Supercurrent and electromotive force generations by the Berry connection from many-body wave functions

概要

Supercurrent and electromotive force generations by
the Berry connection from many-body wave
functions
Hiroyasu Koizumi
Division of Quantum Condensed Matter Physics, Center for Computational Sciences,
University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan
E-mail: koizumi.hiroyasu.fn@u.tsukuba.ac.jp
January 2023
Abstract. The velocity field composed of the electromagnetic field vector potential
and the Berry connection from many-body wave functions explains supercurrent
generation, Faraday’s law for the electromotive force (EMF) generation, and other
EMF generations whose origins are not electromagnetism. An example calculation
for the EMF from the Berry connection is performed using a model for the cuprate
superconductivity.

1. Introduction
The Berry phase first discovered in the context of the adiabatic approximation now
prevails in various fields of physics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. In particular, it is now an
indispensable mathematical tool to detect topological defects in quantum wave functions
[11]. Recently, the Berry connection from many-body wave functions was defined and
its usefulness to calculate supercurrent is demonstrated [12]. A salient feature of such a
formalism is that it provides a vector potential directly related to the velocity field for
electric current. In the present work, we consider the supercurrent and electromotive
force (EMF) generations based on the same formalism [12, 13].
The EMF is expressed using a non-irrotational ‘electric field’, Enon−irrot , whose
origin may not be a real electric field. It is defined as
I
Enon−irrot · dr
(1)
E=
C

where C is a closed electric circuit. This EMF appears due to various causes, such as
chemical reactions in batteries or temperature differences in metals.
One of the important EMF generation mechanisms is the Faraday’s law of magnetic
induction. It is expressed as a total time-derivative of a magnetic flux of the magnetic
field B
Z
d
B · dS
(2)
E =−
dt S

Supercurrent and EMF by Berry connection

2

where S is a surface whose circumference is C. This EMF formula is often called the
R
“flux rule”, since S B · dS is the magnetic flux through the surface S; it has been
claimed curious since it is composed of two different fundamental equations in classical
theory [14], i.e., the Faraday’s law of induction and the Lorentz force. The curiosity
is increased by the fact that one of them is an equation for fields only, and the other
includes particles and is an equation for a force on a particle.
This peculiarity disappears in quantum theory using the vector potential A that is
more fundamental than the magnetic field B [15, 16, 17], and the wave function makes
the velocity of a particle a velocity field [18]. Then, the two contributions in the “flux
rule” are connected by the duality that a U (1) phase factor added on a wave function
describes a whole system motion, and also plays the role of the vector potential when
it is transferred into the Hamiltonian [19].
In the present work, we extend the above vector potential and velocity field
approach for the electric current generation to cases where the vector potential of the
Berry connection from many-body wave functions appears [12]. We show that the EMF
generation other than the electromagnetic field origin, such as those due to chemical
reactions or temperature gradients can be expressed by it.
The organization of the present work is as follows: we explain the velocity field
appearing from the Berry connection from many-body wave functions in Section 2.
We reexamine the Faraday’s EMF generation formula using the velocity field from the
electromagnetic vector potential in Section 3. We examine the EMF generation by the
Berry connection in Section 4, and an example calculation is performed for the Nernst
effect in Section 5. Lastly, we conclude the present work by mentioning implications of
the present new theory in Section 6.
2. The velocity field from the Berry connection form many-body wave
functions and supercurrent generation
The key ingredient in the present work is the Berry connection from many-body wave
functions for electrons given by
Z

1
MB
∗
AΨ (r) =
Re
dσ1 dx2 · · · dxN Ψ (r, σ1 , · · · , xN )(−iℏ∇)Ψ(r, σ1 , · · · , xN )
ℏρ(r)
(3)
where N is the total number of electrons in the system, ‘Re’ denotes the real part, Ψ
is the total wave function, xi collectively stands for the coordinate ri and the spin σi
of the ith electron, −iℏ∇ is the Schr¨odinger’s momentum operator for the coordinate
vector r, and ρ(r) is the number density calculated from Ψ. This Berry connection is
obtained by regarding r as the “adiabatic parameter”[1].
Let us consider the electron system whose kinetic energy operator in the Schr¨odinger

Supercurrent and EMF by Berry connection

3

representation is given by
Tˆ = −

N
X
ℏ2 2
∇j
2m
e
j=1

where me is the electron mass.
For convenience, we also use the following χ defined as
Z r
′
′
χ(r) = −2
AMB
Ψ (r ) · dr

(4)

(5)

