- Email: [email protected]
On the non-commutative two level model for metallic glasses
217—220. Solid State Communications, Vol.35, pp. ‘W~p~J Pergamon Press Ltd. 1980. Printed in Great Britain.
ON THE NON—COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES A. Zawadowski Central Research Institute for Physics Budapest 1525 P.O.B. 49, 1{UNCARY* and Department of Physics University of Illinois at Urbana—Champaign Urbana, IL 61801, USA** and K. Vladar Central Research Institute for Physics Budapest 1525 P.O.B. 49, HUNGARY (Received April 14, 1980 by E. Burstein) The electronic lifetime is calculated for a metallic glass considering a general interaction between electrons and the two level system. It is shown that in the leading logarithmic approximation logarithmic terms occur only if at least two of the coupling parameters do not commute in the momentum space. The first contribution is proportional to the square of the logarithm which results in an increase of the decay rate as the temperature is lowered.
1.
Introduction
of a non—commutative model,
Recently, the possibility that the two level model1’2 characteristic for the low temperature behavior of glasses could be applied to metallic glasses as well has at— tracted a great deal of interest.3 Kondo~’5 has proposed a physical model, where the two level system of the glassy state is symmetric, He pointed out that the potential due to the atom jumping between two positions is screened by the conduction electrons.5 The screening builds up like in the X—ray absorption problem. Thus this problem had to have some similarity to a subsequent series of repeated X—ray ab— sorption processes. Furthermore, he has also shown that this problem is not necessarily so simple like the X—ray absorption problem. In general there exist two different models (i) where the angular distribution of electrons participating in the screening are not redistributed due to the jump of the atom, (ii) where this redistribution takes place. These models will be called commutative and non—commutative models in the following. The second case shows a strong resemblance to the magnetic impurity Rondo problem, where the scattering of an electron on the impurity may result in the change of its internal degree of freedom, thus in its spin. In the present case the role of electron spin is taken over by the angular distribution of the orbital momentum of the electrons. The electron lifetime or the electrical resistivity has been calculated by Kondo5 and 6 In both cases a log2 term appears the fourth order of perturby Gyorffy andinBlack. bation theory. The main differences are (i) Rondo obtained this result only in the case
(ii) the sign of
this term are different at these authors. The aim of the present work is to clarify that point by treating a completely general model. The calculation is carried Out in framework of leading logarithmic approximation. Instead of a brute force calculation of dia— grains the scaling equations valid in leading logarithmic orders are constructed. By systematic iterations of these equations the required results are obtained in a very concise form which makes the general discussion possible. In the treatment presented here it is assumed that the parameters of two level systems like splitting and hopping are small compared with the temperature or energy con— sidered. It is important to emphasize that all of the equations are written in such a form that they are generally valid for the invariant coup— ling or for the renormalized vertexes. These are the same in the leading logarithmic approxi— mation. The results in perturbation theory are obtained by replacing the vertex functions by their bare values in the final expressions. One gets the perturbation series to a given order in this way. 2. Hamiltonian The two positions of the atom can be des— cribed by a psuedospin and the corresponding Pauli operators are denoted by o~(i~’x,y,z). The first part of the Hamiltonian H + H can be written as 1 (1) A0
+
A lx
z
H 0
=
~ Lkak ak
—
T ~
+
T
o
(2)
where A2onal is the to energy the tunneling splitting amplitude and A1 e is~ between the two states and the first part proport
* Permanent address ~* Supported in part by NSF Grant No. DMR78—2lO68
217
218
NON-COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES
corresponds to the conduction electrons using the usual notations. As the spins of the con— duction electrons are not essential in these
spin space, thus V~k, = 0 remains valid. In the following we use the representation where A 1 = 0.
