We have provided microscopic explanations to superconducting flux quantum and (superconducting and normal) persistent current. Flux quantum is generated by current carried by "deep electrons" at surface states. And values of the flux quantum differs according to the electronic states and coupling of the carrier electrons. Generation of persistent carrier electrons does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. Even for or persistent carriers,there should be a build-up of energy of the middle state and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.

1. Microscopic Mechanisms of Superconducting Flux Quantum and
Superconducting and Normal Persistent Currents
In view of energy, according to Drude model, and considerring the kinetic energy
of en carrier electrons as:
2 2
e
2
e
mn m
T
2 2n e
= =
v j
, where en is number density of
electrons and m is electron’s mass, there would be ( Michael Tinkham: Introduction
To Superconductivity, Second Edition, McGraw-Hill, Inc., 1996, sec. 2.5.1):
2
2
e
T m
/ (2 43)
t n e t
σ
∂ ∂
− = = −
∂ ∂
j j
Ej j ，or
2
e
m
/ (2 44)
n e t
σ
∂
− = −
∂
j
E j .
Obviously, when σ → ∞ , (2 44)− becomes London equation. (But here we do
not divide j into superconducting and normal currents.)
Next we discuss superconducting flux quantum. According to Maxwell equation,
for magnetic flux Φ threading a superconducting ring, there is:
c d (3 1)
t
∂Φ
= − −
∂ ∫ E lgÑ
By substituting London equation
t
∂
Λ =
∂
j
E into (3 1)− and integrating with respect to
time, we will obtain:
' c dt d c d (3 2)
t
∂
Φ − Φ = − Λ = − Λ −
∂∫ ∫ ∫
j
l j lg gÑ Ñ
Assuming that the system contains only one superconducting electron (we would
later see that this assumption is not necessary,) the current density could be (see:
Michael Tinkham: Introduction to Superconductivity, Second Edition, McGraw-Hill,
Inc., 1996, sec. 1.5):
2
2* *
( ) (3 3)
2mi mc
e e
φ φ φ φ φ= − ∇ − ∇ − −j A
h
where the wave function φ is Bloch function:
i
e ( ) (3 4)uφ = −kr
r
As London equation can also be in the form of: c= −ΛA j , and as
2
1
m
e Λ
= ,
(3 3)− becomes:
2
2* *
( ) (3 3')
2mi m
e e
φ φ φ φ φ
Λ
− ∇ − ∇ = − ≈ −j j j
h
By substituting (3 3')− and (3 4)− into (3 2)− , with 2
m
sn e
Λ = , we obtain
* * * * *c c
( ) d (2i ) d (3 5)
2 i 2 is s
u u +u u u u
n e n e
φ φ φ φΦ = ∇ − ∇ = ∇ − ∇ −∫ ∫l k l
h h
g gÑ Ñ .
As a trial treatment to “number density of electrons” sn , letting
2
sn φ= (we would
modify this;) thus, the 2nd
and 3rd
terms on the right of (3 5)− become:
1

