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Fluctuations of the relaxation time in one-dimensional systems
Solidi State Communications, Vol. 47, No. 7, pp. 549-553, 1983. Printed in Great Britain.
0038-1098/83 $3.00 + .00 Pergamon Press Ltd.
FLUCTUATIONS OF THE RELAXATION TIME IN ONE-DIMENSIONAL SYSTEMS L.V. Chebotarev Kharkov State University, 310 077 Kharkov, U.S.S.R.
(Received 14 January 1983 by E.A. Kaner) The direct interaction of electrons with impurity field is represented as interelectronic interaction of a special kind which is called "dissipative". This type of interaction is responsible for relaxation process and generally leads to fluctuations of the relaxation time. The probability distribution function for the mean free time of electrons is found in the white-noise approximation for a one-dimensional conductor. It is shown that the dielectric response function can be obtained in this case by averaging relevant expression over the fluctuations of the relaxation time. responsible for those fluctuations. In this paper we show that the relaxation time of electrons really fluctuates and finds the probability distribution function for the relaxation time in the white-noise approximation.
1. INTRODUCTION THE RELAXATION PROCESS in three-dimensional electronic systems is known to be well described by the conventional kinetic equation, provided pol >> 1 (Po is the Fermi momentum and l is the mean free path of electrons). Owing to the existence of the small parameter (pol) -1, in a three-dimensional system all the diagrams of perturbation series which contain intersections of impurity lines are found to be small in terms of this parameter [1 ]. However, in one-dimensional systems the small parameter (pol2f 1 does not appear even ifpol2 >> 1 [2], where l~ is the mean free path of an electron with respect to the backscattering. For this reason, investigation of one-dimensional systems is bound up with the necessity to sum all diagrams of the perturbation series [2, 3]. Physically, the situation is illustrated by the fact that Anderson's localization in a three-dimensional conductor appears only at sufficiently large amplitudes of a random potential, i.e. provided pol <~ 1, whereas in a one-dimensional conductor all the electronic states near the Fermi level are localized even ifpol2 >> 1. In the present paper the distinctive features of one-dimensional systems are explained by a considerable increase of fluctuations of a peculiar kind, which we call "dissipative" and which give rise to fluctuations of the relaxation time and of other dissipative characteristics of electrons. In a three-dimensional system the dissipative fluctuations are present as well, but they are small with pol >> 1 and become appreciable only in highly disordered systems. The reduction of the dimension of a system leads to a growth of all fluctuations in it, including the dissipative ones. The growth of the dissipative fluctuations causes the small parameter (pol) -1 to vanish thus making important a large amount of diagrams conraining intersections of impurity lines, which are
2. THE HAMILTONIAN OF THE "DISSIPATIVE" INTERACTION Consider a system of electrons in a metal which contains randomly disposed impurities. Any physical variable characterizing the electron system can be expressed in terms of electronic Green functions, while its observed value can be obtained by averaging this expression over the impurity configurations. Such averaged values can be calculated if one knows the generating functional for the Green functions, which is averaged over the impurity configurations. The Hamiltonian for the electrons in the random potential field
F(x) = ~ u(x--xa),
(1)
a
which is created by impurities has the form
H = f dx~÷(x)HotP(x)+ f dxF(x)~÷(x)~(x),
(2)
where x is the spatial coordinate of a point in a space of relevant dimension, x a is the coordinate of an impurity atom, ~k÷(x) and ¢(x) are, respectively, the Fermi operators of creation and annihilation of an electron at the point x, Ho is the Hamiltonian of the kinetic energy of the electron. We assume the electrons to be noninteracting. The generating function G(A) for the set of electronic Green's functions, averaged over the impurity configurations, may be written in the form:
549
550 G(A)=
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
const f Dq~ D~+ exp{i f dxdt[i~+(x,t)~(x,t)
--l f dxldtlf
sms(¢+, ~) =
-- t~÷(x, t)no~(X, t) + ~k+(x, t)A(x, t) + A+(x, t)~(x,
t)] +
iSdis(~k+, ~b)}
Vol. 47, No. 7
dx2dt2$+(xl, tl)
X ~b(Xl, tx)iVdis(Xl --X2)~+(X2, t2) x ¢(x2, t2),
(3)
(6)
where the effective potential of binary interaction
1
tSdis(qJ+, if) = -- i In W(-- ~0+ff)
iVdis(Xl --X2)
X ~(X, t)} -- 1) L.
