Friedel Sum Rule, Levinson Theorem,
and the Atiyah - Singer Index
De - Hone Lin
Department of Physics, NSYSU
09 October 2008
介紹兩個和相移有關的定理,它們和
At i yah- Si nger 指標的關係以及磁通
1. Levinson Theorem
的拓樸效應對電荷密度的影響.
Friedel Sum Rule
2. Friedel Theorem 
Friedel Oscillation
Friedel Sum Rule for Chiral Fermions
3. Friedel Theorem for Fermions 
Friedel Theorem for 2d Dirac Fermions
4. Connections among the Friedel Sum,
Levinson Theorem, and the Atiyah-Singer Index
5. Aharonov-Bohm Effect on Friedel Oscillation
1. Levinson Theorem
有多少個束縛態在 potential well 中?
Levinson associated the problem with the phase shifts of scattering states
V (r) 
1
r2
and
V (r)  0
for r  a
for r  a
How many bound states
are there in the well?
Phase shifts
charged particles
N. Levinson in 1949 clarified the relation between the
phase shifts and the number of bound states.
 l (0)  nl , l  1,2,
The phase shift of scattered state
of l channel at the threshold of
zero mometum
The number of bound states in
the angular momentum channel l
Levinson Theorem under an Aharonov-Bohm Effect
Aharonov-Bohm magnetic flux
V (r) 
1
r2
and
V (r)  0
for r  a
for r  a
Bound states therein
charged
particles
Phase shifts
  (0)  n  ,   m  0
Denotes the phase shift
of scattered state at
threshold of zero
momentum.
The total number of bound
states in the fractional angular
momentum channel
m  0,1,2,
0   /  0 with  0  hc / e,
the fundamenta l flux quantum.
D.H. Lin, Phys. Rev. A68, 052705 (2003).
Levinson' s scientific contributi ons and enormous impact on MIT were
admirably summarized by his colleagues and peers as follows (Dec. 17, 1975) :
Norman Levinson was the heart of mathematic s at MIT, a man who combined
creative intellect of the highest order with human compassion and unremittin g
dedication to science and to excellence in its pursuit.
Throughout the mathematic al world, the name of MIT and the name of
Norman Levinson have been synonymous for many years .
To those of us who were fortunate to have him as friend and colleague,
this is entirely fitting, because we are aware that,
with extraordin ary effectiven ess and caring,
he devoted forty six years of his life to mathematic s and to this Institute.
Friedel Sum Rule
2. Friedel Theorem 
Friedel Oscillatio n
雜質對 many - body system 有什麼影響?
Friedel Sum Rule
N 
2

(2l  1) ( E


l 0
The total change of
the number of states
l
f
)
Where  l ( E f ) denotes the phase shift
in the angular-momentum channel l
with energy at the Fermi surface.
The rule is believed to be exact even in interactin g many - particle system.
G.D. Mahan, Many - Particle Physics (Plenum, New York, 2000) P.236.
This formula provides a most useful method for calculatin g
the scattering cross - section of an impurity. 
J.M. Ziman, Principles of the Theory of Solids
(Cambridge , New York,1995)
Friedel Sum Rule 使雜質, 合金的residual resistance , 空位形成能,
和抗磁磁化率的理論計算變的十分簡單.
by 固體物理學大辭典 (馮端主編)
The Friedel theorem has been of major importance for
understand ing the electronic structure of impurities in metals, .....
G. Grosso, and G.P. Parravicin i, Solid State Physics
(Academic Press, New York, 2000)
Friedel Oscillatio n
 
1
4 2 r

l
(2
l

1)(

1)
cos[2k F r   l (k F )]sin[ l ( k F )]
3 
l 0
1. Chaired the consultative committee tothe French government
for scientific and technological research (1978-1980).
2. President of the Societe francaise de physique and
of the European Physical Society.
3. President of the French Academy of Sciences (1992-1994).
3. Friedel Sum Rule for Chiral Fermions in D Space