2. * * *
*
1 1 1
( ) d ( ) d 0 (3 6)
s
u u u u u u
n u u
∇ − ∇ = ∇ − ∇ = −∫ ∫l lg gÑ Ñ ,
thus the superconducting current j in (3 5)− is in the direction of wave vector k .
Since we can always take the integral loop as along j , that is, along xk , there would
be:
*
x
c c
d k L
s
u u
n e e
Φ = =∫ k l
h h
gÑ , where L and xk are the length of the loop and the x
component of the wave vector respectively. According to Born–von Karman boundary
condition: xk 2 s / Lπ= , where s is an integer, there is:
2 c
s (3 7)
e
π
Φ = −
h
,
which is a result of magnetic flux quantization. This result leads to some further
conclusions. First, in an ideal crystal, it is impossible for a loop to be kept along xk ;
but in the initial experiment verifying flux quantum, the ring is formed by deposition
of superconducting material on a cylindrical substrate (B. S. Deaver and W. M.
Fairbank, Phys. Rev. Lett. 7, 43(1961).). Thus, we could understand that the ring
should include a single crystal having a series of dislocations, and the direction of its
xk should have been gradually altered by the dislocations, thereby “loop always along
the direction of xk ” was realized. On the other hand, it seems that these dislocations
did not affect the superconductivity of the samples. Second, the result of (3 7)− is
twice of the experimentally measured value of flux quantum 0
ch
2e
Φ = . So
modification has to be made.
Due to non-zero flux, corresponding perturbation exists. According to existing
perturbation theory, the original degenerated electronic states would linearly combine
to form new zero order quantum states. For example, if the original degenerated states
before disturbance is Block functions:
i
e ( )uϕ ±
±± = k r
k r , where ( 1,0,0)± = ±k , then the
newly combined electronic states after perturbation could be like:
1
( ) (3 8)
2
φ ϕ ϕ± + −= ± − .
Although other states of ( 1,0,0)≠ ±k may also be included, in the absence of external
field the weight of these other states can be negligible due to that the perturbation
corresponding to magnetic field of the order of 0Φ is very weak. According to the
principle of superposition, the measured electronic state of each of the two on states
φ± must be one of ϕ± of (3 8)− . But according to (3 7)− , if one of the two electrons
is measured at ϕ+ , the other electron cannot be measured at ϕ− or the current would
become zero according to (3 3')− and the flux would also become zero. Thus, a single
flux quantum 0Φ has to be provided by a pair of carrier electrons, which are
originally on two degenerated states of ϕ± .
1 2 1 2 2 1
1 1 1 1
( , ) ( ) '( ) ( ) '( ) (3 10)
2 2 2 2
ϕ ϕ ϕ ϕ+ + + +Ψ = − −r r r r r r
Thus, we need to consider the wave function for two electrons. First, we cannot
directly substitute (3 8)− into (3 6)− or the antisymmetrical wave function:
1 2 1 2 1 2( , ) ( ) ( ) ( ) ( ) (3 9)φ φ φ φ+ − − +Ψ = − −r r r r r r
2

3. would become zero when 1
1
( )
2
φ ϕ+= and 2
1
( )
2
φ ϕ+= . The problem is due to that
although both φ± of (3 8)− contain ϕ+ , the two states ϕ+ are not the same state in
fact, because they are of different energy. That is, “approximate” single-fermion wave
functions may not be used to construct the anti-symmetrical wave function of a multi-
fermion system. Thus, we need to differentiate the two ϕ+ states; we do this by
marking one of them as 1'( )ϕ+ r , as:
1 2 1 2 2 1
1 1 1 1
( , ) ( ) '( ) ( ) '( ) (3 10)
2 2 2 2
ϕ ϕ ϕ ϕ+ + + +Ψ = − −r r r r r r
In addition,
2
1 2( , )Ψ r r can no longer represent “number density” sn of two electrons,
as (among other things)
2
1 2( , )Ψ r r cannot be greater than one; we first try to modify it
by multiplying it with the number N of electrons (and as we will see, this is not
correct.) Thus, with 1 2∇ = ∇ + ∇ , we would have:
* *
1 2 1 2 1 2 1 2
2 2
1 2 2 1 1 2 2 1
*
1 2 2 1 1 2 2 1
1
( ( , ) ( , ) ( , ) ( , ) )
1 1 1 1 1
[(2i ) ( ) '( ) ( ) '( ) ( 2i ) ( ) '( ) ( ) '( )
2 2 2 2
1 1 1 1
( u ( )u '( ) u ( )u '( )) ( u ( )u '( ) u ( )u '( )) c.c.]
2 2 2 2
1
s
s
n
n
ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ+ + + + + + + +
+ + + + + + + +
Ψ ∇Ψ − Ψ ∇Ψ
= − − − − +
− ∇ − −
=
r r r r r r r r
k r r r r k r r r r
r r r r r r r r
2
1 22
1 2
*
1 2 2 1 1 2 2 1
*
1 2 2 1 1 2 2 1
1 2
[(4i ) ( , )
2 ( , )
1 1 1 1
( u ( )u '( ) u ( )u '( )) ( u ( )u '( ) u ( )u '( )) c.c.]
2 2 2 2
1 1 1 1
( u ( )u '( ) u ( )u '( )) ( u ( )u '( ) u ( )u '( )) c.c.]
2 2 2 22i
1 1
( ) '( ) (
2 2
ϕ ϕ ϕ
+ + + + + + + +
+ + + + + + + +
+ + +
Ψ +
Ψ
− ∇ − −
− ∇ − −
= +
−
k r r
r r
r r r r r r r r
r r r r r r r r
k
r r
2
2 1) '( )
(3 11)
ϕ+
−
r r
The lass two terms on the right of (3 11)− has the form of 1 2lnf( , )∇ r r , so their loop
integrals are zero (for example, we can let one of 1r and 2r equals to r and let the
other one be R+r , with d d=l r , thereby doing the loop integral.) Substituting
(3 11)− into (3 5)− , and replacing φ with 1 2( , )Ψ r r , we would obtain:
0
c c 2 c
2i d kL s 2 s (3 12)
2 ie e e
π
Φ = = = = Φ −∫ k l
h h h
gÑ
where s is an integer, L is the perimeter of the loop, and there is kL 2 sπ= (with s 1=
in the present example.) But this result still differs from the experimental result of 0Φ
. The problem is with the representation of sn by
2
1 2 NN ( , ,...... )Ψ r r r . First, sn cannot
be
2
i
i
φ∑ , for it miss the cross terms. The scenario of the present case is: two
3