(4)
Here W(-- ~+~) is the characteristic functional for the random field F(x), taken at the "point" (-- ~÷~), the subscript L at the brackets in this formula stands for the linked average of the expression in the brackets calculated with respect to the fluctuations of the random field F(x ). The functional Sais(~+; ~) represents an addition to the action function for the electronic system which describes an additional interaction between the electrons due to the randomly disposed impurities. Such an interaction is known to arise formally in the quantum theory and sometimes it can be described by impurity lines in the diagram techniques [1, 4]. We shall call this type of interaction "dissipative", as it is responsible for the relaxation process in the electronic system. The Hamiltonian Hdi~ of the dissipative interaction, related to S~s(~O+, ~O),is generally non-Hermitian and describes multiparticle interactions of all orders between the electrons rather than only pairwise interaction between them. In case of uniformly and independently distributed impurities the functional W is well known and
=
--ini f dxu(x, --x)u(x2 --x),
(7)
is purely imaginary. The first-order term of the expansion is related to a correction for the chemical potential and therefore it has been omitted. The functional (6) describes the pairwise interaction between electrons, which has been represented in the diagrams techniques in [1] and in [4]. Now we consider a one-dimensional conductor. As is known, in the one-dimensional case the constriction of the Fermi surface to two isolated points in the momentum space lays emphasis on them even in comparison with the Fermi surface in a three-dimensional metal. Therefore, the main part in the interaction functional Sdis belongs to the terms which are due to the vicinities of the Fermi points + Po; the rest of Sdis may be taken as a background for the interaction of electrons. Let a÷(p) and a(p) be the creation and annihilation operators of electrons with the momentum p. In the usual way, we may introduce new field operators to describe the electronic states near the right (ct = 1) and the left (a = 2) Fermi points:
~(x)
=
1
~
~eikXa~(k),
Ikl ~ P o .
(8)
Here al(k) = a ( p 0 + k),a2(k) = a(--po + k). Then the main part of Sdis may be expressed in terms of $~(x) as - - ¢ , 0 ) + .~,(2) follows: Sdis - Oint ~ i n t ' X'O) = ---1 ~int 2
;
dxxdtl
j dx2dt2[qJ+(xa,q)~(Xl,
q)]
× Ux(xx -x2)[ff*(x2, t2)ff(x2, t2)];
The diagrams technique for such multiparticle interaction functional, with the Green functions expanded in powers of concentration ni, is constructed as the "bunch diagrams method", which has been developed in [5], and leads to the same results. On the other hand, when the scattering of electrons by impurities is weak and the potential u(x --x') is small, the diagrams technique may be based on the expansion of the Green functions in powers of u(x -- x ') itself. Then we obtain with an accuracy to the secondorder:
,¢(2) = ~int
__lfdxxdtlf dx2dt2[~+(xl, tl)a+~(x~,tl)] 2 × U2(xl
--x2)[~k÷(x2, t2)a¢(x2, t2)]
I
f dx,dtl f dx2dt2[~+(xx ' t,)a~(xl, 2 X Uf(xx --x2)[~0+(x2, t2)a+¢(x2, t2)].
tl)] (9)
a, a t are two-row matrices: a
~
,
~
lo) ,
(10)
Vol. 47, No. 7
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
and the parentheses in equation (9) embracing two field operators designate the trace with respect to the "spinor" variables only. The functions UI,2 are
l y
UI'2(x) = L q eiqXUl'2(q)'
Ul(q) = -- &ilu(q)12;
[ql~Po,
(11)
U2(q) = --inilu(2po + q)l 2, (12)
and u(q) is the Fourier transform of u(x --xa). The symbol T stands for the transposition procedure. The operator ~O) ° i n t in equation (9) describes the electronic interaction due to the forward-scattering and the operator Si(n2~describes the interaction due to the backscattering of electrons by impurities. The generating functional for Green functions of 1-d electrons now may be constructed as follows:
G(A) = const f Dff D~ + exp {i(~b+Kff) + iSint(~+~//) + i~+A + iA+~}.
Here
(13)
a
K = iat +ivoaax Sint(~ ÷, 4) = ½pU~p + ½p~÷)U2p~-~ + ½p~->U?p<÷), (14) p = (qJ+~O); 0 (+) = (~0÷a+¢); p(-) = (qJ+aq0.
(15)
All the products in equation (13) and in what follows have the matrix sense, v is the Fermi velocity, o3 = a + a - - a a +.