calculations, therefore they will not be labelled. The general form of the interaction between the two state system and the conduction electrons can be written as
ill
=
~ (ak,+ Vk,kak)cr i=x,y,z k,k’
-
=
—
3. Logarithmic Approximation The scattering processes can be depicted in form of time ordered diagrams as shown in Fig. 1. The time flow represents the horizontal dotted line and the solid lines corre— spondto electrons. The contribution of the two basic diagrams of the logarithmic approximation shown in Fig. 1 can be easily calculated if one assumes that every a is large compared with A0
(3)
p 0 log
D
dS k + -~ -÷ -÷ ~_(Vk,~o)(V~ka)
2ip log
~ j—j-
J
~dS~ -~---—-
=
exp(-A{p (x)})(l
-
J
D -1--T
+ p log
(V~,1~ Vkk)E
J
dS k -* -~ -~ -~~_(V~ko)(Vk,~o)
ijk k a
=
(5)
where a is the incoming electron energy, po is the conduction electron density of states for one spin direction near the Fermi level, D is the conduction electron band width cut—off, 1o~=ic a dS is the surface element of the Ferm~jP~ere In expression the finally, integrals operpendicular ofktotal area S(~) and, to the Fermi sphere are performed, thus Iki = Ik’I=kF where kF is the radius of the Fermi sphere. The double indices like i,j,k are summed up. In the following, the scaling due to the energy a~ is considered. In the leading loga— rithinic approximation it can easily be derived
where V~k, is the difference in electron scattering amplitude considering the two x positions of the atom and Vk~, and V~k, des— cribe the electron assisted unneling of the atom. The momentum dependence of this scat— the Hamiltonian is Hermitian the identity terming amplitudes play an important role. As I * I (Vkk,) = Vk,k holds. The most simple model for VXk, and Vkk, is that the tunneling amplitude ~t the atom is influenced by the conduction electron density p(x) near the atomic site, thus the tunneling amplitude is a functional of p(x) exp(-Xfp(x)})
Vol. 35, No. 3
~p(x)) + O((~p)2)
~
(4)
where p (x) is the density in the equilibrium and ~p(~) stands for the deviation from that. system ~A/6p(x) is also real. Thus ~ Choosing real wave functions for the two can have imaginary part only due to the ex— i(k—k’)x + ponential factor in p(x)=klk,e akiak. In the case of a symmetric two level system the complex parts cancel out in expression (4) if the center of the coordinate system and of the two level system coincide. Thus the imaginary part of exp(—A{p(x)}) is due to the asymetry of the two level system and contains a small parameter as the ratio of the barrier width and of the lattice constant. Furthermore, it is an odd function in the variable k—k’. As identity (Vkk,)X
=
Vk,k generally holds, any contri—
Fig.1.
bution to V which is even function of the variable k—k’ must be real and any odd one is pure imaginary. Thus the term considered here can be incorporated into ~x, Therefore, in this model one can assume for the initial couplings that V~, = o. In the conventional treatment of the two level system the parameter A 1 is eliminated
using Anderson’s “poor man’s derivation”7 or by more sophisticated renormalization group approaches.8 Thus the elimination of the phase space due to the reduction in the cut—off D can be compensated by changing the couplings as —
(a)
I
_________
L3bogjaI k
from the Hamiltonian H0 by introducing a new linear combination of the two wave functions of the two level system. As this transforma— tion is a rotation around the y—axis in the
dS
oj SF =
k
i 1 ijs (Vk,~(w)V~k(w))r
(6)
2ip
where tOe couplings are the invariant couplings or the vertex function and they depend on aID for the scaled system. This scaling equation
Vol. 35, No. 3
NON-COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES
is basically equivalent to the summation of the
important to emphasize that this result does
9 Looking for so—called “parquet” diagrams. cross—sections a ~imple consequence of Eq. (6)
not mean the lack of higher order logarithmic terms. Namely, the scalin~ equation (6) shows that the coupling V and V generate a loga—
will be useful
~.1 1 dS
dS
,
(7)J
31V ~I2 ~log~wI
=
219
+ 4ichl5p 0Tr(Vi(a)Vi(a)V5(w))
where the trace is interpreted as a momentum surface integral fdSw/Sy along the Fermi sur— face which is normalFzeB by the total area of the Fermi surface.
to V” in the second order of the perturbation theory. As it will be shown in fourth order of the coupling this con— tribution results in a logarithmic square term
According to the “golden rule” the life— time r of an electron with momentum k is
in the inverse lifetime. In order to get that
Tk’(a)
=
2ap
J
~
rithmic contribution
V~,~,(a)1
(8)
F
where the summation corresponds to the three scattering channels in Hamiltonian given by (3). This equation can be averaged over the direction of the incoming electron
~J
~__~j~__i~_IVi (d13)12 ,
s
Dual
F
F
(7) is differentiated with respect to log~a~and the scaling equation (6) is used to calculate the right hand side of Eq. (7). The straightforward calculation leads to the following result
2
fdS r dS ~ ~j_.J~j _Ji~.