4. coupled electrons, which are at
1
( )
2
φ ϕ ϕ± + −= ± respectively, jointly forms a
constant current, and there are four combinations of electronic states: ϕ ϕ+ − , ϕ ϕ− + ,
ϕ ϕ+ + and ϕ ϕ− − , where only ϕ ϕ+ + can provide a positive and constant current; this
means that all measurements corresponding to any of the other three combination
states are not allowed to be “expressed” (due to limitation of energy conservation
law), and weights corresponding to the three combination states are lost in the
measured current value. Thus, when we represent sn with
2
1 2( , )Ψ r r of (3 10)− , we
miss a factor of 4 (instead of the electron number 2). The physics in it is: during the
time slots of the combination states not allowed to be expressed, physical effects
relating to Λ are still kept valid, so we need to add the sn ’s parts corresponding the
“lost slots” into the representation of sn , and the factor to be multiplied with
2
1 2( , )Ψ r r should be “the total number of micro states divided by the number of the
micro states allowed to be expressed”. Thus, in the present case, we should have
2
1 24 ( , )sn = Ψ r r , and the result of (3 12)− becomes:
0
hc
(3 12')
2e
Φ = = Φ −
which is the same as the experimental results. As indicated by the above operations
and results, the flux quantum of 0Φ is generated by two electrons, which are
originally at k ( 1,0,0)= ± respectively and degenerate, at very weak coupling. This
understanding is consequential. But before further discussing it, we first look at the
situation where superconducting current is in the direction of (1,1)=k . Assuming that
the current is co-generated by the two electrons at
1
( )
2
φ ϕ ϕ± + −= ± respectively,
with x yi i y
e uϕ
± ±
± ±=
k x k
. Now we would have:
x 1 x 2 y 1 y 2
x 1 x 2 y 1 y 2 x 1 x 2 y 1 y 2
i i i y i y
1 2 2 1 1 2
i i i y i y i( y y )
2 1 1 2 2 1
1 1 1 1 1
( ) '( ) ( ) '( ) e ( ) '( )
22 2 2 2
1 1
e ( ) '( ) e [ ( ) '( ) ( ) '( )]
2 2
u u
u u u u u u
ψ ϕ ϕ ϕ ϕ+ + + + + +
+ + + + + +
= − = −
= −
b x + b x + b + b
b x + b x + b + b b x +b x +b +b
r r r r r r
r r r r r r
And
2
1 24 ( , )sn = Ψ r r is still valid. (3 11)− becomes：
x 1 x 2 y 1 y 2
x 1 x 2 y 1 y 2
i( y y )* * * * * *
1 2 2 12
1 2
i( y y )
1 2 2 1
2 2
x y 1 2 x y 1 2
2
1 2
1 1 1
( ) { e [ ( ) '( ) ( ) '( )]
24 ( , )
1
( e [ ( ) '( ) ( ) '( )]) c.c.}
2
2i( ) ( , ) 2i( ) ( , )
4 ( , )
1
2
s
u u u u
n
u u u u
ψ ψ ψ ψ
−
+ + + +
+ + + +
∇ − ∇ = −
Ψ
∇ − −
Ψ + Ψ
= +
Ψ
b x +b x +b +b
b x +b x +b +b
r r r r
r r
r r r r
b + b r r b + b r r
r r
* * * *
1 2 2 1 1 2 2 1
2
1 2 2 1
1
[ ( ) '( ) ( ) '( )] ( [ ( ) '( ) ( ) '( )]) c.c.
2 (3 13).
4 ( ) '( ) ( ) '( )
u u u u u u u u
u u u u
+ + + + + + + +
+ + + +
− ∇ − −
−
−
r r r r r r r r
r r r r
4

