1. Introduction
Dirac considered a string of singularities of a wave function with flux through it; he showed that a magnetic monopole should exists at a terminal end of the string [
1]. The vortex line in a type II superconductor may be considered as a realization of such an object with a magnetic monopole at a terminal end of it at the surface of the superconductor.
The vortex in the superconductor is explained by the emergence of a vector potential
which accompanies the electromagnetic vector potential
, where
ℏ is Planck’s constant divided by
,
is the electron charge, and
is an angular variable with period
. The sum of the electromagnetic vector potential
and
is an effective gauge invariant vector potential existing in superconductors [
2].
The standard theory of superconductivity is the BCS theory [
3]. It was originally developed from the energy gap model of Bardeen [
4], and identified the cause of the energy gap as the electron pair formation. The BCS theory has been successfully predicted the superconducting transition temperature T
c, where T
c is given as the energy gap formation temperature. The appearance of
in
is due to the use of the particle number non-fixed wave function in the BCS theory; namely, it is attributed to the
gauge symmetry breaking, thus it has been believed that the particle number non-fixed formalism is crucial for a superconductivity theory [
5,
6].
Due to the success of the BCS theory, many researchers had thought that superconductivity was a completely solved problem; however, the high temperature superconductivity found in cuprates [
7] has proved it is not so. The superconductivity in the cuprate (the
cuprate superconductivity) is markedly different from the superconductivity explained by the BCS theory (the
BCS superconductivity). Apart from the very high superconducting transition temperature, differences include,
- (1)
The normal state from which the superconducting state emerges is a doped Mott-insulator [
8] although the BCS superconductivity assumes the band metal for the normal state.
- (2)
Local magnetic correlations in the superconducting state and a close relation between the magnetism and superconductivity have bee observed in the cuprate [
9,
10], while the magnetism is harmful for the BCS superconductivity.
- (3)
The superconducting coherence length of the cuprate is in the order of lattice constant (nano-scale) [
11], while it is assumed to be much larger than the lattice constant in the BCS superconductivity.
- (4)
The superconducting transition temperature for the optimally doped cuprate is given by the stabilization temperature of the nano-sized loop currents [
12,
13,
14], while it is given by the energy gap formation temperature in the BCS superconductivity.
- (5)
The hole–lattice interaction is very strong and small polarons and bi-polarons are created in the cuprate [
15,
16,
17,
18], while the BCS superconductivity does not assume such a strong electron-lattice interaction that forms small polarons.
In spite of more than 30 years of research, no widely-accepted theory exists for the mechanim of the cuprate superconductivity. It is very plausible that a drastic departure from the BCS theory is needed for the elucidation of the cuprate superconductivity.
In order to explain the cuprate superconductivity, a new supercurrent generation mechanism where
appears as the Berry connection [
19,
20] has been put forward [
21,
22,
23,
24]. In this theory, a macroscopic supercurrent is generated as a collection of
spin-vortex-induced loop currents (SVILCs), where the SVILC is a superconducting-coherence-length-sized loop current induced by a spin-vortex (SV) created around each doped hole in the CuO
2 plane. It explains a number of experimental results in the cuprate superconductors [
25];
- (1)
Nonzero Kerr rotation in zero-magnetic field after exposed in a strong magnetic field [
26].
- (2)
The change of the sign of the Hall coefficient with temperature change [
27].
- (3)
The suppression of superconductivity in the
static-stripe ordered sample [
9].
- (4)
A large anomalous Nernst signal, including its sign-change with temperature change [
28].
- (5)
The hourglass-shaped magnetic excitation spectrum [
9].
- (6)
Fermi-arc observed in the AEPES [
29].
Actually, the new supercurrent generation mechanism does not require the electron-pair formation for the supercurrent generation (this does not mean that the electron pairing is not relevant to the cuprate superconductivity); however, the resulting supercurrent explains the flux quantum and the voltage quantum (f is the frequency of the radiation field).
Motivated by the above developments, we have reinvestigated the superfluidity problem in general [
30]. Then, we have found that the particle number non-fixed formalism, such as the standard BCS formalism or the Bogoliubov–de Gennes formalism [
31,
32], can be cast in a particle number fixed formalism if the Berry connection is employed. This indicates that the
gauge symmetry breaking origin of
may be replaced by the Berry connection origin.