0

and express the many-electron wave function Ψ as
!
N
iX
Ψ(x1 , · · · , xN ) = exp −
χ(rj ) Ψ0 (x1 , · · · , xN )
(6)
2 j=1
 P

Then, Ψ0 = Ψ exp 2i N
χ(r
)
is a currentless wave function for the current
j
j=1
ˆ
operator
 Passociated
 with T in Eq. (4) since the contribution from Ψ and that from
exp 2i N
j=1 χ(rj ) cancel out. In other words, a wave function is given as a product

 P
χ(r
)
. The total
of a currentless one, Ψ0 , and the factor for the current exp − 2i N
j
j=1
wave function Ψ must be a single-valued function of coordinates. This makes χ as an
angular variable that satisfies some periodicity. This periodicity gives rise to non-trivial
topological integer as will be explained, shortly.
When electromagnetic field is included, the kinetic energy operator becomes
Tˆ′ =

N
X
1
(−iℏ∇j − qA)2
2m
e
j=1

(7)

where q = −e is the electron charge, and A is the electromagnetic field vector potential.
The magnetic field is given by B = ∇ × A.
In the following, we will use the same expression, Ψ, for the total wave function.
Then, the current density for Ψ is given by
j = −eρv

(8)

with the velocity field v given by


ℏ
e
A − ∇χ
v=
me
2e
e
ℏ MB
=
A+
A
(9)
me
me Ψ
The current density in Eq. (8) is known to give rise to the Meissner effect if it
is a stable one due to the fact that it explicitly depends on A [18]. For the stable
current case, ∇χ compensates the gauge ambiguity in A, i.e., the ambiguity that exists
in A for the same electric field E and magnetic field B (we may take A′ = A + ∇f ,
where f is a function, instead) can be absorbed by adopting different ∇χ′ given by
∇f , yielding the same v in Eq. (9). This compensation of ∇f also
∇χ′ = ∇χ + 2e
ℏ
guarantees the local charge conservation.

Supercurrent and EMF by Berry connection

4

If the Meissner effect is realize, the magnetic filed is expelled from the bulk of a
superconductor [18]. Then, the flux quantization is observed for magnetic flux through
a loop C that goes through the bulk of a ring-shaped superconductor
Z
I
B · dS =
A · dr
S
C
I
ℏ
=
∇χ · dr
2e C
h
= wC [χ]
(10)
2e
where wC [χ] is the topological integer ‘winding number’ defined by
I
1
wC [χ] =
(11)
∇χ · dr
2π C
According to Eq. (9), the presence of non-zero wC [χ] means the existence of the
stable velocity field that satisfies
I
h
v · dr =
(12)
wC [χ]
2me
C
In superconductors, the quantized flux persists. This means that the condition
d
wC [χ] = 0
(13)
dt
is realized.
In normal metals, the time-derivative of the velocity field is often expressed as
1
dv
=− v
(14)
dt
τ
using a relaxation time approximation, where τ is the relaxation time.
Combination of this with Eq. (12) yields
d
τ wC [χ] = −wC [χ]
(15)
dt
If the condition in Eq. (13) with nonzero wC [χ] is realized, Eq. (15) means that τ
must be ∞, i.e., an infinite conductivity, or zero resistivity is realized.
3. The vorticity field from the vector potential A and Faraday’s flux rule
In this section, we consider the case where non-trivial AMB
is absent. When AMB
is
Ψ
Ψ
trivial, it satisfies
∇ × AMB
Ψ = 0
Thus, by applying ∇× on the both sides of Eq. (9)
e
B
∇×v =
me
is obtained.
Taking the total time-derivative of the above yields
dv
e
e
∇×
=
∂t B +
(v · ∇)B
dt
me
me

(16)

(17)

(18)

Supercurrent and EMF by Berry connection

5

where the total time-derivative of the field B is given by Eulerian derivatives as
dB
= ∂t B + (v · ∇)B
(19)
dt
Note that the vector potential version for this equation (the one replacing B by A) has
been used for a while [20].
Integrating Eq. (18) over the surface S, we have
I
Z
Z
e
dv
e
· dr =
∂t B · dS +
(v · ∇)B · dS
(20)
me S
me S
C dt
where the Stokes theorem is used to convert the surface integral to the line integral.
Noting that the electromotive force for an electron is given by
I
1
d(me v)
E=
· dr
(21)
−e C dt
the following relation is obtained
Z
Z
(22)
E = − ∂t B · dS − (v · ∇)B · dS
S

