494
Progress of Theoretical Physics, Vol. 42, No. 3, September 1969
Theory of Hall Effect. I
--Nearly Free Electron-Hidetoshi FUKUY AMA, Hiromichi EBISA W A and Yasushi W ADA
(Received March 5, 1969)
A gauge-invariant expression of the Hall conductivity in a weak magnetic field is derived
on the basis of the Kubo formula for nearly free electron systems. In order to calculate
the correlation functions systematically, diagram techniques have been employed. It is shown
that the Hall coefficient is inversely proportional to the square of the density of states at the
Fermi energy, which justifies the conjectures of Mott and Ziman in connection with the
discussion of liquid metals.
§ 1.
Introduction
It is well known that the measurements of Hall effects are effective to know
the electronic properties of solids or liquids. For example the concept of impurity conduction has been introduced by the temperature dependence of the
Hall coefficient. 1l Recently, careful measurements have been performed. Greenfield2l measured the Hall coefficients of several liquid metals and found that
some of them have some deviations from the free-electron value. Alderson, Farrell and Hurd 3 l determined the temperature dependences of the coefficients for
solid Cu, Ag and Au. Their data showed a new behavior at low temperatures.
The absolute magnitudes of the coefficients had an apparent maximum as a
function of temperature. The Hall effect in antiferromagneti c chromium has
4
been observed by Amitin and Kovalevskaya. l The coefficient has a characteristic
temperature dependence. In order to understand these results the effects of
many-body interactions or random array of atoms should be taken into account
in each case. T'he purposes of the present series of papers are to develop a
formalism in which such effects can be systematically investigated and to apply
the discussion to a few typical examples.
A Hall effect is observed because the paths of charged particles are being
bent in a magnetic field due to the Lorentz force. Let us take the direction of
a uniform current with the density Jx as the x-axis. Suppose a uniform magnetic
field Hz is applied perpendicularly along the z-axis. One measures the induced
electromotive force along the y-axis, that is, the electric field Ey, suppressing
the current in the same direction. The Hall coefficient R is then defined by
R=Ey/HzJc.
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DejJartnzent of Physics, University of Tokyo, Tokyo
Theory of Hall Effect. I
495
When the magnetic field 1s weak enough, R is given by 5>
(1·1)
rJ f'V 1s the conductivity tensor which satisfies the relation
Jf'
=
.z= (J
f'VEv •
v
diagonal
so far. 6>
transport
with the
magnetic
Here v is the velocity of the carriers. The fields are assumed to be weak and
the relaxation time is introduced in order to consider the effect of collision term
in an approximate way. The induced current Jx is thus determined and the
Hall coefficient has turned out to be
R= 1/nqc,
(1·2) .
where n is the number density of the carriers. This result is remarkable at the ·
following two points. First it directly tells us the nature of the carriers, since
it has the same sign as their electric charge q. Furthermore we can easily find
their density n. Although the carriers are usually electrons, we can sometimes
discuss the conduction processes in some solids introducing the concept of "holes"
with positive charge. Thus the sign of the Hall coefiicient indicates which of
the two types of carriers is more effective in the processes. In these arguments,
the energy of each carrier is assumed to have a definite relationship with its
momentum. In other words, the system is well represented by an assembly of
quasiparticles. When the interactions become essential, the above assumptions
may no longer hold and we may expect a modification to the simple formula
for the Hall coefficient (I· 2). Let us refer to two typical examples to which
the simple theory is not applicable. One is the impurity conduction processes.
The other is the cases where many-body correlations are important. In order
to discuss these cases quantitatively, we consider how the quantum mechanical
density matrix varies owing to the weak electric and magnetic fields. The density matrix may be developed in a power series of these fields. When the
magnetic field is uniform, the vector potential becomes large in the regions far
from the relevant area. Therefore it is necessary to take into account carefully
the contributions from these regions. It is rather a clumsy procedure. Kubo7)
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We shall confine our discussion in this limit of the weak field. The
elements of the tensor r5.11 x and rJ YY have been extensively discussed
Therefore, our main concern is in the off-diagonal element (J xy·
The simple theory for the Hall effect makes use of the Boltzmann
equation. 5> The velocity distribution in a system of charge carriers
electric charge q is modified by the force F due to the electric and
fields, E and I-I,
496
H. Fukuyama, H. Ebisawa and Y. W ada
Using these results, we take the limit of long wave length for the vector
potential in the expression for the off-diagonal clement of the conductivity tensor
r5xy·
Finally, the frequency of the oscillating electric field is set to be zero.