Since the persistent current in topological insulators can be attributed to the Berry connection [
33], the Berry connection may be the unified ingredient for persistent current generation in superconductors and topological insulators.
In the present work, we put forward a possible appearance of
in the BCS superconductor from the view point of the Berry connection origin. In this theory, the nontrivial Berry connection arises from the spin-twisting itinerant motion of electrons. Such a motion is realized by the spin-orbit interaction [
34]
where
is the electron spin angular momentum,
m is electron mass, and
is an electric field. When this interaction affects conduction electrons, it is called the Rashba spin-orbit interaction [
35]. Since the internal electric field
exists more or less in any materials, the Rashba interaction exists more or less in any materials. In this work, we consider the case where the Rashba interaction energy is much smaller than the energy gap created by the electron-pairing.
The organization of the present work is following: in
Section 2, Bloch electrons under the influence of a magnetic field and the Rashba spin-orbit interaction are investigated. It is shown that the quantization of orbits is realized even without an external magnetic field due to the existence of
arising from the spin-twisting itinerant motion of electrons. In
Section 3, the energy gap for the BCS model is obtained for the case where the electron pairing occurs between
and
, where
is the wave vector and
is the spin for the electron that depends on the coordinate
due to the spin-twisting itinerant motion of electrons. In
Section 4, the kinetic energy gain by the spin-twisting itinerant motion is investigated. The energy reduction is shown to be optimum when the Berry connection is given by
, and the Meissner effect is realized if the nontrivial
is stable. In
Section 5, the Berry connection from the many-body wave function [
30],
, is calculated for the present model. It is shown to be identified as
. According to our previous work [
30],
is stabilized by the electron-pairing, meaning that the
is stabilized by the electron-pairing; thus, a stable nontrivial
necessary fo superconductivity is realized. In
Section 6, we succinctly summarize part of our previous work that utilizes the particle-number fixed version of the BCS ground state [
30], and conclude the present work by discussing implications for it.
2. Appearance of Spin-Twisting Itinerant Motion of Bloch Electrons under the Influence of the Rashba Spin-Orbit Interaction
In this section we consider Bloch electrons under the influence of the Rashba interaction and magnetic field. We calculate the quantized energy for them using the method of the periodic-orbit quantization. It is known that the quantized energy obtained in this way gives an accurate energy [
36].
In order to obtain the periodic orbit, we use the wave-packet dynamics formalism [
37]. In this formalism, the motion of the center of the wave packet corresponds to the classical motion of the electron. The force for the classical motion can be evaluated using the wave packet localized both in the real coordinate space
and the wave vector space
under the constraint of the Heisenberg uncertainty condition [
38].
Let us consider electrons in a single band and denote its Bloch wave as
where
is the wave vector and
is the periodic part of the Bloch wave.
It satisfies the Schrödinger equation,
where
is the zeroth order single-particle Hamiltonian for an electron in a periodic potential.
According to the wave packet dynamics formalism,
is modified as
in the presence of the magnetic field
.
Using the Bloch waves, a wave-packet centered at coordinate
and central wave vector
is constructed as
where
is a distribution function, and
is a spin function. In the present model, we assume the situation where the band is approximated by the extended zone scheme for the nearly free electron model as has been done in the original BCS derivation [
3]. Then, the single-band approximation above is adequate. However, in reality, the construction of the wave packet may need many-band Bloch states.
An important point is that we use the spin function that depends on the coordinate
given by
where
and
are the polar and azimuthal angles of the spin-direction, respectively. This coordinate dependence is necessary to describe the spin-twisting itinerant motion. The expectation value of spin
is given by
In the following argument, the single-valued requirement of the wave function as a function of the coordinate is a crucial condition. This is the postulate adopted by Schrödinger [
39], and we impose this condition on the wave packet.
The phase factor in is introduced to impose the single-valued condition. Let us consider an example case where the electron performs spin-twisting itinerant motion in which occurs after a circular transport along a loop in the coordinate space. This leads to the sign-change in , and which is compensated by , for example, this sign change is compensated if occurs for the same circular transport. The explicit condition for this compensation will be given, later.
The distribution function
satisfies the normalization
and the localization condition in
space,
The distribution of is assumed to be narrow compared with the Brillouin zone size so that can be regarded as the central wave vector of the wave packet.