S

This is equal to the Faraday’s formula in Eq. (2).
In the situation where the circuit C moves with a constant velocity v0 , we have the
following relation
(v0 · ∇)B = ∇ × (B × v0 ) + v0 (∇ · B)
= ∇ × (B × v0 )
due to the fact that B satisfies ∇ · B = 0 [21].
As a consequence, the well-known EMF formula
Z
I
E = − ∂t B · dS + (v0 × B) · dr
S

(23)

(24)

C

is obtained. The first term in it is attributed to the Faraday’s law of induction, and
the second to the Lorentz force. This formula is composed of two different fundamental
equations in classical theory [14]. However, in the quantum mechanical formalism, two
contributions stem from a single relation in Eq. (9).
4. The EMF generation by the Berry connection
The velocity field in Eq. (9) contains the vector potential AMB
in addition to the
Ψ
MB
electromagnetic vector potential A. Just like A, AΨ will also give rise to the EMF.
We now consider a general case where the Berry connection arises from a set of
states {Ψj } and given by
X
(25)
AMB =
pj AMB
Ψj
j

where pj ’s are probabilities satisfy
X
pj = 1
j

(26)

Supercurrent and EMF by Berry connection
and AMB
Ψj is obtained from Eq. (3) by replacing Ψ with Ψ j .
We express AMB using the following density matrix
X
dˆ =
pj |Ψj ⟩⟨Ψj |

6

(27)

j

ˆ MB is defined through the relation
where the operator A
ˆ MB |Ψj ⟩ = AMB
⟨Ψj |A
Ψj

(28)

From now on, we allow the time-dependence in Ψj . When Ψj is time-dependent, AMB
Ψj
is also time-dependent. The distribution probability pj can be also time and coordinate
dependent.
ˆ MB , the vector potential from the
Using the density operator dˆ and the operator A
Berry connection is given by


MB
MB
ˆ
ˆ
A = tr dA
(29)
We define BMB by
BMB = ∇ × AMB

(30)

Then, the EMF from the Berry connection is given by
Z
Z
ℏ
ℏ
MB
MB
∂t B · dS −
(v · ∇)BMB · dS
E
=−
(31)
e S
e S
The first term in the right hand side can arise from the time-dependence of pj . This
means that if pj varies with time due to chemical reactions, photo excitations, or etc.
it will give rise to the EMF. The second term will be nonzero if the temperature
depends on the coordinate, T (r) with ∇T (r) ̸= 0 , and pj contains the Boltzmann
E
factor exp(− kB Tj(r) ), where Ej is the energy for the state Ψj ; in this case, terms like
−

Ej

E ∇T (r) −

Ej

∇e kB T (r) = kBj T 2 (r) e kB T (r) arise. The contribution from the second term also arises
when pj depends on the coordinate due, for example, to the concentration gradient of
chemical spices.
Now we consider the case where the circuit moves with a constant vector v0 . The
circuit in this case should be regarded as a region of the system which flows due to the
flow existing in the system. Such a motion may arise from a temperature gradient or
concentration gradient in the system. In this case, we have the following relation,
(v · ∇)BMB = −∇ × (v0 × BMB )

(32)

due to the fact that ∇ · BMB = ∇ · (∇ × AMB ) = 0.
The equation (31) can be cast into the following form
I


ℏ
MB
E
=−
∂t AMB − v0 × (∇ × AMB ) · dr
(33)
e C
that only contains AMB . However, the above formula may not be convenient to use due
to the fact that AMB contains topological singularities. A convenient one may be the
following
Z
ℏd
MB
BMB · dS
(34)
E
=−
e dt S
where B in the Faraday’s law in Eq. (2) is replaced by BMB .

Supercurrent and EMF by Berry connection

7

5. Nernst effect
In this section, we examine the Nernst effect observed in cuprate superconductors
[22, 23, 24]. We examine this phenomenon using Eq. (34). A theory of superconductivity
in the cuprate predicts the appearance of spin-vortices in the CuO 2 plane around doped
holes that become small polarons [25, 26, 27]. The spin-vortices generate the vector
potential
1
AMB = − ∇χ
(35)
2
where χ is an angular variable with period 2π. This angular variable appears due to
the requirement that the wave function to be a single-valued function of coordinates
in the situation where itinerant motion of electrons around the small polaron hole is a
spin-twisting one.
We can decompose χ as a sum over spin-vortices
χ=

Nh
X

χj

(36)

j=1

where χj is a contribution form the jth small polaron hole, and Nh is the total number
of holes that become small polarons.
Each χj is characterized by its winding number
I
1
wj =
∇χj · dr
(37)
2π Cj
where Cj is a loop that only encircles the center of the jth spin-vortex. We can assume
wj to be +1 or −1; only odd integers are allowed due to the spin-twisting motion. The
numbers ±1 are favorable from the energetic point of view.