1 
ΔN R =  g l  D  δ κ  EF  + δ κ  -E'F  - 2δ κ  0 
π l=0
Where   (2l  D  1)/2,
  0  is phase shift at zero-momentum,
 D  l  2
and g l  D   2 
 is the degeneracy for each l.
l


d
The change of the number of states
of right-handed fermions
The right hand side involves three physical quantities and
tow important important concepts.
Three quantities:
(1) the orbital  spin angular momentum;
(2) the energy of Fermi surface;
(3) the energy of zero-momentum.
Two concepts:
(1) The phase shift of matter wave;
(2) the anti-particles.
Does the relativist ic theorem is meaningful
in the condensed system?
structure of graphene
Novel Phenomenon in Condensed Matter
Friedel Theory for the Relativistic Spin-1/2 Systems
in Two Dimensions
The Friedel Sum Rule
1
ΔN =
[δ j (Ef ) - δ j (μ) + δ j (-Ef ) - δ j (-μ)]

π j= 1/ 2,3/ 2,
The relation w ith the relativist ic Levinson t heorem
N 

nj  0
j  1 / 2 ,..
Friedel Oscillation of the Relativistic Spin-1/2 System in Two Dimensions
δρ =
1
1
κ+1 ε Eμ 1
(-1)
sin[2k
r
+
δ
(k
)]sinδ
(k
)
+
O(
),
f
j
f
j
f
2 
2
3
(2π) j
π r
r
where   (j  1/ 2), and  E  1 ( 1) for fermions (antifermions).
4. Connection s among the Friedel Sum,
Levinson Theorem, and the Atiyah - Singer Index
Relation Between th e Friedel Sum, and Levinson Theorem

 g  D Z   
l 0
l
m
 N
R,L
R,L
0
where Z  {m} is the number of zero - modes
for a given angular momentum channel ( ,{m}).
The change of the number
of scattering states
Connection s among the Friedel Sum, Levinson Theorem,
and the Atiyah - Singer Index
index D
 A   Z R  Z L  N L  N R
total zero-modes of
right-handed chiral
fermion
The Euclidean Dirac Operator
Proof of the Friedel Sum Rule for Chiral Fermions
This green slate memorial is engraved
at Westminster Abbey in Cambridge, UK.
Proof of the Friedel Sum Rule for Chiral Fermions
Dirac Equation :
D

  a pˆ a  V  r    r   E   r 
 a 1

Dirac : the equation
govern most of physics
and the whole of chemistry.
The chiral representa tion of  a in (2d  1) space is given by
 a
 a  
0
0 
,
- a 
 2 m-1  I
  

3    3,

I   1  


m -1
d-m
 2 m  I
  

   3,

I   1  
3

m -1
 2d 1   3   3       3 .
d -m
The Dirac matrices satisfy th e algebra
 a  b   b a  2 ab , a, b  1,2,..., 2d  1,
 a2  I,
 a  0   0 a  0,
Here the chiral representa tions of  0 in D space is given by
 0 - I

 0  
- I 0 
Letting   ( R ,  L ) T with T the operation of transposi tion,
we obtain the chiral decomposit ion of Dirac equation
D

a 1
a
p̂ a  R r   E   Vr  R r ,
a
p̂ a  L r   E   Vr  L r ,
D

a 1
D
where   a p̂ a is the helicity opreator.
a 1
Because of the fact that chiral fermions or antifermio ns appear
only with a definite helicity, we must require that in the multicompo nent
descriptio n half of the components vanish. This is achieved by applying
the projection operator P  (1   5 ) / 2 to  , where
I 0 
 is chirality operator, such that R , L  (1   5 ) / 2.
 5  
0 - I
One can show that
D




p̂

V
r
 a a
 R, L r   E  R, L r .
 a 1

There R , L are the eigenfunct ions of the Hamiltonia n, the helicity operator,
and of  5 .
Due to the nature of parity violation of chiral fermions,
the angular momentum representation of eigenstates
for chiral fermions and antifermions contain spin spherical
harmonics  m . The angular momentum representation
can be found by the unitary matraix
1  I I
U