5. The loop integral of the 2nd
term on the right of (3 13)− is obviously zero. The
result corresponding to that of (3 12)− is:
x y
c
i( ) d (3 14)
2 ie
Φ = −∫ b + b l
h
gÑ .
That the direction of loop SL is always along vector x x( )b + b can be realized in the
presence of dislocations, as long as the latter do not destroy the periodic potential
substantially; then, with
x y
x y
d dl=
b + b
l
b + b
, (3 14)− becomes:
x y
x y x y x y
x y
( )c c c
i( ) d ( ) dl L (3 15)
2 i 2 2e e e
Φ = = = −∫ ∫
h h h
g gÑ Ñ
b + b
b + b l b + b b + b
b + b
.
where x yL N= a + a is the perimeter of the loop, xa and ya are the base vectors
along x and y directions respectively, x
x
1
Na
=b , and x
y
1
Na
=b , then:
1
2 2 1/2 2 2 2
x y x y x y2 2
x y x y
2 2 2
x y x y
x y x y
1 1 1 1
L [( ) ( ) ] N [( )(a a )]
Na Na a a
a a (a a )
2 (3 16)
a a a a
= + = + +
+ −
= = + −
b + b a + a
when x ya a; , (3 16)− and (3 15)− lead to:
x y 0
c c
L 2 (3 17)
2e e
Φ = = Φ −b + b
h h
; ;
that is, when the current in the ring is carried by the pair of electrons at (1,1)=k and
( 1, 1)= − −k , the flux quantum is 0 / 2Φ , which is the same as was concluded
elsewhere (“Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in
Superconducting Loops” ， Loder, Florian; Kampf, Arno P.; Kopp, Thilo.
arXiv:1206.1738.) For such “diagonal” current, an exemplary integral loop could be
arranged as: the “ring” is in the form of a long cylinder, with y axis extending along
the axial direction of the cylinder, z axis being perpendicular to the surface, and x axis
being along the circumferential direction of the surface of the cylinder; the x axis is
kept to be along the tangential by displacements, and the z axis is also kept
perpendicular to the cylinder surface by displacements; the integral loop starts from
the middle of the cylinder and diagonally extends in opposite directions, forming half
of the loop, and joints the other half formed similarly on the opposite side of the
cylinder to form a complete loop; the perimeter along x direction is x xN a , the length
along y direction is y yN a , and the completeness of such a loop, as well as a result
close to that of (3 16)− , could be ensured with y yN a being somewhat greater than
x xN a .
Apparently, determination of flux of a superconducting ring depends on the electronic
states and sn of the pair of carriers. The effect of sn is remarkable, which depends on
“the total number of micro states divided by the number of the states allowed to be
expressed”; when the number of pairs engaging in coupling increases, the quotient
increases and the flux quantum decreases accordingly. For example, when the four
electrons at ( 1, 1,0)= ± ±k are in “full coupling” to form a “carrier team” of four
5

6. electrons and carrying a current corresponding to x2k in x direction, the quotient is
16, and the four electrons carry a current corresponding to a flux quantum of 0 / 8Φ .
But whether sn is multiplied by the same quotient among electron pairs having very
weak or substantially no coupling, the answer seems “no”. Perhaps some limitation
based on intensity of coupling should be introduced; but the cases discussed so far,
like (3 8)− , are in full coupling. The author still could not decide on this. But it could
be sure that sn needs to be at least doubled for two pairs of electrons.
The understanding, discussion and results so far, while being in agreement with some
existing experimental evidences, lead to new questions. First, electrons with xk 1=
are mostly not near FE , indicating that these electrons are “deep electrons”, contrary
to the understanding that carrier electrons should be near FE , and that carrier
electrons should be participants of phonon process leading to electrical resistance and
energy dissipation. If at least some carriers are “deep electrons”, the dual functions of
electrical conductance and electrical energy dissipation are respectively undertaken by
deep electrons and surface electrons. Such understanding seems more reasonable, as
screening to phonon interaction by binding energy of electron pairing, albeit
remarkable, seems not sufficient to supports things like persistent current, whose
mechanism will be discussed later.
Another question relates to mechanism of electricity. Symmetrical distribution of
electrons in k space would eliminate current, so generation of current should relate to
destruction of such symmetry. The discussion relating to flux quantum provides a
mechanism of such destruction, as well as some explanation of construction of sn ,
and provides some explanations to flux quantum. The expression of (3 2)− was
previously discussed by some people, but flux quantum was not derived from it, with
the understanding that current j was zero within the body of superconductor. 0=j is
indeed consequential, and we have not fully addressed it so far. But our derivations
relating to (3 12)− and (3 12')− are correct. The key lies in sn ; the loop integral of
(3 2)− is with respect to Λj rather than j ; thus, there is the possibility that even if
0→j the integral on Λj still has a finite value. In fact, as j is finite at the surface
and 0→j within the body of superconductor due to magnetic field z
e λ−
B : (where
λ is the penetration depth and z is the coordinate perpendicular to the surface,) the
current ought to have similar relation z
e λ−
j : ; as such, to maintain 2
d
φ
∫
j
lgÑ at a
finite value for all integral loops within the body of the material, there must be:
2 z
e (3 18)λ
φ −
−: .
Such an electronic state does exist, and it is the surface state(s) of crystal. We thus
could conclude that superconducting carrier electrons are those of surface states.
The results that a pair of electrons at ( 1,0)= ±k generate flux 0 / 2Φ and a team
of four electrons on ( 1, 1,0)= ± ±k states generate flux 0 / 8Φ is due to that the more
microstates are involved, the smaller the ratio of allowed microstates in them, and the
greater the ratio of “the lost slots”.
Clearly, the “lost slots” (corresponding to “measurement collapse of wave
6