The next important step to transform G(A) is to apply to the interaction functional Sint(~b÷, ~b) a procedure which might be called "extraction of the stochastic square root" from the integral operations U1,2. The procedure consists in representing exp (iSint) in the form of continual integral over new Bose variables ~/(x), ~'(x) and ~'+(x) exp (iSint(ff +, ~b)) : const f DO f Dr D~"+ x exp [~ r~Ullr/+ i~+U;l~ + i~?(~+~) + i~+(~/+a~) + i~(~+a+~) I ,
(16)
where U~.~ are the operations inversed with respect 1 to U~,~. To define these operations as one-valued ones, it is convenient to determine the integration space in equation (16) by imposition of the periodicity conditions on the border of the system: r/( L ) =
r/(+L);
~.(L)
= ~'(+2)"
(17)
551
The field r~(x) may be taken as real, whereas the field ~'(x) must be complex to ensure the true structure of the functional (13). That the field ~'(x) is one of the Bose type follows from the fact that the two terms in equation (14) containing/-]2 have the same sign. After substituting the expression (16)into equation (13) we get in the exponent the functional to be bilinear with respect to the Fermi fields: G(A)=
constf D¢ D¢+f DnD~D~+exp {i(¢+K6) i + i~'(ff+a+~k) + i~b+A + iA+~bI .
(18)
In this formula the direct dissipative interaction between electrons is represented as a result of their interaction with the new "quantized" Bose-fields 77 and ~'. The excitations of these fields bear the "quanta" of dissipative characteristics of the system rather than those of its dynamic values, as do the usual quantized fields. The fact that the generating functional for Green functions of the electronic system, which defines all the features of its dynamics, has been expressed in the form (1 8), being so much close to the generating functionals in the quantum fields theory, has far going consequences. Namely, the conclusion is to be drawn that fluctuations are intrinsic to the nature of relaxation processes as much as they are intrinsic to the nature of quantum motion. In other words, the values characterizing the irreversible process (the relaxation time etc.) are not something rigidly set for the system under consideration. On the contrary, those values are produced in the very course of the relaxation process under the influence of fluctuations of a special kind, i.e. dissipative fluctuations, which have an origin to be common with the genesis of the irreversibility itself. The effect of dissipative fluctuations on the formation of dissipative values is reflected in the fact that mathematical expressions for such values always include an averaging over configurations of a random field. It should be mentioned that the dissipative fluctuations have no direct connection with the dimension of the system. Such fluctuations exist in one-dimensional systems as well as in three-dimensional ones. The distinction of one-dimensional systems lies in the fact that the dissipative fluctuations in a three-dimensional conductor are mostly small, provided (pol) >> 1. It may be shown that the formation of the relaxation time in this case is affected especially by the binary correlations of a random field, whereas the higher order correlations give corrections to be proportional to (pol) -1. This fact
552
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
manifests itself in the diagram techniques as smallness of all diagrams which include intersections of impurity lines [l ]. On the contrary, in a one-dimensional system the dissipative fluctuations become much increased. The anomalous growth of fluctuations in general is a typical feature of one-dimensional systems. In this case, not only the binary correlations of a random field contribute to the formation of the relaxation time, but so do also the higher order correlations to the effect of vanishing of the small parameter (pol) -1. Correspondingly, increases the number of important diagrams with intersections of impurity lines, which are responsible for those fluctuations. 3. THE DIELECTRIC RESPONSE FUNCTION FOR A ONE-DIMENSIONAL CONDUCTOR IN A STEADY FIELD Now we shall show that the concept of the fluctuations of the relaxation time is by no means a quite formal one. To shed light upon this fact, let us consider a one-dimensional conductor in an applied electric field varying in space and time. The dissipative fluctuations are found to be large in the low-frequency limit oo72 ~ 1, where r2 is the mean free time of electron with respect to the back scattering. In this case Anderson's localization takes place and a one-dimensional electronic system is an insulator rather than a conductor. Taking advantage of the generating functional (18), we can obtain the following expression for the dielectric response function of the system:
e(co, K) = 1 + (%,r2 cosot)Z[Q(00, K) + Q ( 0 0 , - K)]. (19) Here 00p is the plasma frequency of electrons, a is the angle between the high conductivity direction (z) and the wavevector q of an applied field, 00 and K are, respectively, the dimensionless frequency and dimensionless component of q along the direction z. In order to obtain dimensional units one has to put 00 ~ 00r2, K + qzl2. Only the conduction electrons have been taken into account to obtain the formula (1 9). Neglecting a small delocalization of electronic wavefunctions, the function Q(00, K) is found in the white-noise approximation to be quite the same as it was defined by Berezinsky [3]:
Q(00, ~) = 2 J dxW(x)Ko(ZV~).
