J. Mod. Ph
y
s
.
doi:10.4236/jm
p
Copyright ©
2
A
M
Abstract
Some aspec
t
of the Hall
c
equation for
m
a single non
l
Keywords:
A
1. Introdu
c
The quantu
m
now well kn
o
dimensional
e
p
henomena
w
magnetic fiel
d
the integer q
u
and was sub
s
effect [3,4].
R
tum Hall effe
tally to exhibi
where
f
is
tion. Thus (1
)
integer case
,
1,2, 3,
, an
d
case appear
i
The two effe
c
differences i
n
electrons and
for the forma
t
At the transit
i
teger quantu
m
observed. Th
e
to solve the
defined by th
e
j
e
1
2m
H

.
, 2010, 1, 158
p
.2010.13023
P
2
010 SciRes.
M
odel
f
D
Receive
d
t
s of anyon
p
c
onductance.
m
ulated fro
m
l
inear differe
n
A
nyon, Hall
c
tion
m
Hall effect
i
o
wn phenome
n
e
lectron syste
m
w
hen subject
e
d
[1]. First en
c
u
antum Hall e
ff
s
equently foll
o
R
eferring to t
h
ct, the Hall r
e
t plateaus at t
h
H
R
either an inte
g
)
incorporates
,
f
takes
d
some promin
e
i
n sequences
c
ts show rema
r
n
origin. In bo
quasiparticles
t
ion of the pla
t
i
ons between
s
m
Hall effect
e
oretically, th
e
many-
b
ody
q
e
many body
H
jj
e
()
e
p
Ar
c







-162
P
ublished Onlin
e
f
or the
Q
in th
D
epartment o
f
d
January 25,
2
p
hysics are r
e
A single p
a
m
the anyon p
n
tial equatio
n
Resistance,
C
i
s a recently
n
on which ap
p
m
which exhi
b
e
d to an int
e
c
ountered exp
ff
ect has recei
v
owed by the
h
ese two effe
c
e
sistance is fo
u
h
e quantized v
a
2
h
f
e
g
er or a simp
l
both of these
on integer
e
nt fractions
fo
such as
f

r
kable similar
i
th effects, th
e
s
is believed t
o
t
eaus in the H
a
s
uccessive pl
a
, scaling beh
a
e
aim in unde
r
q
uantum mec
h
H
amiltonian gi
v
22
jk
j
k
1e
2r






e
August 2010 (
h
Q
uanti
z
e Qua
n
Pa
u
f
Mathematics
,
r
2
010; revised
F
e
viewed with
a
rticle Schrö
d
icture. The
S
n
and solutio
n
C
onductance
,
discovered a
n
p
ears in a tw
o
b
its spectacul
a
e
nse transver
s
erimentally [
2
v
ed much stud
y
fractional H
a
c
ts as the qua
n
u
nd experime
n
a
lues
(
1
l
e rational fra
c
effects. For t
h
values
f
n

fo
r the fraction
21
n
n, 4
1
n
n

i
ties despite t
h
e
localization
o
o
be responsib
a
ll conductivit
y
a
teaus in the i
n
a
vior has be
e
r
standing this
h
anical proble
m
v
en by
j
j
k
U( r)

(
2
h
ttp://www.scir
p
z
ation
o
n
tum H
a
u
l Bracken
,
University o
f
m
F
ebruary 26,
2
the intentio
n
d
inger mode
l
S
chrödinger e
n
s for the m
o
Composite
P
n
d
o
-
a
r
s
e
2
],
y
,
a
ll
n
-
n
-
1
)
c
-
h
e

al
1
.
h
e
o
f
l
e
y
.
n
-
e
n
is
m
2
)
The
energy
field; t
h
and th
e
form p
to mov
It is
versio
n
tures o
c
are det
e
a har
m
model
constr
u
tions
c
some
s
p
hysic
a
lished
f
the Ha
l
up the
introd
u
2. Set
Let
Ψ
though
t
numbe
r
an ope
r
The
p
.org/journal
/
j
m
o
f the
H
a
ll Eff
e
f
Texas, Edinb
u
m
.edu
2
010; acc e p te
d
n
of establis
h
l
is introduc
e
quation-con
s
o
del can be p
r
P
articles
first term o
n
in the prese
n
h
e second ter
m
e
third term i
s
ositive backg
r
e in the two-d
i
the intention
n
of (2) subje
c
c
cur often in
t
e
rmined by s
o
m
onic oscillato
which emph
a
u
cted for a H
a
c
an be found.
s
pecific assu
m
a
l properties
t
f
rom the mod
l
l resistance,
(
model some
u
ced. Let us pr
o
tin
g
up the
Ψ
()x