7. function from such as
1
( )
2
φ ϕ ϕ± + −= ± to
1
2
φ ϕ± += ”) and associated effect have
certain inherent association and similarity to the “measurements of transitions in
nonstationary state” discussed with respect to AC Josephson effect; but there is a
difference, the collapse effect here happens in stationary effect, while in the AC
Josephson case it happens in non-stationary states. This difference is also associated
with the matrix elements, as the matrix for degenerate perturbation of two electrons is:
1 1
1 1
1
2 −
 
 ÷
 
, and that for “non-stationary stimulated transitions between two energy
levels” would be
0 1
1 0
 
 ÷
 
; the former has matrix elements
1
2
while the latter has
matrix elements of 1, that is why “lost slots” appear in the former but not in the latter.
And it is due to the “lost slots” that more electrons in coupling carry a smaller current
than fewer electrons. It is also due to that no slot is lost in non-stationary
measurements of transitions that the peak in the “peak-dip-hump” in ARPES results of
superconducting cuprates (such as B2212 system) can be observed.
Moreover, we need to distinguish the ϕ+ states of the two electrons in (3 10)− , in
agreement with our proposal “two measured states of the same energy and wave
vector in non-stationary context are not the same state ”. In particular, Single-fermion
wave functions in their “approximate” representation may not be used to construct the
anti-symmetrical wave function of a multi-fermion system.
The conclusion that carriers can be “deep electrons” is in agreement with the
experimental evidences of normal persistent cuurent (Persistent Currents in Normal
Metal Rings,Phys. Rev. Lett. 102, 136802 – Published 30 March 2009,Hendrik
Bluhm, Nicholas C. Koshnick, Julie A. Bert, Martin E. Huber, and Kathryn A.
Moler.).
We are now going to establish a model of normal and superconducting persistent
currents. As indicated schematically in Fig. 3, assuming that two state of wave vectors
( 1,0)= −k and (1,0)=k respectively are indicated by D' and D, and that the two
states couple to form states in the form of:
1
( ) (3 8)
2
φ ϕ ϕ± + −= ± −
When the two electrons of (3 8)− are both measured at ϕ+ , current in the form of
2
2* *
( ) (3 3')
2mi m
e e
φ φ φ φ φ
Λ
− ∇ − ∇ = − ≈ −j j j
h
is formed, and E establishes corresponding magnetic field and energy and also stores
current energy
2
2
e
m
2n e
j
, which corresponds to momentum:
2 2 2
2
e e e e
m m T
(3 19)
2n e 2n n 2mn
= = = −
j v p
,
where en is the density of carrier number. Thus, when en is determined, flux and j
could be estimated from:
' c dt d c d (3 2)
t
∂
Φ − Φ = − Λ = − Λ −
∂∫ ∫ ∫
j
l j lg gÑ Ñ and
7