Our aim is to get the log
the equation
dS
i=x,y,z
r~1(w) = 2iip
result
~
2
Vk ,(a)(
3log
k
1u1
2
dependence of
[Fk~
=
2 i —24 p (~r(Vi~~~iVi) — Tr(V1V~V~V~
=
12 P~
~
~kJ
IFkk~(w)l2
5F
=
(10)
where the notation
(a)
=
J~-~
(V~(a)V~k,(a)- V~(a)V~k,(a))
ijx the inverse lifetime, where a is the energy of the electron. If lal
and the identity (Fkk,
ij =
_Fk,
have been
used. It is important to emphas~zethat the quantity calculated is positive and it is zero
pressions can be obtained, only a~must be replaced by T. (lal or T>A 0 are assumed.) This dependence can be obtained by substituting the solution of the Eq. (6) for the vertex into Eq. (9). It is, however, extremely difficult to solve Eq. (6) in general, thus one should use perturbation theory or, in other words, that equation must be iterated. The simple logarithmic contribution to the right hand side of Eq. (9) must be proportional to the 2r~j i— —l f dSk r(a) = right hand side of Eq. (7), where the bare couplings are inserted. If one of the three couplings is zero then this expression vanishes, In this way, we have shown that in the model investigated the inverse lifetime does not contain linear logarithmic terms because for the bare coupling V~k, 0 has been assumed. This general result agrees with Kondo’s calculation for his particular model.5 It is
)
(11)
only if all of the couplings commute. In the case where two couplings e.g. VX and vZ are not commuting that expression is positive definite. In order to calculate the fourth order contribution to the inverse lifetime the golden rule given by Eq. (9) is combined with Eq. (10) and then one obtains
jI
—~--—-
dSk,
(
Va,, + 6plog i 2 2 2
i-:iD
ij Fkk, ) ij 2
(12)
in the leading logarithmic approximation, where in the right hand side the vertexes are replaced by the bare couplings in order to get perturba— tion expansion. This result shows that the decay rate in— creases as the electron energy tends to the Fermi energy, a ~ 0. For completeness the ex— pression for the transport relaxation time will be given. The definition of the transport
220
NON—COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES
relaxation time is
~
J~—j
dSkI dSk,
T_l()
=
tr
2ap
,(w) 2(1
—
- 1(a) Ttr
=
2up o
dS
dS
(
5F k
SF k
[~kk~kk)
J __~I_~ J dS
2 2 + 2plog
u D
(iii)
The log
(13)
~
where 4kk’ is the angle between k and k’. It is an important feature of the calculation that there are two contributions to VkkH2, (i) each coupling is calculated in the second order in the perturbation theory, (ii) one in third and one in first order. The result can not be written in such a compact form as in Eq. (12), but rather in a form corresponding to the first expression in Eq. (10) ______
0.
cos4kk,)
Vol. 35, No. 3
2Jpja~ term has always a
J1k3 S~
~p~itive coefficient in T and very likely in —l Tth’ The statement (i) does not hold in the next to the leading logarithmic approximation, but these terms cancel out in the resistivity.10 Rondo5 has calculated resistivity for a realis— tic model and his results for that model are in agreement with
(i), (ii) and (iii).