The Hall coefficient R turns out to be modified from the expression (1· 2) by
the square of a ratio between the unrenormalized and renormalized density of
states at the Fermi energy. This renormalization is not the same with the mass
renormalization in the electronic specific heat. 15 ) The latter comes from the
energy dependence of the self-energy part whereas the former comes from the
momentum dependence. This result agrees with one of Ziman's conjectures 16 )
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pointed out a possibility to circumvent this problem by using the technique of
Wigner representations for the density matrix and the dynamical variables. His
result is characteristic in classifying the contributions to the "classical" terms
and the "quantum-mechanical" corrections. However, this classification turns out
to be not very convenient to discuss the correlation effects. 8l Meanwhile Evans 9l
and Springer10 ) investigated the Hall effect in a system with a static perturbing
potential. Their treatments depend strongly on the choice of the gauge of vector
potential, and the systematic treatment of the correlation effects looks rather
difficult.
In our formulation we first assume for the vector potential to have a form
of a plane wave in order to avoid unphysical contributions from distant regions.
An expression for the current density is given by the linear response theory
when the uniform electric field is weak enough. The electric field is first assumed
to oscillate with a frequency w. That is the so-called Kubo formula. 11 l This
expression is expanded as a power series of the magnetic field and we keep
only the linear term. The obtained result is rewritten in terms of the thermal
Green functions and the vertex parts.
In this article we shall consider a simple case, that is, a system composed
of nearly free electrons. The effects of the lattice periodicity will be neglected.
We know the electron self-energy parts fairly well for some particular cases.
For instance, in discussing the residual resistances, it is considered that the
impurity potential is of short range and that the electron mean free path is long
in comparison with the Fermi wave length. 6l Then the electron self-energy part
is obtained by the self-consistent treatments of the second order self-energy part.
The same conclusion holds for the electron-phonon system because of the large
difference betvveen electron velocity at the Fermi level and the velocity of sound. 12 )
Ziman's 13 ) treatment of liquid metals leads to the same situation. Therefore, it
is quite worthwhile investigating these simple cases first.
Since the vector potential can have an arbitrary gauge, our discussions have
to be performed in a gauge-invariant way. This condition requires that the approximations for the current vertex part have to be consistent with those for
the self-energy part. 14l It is easy to find the consistent approximations for the
vertex parts.
Theory of Hall Effect. I
497
for the Hall effect in liquid metals. The possibility that the Hall coefficient
should depend on the density of state has been pointed out by Mott. 17>
In § 2 the general expression for (J xy will be derived. It is estimated in
the static limit and the Hall coefficient R is given in § 3. Some discussions
will be given in § 4.
§ 2.
Hall conductivity
A (r) = Aqeiqr
(2 ·1)
Is given by 11 >
where
and (J)A-=2n'AT (A: integer, T: temperature). Tin the bracket is the time-ordering
operator. The operator fu(k) is defined by
where
(2 ·2)
and
(2 ·3)
The time evolution and the thermal average are determined by the Hamiltonian
1 /'.
!}{II= .!J( - -
c
J ( - q) Aq .
(2·4)
Equation (2 · 4) is sufficient as we are concerned with a magnetic field only up
to the linear order and H in this equation denotes all the energy operators m
the absence of a magnetic field. Expanding the expression for (J p,v in terms of
Aq, we have the following formula for off-diagonal component (!1 = x, v = y):
(2. 5)
where
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The conductivity tensor in the presence of a vector potential
H. Fukuyama, H. Ebisawa and Y. liVada
498
2
f{fi~(q, w) =--r:_--o"aCLfo(q, w+io) -Lfo(q, O+io))
me
+ l_(Lfi~(q, w+io) -Lfi~(q, O+io)),
(2·6)
c
r~dr i' dr' exp [iw).. (r- r')] <T Jfo (q, r) p(- q, r')) ,
8
Lfo (q, iw)..) = _l_
(3
iw)..) = l_
(3
(2 · 7)
Jo
1;3dr s(3 d r' ~~ dr" exp [iw).. (r- r'")]
Jo
o Jo
(2 ·8)
Hereafter the thermal average and time evolution of the operators are defined
by the Hamiltonian !J(.