x0 x0+v0Δt

v0
Ly
y
x

C(t)

C(t+Δt)

Figure 1. A schematic picture for the EMF appearing from the Berry connection
generated by spin-vortices. The Berry connection creates the vector potential
proportional to ∇χ, which creates vortices (loop currents) denoted by circles with
arrows. We consider two loops C(t) and C(t + ∆t), where t and t + ∆t denote two
times with interval ∆t. The loop moves with velocity v0 in the x-direction due to the
temperature gradient in that direction. A constant magnetic field is applied in the zdirection. A voltage is generated across the y-direction. The sample exists 0 ≤ y ≤ Ly .
The left edge of the loop at time t is x0 and that at time t + ∆t is x0 + v0 ∆t.

Supercurrent and EMF by Berry connection

8

Let us consider the situation depicted in Fig. 1. We neglect the contribution from
A assuming that it is small. The EMF generated across the sample in the y-direction
is given by
Z

Z
ℏ 1
MB
MB
MB
E
= −
B · dS −
B · dS
e ∆t S(t+∆t)
S(t)
I

I
ℏ 1
MB
MB
= −
A · dr −
A · dr
e ∆t C(t+∆t)
C(t)
I
ℏ 1
=
AMB · dr
(38)
e ∆t ∆C
where S(t+∆t) and S(t) are surfaces in the xy-plane with circumferences C(t+∆t) and
C(t), respectively; ∆C is the loop encircling the area x0 ≤ x ≤ x0 + v0 ∆t, 0 ≤ y ≤ Ly ,
with the counterclockwise direction.
H
We approximate ∆C AMB · dr by
I
I
1
MB
∇χ · dr
A · dr = −
2 ∆C
∆C
1
≈ − 2π(nm − na )Ly v0 ∆t
(39)
2
where nm and na are average densities of wj = 1 (‘meron’) and wj = −1 (‘antimeron’)
vortices, respectively. Thus, nm Ly v0 ∆t and na Ly v0 ∆t are expected numbers of wj = 1
and wj = −1 vortices within the loop ∆C, respectively.
From Eqs. (38) and (39) , the approximate E MB is given by
hv0
(na − nm )Ly
(40)
2e
Thus, the electric field generated by E MB in the y-direction is given by
hv0
(na − nm )
(41)
Ey ≈
2e
In our previous work, na is denoted as nd indicting that it yields a diamagnetic
current, and nm as np indicting that it yields a paramagnetic current [26, 27]. Using nd
and np , the Nernst signal is obtained as
E MB ≈

eN =

Ey
hv0 (nd − np )
=
|∂x T |
2e|∂x T |

(42)

The same formula was obtained previously for the situation where spin-vortices move
by the temperature gradient [26, 27]. Here, the situation is different; the spin-vortices
do not move, but the electron system affected by ∇χ moves. Considering that the
small polaron movement is negligible at low temperature, the present situation is more
realistic than the previous one. The temperature dependence is the same as the one
that qualitatively explains the experimental result [27].
Note that experiments indicate the presence of loop currents different from ordinary
Abrikosov vortices [28] in the cuprate [29]. The present result suggests that the observed
Nernst signal can be explained by the presence of spin-vortex-induced loop currents.

Supercurrent and EMF by Berry connection

9

6. Concluding remarks
Since the EMF by the Berry connection is not the electromagnetic field origin, it may
be more appropriate to call it the Berry-connection motive force (BCMF) given by
F

BCMF

= −eE

MB

d
=ℏ
dt

Z
BMB · dS

(43)

S

The BCMF will also arise from quantum mechanical dynamics of particles other
than electrons, for example, from proton dynamics through chemical reactions. The
non-trivial Berry phase effect has been predicted [30], and observed in the hydrogen
transfer reactions [31]. Quantum mechanical effects are important in such reactions due
to the relatively light mass of protons [32, 33]. It is known that the EMF generated by
the proton pumps is a very important chemical process in biological systems [34], and
the BCMF may play some roles in the working of them. It may be also useful to invent
high performance batteries.
In order to perform realistic calculations for F BCMF in various systems, we need to
develop methods to obtain BMB for them. The key point of such a development will
be to find methods to locate singularities of wave functions and calculate associated
winding numbers [35]. When such information is combined with the conservation of
charge will yield AMB [13]. For an infinite system case, the periodic boundary condition
may be employed.
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2022
Physics
Letters
A
450
128367
ISSN
0375-9601
URL
https://www.sciencedirect.com/science/article/pii/S0375960122004492
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