We shall try the solution of this equation in the form of the Kontorovich-Lebedev transformation: 1 W(x) =
f dlaKiu(2X/x)q~(t~),
(22)
0
which satisfies proper boundary conditions, W(x) being integrable at x ~ + 0 and decreasing at x ~ + oo. After substituting this expression into equation (21), we get the equation oo
f dtaKiu(Zx/x)[v(la) o
i(00 -- .K)]~b(/a) = 2x/~Ko(2X/~), (23)
where u(g) = (1 +/a2)/4. The solution of this integral equation is
2_
e~
U Sh (7r/1)
( dx
Kiu( 2X/x)Ko(2X/~). (24)
Making use of the formulas (20), (22) and (24), we obtain n2 oo
Sh n/.t2
1
Q(00, K) = -~ f dlala 7rlav(#)--i(00--K) . o ch a - 2
(25)
Hence, v(kt) is in fact the collision frequency of electrons and ~ is the corresponding "quantum number". The expression (25) means averaging the typical energy denominator (v -- i00 + iK)-1 over the fluctuations of v, which has the probability distribution function (going back to dimensional units)
w(v) = r?r=O(vr= - ~) Sh rr(vr= - ~[)1/2 ch 3 rr(vr2 ~)i/2 , -
-
(26)
where O(x) = 0 at x < 0 and O(x) = 1 at x > 0. It is easily seen that w(v) is non-negative and duly normalized. It follows now from equations (19) and (25) that the expression for the dielectric response function in a steady field (w = 0) is oo
e(K) = 1 + (copr2 coso02 f dv 00(v) v2 2v+~:2.
(27)
In a uniform field K = 0, and we obtain e(0) = 1 + 4~'(3)(00pr2 cos a) 2,
(28)
(20)
0
Here Ko(z) is the Macdonald function, and W(x) is to be found by solving the differential equation I-- x 2 -d- ~ -- 2x d + ] W(x) -- i(00 -- K)W(x) dx 2 G = 2Ko(2x/-x).
Vol. 47, No. 7
(21)
where ~'(x) is the ~'-function of Riemann. This expression is in full agreement with the known result [2, 6]. It is worthwhile to make note of the fact that the operator in the brackets in the 1.h.s. of equation (21) may be called "collision frequency operator" in a sense to be very close to the quantum mechanical one. Indeed, the eigenvalues of the operator give possible values of
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
Vol( 47, No. 7
the collision frequency u, while the formula (22) corresponds to an expansion of IV(x) into the complete set of orthogonal and normalized eigenfunctions of this operator: Cu(x) =
f
Sh n#]
Kiu(2V%),
'
o
,
in compliance with conventional expansions in the quantum mechanics as well. So we are led once more to the fact that the dissipative fluctuations show resemblance to the quantum ones [8].
(29)
REFERENCES 1.
d~t~u(x)~u(x ) = 6 ( x - - x ); (30)
2.
dx ¢u(x)Ov(x) = 8(la -- v).
3.
0
The first relation (30) has been proved in [7], and the second one is to be found in physical literature and may be proved as well using the Nicholson formula known in the theory of Bessel's functions. The Green function of the equation (21) proves to be: G ( x , x ' ) = _( du 0
gPt~(x)cku(x') v(la) -- i(¢o -- K) '
(31)
553
4. 5. 6. 7. 8.
A.A. Abrikosov, L.P, Gor'kov & I.E. Dzjaloshinsky, Methods o f the Quantum FieM Theory in the Statistical Physics, Nauka, Moscow (1961). A.A. Abrikosov & I.A. Ryzhkin, Adv. Phys. 24, 148 (1978). V.L. Berezinsky, Soy. Phys. - JETP 65, 1251 (1973). S.F. Edwards, Phil. Mag. 3, 1020 (1958). F. Yonezawa & T. Matsubara, Progr. Theor. Phys. 35,759 (1966). A.A. Gogolin, V.I. Melnikov & E.I. Rashba, Soy. Phys. - JETP 69,327 (1975). N.N. Lebedev, Soy. Phys. Appl. Math. Mech. 13, 465 (1949). L.V. Chebotarev, Solid State Commun. 45,553 (1983).