Equations (2 · 6), (2 · 7) and (2 · 8) are exact but we can calculate these correlation function, LP and ~LJI~, only approximately making use of a particular nature
of the system. The self-consistency of such approximations for L,J, and Lfi~ is
necessary to ensure the gauge invariance of the results which we now discuss.
For the case of nearly free electron system it is convenient to take the plane
wave function as the basis. We introduce a one-particle thermal Green function as
g
(p, 8n) =
X
---~--
{3V
s
drdr'
r~dr r~dr' exp [ -ip (r- r') + ien(r -r') J
Jo
Jo
<T exp [ (!J(- pi\!) (r- r') J¢ (r) exp [ (!J(- 11N) (r'- r) J¢t (r')) ,
where Bn = (2n + 1) nT, J1 is the chemical potential and N the total number of
electrons. A self-consistent treatment of a second order perturbation diagram
for the self-energy of the one-particle Green function gives*)
(2·9)
and the Green function turns out to be
g (p, 8n) =[iBn- E p
-1,' (p,
8n) ]- 1,
where Bv = p2/2m- Jl. To be specific we discuss explicitly the case of scattering
by a static potential. Then
V(p-p',8n-8n') =V(p-p')On,n'.
(2 ·10)
The arguments are, however, applicable to the cases of dynamical interactions,
too. Equations (2 · 9) and (2 ·10) lead, through gauge iuvariance arguments, to
the ladder correction for the vertex function of the current operator: 14 )
*)
agator.
In the case of electron-phonon interaction, V(p- p', En -En') is essentially the phonon propIn the case of static potential it is in second order of the perturbing potential.
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Lfi~ (q,
Jo
499
Theory of Hall Effect. I
X J( p I+ ,en;
p 1- , en- )
f? (
::d
p 1- , Cn- )
,
(2 ·11)
where
XJa ( p - , Sn- ; p + , Bn- )
acp
5d
+,
en- )Jv ( p + , en- ; p + , en)
(2 ·12)
which are shown in Fig. 3. It can be proved that the gauge invariance is not
broken for a finite q (Appendix A). However, as we are concerned with a uniform
magnetic field, we first expand the contributions in Fig. 3 in terms of q and
~(P-P~)
Z(P,cnl=
~
P',
En
Fig. 1. Self-energy diagram for one-particle
propagator. A line with p' and En is
il(p',
En).
1
c
=
·#;~),+·
,_
~
lAp'+
P,cn.Q ,en
Fig. 2. Integral equation for current vertex
function,Jf'(p+, En; p-, En-). A dot with
epfbi m is a bare matrix element of current
operator.
Fig. 3.
•
En.~P'
p-~t'n
J.
(J-c
J"'
"
n
.
Contributions to K~v (q, iwA.).
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Equations (2 · 9), (2 ·10) and (2 ·11) are shown graphically In Figs. 1 and 2.
Under the present approximation, contributions to KfJ.~' Eq. (2 · 6), can be given
in the form
500
H. Fukuyama, H. Ebisawa and Y. W ada
2
-21 · me
-e · q.
'.!
(a
l
(b)
Fig. 4. Contributions from the expansion of propagators of the first diagram in Fig. 3.
