\prime}}V_{nm;n^{\prime}m^{\prime}}\langle{\rm
Gnd}|e^{-i\hat{X}}c_{n^{\prime}-m^{\prime}\downarrow}c_{n^{\prime}m^{\prime}\uparrow}|{\rm
Gnd}\rangle$ (42) $\displaystyle=$
$\displaystyle-\sum_{n,m,n^{\prime},m^{\prime}}V_{nm;n^{\prime}m^{\prime}}u_{n^{\prime}m^{\prime}}v_{n^{\prime}m^{\prime}}$

Note that the it is calculated with keeping the particle number constant.

As in the BCS theory, we assume $V_{nm;n^{\prime}m^{\prime}}=-V$ if
$|\tilde{E}_{nm}|,|\tilde{E}_{n^{\prime}m^{\prime}}|\leq\hbar\omega$ and zero
otherwise with a cut-off energy $\hbar\omega_{c}$. By following the BCS
approximation [1], we obtain

$\displaystyle u_{nm}^{2}={1\over
2}\left(1+{{\tilde{E}_{nm}}\over{\sqrt{\tilde{E}_{nm}^{2}+\Delta^{2}}}}\right),\quad
v_{nm}^{2}={1\over
2}\left(1-{{\tilde{E}_{nm}}\over{\sqrt{\tilde{E}_{nm}^{2}+\Delta^{2}}}}\right),\quad\Delta\approx
2\hbar\omega_{c}e^{-{1\over{N(0)V}}}$ (43)

where $N(0)$ is the density of states at the Fermi energy.

The total energy is given by

$\displaystyle E_{\rm tot}$ $\displaystyle=$ $\displaystyle\int
rdrd\phi{{\rho(r)\hbar^{2}}\over{2m_{e}}}\left({1\over
2}{{d\chi}\over{d\phi}}\right)^{2}+2\sum_{mn}\tilde{E}_{nm}v_{nm}^{2}-{{\Delta^{2}}\over
V}$ (44) $\displaystyle=$ $\displaystyle\int
rdrd\phi{{\rho(r)\hbar^{2}}\over{2m_{e}}}\left({1\over
2}{{d\chi}\over{d\phi}}\right)^{2}-{1\over 2}N(0)V\Delta^{2}$

where the number of electrons in the collective mode is calculated as

$\displaystyle\int rdrd\phi\rho(r)=\sum_{\tilde{E}_{nm}\leq
0}u^{2}_{nm}=2N(0)\int_{-\hbar\omega_{c}}^{0}{1\over
2}\left(1+{{x}\over{\sqrt{x^{2}+\Delta^{2}}}}\right)dx$
$\displaystyle=N(0)\left(\hbar\omega_{c}+\sqrt{\Delta^{2}}-\sqrt{\hbar^{2}\omega_{c}^{2}+\Delta^{2}}\right)\approx
N(0)\left(\Delta-{1\over{2\hbar\omega_{c}}}\Delta^{2}+{1\over{8\hbar^{3}\omega_{c}^{3}}}\Delta^{4}\right)$
(45)

If the energy gap formation makes the current carrying state lower in energy
than the currentless state, the superconducting state is realized.

A salient feature of the present theory is that the supercurrent is generated
by the collective mode whose kinetic energy is separately given with mass
$m_{e}$. The number of electrons in this mode is nonzero when the pairing gap
is formed as seen in Eq. (45), thus the supercurrent carrying state and the
nonzero pairing gap state coincide.

If we consider a more general setting by including the potential energy from
the underlying ion lattice and effective field from other electrons, the spin-
twisting itinerant motion occurs as the circular motion around a section of
the Fermi surface of the metal [9].

When a magnetic field exists, the vector potential from magnetic field ${\bf
A}^{\rm em}$ appears in addition to the gauge field ${\bf A}^{\rm MB}_{\Psi}$.
Then, the kinetic energy of the collective mode is given by

$\displaystyle E_{\chi}=\int d^{3}r{{\hbar^{2}\rho({\bf
r})}\over{2m_{e}}}\left({1\over 2}\nabla\chi+{e\over\hbar}{\bf A}^{\rm
em}\right)^{2}$ (46)

Then, the supercurrent density is given by

$\displaystyle{\bf j}=-{{\partial E_{\chi}}\over{\partial{\bf A}^{\rm
em}}}=-{{e^{2}\rho({\bf r})}\over{m_{e}}}\left({\hbar\over{2e}}\nabla\chi+{\bf
A}^{\rm em}\right)$ (47)

This is a diamagnetic current that explains the Meissner effect. The presence
of the angular variable $\chi$ yields the flux quantum ${h\over{2e}}$.

The velocity field associated with the above supercurrent is

$\displaystyle{\bf
v}_{s}={{e}\over{m_{e}}}\left({\hbar\over{2e}}\nabla\chi+{\bf A}^{\rm
em}\right)$ (48)

The velocity field generated inside the superconductor by rotating it with an
angular velocity ${\bm\omega}$ is given by ${\bf v}_{\rm
rot}={\bm\omega}\times{\bf r}$.

Since supercurrent electrons move with the body to shield the electric field
from the ion core, the condition ${\bf v}_{s}={\bf v}_{\rm rot}$ is satisfied.
Then, the magnetic field ${\bf B}^{\rm em}={{2m_{e}}\over e}{\bm\omega}$ is
generated inside of the superconductor. This formula has the free electron
mass in accordance with the experimental results.

##  5 Conclusion

In the present theory, the supercurrent is not the linear response current,
but the flow of electrons caused by the velocity field generated by the Berry
connection from many-body wave functions. It predicts the carrier mass that
agrees with the London moment measurement.

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