The calcu—
+
dS j
SF —~-
1 V3 V~ V3 (2(1— k [V 1k2 k2k3 k3k4 k4k1 cos4kk)
—V~ V3 V3 V1 2 _cos4kk) k1k2 k2k3 k3k4 k4k1~
+ (l_cos4kk))]}
there is an invariant combination of the couplings, thus, if the initial value of the coupling is small then the fixed point value is also small. This is the situation in the X—ray absorption problem and in the commutative case of the present problem.6’10 For the other cases the following possibilities remains: the fixed point is zero or not small which can be large or infinite. In the case of arbitrary k— dependence it is hard to solve the scaling equation. There is, however, a simple model which can be treated11 and that treatment shows that at least that model scales into the strong coupling region or according to Anderson’s general argument7 rather into the infinite strong coupling region. The magnetic field like term characterized by the parameter A may stop that scaling on the way. Finally? it is important to note that the results obtained are valid in two and three dimensions as well. The main results of the present work are as follows: (i) At least two of the V1 , matrixes must not commute to get logari~3~mic contribution in the leading logarithmic approxi— mation. (In realistic physical models VX and do not commute.) (ii) Linear logarithmic term does not appear in the inverse lifetime if
+
(l_cos4kk))
—
(14)
lation of Black and GyHrffyt contradicts with (1) but agrees with (ii). In the non—commutative case the leading logarithmic approximation provides logarithmic terms and scaling exists. If there is no small parameter in a problem like in the case of 4—c expansion for phase transition then the fixed point, reached by scaling may belong to differ— ent classes. The first typical case is, where Having studied the first terms of the per— turbation theory, we can draw the conclusion that in the non—commutative model the angular distributions of the momenta of the scattered electrons keep some memory of the subsequent order of the different scattering processes, therefore, the problem i~somewhat similar to the Rondo problem, where this memory is built in the spin distribution of the conduction electron. Acknowledgment——The author expresses his grati— tude to K. Fisher, W. Götze, B. Gyorffy and J. S~lyom for valuable discussions and to D. Pines for his hospitality in the Physics Department at the University of Illinois, where this work was completed.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
P. W. ANDERSON, B. I. HALPERIN and C. M. VARNA. Philosophical Magazine 25, 1 (1972). W. A. PHILLIPS. Journal of Low Temperature Physics L~351 (1972). See for a review J. L. BLACK. To appear in Metallic Glasses edited by H. J. Giintherodt, Springer—Verlag, N.Y. J. RONDO. Physics 84B, 40 (1976). J. KONDO. Physics 84B, 707 (1976). J. L. BLACK and B. L.GY~RFFY. Physical Review Letters 41, 1555 (1978). P. W. ANDERSON. Journal of Physics C 3, 2346 (1970). M. FOWLER and A. ZAWADOWSKI. Solid State Communications 9, 471 (1971). A. A. ABRIKOSOV. Physics 2, 5 (1965). A. ZAWADOWSKI and K. VLADAR. To be published. A. ZAWADOWSKI. To be published.
ON THE NON—COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES A. Zawadowski Central Research Institute for Physics Budapest 1525 P.O.B. 49, 1{UNCARY* and Department of Physics University of Illinois at Urbana—Champaign Urbana, IL 61801, USA** and K. Vladar Central Research Institute for Physics Budapest 1525 P.O.B. 49, HUNGARY (Received April 14, 1980 by E. Burstein) The electronic lifetime is calculated for a metallic glass considering a general interaction between electrons and the two level system. It is shown that in the leading logarithmic approximation logarithmic terms occur only if at least two of the coupling parameters do not commute in the momentum space. The first contribution is proportional to the square of the logarithm which results in an increase of the decay rate as the temperature is lowered.
1.
Introduction
of a non—commutative model,
Recently, the possibility that the two level model1’2 characteristic for the low temperature behavior of glasses could be applied to metallic glasses as well has at— tracted a great deal of interest.3 Kondo~’5 has proposed a physical model, where the two level system of the glassy state is symmetric, He pointed out that the potential due to the atom jumping between two positions is screened by the conduction electrons.5 The screening builds up like in the X—ray absorption problem. Thus this problem had to have some similarity to a subsequent series of repeated X—ray ab— sorption processes. Furthermore, he has also shown that this problem is not necessarily so simple like the X—ray absorption problem. In general there exist two different models (i) where the angular distribution of electrons participating in the screening are not redistributed due to the jump of the atom, (ii) where this redistribution takes place. These models will be called commutative and non—commutative models in the following. The second case shows a strong resemblance to the magnetic impurity Rondo problem, where the scattering of an electron on the impurity may result in the change of its internal degree of freedom, thus in its spin. In the present case the role of electron spin is taken over by the angular distribution of the orbital momentum of the electrons. The electron lifetime or the electrical resistivity has been calculated by Kondo5 and 6 In both cases a log2 term appears the fourth order of perturby Gyorffy andinBlack. bation theory. The main differences are (i) Rondo obtained this result only in the case