In the remammg half of this section, we represent graphically the procedures
to obtain such an expression, while an analytic derivation is given in detail in
Appendix B. First, we consider the contributions coming from the expansion
of the momentum arguments of one-particle propagators. Those from L11~ just
cancel out (Appendix C) and there remain those from L"' which are shown in
Fis. 4. The line of propagator with a cross X p means (8 j8kP) g (k, en) =8 Pg (k, en),
and the small circle OJ"' denotes the vertex function with q = 0 in Eq. (2 ·11),
(2 ·13)
Next we evaluate the contributions from the q-linear term of the current vertex
functions. One of them, which couples with the magnetic field, Jcn satisfies
Eq. (2 ·11) with W>... = 0. In this case, it has the symmetry under the interchange
between p+ and p-, so that
Ja (p-, en; p+, en)
=
Ja (p, en)+ 0 (q 2) '
(2 ·14)
and moreover we can write
(2 ·15)
The identity (2 ·15) is discussed in Appendix A.
electric field, Jv, can be expanded as
The vertex coupled with the
(2 ·16)
We consider the quantity
J/
defined by
J"'(p+, en;p-, en_) =J"'(p, em (J)A.) +J/,
(2 ·17)
which satisfies an integral equation derived by the expansion of Eq. (2 ·11) in
terms of q, and is shown in Fig. 5, where J/ is represented by a square o.
By Eqs. (2 ·15), (2 ·16) and (2 ·17) the total contributions to K11~ are shown in
Fig. 6. In this figure, the current vertex with a cross X p means (8/8kp) J(k,
em(!)>...). The diagrams (d) and (e) in Fig. 6 can be rewritten as in Figs. 7
and 8 respectively in virtue of the integral equation for J/. The contributions
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retain only the linear term. In this case the gauge invariance can be ascertained
if we get the following q-dependence of K11~,
501
Theory of Hall Effect. I
of (a) in Fig. 4, (h) m Fig. 7 and (k) in Fig. 8. are gathered together to be
(I) in Fig. 9. Here Eq. (2 ·11) with q = 0 and the relation
e2
me'
OJ;
Integral equation for
+
g. [
8110(.
J~.
a(j oa
+
(e)
(d)
(c)
e
+-·
2C
qp·l
oa j
a(j
fJX
j)X
J'/1
( 9)
(f )
Fig. 6.
J
Expansion of current vertices of the triangular diagrams in Fig. 3.
=
(d)
:c . ~- [
i!f :0]
+!!_
c
Jv
(h)
atJ
(j )
( i)
Fig. 7.
(e
l
2~
•
qf' ·[
(j: -p() ]
Jv
( k)
(l)
Fig. 8.
+
e
c
e:
(m)
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Fig. 5.
H. Fukuyama, H. Ebisawa and Y. vVada
502
(I)
=
e
2C
qp . [
8a (~ V(p- p') _q (p', en)
p'
X
x Jv
Jv (p', em W;..) g (p', en_))
(2 ·18)
are used. Similarly from (b), (i)
and (1) and from (c), (j) and (m)
we obtain (2) and (3) in Fig. 9
respectively. Diagrams (o), (q)
and (s), as a whole, cancel with
(r) under the relation of Fig. 5
and we now have only four; (f),
(g), (n) and (p). From the symmetry of the isotropic system
qP·oP can be expressed as
(2}
e
2C
.qp·[-?
a
Xa
( n}
(o)
+
Jl)
Xa
~
(q)
a 0
. [
qp·op=o"aqp,op,+op,aq"o",
Jv
so that the sum of the diagrams
(f) and (p) yields the desired
gauge-invariant result. It is also
the case for (g) and (n). Finally we obtain,
q,
J
Xa
(P}
(31
J
Xa
Xa
( r)
(S}
j
Fig. 9.
111
the limit of small wave number
(2 ·19)
where
g=g(p, en),
g (-)
=
g (p, en_),
and each Oa operates only on the function next to this symbol. We have neglected
any effects of electron spin except a multiplicity factor 2 in Eq. (2 ·19). In an
isotropic system, J is proportional to (e/ m) p, so that it takes a form
(2. 20)
Using this expressiOn for J, we obtain
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~ ~y
(j ?0
503
Theory of Hall Effect. I
2
____, "L....J X2 ( p, ZSm
•
•
) P.'lJ
e 7 , ,L....J
rJ xy (W) -_ -WeZW"A
w
n
m
p
(2. 21)
§ 3.
Explicit evaluations
As far as we are concerned with a static response of a weakly perturbed
electron system which is in normal state, the electromagnetic properties of a
metal are characterized by the behavior of quasiparticles near the Fermi surface.