b
e the
e
t
of as a flux
c
r
. A composi
t
r
ator phase tra
n
(
)
x


phase field
Θ
Θ()
x


m
p)
H
all Re
s
e
ct
u
rg, USA
d
March 20, 2
0
h
ing a model
e
d and coupl
s
traint syste
m
r
oduced.
n
the righ
t
-ha
n
n
ce of a cons
t
m
is the Coulo
m
s
a one-
b
ody
p
r
ound. The el
e
i
mensional 

here to set
u
c
t to a physic
a
t
his area, for
e
o
lving the Sch
r
r potential. T
h
a
sizes geome
t
a
ll system and
A wavefunc
t
m
ptions. It w
i
t
hat are very
r
el; in particul
a
(
1) can be ob
t
more physica
o
ceed to this [
Model-Co
m
e
lectron fiel
d
c
arrying a bos
o
t
e-
p
article fiel
n
sformation
Θ()
)
im x
e
x






Θ
()
x

is defin
e
2
()dyx y




s
istanc
e
0
10
for the qua
n
ed with a c
o
m
can be con
v
n
d side is th
e
t
ant external
m
m
b interactio
n
p
otential due
t
e
ctrons are co
n

-plane.
u
p and solve
a
a
l constraint.
S
e
xample Land
a
r
ödinger equa
t
h
us a simple
t
ry in the pr
o
it is shown t
h
t
ion is obtain
e
i
ll be seen t
h
r
elevant can
b
a
r, the quanti
z
t
ained. To be
g
l concepts ne
e
5,6].
m
posite Pa
r
d
. An anyon
o
n or fermion
q
d ()x


is de
)
x
.
e
d by
()y


,
JMP
e
n
tization
o
nstraint
v
erted to
e
kinetic
m
agnetic
n
energy;
t
o a uni-
n
strained
a
simple
S
uch pic-
a
u levels
t
ion with
physical
o
blem is
h
at solu-
e
d under
h
at some
b
e estab-
z
ation of
g
in to set
e
d to be
r
ticles
may be
q
uantum
fined by
(3)
(4)

P. BRACKEN 159
where  in (3) is an integer and ()
x
y  

in (4) is the
angle made between the vector
x
y

and the -axis;

represents anyon density. The effect of the operator
phase transformation (3) is to attach m flux quanta to
each electron. Composite particles experience the effec-
tive magnetic field B()
eff
x

described by the potential
Α()
j
x
, where Α()
j
x
 depends on the external vector
potential ()
ext
j
A
x

and a field ()
k
Cx

, which is an aux-
iliary field determined solely by the density ()
x


,
()() ()
ext
jj
A
xAxCx

. (5)
Therefore, from (5), it follows that
() ()()
effij ijD
BxAx Bmx 

 
 
 , (6)
and so the effective magnetic flux is the sum of the real
magnetic flux and a term which can be regarded as a
Chern-Simons flux.
Now suppose that ()
j
A
x

in (4) satisfies the Coulomb
gauge condition
() 0
jj
Ax

. (7)
It is possible to express ()
j
A
x

in terms of a scalar
field ()
A
x

as
() A()
jjkk
A
xx
e
 


 . (8)
This conclusion is only possible in a planar geometry.
Substituting (8) into ()
eff
Bx

, the field A()
x

can be
regarded as the scalar potential of the effective magnetic
field,
2
() A()
eff
Bx x
e
 

. (9)
This is basically the type of constraint we would like
to apply in order to solve (2); that is, by taking a particu-
lar reasonable form for ()
eff
Bx

.
The state vector Ψ is assumed to fully or very nearly
characterize the electronic state of the system. The total
free charge is given by
22
s
Qe dx
. (10)
The steady state time-independent wavefunction is
given by
/iEt
e

 
,
where 0 Ψ is time-independent and will have to satisfy
the time-independent Schrödinger equation
HE

 . (11)
Let us incorporate an additional assumption into the
construction of this model here. Let us suppose that we
can write () ()
eff
Bx Bx


in the following form
2
()Bx k



, (12)
where k is related to the total magnetic flux through
the surface; that is, the number of flux quanta of the
magnetic field and other constants. The magnetic flux
density affects the electronic states as it modifies the
Hamiltonian. Of course, the Hamiltonian is modified by
the vector potential, which in a simply-connected domain
is given by the usual formula

A
Bx . For exam-
ple, suppose we write and use (12) in the form
2
0
()Bxa



, (13)
and a is a constant which satisfies
22
0
Φ = ()
ss
Q
Bxdxadx aaN
e
 


(14)
In (14), N is the number of relevant current carrying
charge quanta. Moreover, let
M
denote the number of
magnetic flux quanta, which means the total flux can be
written as
Φ
2
h
Me . (15)
When the flux and charge are quantized, these results
imply that a is a fraction which can be expressed in
terms of the flux quantum
2
M
h
aNe . (16)
On a simply connected region, the vector potential can
be represented as a one-form given in terms of a single
function