(ii) the sign of
this term are different at these authors. The aim of the present work is to clarify that point by treating a completely general model. The calculation is carried Out in framework of leading logarithmic approximation. Instead of a brute force calculation of dia— grains the scaling equations valid in leading logarithmic orders are constructed. By systematic iterations of these equations the required results are obtained in a very concise form which makes the general discussion possible. In the treatment presented here it is assumed that the parameters of two level systems like splitting and hopping are small compared with the temperature or energy con— sidered. It is important to emphasize that all of the equations are written in such a form that they are generally valid for the invariant coup— ling or for the renormalized vertexes. These are the same in the leading logarithmic approxi— mation. The results in perturbation theory are obtained by replacing the vertex functions by their bare values in the final expressions. One gets the perturbation series to a given order in this way. 2. Hamiltonian The two positions of the atom can be des— cribed by a psuedospin and the corresponding Pauli operators are denoted by o~(i~’x,y,z). The first part of the Hamiltonian H + H can be written as 1 (1) A0
+
A lx
z
H 0
=
~ Lkak ak
—
T ~
+
T
o
(2)
where A2onal is the to energy the tunneling splitting amplitude and A1 e is~ between the two states and the first part proport
* Permanent address ~* Supported in part by NSF Grant No. DMR78—2lO68
217
218
NON-COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES
corresponds to the conduction electrons using the usual notations. As the spins of the con— duction electrons are not essential in these
spin space, thus V~k, = 0 remains valid. In the following we use the representation where A 1 = 0.
calculations, therefore they will not be labelled. The general form of the interaction between the two state system and the conduction electrons can be written as
ill
=
~ (ak,+ Vk,kak)cr i=x,y,z k,k’
-
=
—
3. Logarithmic Approximation The scattering processes can be depicted in form of time ordered diagrams as shown in Fig. 1. The time flow represents the horizontal dotted line and the solid lines corre— spondto electrons. The contribution of the two basic diagrams of the logarithmic approximation shown in Fig. 1 can be easily calculated if one assumes that every a is large compared with A0
(3)
p 0 log
D
dS k + -~ -÷ -÷ ~_(Vk,~o)(V~ka)
2ip log
~ j—j-
J
~dS~ -~---—-
=
exp(-A{p (x)})(l
-
J
D -1--T
+ p log
(V~,1~ Vkk)E
J
dS k -* -~ -~ -~~_(V~ko)(Vk,~o)
ijk k a
=
(5)
where a is the incoming electron energy, po is the conduction electron density of states for one spin direction near the Fermi level, D is the conduction electron band width cut—off, 1o~=ic a dS is the surface element of the Ferm~jP~ere In expression the finally, integrals operpendicular ofktotal area S(~) and, to the Fermi sphere are performed, thus Iki = Ik’I=kF where kF is the radius of the Fermi sphere. The double indices like i,j,k are summed up. In the following, the scaling due to the energy a~ is considered. In the leading loga— rithinic approximation it can easily be derived
where V~k, is the difference in electron scattering amplitude considering the two x positions of the atom and Vk~, and V~k, des— cribe the electron assisted unneling of the atom. The momentum dependence of this scat— the Hamiltonian is Hermitian the identity terming amplitudes play an important role. As I * I (Vkk,) = Vk,k holds. The most simple model for VXk, and Vkk, is that the tunneling amplitude ~t the atom is influenced by the conduction electron density p(x) near the atomic site, thus the tunneling amplitude is a functional of p(x) exp(-Xfp(x)})
Vol. 35, No. 3
~p(x)) + O((~p)2)
~
(4)
where p (x) is the density in the equilibrium and ~p(~) stands for the deviation from that. system ~A/6p(x) is also real. Thus ~ Choosing real wave functions for the two can have imaginary part only due to the ex— i(k—k’)x + ponential factor in p(x)=klk,e akiak. In the case of a symmetric two level system the complex parts cancel out in expression (4) if the center of the coordinate system and of the two level system coincide. Thus the imaginary part of exp(—A{p(x)}) is due to the asymetry of the two level system and contains a small parameter as the ratio of the barrier width and of the lattice constant. Furthermore, it is an odd function in the variable k—k’. As identity (Vkk,)X
=
Vk,k generally holds, any contri—
Fig.1.