For such a system, the perturbational approach can be used, and the results of
the preceding section are applicable.
The diagonal element of the conductivity tensor rJ.7Jx IS giVen by
2
P
. ) -- - 2 e 2'T"
. ZSn. ) :dC( p, Sn)g( p, Sn- ) .
(/) xx ( tw"A
L....J "''.X
L....J ---2-X ( p, ZSm
n
p m
(3 ·1)
Equations (2 · 21) and (3 ·1) determine the Hall coefficient within the framework
of the present approximation.
Representing the summands in Eqs. (2 · 21) and (3 ·1) by 9! (isn, isn_) we.
perform n-summation over all integers by transforming it into an integral along
the contour C in Fig. 10. 18 )
(3·2)
c
where
f(z)
Fig. 10. Contours of integration in z-plane. Crosses are
on iEn = i (2n + 1) reT.
=
[e.Sz + 1]-I,
(3. 3)
and F(z) is a function of a complex argument z
defined by the analytic continuation from F(isn).
As the Green function G (p, z) has a cut along
Imz = 0 and the vertex correction X (z, z- iw"A) has
cuts where the imaginary part of each energy argument is zero, 18 ) the singularities of 9! (z, z- iuh) are
located along Im z = 0 and Im z = w"A. The contour
C is transformed to Cr + C 2 + C 3 + C4 along these cuts
as is shown in Fig. 10, though C 1 + C 4 give a negligibly small contribution in the present case. For
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where we= lellljmc is a cyclotron frequency. Although Eq. (2·21) is derived
in the case of static perturbing potential, this is a quite general expression as
far as the second order perturbation diagrams are self-consistently treated.
H. Fukuyama, H. Ebisawa and Y. W ada
504
C2 + Cs we make suitable transformations for integral variables and perform the
analytic continuation, iw>- ~w + io, then the RHS of Eq. (3 · 2) turns out to be
- ~ Joo dxf(x) [F(x + io,
2nz
x- w- io) - F(x + w + io, x- io)].
(3·4)
-oo
Confining ourselves to the static response, we need only w-linear term in the
bracket of Eq. (3 · 4), and a partial integration leads to
(3· 5)
Some words should be added about this result: Not only the diagonal but
also off-diagonal components of the conductivity tensor have the main contribution from the regions near the Fermi energy. The diagonal part is dissipative
and, correspondingly, it can easily be verified from Eq. (3 ·1) that it has a factor
- df/ dx in any case even if the contribution from the path cl + c4 is included.
But the off-diagonal one is dispersive and then the expression for oxv has a factor
f(x) if we consider C1 + C4. This is the characteristics of the effect of the
magnetic field. The contributions from the terms that include f(x) come from
the deviation of the distribution function of the system by the influence of the
magnetic field.
Thus, in the degenerate case the static conductivity tensor 1s g1ven by
Oxy=Wc~ ~
2m
P
PxX 2 (p) {GR(p, O)axGA(p, 0) -axGR(p, O)GA(p, 0)},
m
(3·7)
where GR and GA are retarded and advanced Green functions respectively (e.g.
GR(p, ien) =g (p, en)' en>O)'
GR(p, x)
=
[x-sp-l'' (p, x) +il'" (p, x)]- 1 •
(3·8)
Here 2' and .I;" are real and imaginary parts of the self-energy function respectively (l'">O). X (p) is defined by
(3·9)
If 2" is much smaller than the chemical potential, the Green function may
be considered as a function of Lorentzian type regarding cp or x. Expanding
.I;' (p, x) in terms of cp and x and neglecting the cp and x dependence of .I:",
we obtain
(3 ·10)
where
(3 ·11)
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- iw JC<J dx(- df(x)_)FCx+io, x-io).
2rr -oo
dx
Theory of Hall Effect. I
505
(3 ·12)
and Po is the Fermi momentum.
EP is a solution of
The velocity v (p) is defined by dEp/dp and
N(O) = _ _!__
TC
~ Im GR(p, 0) =N(O)b,
(3 ·13)
P
where N(O) is the density of states at the Fermi energy of a free electron
system.