The wave packet is also localized in
space around the central position
,
The localization in the
and
spaces is assumed to satisfy the Heisenberg uncertainty principle [
38].
We include the following Rashba interaction term in the Hamiltonian
where
is the spin-orbit coupling vector (its direction is the internal electric field direction),
is the momentum operator, and
is electron charge [
35].
Let us construct the Lagrangian
using the time-dependent variational principle [
40],
where
H is composed of the Hamiltonian for the band electron that gives the band dispersion
and the Rashba interaction
.
For convenience sake, we introduce another Lagrangian
L that is related to
as
where
is the phase of
.
By following procedures for calculating expectation values for operators by the wave packet [
37],
L is obtained as
where
is the expectation value of spin for the wave packet centered at
given by
and
where
is the Berry connection arising from
given by
We introduce the following wave vector
,
and change the dynamical variables from
to
[
37].
Then, the Lagrangian with dynamical variables
, and
is given by
Using the above Lagrangian
L, the following equations of motion are obtained,
where
is the Berry curvature in
space defined by
and
is the effective magnetic field,
In the following, we consider the case where
. Then, Equation (
22) becomes
Using Equations (23) and (
26) becomes,
Equations (
26) and (
27) indicate that the wave packet exhibits cyclotron motion for the electron in the band with energy
By following the Onsager’s argument, let us quantize the cyclotron orbit [
41]. From Equation (
21), the Bohr–Sommerfeld relation becomes
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 Equation (
27), we have
Thus, Equation (
29) becomes
Note that in the usual quantization condition,
is absent.
Now we consider the case where
is present. The above quantization condition is satisfied even if the magnetic field is absent. In this case, the first term is zero, and we have
which can be satisfied by the following two sets of conditions; one is
,
, and
; and the other is
,
, and
, where
is the winding number of
along loop
C.
We will show later that the condition is achieved by the condition of the kinetic energy gain if the electron-pairing occurs.
The condition
leads to the following requirements
for the single-valued condition for the spin function
as a function of the coordinate
. If this condition is satisfied, the product of the phase factors
and
in Equation (
8) become single-valued.
The condition requires that must be odd, thus, is not zero. The nonzero value of means that electrons perform spin-twisting itinerant motion. This indicates that the quantized cyclotron motion may occur without an external magnetic field when the itinerant motion is accompanied by the spin-twisting.
3. The Pairing Energy Gap
The results in the previous section indicate that due to the presence of the Rashba interaction, the band energy becomes the one in Equation (
28), and Bloch electrons may perform the spin-twisting itinerant motion.
In this section, we consider a modified BCS model where the pairing between single particle states and occurs, instead between and .
Since we use the results of the BCS theory below, let us briefly review it first [
3]. The Hamiltonian for the BCS model is given by
, where
is the kinetic energy given by
is the energy measured from the Fermi energy
given by
and
is the interaction energy given by
The electron pairing occurs between electrons near the Fermi surface due to an attractive that exists in that region. In the BCS model, is nonzero () only when ( is the Debye frequency) is satisfied. Then, becomes independent of , will be expressed as .
The superconducting state is given by the following state vector,
This state exploits the attractive interaction between electron pairs
and
, and the following energy gap equation is obtained,
where
and
are parameters given using
and
as
and
respectively.
The total energy by the formation of the energy gap is given by
where
is the normal state energy, and
is the density of states at the Fermi energy
[
3].
Now, consider the pairing of
and
, and also
and
. We divide the system into coarse-grained cells of volume 1 to take into account the coordinate dependence of the band energy in Equation (
28), assuming that its coordinate dependence is very slow. Then, the ground state in the cell with the central position
is given by
where
and
are given by
with
Then, the gap function
is the solution of the gap equation given by
where
is the density of states at the Fermi energy in the coarse grained cell with the central position
, and
.
From the above relation, we obtain the following,
where
is the gap value without the spin-orbit interaction; here, it is assumed that
holds. The gap
is reduced by the spin–orbit interaction. However, if the spin–orbit interaction parameter
is significantly smaller that
, the gap is almost the same as the original one. In the following we assume such a case.
4. Kinetic Energy Gain by the Spin-Twisting Itinerant Motion
In this section, we consider the appearance of from the view point of the kinetic energy gain.