bution to V which is even function of the variable k—k’ must be real and any odd one is pure imaginary. Thus the term considered here can be incorporated into ~x, Therefore, in this model one can assume for the initial couplings that V~, = o. In the conventional treatment of the two level system the parameter A 1 is eliminated
using Anderson’s “poor man’s derivation”7 or by more sophisticated renormalization group approaches.8 Thus the elimination of the phase space due to the reduction in the cut—off D can be compensated by changing the couplings as —
(a)
I
_________
L3bogjaI k
from the Hamiltonian H0 by introducing a new linear combination of the two wave functions of the two level system. As this transforma— tion is a rotation around the y—axis in the
dS
oj SF =
k
i 1 ijs (Vk,~(w)V~k(w))r
(6)
2ip
where tOe couplings are the invariant couplings or the vertex function and they depend on aID for the scaled system. This scaling equation
Vol. 35, No. 3
NON-COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES
is basically equivalent to the summation of the
important to emphasize that this result does
9 Looking for so—called “parquet” diagrams. cross—sections a ~imple consequence of Eq. (6)
not mean the lack of higher order logarithmic terms. Namely, the scalin~ equation (6) shows that the coupling V and V generate a loga—
will be useful
~.1 1 dS
dS
,
(7)J
31V ~I2 ~log~wI
=
219
+ 4ichl5p 0Tr(Vi(a)Vi(a)V5(w))
where the trace is interpreted as a momentum surface integral fdSw/Sy along the Fermi sur— face which is normalFzeB by the total area of the Fermi surface.
to V” in the second order of the perturbation theory. As it will be shown in fourth order of the coupling this con— tribution results in a logarithmic square term
According to the “golden rule” the life— time r of an electron with momentum k is
in the inverse lifetime. In order to get that
Tk’(a)
=
2ap
J
~
rithmic contribution
V~,~,(a)1
(8)
F
where the summation corresponds to the three scattering channels in Hamiltonian given by (3). This equation can be averaged over the direction of the incoming electron
~J
~__~j~__i~_IVi (d13)12 ,
s
Dual
F
F
(7) is differentiated with respect to log~a~and the scaling equation (6) is used to calculate the right hand side of Eq. (7). The straightforward calculation leads to the following result
2
fdS r dS ~ ~j_.J~j _Ji~.
Our aim is to get the log
the equation
dS
i=x,y,z
r~1(w) = 2iip
result
~
2
Vk ,(a)(
3log
k
1u1
2
dependence of
[Fk~
=
2 i —24 p (~r(Vi~~~iVi) — Tr(V1V~V~V~
=
12 P~
~
~kJ
IFkk~(w)l2
5F
=
(10)
where the notation
(a)
=
J~-~
(V~(a)V~k,(a)- V~(a)V~k,(a))
ijx the inverse lifetime, where a is the energy of the electron. If lal
and the identity (Fkk,
ij =
_Fk,
have been
used. It is important to emphas~zethat the quantity calculated is positive and it is zero
pressions can be obtained, only a~must be replaced by T. (lal or T>A 0 are assumed.) This dependence can be obtained by substituting the solution of the Eq. (6) for the vertex into Eq. (9). It is, however, extremely difficult to solve Eq. (6) in general, thus one should use perturbation theory or, in other words, that equation must be iterated. The simple logarithmic contribution to the right hand side of Eq. (9) must be proportional to the 2r~j i— —l f dSk r(a) = right hand side of Eq. (7), where the bare couplings are inserted. If one of the three couplings is zero then this expression vanishes, In this way, we have shown that in the model investigated the inverse lifetime does not contain linear logarithmic terms because for the bare coupling V~k, 0 has been assumed. This general result agrees with Kondo’s calculation for his particular model.5 It is
)
(11)
only if all of the couplings commute. In the case where two couplings e.g. VX and vZ are not commuting that expression is positive definite. In order to calculate the fourth order contribution to the inverse lifetime the golden rule given by Eq. (9) is combined with Eq. (10) and then one obtains
jI
—~--—-
dSk,
(
Va,, + 6plog i 2 2 2
i-:iD
ij Fkk, ) ij 2
(12)
in the leading logarithmic approximation, where in the right hand side the vertexes are replaced by the bare couplings in order to get perturba— tion expansion. This result shows that the decay rate in— creases as the electron energy tends to the Fermi energy, a ~ 0. For completeness the ex— pression for the transport relaxation time will be given. The definition of the transport
220
NON—COMMUTATIVE TWO LEVEL MODEL FOR METALLIC GLASSES
relaxation time is
~
J~—j
dSkI dSk,
T_l()
=
tr
2ap
,(w) 2(1
—
- 1(a) Ttr
=
2up o
dS
dS
(
5F k
SF k
[~kk~kk)
J __~I_~ J dS
2 2 + 2plog
u D
(iii)
The log
(13)
~
where 4kk’ is the angle between k and k’. It is an important feature of the calculation that there are two contributions to VkkH2, (i) each coupling is calculated in the second order in the perturbation theory, (ii) one in third and one in first order. The result can not be written in such a compact form as in Eq. (12), but rather in a form corresponding to the first expression in Eq. (10) ______
0.