Inserting Eq. (3 ·10) into Eqs. (3 · 6) and (3 · 7) and considering that the
function X de_r:;ends weakly on the momentum variable, we obtain
(3 ·14)
(3 ·15)
where n is the number density of electrons n = p 0 3 /3rc 2 • The transport relaxation
time !"tr is X(Po) /21:", which coincides with the usual definition. The reason why
b does not appear in (J xy is that the current vertex Ja coupled with a magnetic
field has a character different from other vertices, so that it is expressed, owing
to a "generalized Ward's identity" ,1 4> as a derivative of the Green function with
respect to momentum variable.
Then we get the Hall coefficient in nearly free electron systems as
R-
(jxy
_
Hz(J;:x-
1 (N(0))
nee
Nco)
2
(3 ·16)
The mass renormalization, which is important for an electronic specific heat,
appear neither in (J xx 19> nor in (J xy, so that if the self-energy does not have a
momentum dependence as in the case of electron-phonon interactions, we obtain the
same results for (J xJ; and rJ xy (and also R) as those given by the Boltzmann
transport equation. The momentum dependence of self-energy, however, affects
only (J xx, so that the Hall coefficient is inversely proportional to the square of
the density of states N(O) at the Fermi energy.
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The quantities a and b are energy renormalization factors reflecting the energyand momentum-dependent interactions between electrons or electrons and external
systems. For example the electron-phonon interactions enhance the electronic
specific heat at low temperatures by a factor a- 1 • 15 > On the other hand b is the
renormalization factor of the density of states in the sense that
506
H. Fukuyama, I-I. Ebisawa and Y. lVada
§ 4.
Conclusion
Acknowledgements
The authors wish to express their hearty thanks to Professor R. Kubo for
some useful advice. Thanks are also due to Mr. K. Kawamura for helpful discussions and to other colleagues. Comments by Professor M. Tanaka are gratefully acknowledged. The authors thank to Professor T. Matsubara for sending
a manuscript prior to publication.
Appendix A
The gauge invariance is required by the fact that Jf' should not vary by
the gauge transformation of the vector potential as
(A·1)
What we are to show 1s that
KfJ.~
in the present approximation satisfies
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We have derived an expression for the off-diagonal component of the conductivity tensor as Eq. (2 · 21) in the presence of a magnetic field to its linear order
by use of the Green function. The derivation has been performed for the free
electron system with weak perturbations. The explicit calculation shows that
even in such cases the Hall coefficient differs from the value given by the Boltzmann transport theory, when the interactions are momentum-dependent. The
present result, Eq. (3 ·16), applies to the liquid metals if we follow Ziman. 16 )
In liquid metals this dependence of the self-energy function on the momentum
variable comes from that of both the pseudopotential and the structure factor.
Ziman has argued that the Hall coefficient should depend on the density of states,
and that some evidences exist as regards this point. Liquid mercury shows the
free electron value for the Hall coefficient in spite of the fact that other properties indicate a strong modification in the density of states. 20 ) This may mean
that Ziman's approach might not be sufficient for Hg and that the current vertex
part might be varied from ep/m. On the other hand, the experiments on the
Knight shife 1) and on the Hall coefficiene 2) of liquid Hg-In alloys can consistently
be interpreted, if we are allowed to treat these as nearly free electron systems.
The qualitative feature that R is larger in the system with smaller density of
states at the Fermi energy is manifested in liquid semiconductors, too. 23 )
Matsubara and Kaneyoshi 24) discussed the Hall effect in impurity conduction
processes under the tight-binding approximation. They obtained a result different from ours. It is due to the different definition of density of states.
The temperature dependence of R in the noble metals cannot be understood
within the present discussions. However, it looks fairly straightforward to take
into account the possible effects of phonon drags along the line developed in
this work.
507
Theory of Hall Effect. I
~ K11~qa=O,
(A·2)
a=.J;,y,z
so that the results are gauge-invariant. We no\v treat this problem similarly as
the arguments of gauge invariance in the many-body theory of superconductivity
by Nambu/ 4) and first introduce a "generalized Ward's identity" (GWI) in the
special case of zero external frequency
(A·3)
From (A·3) we obtain
which leads immediately to the identity (~ ·15).