We consider the case where the pair
and
, and another pair
and
, are both occupied. We assume that
for the first pair arises from the spin function
in Equation (
8), and
for the second pair arises from the spin function
given by
Note that and are orthogonal.
The fictitious vector potential
from
is calculated as
and the effective vector potential is given by
The single-particle energy for the pair
and
is
, and that for the pair
and
is
in Equation (
46).
Then, the kinetic energy for the cell at
is given by
For simplicity, we approximate the above energy using the Fermi distribution functions
(
is Boltzmann’s constant) and density of states
as
At temperature
,
; thus, we have
The first term in Equation (
54) may be approximated as
assuming that the term linear in
cancels out due to the time-reversal and/or inversion symmetry. Here
is the number density of electrons (later, we consider it as the number density of electrons participating in the collective mode
).
The second term in Equation (
54) may be approximated as
assuming that the term linear in
cancels out.
From the condition for minimizing the kinetic energy,
is chosen to satisfy
We assume
in the coarse-grained cell at
to be uniform in the
z-direction; then, the optimal
that satisfies the above condition lies in the
plane. Thus,
in
and
is taken to be
, yielding common
for
and
. As a consequence, we have the common effective potential given by
for
and
.
The kinetic energy increase given by the appearance of
in Equation (
55) is calculated as
This indicates that the optimum is the one that gives if this choice is possible. If we adopt when a magnetic field is zero, the condition yields , i.e., the absence of the spin-twisting itinerant motion. When , however, the optimal will be the one for the presence of the spin-twisting itinerant motion.
From the kinetic energy, we can calculate the current density as
This is the London equation, and the system should exhibit the Meissner effect. Thus,
is realized in the bulk. If the system is a ring-shaped, it will lead to the quantization of magnetic flux in the units
. The Equation (
56) indicates the occurrence of the energy reduction in the order of
if the surface energy is negligible compared to the bulk energy. In other words, when a magnetic field is applied the spin-twisting itinerant motion occurs, and gives rise to
that causes the Meissner effect and the flux quantization in
.
5. Berry Connection for Many-Body Wave Functions and
We consider
from the view point of the
Berry connection for the many-body wave functions (dented as
) introduced in our previous work [
30] in this section.
Let us denote the wave function of a system with
N electrons as
where
denotes the coordinate
and spin
of the
jth electron.
The Berry connection [
19] associated with this wave function is called the “
Berry connection for many-body wave function” [
30]. In order to calculate this Berry connection, we first prepare the parameterized wave function
with the parameter
,
where
is the normalization constant given by
Using
, the
Berry connection for many-body wave function is given by
Here,
is regarded as the parameter [
19].
When the origin of
is not the ordinary magnetic field one, i.e.,
it can be written in the pure gauge form,
where
is a function which may be multi-valued.
Let us consider the case where is given as a Slater determinant of spin-orbitals , ⋯, , and , where and are time-reversal partners and N is assumed to be even.
Then,
is calculated as
where “
” indicates the imaginary part, and the fact that
is real (due to the fact that
and
are time-reversal partners) is used.
As is shown in the previous sections, optimal
and
are given by
. In this case, we have
thus
is identified as
with factor
. We may identify
as
.
The kinetic energy part of the Hamiltonian is given by
where
m is the electron mass and
is the gradient operator with respect to the
jth electron coordinate
.
Using
and
, we can construct a currentless wave function
for the current operator associated with
Reversely,
is expressed as
using the currentless wave function
.
Due to the spin-twisting the winding number of
is non-zero, thus, a line of singularities exist within the loop around which the winding number is non-zero. The flux threaded through the line of singularities can be calculated using
, and yields
; thus, the line of singularities is the
-flux Dirac string. The wave function in Equation (
71) indicates that a collective mode described by
produces the persistent current, and it arises from the Dirac string.
In the present formalism, the superconducting state is the one with nontrivial
. It plays dual roles; it is a Berry connection that enables the comparison of the phase of the wave function at different spatial points and gives rise to the macroscopic quantum interference effects. At the same time, it is the collective mode
of electrons with a long range order of the average momentum [
42].
6. Concluding Remarks
In the present work, we have shown that the spin-twisting itinerant motion occurs for the conduction electrons of metals due to the Rashba spin-orbit interaction, and it generates . When the energy gap formation by electron pairing described by the BCS theory occurs, is stabilized, and the superconducting state is realized.