, which stands for
A
here, as
xy
A
dy dx



. (17)
Using (17), the magnetic field can be calculated and
then (13) yields a constraint equation
2
0
Φ
xxyy a
  . (18)
3. Solution of the Schrödinger Equation
The main objective here is to solve the time-independent
Schrödinger equation coupled with Equation (18) to ob-
tain Ψ. Of course, vector potential (17) appears in the
Schrödinger equation, as can be clearly seen from (2).
This procedure will lead to a nonlinear equation; howev-
er, it will be found that solutions with the correct physi-
cal properties can be determined in closed form. Keeping
the first term in (2), the left hand side without the overall
multiplicative constant applying (17) leads to



2
222 2
00,x0,y 0
2
2i
xyy xxy
ee
    




P. BRACKEN 160
Therefore (11) written out in full takes the form,



2
222 22
00,x0,y 0
2
0
2i
2.
xyy xxy
ee
mE
 

 







(19)
Now the problem takes the form of finding solutions
to (19) subject to the condition (18) This will not be done
in a completely general way, but with some assumptions
which will lead to a physically relevant result.
Suppose the electron system describes a rectangular
geometry in the xy plane. Moreover, let 0 Ψ have a
plane wave dependence in the x direction, so solutions
which have the structure




0
Ψ,.
ikx
x
ye y   (20)
is sought where


y
 is a real function of y. Let us
take the function in the vector potential to be indepen-
dent of
x
,
()
y
 (21)
The derivatives of 0 Ψ can be calculated based on (20)
and then substituted into (19),
 

 

22 22
yy 2e
2.
y
kyykyyey
mE y
 

 


(22)
This takes the form of a second order equation for


y

, but it is coupled to


y

in (17),
   
2
yy 2. y
ykey mEy    (23)
If

is assumed to have the form (21), then 0
xx


and (18) assumes the simple form


2
yy .ay
  (24)
Since the right-hand side of (24) depends only on y,
(24) can be integrated once to obtainy

, which appears
in (23), in terms of


y
 as
 
0
2
y.
y
y
yad

   (25)
Imposing


0
0
y
y
 . Substituting (25) into (23), this
coupled system is reduced to the following nonlinear
eigenvalue problem
  
0
22
yy 2
2
() .
y
y
emE
ykad yy   

(26)
Therefore, the dependent variable in (26) is


y

. In
addition to (26), it is useful to write down a decoupled
version which is obtained by introducing a new variable


y

given by
 
0
2.
y
y
e
yk ad

   
 (27)
Equation (26) can be written in the form of a pair of
equations as follows,
 
22
yyy 2
2
,.
emE
yayy yyy    

(28)
The Hall resistance for this two-dimensional system
can be calculated based on (28), in fact it can be written
in terms of


y
 . The geometry is that of a rectangular
plate with edges which are parallel to the
x
and -axes.
To be consistent with (20), where the
x
-dependence in
Φ is assumed to be a plane wave, only the y dimen-
sion will be of significance here. The terminations for
integration localized at fixed -coordinates, are termed
the left (L) and right (R) edges of the geometry. The Hall
potential is defined as the difference of potentials be-
tween these two edges of the rectangle. In fact, the Hall
potential can be obtained from (26), or better in terms of
the solution for


y

by means of
 

2
22
H,
2
VRL
me
 
 (29)
where R and L refer to right and left. Only the longitu-
dinal
x
or plane wave component of the current density
contributes

 
0
22
x0 0
Re A.
y
y
eee
jiekady
mm
  


 


(30)
The potential H V is transverse to the current. From
(28), since 2

can be related to y

, the current density
can be represented entirely in terms of the variable

as

22
22
xy
.
2
y
eh
jmamam    

Integrating x
j and using the definition of H V given
in (29), x
I
can be related to H V as follows,

 

22
222
xx H.
22
LL
y
RR
e
IjdydyRL V
am ama
  


(31)
By means of (16), the quantity a can be eliminated
from (31) to produce the following remarkable formula,
2
xH
2. Ne
I
V
Mh
 (32)
The result in (32) immediately implies the Hall resis-
tance is quantized according to,
H
H2
x
.
2
V
M
h
RIN
e
 (33)

P. BRACKEN 161
Finally, it will be shown that a wavefunction  can
be determined based on the coupled system (28). In fact,
the coupled equations in (28) can be combined into a
single nonlinear differential equation for the function


y

, from which


y

can be determined. To begin
to do this, differentiate the first equation in (28) and then
divide this by y

to obtain
yy y
y
2.