0038-1098/83 $3.00 + .00 Pergamon Press Ltd.
FLUCTUATIONS OF THE RELAXATION TIME IN ONE-DIMENSIONAL SYSTEMS L.V. Chebotarev Kharkov State University, 310 077 Kharkov, U.S.S.R.
(Received 14 January 1983 by E.A. Kaner) The direct interaction of electrons with impurity field is represented as interelectronic interaction of a special kind which is called "dissipative". This type of interaction is responsible for relaxation process and generally leads to fluctuations of the relaxation time. The probability distribution function for the mean free time of electrons is found in the white-noise approximation for a one-dimensional conductor. It is shown that the dielectric response function can be obtained in this case by averaging relevant expression over the fluctuations of the relaxation time. responsible for those fluctuations. In this paper we show that the relaxation time of electrons really fluctuates and finds the probability distribution function for the relaxation time in the white-noise approximation.
1. INTRODUCTION THE RELAXATION PROCESS in three-dimensional electronic systems is known to be well described by the conventional kinetic equation, provided pol >> 1 (Po is the Fermi momentum and l is the mean free path of electrons). Owing to the existence of the small parameter (pol) -1, in a three-dimensional system all the diagrams of perturbation series which contain intersections of impurity lines are found to be small in terms of this parameter [1 ]. However, in one-dimensional systems the small parameter (pol2f 1 does not appear even ifpol2 >> 1 [2], where l~ is the mean free path of an electron with respect to the backscattering. For this reason, investigation of one-dimensional systems is bound up with the necessity to sum all diagrams of the perturbation series [2, 3]. Physically, the situation is illustrated by the fact that Anderson's localization in a three-dimensional conductor appears only at sufficiently large amplitudes of a random potential, i.e. provided pol <~ 1, whereas in a one-dimensional conductor all the electronic states near the Fermi level are localized even ifpol2 >> 1. In the present paper the distinctive features of one-dimensional systems are explained by a considerable increase of fluctuations of a peculiar kind, which we call "dissipative" and which give rise to fluctuations of the relaxation time and of other dissipative characteristics of electrons. In a three-dimensional system the dissipative fluctuations are present as well, but they are small with pol >> 1 and become appreciable only in highly disordered systems. The reduction of the dimension of a system leads to a growth of all fluctuations in it, including the dissipative ones. The growth of the dissipative fluctuations causes the small parameter (pol) -1 to vanish thus making important a large amount of diagrams conraining intersections of impurity lines, which are
2. THE HAMILTONIAN OF THE "DISSIPATIVE" INTERACTION Consider a system of electrons in a metal which contains randomly disposed impurities. Any physical variable characterizing the electron system can be expressed in terms of electronic Green functions, while its observed value can be obtained by averaging this expression over the impurity configurations. Such averaged values can be calculated if one knows the generating functional for the Green functions, which is averaged over the impurity configurations. The Hamiltonian for the electrons in the random potential field
F(x) = ~ u(x--xa),
(1)
a
which is created by impurities has the form
H = f dx~÷(x)HotP(x)+ f dxF(x)~÷(x)~(x),
(2)
where x is the spatial coordinate of a point in a space of relevant dimension, x a is the coordinate of an impurity atom, ~k÷(x) and ¢(x) are, respectively, the Fermi operators of creation and annihilation of an electron at the point x, Ho is the Hamiltonian of the kinetic energy of the electron. We assume the electrons to be noninteracting. The generating function G(A) for the set of electronic Green's functions, averaged over the impurity configurations, may be written in the form:
549
550 G(A)=
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
const f Dq~ D~+ exp{i f dxdt[i~+(x,t)~(x,t)
--l f dxldtlf
sms(¢+, ~) =
-- t~÷(x, t)no~(X, t) + ~k+(x, t)A(x, t) + A+(x, t)~(x,
t)] +
iSdis(~k+, ~b)}
Vol. 47, No. 7
dx2dt2$+(xl, tl)
X ~b(Xl, tx)iVdis(Xl --X2)~+(X2, t2) x ¢(x2, t2),
(3)
(6)
where the effective potential of binary interaction
1
tSdis(qJ+, if) = -- i In W(-- ~0+ff)
iVdis(Xl --X2)
X ~(X, t)} -- 1) L.