cos4kk,)
Vol. 35, No. 3
2Jpja~ term has always a
J1k3 S~
~p~itive coefficient in T and very likely in —l Tth’ The statement (i) does not hold in the next to the leading logarithmic approximation, but these terms cancel out in the resistivity.10 Rondo5 has calculated resistivity for a realis— tic model and his results for that model are in agreement with
(i), (ii) and (iii).
The calcu—
+
dS j
SF —~-
1 V3 V~ V3 (2(1— k [V 1k2 k2k3 k3k4 k4k1 cos4kk)
—V~ V3 V3 V1 2 _cos4kk) k1k2 k2k3 k3k4 k4k1~
+ (l_cos4kk))]}
there is an invariant combination of the couplings, thus, if the initial value of the coupling is small then the fixed point value is also small. This is the situation in the X—ray absorption problem and in the commutative case of the present problem.6’10 For the other cases the following possibilities remains: the fixed point is zero or not small which can be large or infinite. In the case of arbitrary k— dependence it is hard to solve the scaling equation. There is, however, a simple model which can be treated11 and that treatment shows that at least that model scales into the strong coupling region or according to Anderson’s general argument7 rather into the infinite strong coupling region. The magnetic field like term characterized by the parameter A may stop that scaling on the way. Finally? it is important to note that the results obtained are valid in two and three dimensions as well. The main results of the present work are as follows: (i) At least two of the V1 , matrixes must not commute to get logari~3~mic contribution in the leading logarithmic approxi— mation. (In realistic physical models VX and do not commute.) (ii) Linear logarithmic term does not appear in the inverse lifetime if
+
(l_cos4kk))
—
(14)
lation of Black and GyHrffyt contradicts with (1) but agrees with (ii). In the non—commutative case the leading logarithmic approximation provides logarithmic terms and scaling exists. If there is no small parameter in a problem like in the case of 4—c expansion for phase transition then the fixed point, reached by scaling may belong to differ— ent classes. The first typical case is, where Having studied the first terms of the per— turbation theory, we can draw the conclusion that in the non—commutative model the angular distributions of the momenta of the scattered electrons keep some memory of the subsequent order of the different scattering processes, therefore, the problem i~somewhat similar to the Rondo problem, where this memory is built in the spin distribution of the conduction electron. Acknowledgment——The author expresses his grati— tude to K. Fisher, W. Götze, B. Gyorffy and J. S~lyom for valuable discussions and to D. Pines for his hospitality in the Physics Department at the University of Illinois, where this work was completed.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
P. W. ANDERSON, B. I. HALPERIN and C. M. VARNA. Philosophical Magazine 25, 1 (1972). W. A. PHILLIPS. Journal of Low Temperature Physics L~351 (1972). See for a review J. L. BLACK. To appear in Metallic Glasses edited by H. J. Giintherodt, Springer—Verlag, N.Y. J. RONDO. Physics 84B, 40 (1976). J. KONDO. Physics 84B, 707 (1976). J. L. BLACK and B. L.GY~RFFY. Physical Review Letters 41, 1555 (1978). P. W. ANDERSON. Journal of Physics C 3, 2346 (1970). M. FOWLER and A. ZAWADOWSKI. Solid State Communications 9, 471 (1971). A. A. ABRIKOSOV. Physics 2, 5 (1965). A. ZAWADOWSKI and K. VLADAR. To be published. A. ZAWADOWSKI. To be published.