Using the GWI, (A· 3), we obtain from Fig. 3
where abbreviations J"'(p) and Jv(p+) represent J"(p+, en;p-, en_) and Jv(p+,
en-; p+, en). Substituting Eq. (2 ·11) for Jv (p+, en-; p+, en) into the second term
of (A· 5), we obtain
!!__y~ ~ Jl'(p)fl(p+, en) r~fYv++ L: V(p-p')fl(p'+, en)Jv(p'+)
np
C
p'
111-
(A·7)
and into the first term of (A· 6),
- _!!__ T
C
~ ~ J" (p) _q (p+,
n
en) [_!!__ Pv- + ~ V (p- p') f) (p'-, en) Jv (p'-)
P
111-
P'
X fJ (p'-,
J
en_) fJ (p-, en_) .
(A· 8)
The first terms of (A· 7) and (A· 8) are cancelled by (A· 4), because
e
+
e
_
e
-p - - p =-q.
m
m
m
(A·9)
By the similar procedures, the first term of (A· 5) and the second of (A· 6)
lead to
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which IS valid for Eqs. (2·9)and (2·11) with (J)~=O.
in the limit q->0
H. Fukuyama, H. Ebisawa and Y. W ada
508
X
g (p+, en) J" (p+) g
(p+, en_)
+£TL: ~[£p'"'+ L: V(p-p')g(p'-,
c
np?n
(A ·10)
en_)J'"'(p')Q(p'+, en)]
p'
(A ·11)
Appendix B
The procedures of deriving Eq. (2 ·19) from Eq. (2 ·12) are as follows.
First the contributions coming from the expansion of propagators is given by
1 e2
- --O"aT~
2
?nc
n
~ qp ~{f)pg(p, en)J#(p, em {J)")g(p, en_)
p
(a)
p
as is shown in Fig. 4. Those from the q-linear term of matrix elements of
current operator, J/, are
(c)
and
(d)
(e)
The expansion of current vertex J" yields
J/
satisfies the following integral equation,
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We can see that the second terms of (A ·10) and (A ·11) cancel with the second
terms of (A· 7) and (A· 8) respectively, and that the first terms of (A ·10) and
(A ·11) together lead to the one which vanishes by the isotropy of the system.
Thus all the terms vanish.
Theory of Hall Effect. I
509
(B·l)
which is shown in Fig. 5.
2::
V (p- p') g (p', en) J/ g (p', en_) } ,
p'
(j)
and (e) can be written as follows,
(m)
The sum of the diagrams (h) and (k) includes
Oa [Q (p, Cn-) J" (jJ, em W>c) g (p, en) J
- g (p, en_) OaJ" (p, em W>c) g (p, en).
Partial integration over p and the equation
transform the sum of the contribution (a), (h) and (k) as follows:
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X
By this equation, the term (d) can be rewritten as
H. Fuhuyama, H. Ebisawa and Y. 1Vada
510
Similarly, from (b), (i) and (1) and from (c), (j) and (m) we get
_l_
!!_____
C
T
~ ~ qP ~ g
n
P
(p, en) J'" (p, en,
()),J aPg (p, en_) aaJv (p, em uh)
(p)
P
and
(r)
(s)
respectively. The expressions from (n) to (s) are sho,vn in Fig. 9. Among these
contributions, (o), (q), (s) cancel with (r) as is easily verified by use of Eq.
(B ·1). Remaining four terms (£), (g), (n) and (p), gives the final expression,
Eq. (2·19).
Appendix C
We show here that we have no q-linear contribution to L11~ from the expansion of the Green functions in Fig. 3. There are apparently three pairs of
contributions.
(1) Expanding g (p+, en) of the second and g (p-, en_) of the third diagrams
in Fig. 3 and using (2 ·14), we obtain the contribution as shown in Fig. 11,
which turns out to vanish by the symmetry of the isotropic system.
(2) Expansion of the remaining two Green functions in the second diagram jn
Fig. 3 just cancel out.
(3) Similarly we obtain no contribution from the two Green function of the
last diagram in Fig. 3.
2~.
qp" [
aVp _pOaJ
Jll
Fig. 11.
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2
Theory of 1-Iall Effect. I
511
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