.
2  -I I 
It transforms the Dirac spinor in standard representation as
1
  m  r   d
r
 F  r   m 


 iG   r  - m 


to right - handed and left - handed chiral fermions
 R, m r   1   5 / 2U m
1 1
F r  m  iG  r - m ,

d
2r
 L, m  1 -  5 / 2U m 
1 1
- F r  m  iG  - m .
d
2r
The radial spinors F and G  satisfy th e equations
d

G   G   E  - Vr F ,
dr
r
(1)
d

 F  F  E  - Vr G  .
dr
r
(2)
where   (2l  D  1) / 2.
The asymptotic behaviors of normalized solutions
for E   0 and   0 are

F ,  r  r




cos kr    k  ,

2


1



G  ,  r  

sin kr    k  .
 
2

For E   0 and   0, the asymptotic solution are
r 

F ,  r  r
 -
G , 
1



cos kr    k  ,

2


1



r   sin kr -    k  .
 
2

r 
1
For the quantum number   0, from Eqs. (1) and (2),
the solutions for spinors
G  ,-  r    cF ,  r  ,
F ,-  r   cG  ,  r  .
Accordingl y, it can be proved that the states of chiral fermions
with κ  κ and κ   κ are not linearly independen t.
 R,   ic  R,  ,
 L,  ic  L, .
Friedel sum rule for right - handed chiral fermion
To prove the FSR, it is convenient to write the spin state
in a form

 R r    c  m R, m r 
l  0 m
The change of the number of states around the barrier is given by


0
EF

N R    '   dE  lim  d  R r  R' r  -  R0  r  R'0  r 
0
E  ' E 
 -E F

The second integral can be expressed in terms of the summation
of the change in the phase shifts
lim  d 
E ' -E 

R
 r   R'  r  - R
 0
E

 r   R'  r  

 0


l  0 m
d   E  
dE 
The FSR for right - handed chiral fermions is then found to be
N R 
1






'




g
D

E


E
 l
F
F - 2  0 


l 0
with   (2l  D  1) / 2 and
 D  l - 2

g l D   2 
l


being the degeneracy for each l which can be calculated from
d
the Weyl' s dimension formula for the spin representa tion
of SO(2d  1).
Friedel sum rule for left - handed chiral fermion
The spin state is taken as

 L r    c m L, m r 
l 0 m
The change of the number of states around a barrier is expressed as


0
EF

N L    '   dE   lim  d  L r  L' r  -  L0  r  L'0  r 
E  ' E 
 -E F 0 
The difference of the density of state is given by
E

0 
0 
lim  d  L r  L' r  -  L r  L' r  
E  E 
2
'

d 

  dE
l 0 m

This gives the Friedel sum rule for left - handed chiral fermions
N L 
where
1






'




g
D

E


E
 l
F
F - 2  0 


l 0
  (2l  D  1) / 2 and
 D  l - 2

g l D   2 
l


d
The degeneracy for each l.
The Levinson theorem for chiral fermions
The number of zero - mode for the definite quantum state ( ,{m})
can be counted by way of the FSR and the completene ss of relativist ic
states

 n  m r  n  m r'

zero - modes

0
EF



    '   dE     m r    m r'
-E F
0

l  0 m 
  r - r'
Subtractin g the relation from the completene ss of free - particle states,
 
setting r  r , taking trace, and integratin g over (2d  1) space, we find
the equality

ZR,L   g l  D  Z  m
l 0
 N R,L
R,L
The total number of discrete zero - modes is then given by
Z R, L 
g D 2  0 -   E  -   E  -   - E' 