8. 0
c c 2 c
2i d kL s s (3 12)
2 ie e e
π
Φ = = = = Φ −∫ k l
h h h
gÑ ,
and kinetic energy T and momentum p could be estimated from (3 19)− .
As such, however,it seems that when both the electron are measured at ϕ+ , it
would correspond to that the electron initially at ( 1,0)= −k now stays on a state C,
whose energy E is increased by C D'E E E-∆ = corresponding to “energy of the
magnetic field plus T”, and whose xk is increased by C D 'k k k-∆ = corresponding to
p . Thus, three effects emerge. First, microscopically, the electron state at C is
unstable, as the electron might jump back to state ( 1,0)= −k by a single or multiple
phonon process corresponding to ( E, k)∆ ∆ , so the initial energy of electric field is
turned to phonon energy, as in a resistance process. Second, E∆ would correspond to
an increase of the system’s internal energy, and a corresponding increase of the
system’s Gibbs function, with which the system might leave superconducting phase.
Third, most importantly, the momentum of the electron cannot be determined as to
whether it is Dk or Ck , and such a situation might not be allowed to exist (particularly
after field E disappears), for the momentum xk of an electron in a periodic potential
could only be the value of an eigen states kϕ while ( E, k)∆ ∆ are not eigen state
generally.
As such, the only possible situation seems that when the two electrons are both
measured as being at ϕ+ , the electron initially at ( 1,0)= −k does not acquire the
energy and momentum of ( E, k)∆ ∆ , and that one or more phonons are produced in
the process in which the electron is stimulated by electric field to form current, which
phonon(s) has a (total) energy and momentum of D( E, k 2k )∆ ∆ − , and the electron
initially at ( 1,0)= −k only has its momentum changed from Dk− to Dk . Thus, when
the current of the system changes from 0 to D2k , while the energy of the system
increases by E∆ , this increment is not in the electron system; rather, it is in the
phonon system, which also has a momentum change of Dk 2k∆ − ; the electron system
only has a momentum change of D2k . Since E∆ corresponds to a deviation from
equilibrium by the phonon system, it would be transferred to the environment by
thermal equilibrium process, allowing the system to discharge energy E∆ and
maintaining the internal energy of the system to be the same as that in zero current
state. But in such a situation, either of the two electrons at ϕ+ may return to state
( 1,0)− any time, nullifying the current, without any change of the total energy of the
system. So persistent current could not be formed in such a context.( But when the
electron’s transition to state C is virtual, there would be no phonon being produced.)
So we need to explain persistent current with respect to another scenario. As
shown in Fig. 4, the horizontal axis is yk , and xk direction (that of the total current)
is perpendicular to the plane of the paper. Letting state D' has D ' y1( 1,k )= −k and
y1E( 1,k )− , and state D has D y2(1,k )=k and y2E E(1,k )= ,and
y1 y2E( 1,k ) E(1,k ) Eδ− = + ，where E 0δ > , that is, energy of D' is higher than that of
state D. Assuming that the electron at D' is energized by electric field E and
virtually transits to state C; when states C and (one of the two states
1
2
φ ϕ± += at) D
8

9. are matched by one phonon mode CD CD( , )ω qh and states D' and D are matched by
another phonon mode D'D D 'D( , )ω qh , then as discussed previously with respect to
“non-stationary stimulated transitions between two energy levels”, the electron
virtually transiting to state C, by (virtually) emitting a phonon CD CD( , )ω qh ,
“condenses” to (the
1
2
φ ϕ± += state at) D. (By “condense” we mean that two
electrons in non-stationary states both have the measurement results corresponding to
one stationary state.) However, as C is an intermediate state of a multiple-process,
there could be neither emission of any CD CD( , )ω qh phonon nor absorption of any
energy from field E , instead only a real phonon D'D D 'D( , )ω qh is emitted in the whole
process.
As such, there would be two electrons both having measurement results
corresponding to state
1
2
φ ϕ± += , and accordingly no electron could be measured at
the state at D'. Thus, a current contribution of D y2(1,k )=k is formed, which has a
non-zero y component y1 y2( k k )− + because state y1E( 1,k )− lacks one electron while
state D y2(1,k )=k has two electrons. Obviously, in order to form a net total current in
x direction, there must be an electron of y1( 1, k )= − −k undergoing the similar process
with respect to the state of y2(1, k )= −k , whereby the x component of current
increases by Dk while its y component increases by y1 y2(k k )− , so the electron
distribution in y direction resumes symmetrical and the current along y direction
cancels; thus, a net current of D2k in x direction is formed. Most importantly,
however, now each of the two carrier electrons obtains a binding energy of Eδ . In the
situation as above, where one phonon mode CD CD( , )ω qh matches states C and D and
another phonon mode D 'D D 'D( , )ω qh matches states D' and D, the process has the
greatest probability of occurrence; states D' and C are coupled by interaction of
electric field E and does not required phonon matching. If states C and D and/or
states D and D' cannot be matched by one phonon mode, the probability of
occurrence of the process will be greatly reduced, and it would be likely that
corresponding persistent current carrier cannot be obtained. Also, the state D, which is
matched by the electron virtually transiting to state C by phonon mode CD CD( , )ω qh ,
could have a random distribution, and particularly could have a negative xk
component x xk k= − , so as to form a persistent current in –x direction, which is in
agreement with the known results that the normal persistent current could be random.
（ Persistent Currents in Normal Metal Rings,Phys. Rev. Lett. 102, 136802 –
Published 30 March 2009,Hendrik Bluhm, Nicholas C. Koshnick, Julie A. Bert,
Martin E. Huber, and Kathryn A. Moler.）
Whether an electron could “condense” should also depends on attributes of the
initial state (C), the final state (D) and the mediating phonon mode CD CD( , )ω qh ; the
specific relationships of are unknown; but it seems that the more stable the final state
(D) is, the more unstable the middle state (C) is, and the greater the energy of the
mediating phonon mode CD CD( , )ω qh , the more likely “condensation” would occur
9