The stabilization of
by the electron-pairing is explained in our previous work [
30]. The ground state with the total number of particles
N at a coarse-grained cell of volume 1 with the central position
is given by
where
is the number changing operator that satisfies
and
is the state vector for the condensate which is related to the wave function
in Equation (
70) by
The number changing operator
changes the number in the condensate
by two, and also adds the phase factor
[
30].
The ground state in the new theory
corresponds to the BCS ground state with the phase factor
given by
Note that the BCS ground state here is not the original one, but the one that varies slowly with coordinate
due to the coordinate dependence of the parameters,
, and
. This coordinate dependence is necessary to have nontrivial
. A more powerful method to deal with this coordinate dependence is the Bogoliubov-de Gennes method [
31,
32]; the corresponding version for
will be found in our previous work [
30].
The two ground states,
and
, look very similar, and the mathematical structures arising from them are also very similar. However, there exists a serious difference in concerning the ac Josephson effect. This difference indicates the new formalism is more in accordance with the observed ac Josephson effect [
21,
24]. Let us explain this point below.
A serious misfit was found in the recent re-derivation of the ac Josephson effect [
43]. The boundary condition considered by Josephson and the one employed in the experiment are actually different; if the experimental boundary condition is employed with taking into account the gauge invariance, the observed Josephson relation indicates that the charge on the particle is
not
[
21,
24]. The new formalism can explain the experimentally observed ac Josephson effect with
since the role of the electron pairing is the stabilization of
and the charge
supercurrent flow is possible.
The charge
supercurrent flow through the Josephson junction is explained in the new formalism as follows. We first introduce the following particle number conserving Bogoliubov transformation [
30],
where
is the spin,
for ↑, and
for ↓;
and
are annihilation and creation operators for electrons with spin
at the
ith site;
and
are the the particle number conserving Bogoliubov operators;
and
are parameters obtained by solving the Bogoliubov–de Gennes equations;
is the number changing operator that reduces the number of particle in the condensate by one at the
ith site, thereby, introduces a phase factor
.
We use the following electron transfer Hamiltonian for the Josephson junction,
Labels“L” and “R” refer to quantities for the left superconductor and right superconductor , respectively.
Using the Boboliubov transformation in Equation (
76) and including the electromagnetic field by the Peierls substitution,
is rewritten as
We treat two superconductors in the junction as a single system given by the state vector
, where
satisfies
Note also that
satisfies,
Taking expectation value of
using
, we have the energy for the junction
where
Note that usually, the transfer of electron pairs is considered between the two superconductors in the Josephson junction using the second order perturbation theory with respect to
; in this case, the supercurrent that flows without Bogoliubov excitations requires electron-pair tunneling, and the energy for the junction is given by
where
and
are constants. This formula gives rise to the Ambegaokar–Baratoff relation [
44] for the dc Josephson effect [
43], and is valid for a weakly-coupled junction.
In the new formalism,
electron transfer is possible if the two superconductors in the junction is in a close contact and Equation (
81) is valid; however, it is not possible in the standard formalism. Thus, from the view point of the re-derived result for the ac Josephson effect [
21,
24], the new formalism is more in accordance with the experiment.
In order to clarify the above point, re-investigations on the contact effect for the ac Josephson effect is necessary. In the new formalism, both
and
are possible, thus, there will be crossover of the junction properties whether it is described by
or
depending on the contact is weak or close. In this respect, it is noteworthy that un-paired electrons are more abundant than the standard theory in a Cooper pair box [
45].
Lastly, we would like to point out a problem in the standard formalism from the view point of the decoherence by interaction with environment considered by Zurek [
46]. The
gauge symmetry breaking origin of
in the standard theory relies on the breaking of invariance for the global
phase change
(
is a real constant) in
. This breaking occurs because
is a linear combination of states with different particle numbers. However, such a state eventually becomes a mixed state due to the decoherence induced by the environment [
46] since the relevant Hamiltonian conserves the particle number. In the resulting mixed state, the phase differences between the different particle number states are meaningless, thus, a physically meaningful phase
or
disappears sooner or later. Considering the fact that the supercurrent in a superconducting ring persist indefinitely, the physically meaningful
should persist indefinitely; however,
does not have such
. On the other hand,
with the stabilized
has it.