1

l 0
l
F
F
F
The number of zero - mode for a definite channel
of angular momentum channel ( ,{m}) is found to be
Z  m 

2

1

0 -   E F  -   - E' F 
If the potential barrier do not allow any zero - modes therein, from FSR,
we find a duality relation of the phase shifts between th e high - energy
and low - energy limits
 g D  0   g D  E     - E' 

l0

l
l0
l
F
F
For system that only contains a kind of carrier,
the duality relation becomes


 g D  0   g D  E 
l
l 0

l
l0
F

 g D  0   g D  - E' 
l0
l
l 0
l
F
4. The connection amoung the Friedel sum rule,
Levinson t heorem, and Atiyah - Singer index
The Atiyah - Singer index can be expressed as
Index D
 A   n  A  - n  A  ,
where D
 A  is the Dirac operator, and
n  (n  ) is the number of zeromodes
with positive (negative) chirality.
Therefore, the connection s amoung the Friedel sum rule,
Levinson t heorem, and the Atiyah - Singer index
Index D
 ( A )  Z R  Z L  N L  N R
Particular ly, it is of interest for an interactio n
described by an arbitrary short - range central force.
index D
 short-range centralforce  0
5. Aharonov-Bohm Effect on the Friedel Oscillation
Aharonov-Bohm magnetic flux
V (r) 
1
r2
and
V (r)  0
for r  a
for r  a
Bound states therein
Radiusr
charged
particles
Phase shifts
The topological effect of a single flux on the Friedel oscillation of 2d electron gas.
It is remarkable to note that the quantum interferences of electrons with the AB effect
result in the amplification and reduction of the oscillation.
The topological effect of an AB magnetic flux on
the variances of charge density aroung an impurity
Nature, Vol. 407, 55 (2000).
Thank you!
研究生: 李琮懿
幫忙打方程式的powerpoint
The total difference of the number of states
(The Relativist ic Spin - 1/2 Friedel Sum Rule)
N 
1


  ,
 
0
2 |  |{[ ( E f )   (  )    ( Ef )    (  )]
 (1)| |
2
[sin 2  (  )  sin 2   (  )]}
Z.Q. Ma, and G.J. Ni, Phys. Rev. D31, 1482 (1985).
The relation w ith the relativist ic Levinson t heorem
n m j
(1)| |
 [ (  )   ( )]   
[sin 2  (  )  sin 2  ( )]

2
1
Z.Q. Ma, and G.J. Ni, Phys. Rev. D31, 1482 (1985).
The upper bounds of phases are ruled out by a relation   ()    ()  0.
However, the equality does not hold due to restrictio ns of Fermi surfaces.
The Levinson t heorem may be modifed by
n m j 
1

[ (  )   ( E f )   ( E f  )   (  )]
(1)| |
 
[sin 2   (  )  sin 2  (  )]
2
Comparing with the relativist ic Friedel sum, one finds the relation
between th e relativist ic Friedel sum and Levinson t heorem
N 


  ,  0
2 |  |n m j  0
A fundamenta l result based on the Dirac equation.
Density oscillatio n in relativist ic systems

N  4  r 2 dr[  (r )  0 (r )]
0
where (  -  0 )   is the difference of the density of states
 

k k f
d 3k
2
(0)
2
[|

(
r
)
|

|

(
r
)
|
]


3
(2 )
which leads to the relativist ic Friedel oscillatio n
At large distances, one finds
 
1
2
2
| |
k
dk
[(

1)


sin  sin(2kr   )]


E k
3 
2
 kr
  ,  0 m j  j (2 ) k  k f

j
which leads to the relativist ic spin - 1/2 Friedel oscillatio n
j

  k f 1
1
1
 1 E k
 
(

1)
cos[2
k
r


(
k
)]sin

(
k
)

O
(
)


f

f

f
3
3
4
(2 )  , 0 m j  j

r
r
研究生: 李琮懿
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