 (34)
Differentiating both sides of this, there follows
22
yy yyy
22
22 .
y
yy
y




 (35)
Squaring both sides of (34), an additional expression
for 22 /
y

 is obtained. Substituting this into the right
hand side of (35),
2
yy yyyyy
yy
1.
24
 
 





(36)
From the second equation in (28), upon dividing by

,
it follows that
yy 2
2
2mE .



 (37)
Substituting (37) into (36), a third order nonlinear eq-
uation in terms of the independent variable

results,
2
yyy yy 2
yy
122,
2E








(38)
where we put
2
2mE .
E

 (39)
A general solution to (38) may not be possible, how-
ever, something can be done. Note that upon omitting
2
(22)E   from (38), the equation can be integrated.
Thus, we have




yy y
1
lnln 0
2
yy
 , and inte-
grating gives  0. This can be integrated
as well to give
 
3
123
1
1
yy
3cc c c
 
. A specific
physically realistic solution to the general form of (38)
can be approached as follows. The first equation in (28)
implies that the sign of


y 
is determined by a,
therefore, when

does not vanish,

must be a mo-
notonic function. Consequently, one way in which a
class of solution can be obtained is to consider the case
in which y

is only a function of

,

y.w

  (40)
In fact,


g

can be determined explicitly. Diffe-
rentiating both sides of (40) with respect to , we get

2
yy yyyy
w,w .ww www
 
   (41)
Substituting (40) and (41) into (38) gives rise to the
following equation for w,
22
1
w2,
2
wwE
 

 



(42)
Clearly (42) is nonlinear, however, there is a way to
produce a solution which is physically reasonable. There
exists a quadratic polynomial solution for w which can
be expressed in terms of

as
2
w( ) 

 
These constants can be specified upon substitution in
(42), and it will constitute a solution provided that
0


and

2
1
w2.
2E
   (43)
Taking (43) and replacing the result in (40), it is clear
the resulting equation can be separated to give
2
2yc.
2
d
E




 (44)
The negative sign gives a tangent function solution
which will be prone to have poles and can be written
 

2tan 2yc.yE   

However, the other choice of sign in (44) gives rise to
the result,
1arctan()y c.
2EE




This can be solved explicitly for the function


y

,
 
2y
2y
1C
2tanh yc2.
1C
E
E
e
yEE E
e




 




 
(45)
By differentiating (45), an expression for


2y
 is
obtained. The function


y
 which we need to write
the wavefunction (20) is found from the square root of
this, namely

1
2
1y
4
1
2y
1
22 C.
1C
E
E
Ee
yea e 




 





 (46)
The wavefunction is then determined using (46) by
means of,

P. BRACKEN 162

/
Ψ.
iEt ikx
ee y 


This is a bounded function on any right half axis and
square integrable over the rectangular area. Thus there
exists a solution with the desired physical properties.
Therefore, it has been seen how (1) emerges and that
physical classes of solutions to (2) can be investigated.
Most importantly, a link between the wavefunctions im-
plied by the model and the calculation of a corresponding
resistence for the model has been shown.
4. Conclusions
An elementary model for the quantum Hall effect has
been developed. It is known in this field that simple
models based on Schrödinger equations can be very use-
ful in studying the effect. For example, the equation is
solved with the harmonic oscillator potential to describe
and obtain the energies of Landau levels. The model
emphasizes several aspects of the geometry of the system
in obtaining the results (32,33). It is quite interesting that
a single particle Schrödinger equation can be obtained
and solved in closed form, and which incorporates a sig-
nificant amount of the physics involved.
5. References
[1] J. K. Jain, “Composite Fermions, Perspectives in Quan-
tum Hall Effects,” In: S. D. Sarma and A. Pinczuk Eds., J.
Wiley and Sons, New York, 1987.
[2] K. von Klitzing, G. Dorda and M. Pepper, “New Method
for High-Accuracy Determination of the Fine Structure
Constant Based on Quantized Hall Resistance,” Physics
Review Letters, Vol. 45, No. 6, 1980, pp. 494-497.
[3] R. B. Laughlin, “Quantized Hall conductivity in Two
dimensions,” Physical Review B, Vol. 23, No. 10, 1981,
pp. 5632-5633.
[4] R. E. Prange and S. M. Girvin, “The Quantum Hall Ef-
fect,” Springer Verlag, New York, 1987.
[5] Z. F. Ezawa, “The Quantum Hall Effects, Field Theoreti-
cal Approach and Related Topics,” World Scientific,
Singapore, 2000.
[6] R. B. Laughlin, “Fractional Quantization,” Reviews of
Modern Physics, Vol. 71, No. 4, 1999, 863-874.