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An electron in a random potential: A representation of the one-particle green function
Volume 92A, number 7
PHYSICS LETTERS
29 November 1982
AN ELECTRON IN A RANDOM POTENTIAL:. A REPRESENTATION OF THE ONE-PARTICLE GREEN FUNCTION
Klaus ZIEGLER1
Institut fl~rTheoretisehe Physik, Universitilt Heidelberg, Heidelberg, Fed. Rep. Germany Received 9 July 1982 Revised manuscript received 30 August 1982
A new representation of the average one-particle Green function of the n-orbital model with local unitary symmetry (PIE) is constructed by means of a nonlinear transformation.
The behavior of an electron in a random potential can be described in a Green function formalism. The evaluation of the average Green functions is, in most of the studied models, only possible in a perturbative sense. Adirect method is an expansion of the Green function with a special potential which is followed by an averaging of the expansion series over all the possible potentials. However, the characteristic features of the random system can only be obtained by partial summations of the averaged expansion terms, e.g. by the average t-matrix approximation (ATA), by the coherent potential approximation (CPA) [1,2], or by the large n limit [3]. This inconvenience has been overcome by several authors by introducing generating functionals for the Green functions which can be averaged [4,5]. Now the averaged generating functionals
ensemble (PIE) [3], which is governed by the tight binding hamiltonian ,~
H=
~
~—1’2
)f
Ir pi
~‘
~
‘I
(r’
(1)
~P
,p,,’p’
~
(r is the coordination number of a lattice site,p = 1, n is an orbital number) with the probability distribution ...,
/
P[f] =N’exp(—
~E (w~Y’ E
~ r,r’ (f is hermitean: frp r’p’
=
If~~ r’p’12)
(2)
p,p’ f~~’p’ rp’ w1 is symmetric),
the diagonal average one-particle Green function 1
G
1 (r, r;z) = (r, p I (z H) fr, p> (Im z * 0) can be represented as a functional:
(3)
—
can be evaluated by standard arguments of the perturbation theory, as the mean field theory of renormalization group methods [4,5]. But there is still an inconvenience in the representation of the generating functional. The averaging is only practicable if its partition function is independent of the random potential. The latter can be achieved by the replica trick [6] or by introducing of anticommuting vanables [7]. It has been shown in a recent paper [8] that in the case of the n-orbital model of a phase invariant Supported by the Deutsche Forschungsgemeiiischaft through the Sonderforschungsbereich 123.
0 031-91 63/82/0000—0000/$02.75 © 1982 North-Holland
G1 (r, r;z) =
—
E WrP~(QIr’~
(4)
‘
F’
(...)
x
=
f
[Q1,Q2]
C7~
... det [n(w
—
D(z))/2ir]
exp [—L(z)}
(5)
with the lagrangian n L(z)~
2
/
wrr~Q~rQnr~+n ~ ~ 1 r, r’
F
+~
l~ ~1,r “
~2, r (6)
339
Volume 92A, number 7
PHYSICS LETTERS
and the matrix Drr’(z)
~~rr’/(Z +Q 1 ,r )(z + iQ2 ,r ‘)
=
(7)
.
The Qi,r and the Q2,r integrations are along the real axis, A nonlinear transformation of the variables is defined by 2(Q ~ a,r cs,r a,r + ~a AFJ~ Q Q’ n~/ = 1/2 c 2 = —i/2 ‘.
29 November 1982
The advantage of this representation for the average one-particle Green function in comparison with the other representations used lies in its simplicity and compactness. There are no “artificial” things like zero times replicated variables or anticommuting variables. On the other hand all the well-known results of the model with translational invariance can be reproduced from (1) by iteration, for example, the n = oo limit (mean-field theory)as, [3,51, the 1/n-expansion .
—~
,
Ar
2
=
~W~)(Qir’
—
~Q2,r’Y’
lii
F
~ + Q1 ,~ + jfl “~ ~2,r’
)
,
(8)
if the Q2,r integration is chosen such that the integration path does not contain the singularity of the loganithm at Q2,r = iz. The lagrangian is quadratic in these variables: L
E Wrr’Q~rQ~s
(9)
p
=
[9], and the scaling law for large values of n [8]. In a subsequent paper it will be shown, starting from this representation, that a critical dimensiond~ = 6 exists, above which the infrared singularities of the 1/n-expansion at the n = band edges does not occur. This is different from the Lagrange model for the two-particle Green function where d~= 8 [5]. Moreover,there is also hope that one can obtain a better insight into other approximation methods from the representation (14).
a r,r’
To the transformation belongs the jacobian J: det F
=
References
det1~~-\
[1] J.M. Ziman, Models ofdisorder (Cambridge Univ. Press,
\8Q1 r
/
‘~-i—1detI-~-(w—D(z))I r 1—wII .
L det(-\21T /J
J
(10)
L21T
Thus the Q-dependent determinant in the functional (5) is removed by the nonlinear transformation (8): C..>
=
~)f
det(~_.
c~[Q~,Q~} exp(—L),
o1
...
1 r by
solving 12Q~,~~ —
340
n~
.
~
43 (1979) 744.
[9] R. Oppermann and F. Wegner, Z. Phys. B34 (1979) 327.
and it remains the evaluation of the function Q
=
Cambridge, 1979). R.J. Elliott, J.A. Krummhansl and P.L. Leath, Rev. Mod. Phys.46(1974)465. F. Wegner, Phys. Rev. B19 (1979) 783. L. Schafer and F. Wegner, Z. Phys. B39 (1980) 281. A.B. Harris and T.C. Lubensky, Phys. Rev. B23 (1981) 2640. [6] S.F. Edwards, Proc. Phys. Soc. 85 (1965) 613.