10. and more stable the latter would be. Persistent current carriers are those electrons
which could condense and obtain binding energy Eδ ; electrons that could not
condense or that could not obtain adequate binding energy become normal current
carriers.
When both processes D' C→ and C D→ are virtual, the overall process does
not absorb any energy from the electric field E , and no CD CD( , )ω qh phonon is
emitted; instead, only one D'D D 'D( , )ω qh phonon is emitted. Phonon mode CD CD( , )ω qh
merely mediates the virtual transition from the middle state C to the final state D, and
it also mediates the “non-stationary interactions” (condensation) between states C and
D after the transition takes place, which at low temperature the mediation is basically
carried out by the zero-point of the phonon mode CD CD( , )ω qh . As such, the generation
of persistent current carriers itself would not dissipate energy of the electric field, and
instead a phonon D'D D 'D( , )ω qh , the energy of which would be released by thermal
equilibrium to the environment, allowing the energy of the present system to be kept
at a accordingly low level. On the other hand, the generation of persistent current
carriers includes build-up of energy of the middle state C as well as build-up of the
probability of virtual transition of electrons to the latter, so corresponding relaxation
should exist.
Since many (N) electrons participate the process relating to electric field, the
energy of the middle state C which each carrier electron can reach depends on N. The
smaller N is, the greater the energy of C is, and the greater the energy of phonon
CD CD( , )ω qh . As interaction of the electric field might lead to a random distribution of
middle states and final states, the resulted normal electron pairs and superconducting
electron pairs would also have their distribution, which should correspond to the
shape and/or symmetry of the energy band and the phonon spectra. Clearly, smaller
ring allows greater energy per electron transiting to the above middle states C, and the
limitation as (3 12)− would tend to reduce the order of carrier number by 1/3 or 1/2,
which all lead to increase of each carrier candidate’s energy.
With respect to reduction of carrier number, superconducting phase has a clear
advantage in that superconducting carrier electrons must be in surface states, while
normal carrier do not have such a limitation. The experiments like that of Bluhm et.
al. used thin film samples, which is equivalent to limitation of surface states. Low
temperature reduces the number of phonons, leading to sufficiently low probability
that the binding energy Eδ is released by multiple phonon process.
Thus, we have provided a microscopic explanation to (superconducting and
normal) persistent current. The charging of superconducting current is accompanied
by resistance dissipation, which is a necessary outcome of “two fluid” model.
According to our present model, the generation of persistent carrier electron does not
dissipate energy; instead there would be emission of real phonons and release of
corresponding energy into the environment; but the normal carrier electrons involved
still dissipate energy. There should be a build-up of energy of the middle state C and a
build-up of the probability of virtual transition of electrons to the middle state, and the
corresponding relaxation should exist accordingly.
10

11. Fig. 3
Fig. 4
A
C
D
Δ
E
xk
B
D’
( , )ω qh
C
D
E
yk
Eδ
D’
CD CD( , )ω qh
y1k
D'D D'D( , )ω qh
11