[2] [3] [4] [5]
(12)
PHYSICS LETTERS
29 November 1982
AN ELECTRON IN A RANDOM POTENTIAL:. A REPRESENTATION OF THE ONE-PARTICLE GREEN FUNCTION
Klaus ZIEGLER1
Institut fl~rTheoretisehe Physik, Universitilt Heidelberg, Heidelberg, Fed. Rep. Germany Received 9 July 1982 Revised manuscript received 30 August 1982
A new representation of the average one-particle Green function of the n-orbital model with local unitary symmetry (PIE) is constructed by means of a nonlinear transformation.
The behavior of an electron in a random potential can be described in a Green function formalism. The evaluation of the average Green functions is, in most of the studied models, only possible in a perturbative sense. Adirect method is an expansion of the Green function with a special potential which is followed by an averaging of the expansion series over all the possible potentials. However, the characteristic features of the random system can only be obtained by partial summations of the averaged expansion terms, e.g. by the average t-matrix approximation (ATA), by the coherent potential approximation (CPA) [1,2], or by the large n limit [3]. This inconvenience has been overcome by several authors by introducing generating functionals for the Green functions which can be averaged [4,5]. Now the averaged generating functionals
ensemble (PIE) [3], which is governed by the tight binding hamiltonian ,~
H=
~
~—1’2
)f
Ir pi
~‘
~
‘I
(r’
(1)
~P
,p,,’p’
~
(r is the coordination number of a lattice site,p = 1, n is an orbital number) with the probability distribution ...,
/
P[f] =N’exp(—
~E (w~Y’ E
~ r,r’ (f is hermitean: frp r’p’
=
If~~ r’p’12)
(2)
p,p’ f~~’p’ rp’ w1 is symmetric),
the diagonal average one-particle Green function 1
G
1 (r, r;z) = (r, p I (z H) fr, p> (Im z * 0) can be represented as a functional:
(3)
—
can be evaluated by standard arguments of the perturbation theory, as the mean field theory of renormalization group methods [4,5]. But there is still an inconvenience in the representation of the generating functional. The averaging is only practicable if its partition function is independent of the random potential. The latter can be achieved by the replica trick [6] or by introducing of anticommuting vanables [7]. It has been shown in a recent paper [8] that in the case of the n-orbital model of a phase invariant Supported by the Deutsche Forschungsgemeiiischaft through the Sonderforschungsbereich 123.
0 031-91 63/82/0000—0000/$02.75 © 1982 North-Holland
G1 (r, r;z) =
—
E WrP~(QIr’~
(4)
‘
F’
(...)
x
=
f
[Q1,Q2]
C7~
... det [n(w
—
D(z))/2ir]
exp [—L(z)}
(5)
with the lagrangian n L(z)~
2
/
wrr~Q~rQnr~+n ~ ~ 1 r, r’
F
+~
l~ ~1,r “
~2, r (6)
339
Volume 92A, number 7
PHYSICS LETTERS
and the matrix Drr’(z)
~~rr’/(Z +Q 1 ,r )(z + iQ2 ,r ‘)
=
(7)
.
The Qi,r and the Q2,r integrations are along the real axis, A nonlinear transformation of the variables is defined by 2(Q ~ a,r cs,r a,r + ~a AFJ~ Q Q’ n~/ = 1/2 c 2 = —i/2 ‘.
29 November 1982
The advantage of this representation for the average one-particle Green function in comparison with the other representations used lies in its simplicity and compactness. There are no “artificial” things like zero times replicated variables or anticommuting variables. On the other hand all the well-known results of the model with translational invariance can be reproduced from (1) by iteration, for example, the n = oo limit (mean-field theory)as, [3,51, the 1/n-expansion .
—~
,
Ar
2
=
~W~)(Qir’
—
~Q2,r’Y’
lii
F
~ + Q1 ,~ + jfl “~ ~2,r’
)
,
(8)
if the Q2,r integration is chosen such that the integration path does not contain the singularity of the loganithm at Q2,r = iz. The lagrangian is quadratic in these variables: L
E Wrr’Q~rQ~s
(9)
p
=
[9], and the scaling law for large values of n [8]. In a subsequent paper it will be shown, starting from this representation, that a critical dimensiond~ = 6 exists, above which the infrared singularities of the 1/n-expansion at the n = band edges does not occur. This is different from the Lagrange model for the two-particle Green function where d~= 8 [5]. Moreover,there is also hope that one can obtain a better insight into other approximation methods from the representation (14).
a r,r’
To the transformation belongs the jacobian J: det F
=
References
det1~~-\
[1] J.M. Ziman, Models ofdisorder (Cambridge Univ. Press,
\8Q1 r
/
‘~-i—1detI-~-(w—D(z))I r 1—wII .
L det(-\21T /J
J
(10)
L21T
Thus the Q-dependent determinant in the functional (5) is removed by the nonlinear transformation (8): C..>
=
~)f
det(~_.
c~[Q~,Q~} exp(—L),
o1
...
1 r by
solving 12Q~,~~ —
340
n~
.
~
43 (1979) 744.
[9] R. Oppermann and F. Wegner, Z. Phys. B34 (1979) 327.
and it remains the evaluation of the function Q
=
Cambridge, 1979). R.J. Elliott, J.A. Krummhansl and P.L. Leath, Rev. Mod. Phys.46(1974)465. F. Wegner, Phys. Rev. B19 (1979) 783. L. Schafer and F. Wegner, Z. Phys. B39 (1980) 281. A.B. Harris and T.C. Lubensky, Phys. Rev. B23 (1981) 2640. [6] S.F. Edwards, Proc. Phys. Soc. 85 (1965) 613.
[2] [3] [4] [5]
(12)