(4)
Here W(-- ~+~) is the characteristic functional for the random field F(x), taken at the "point" (-- ~÷~), the subscript L at the brackets in this formula stands for the linked average of the expression in the brackets calculated with respect to the fluctuations of the random field F(x ). The functional Sais(~+; ~) represents an addition to the action function for the electronic system which describes an additional interaction between the electrons due to the randomly disposed impurities. Such an interaction is known to arise formally in the quantum theory and sometimes it can be described by impurity lines in the diagram techniques [1, 4]. We shall call this type of interaction "dissipative", as it is responsible for the relaxation process in the electronic system. The Hamiltonian Hdi~ of the dissipative interaction, related to S~s(~O+, ~O),is generally non-Hermitian and describes multiparticle interactions of all orders between the electrons rather than only pairwise interaction between them. In case of uniformly and independently distributed impurities the functional W is well known and
=
--ini f dxu(x, --x)u(x2 --x),
(7)
is purely imaginary. The first-order term of the expansion is related to a correction for the chemical potential and therefore it has been omitted. The functional (6) describes the pairwise interaction between electrons, which has been represented in the diagrams techniques in [1] and in [4]. Now we consider a one-dimensional conductor. As is known, in the one-dimensional case the constriction of the Fermi surface to two isolated points in the momentum space lays emphasis on them even in comparison with the Fermi surface in a three-dimensional metal. Therefore, the main part in the interaction functional Sdis belongs to the terms which are due to the vicinities of the Fermi points + Po; the rest of Sdis may be taken as a background for the interaction of electrons. Let a÷(p) and a(p) be the creation and annihilation operators of electrons with the momentum p. In the usual way, we may introduce new field operators to describe the electronic states near the right (ct = 1) and the left (a = 2) Fermi points:
~(x)
=
1
~
~eikXa~(k),
Ikl ~ P o .
(8)
Here al(k) = a ( p 0 + k),a2(k) = a(--po + k). Then the main part of Sdis may be expressed in terms of $~(x) as - - ¢ , 0 ) + .~,(2) follows: Sdis - Oint ~ i n t ' X'O) = ---1 ~int 2
;
dxxdtl
j dx2dt2[qJ+(xa,q)~(Xl,
q)]
× Ux(xx -x2)[ff*(x2, t2)ff(x2, t2)];
The diagrams technique for such multiparticle interaction functional, with the Green functions expanded in powers of concentration ni, is constructed as the "bunch diagrams method", which has been developed in [5], and leads to the same results. On the other hand, when the scattering of electrons by impurities is weak and the potential u(x --x') is small, the diagrams technique may be based on the expansion of the Green functions in powers of u(x -- x ') itself. Then we obtain with an accuracy to the secondorder:
,¢(2) = ~int
__lfdxxdtlf dx2dt2[~+(xl, tl)a+~(x~,tl)] 2 × U2(xl
--x2)[~k÷(x2, t2)a¢(x2, t2)]
I
f dx,dtl f dx2dt2[~+(xx ' t,)a~(xl, 2 X Uf(xx --x2)[~0+(x2, t2)a+¢(x2, t2)].
tl)] (9)
a, a t are two-row matrices: a
~
,
~
lo) ,
(10)
Vol. 47, No. 7
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
and the parentheses in equation (9) embracing two field operators designate the trace with respect to the "spinor" variables only. The functions UI,2 are
l y
UI'2(x) = L q eiqXUl'2(q)'
Ul(q) = -- &ilu(q)12;
[ql~Po,
(11)
U2(q) = --inilu(2po + q)l 2, (12)
and u(q) is the Fourier transform of u(x --xa). The symbol T stands for the transposition procedure. The operator ~O) ° i n t in equation (9) describes the electronic interaction due to the forward-scattering and the operator Si(n2~describes the interaction due to the backscattering of electrons by impurities. The generating functional for Green functions of 1-d electrons now may be constructed as follows:
G(A) = const f Dff D~ + exp {i(~b+Kff) + iSint(~+~//) + i~+A + iA+~}.
Here
(13)
a
K = iat +ivoaax Sint(~ ÷, 4) = ½pU~p + ½p~÷)U2p~-~ + ½p~->U?p<÷), (14) p = (qJ+~O); 0 (+) = (~0÷a+¢); p(-) = (qJ+aq0.
(15)
All the products in equation (13) and in what follows have the matrix sense, v is the Fermi velocity, o3 = a + a - - a a +.
The next important step to transform G(A) is to apply to the interaction functional Sint(~b÷, ~b) a procedure which might be called "extraction of the stochastic square root" from the integral operations U1,2. The procedure consists in representing exp (iSint) in the form of continual integral over new Bose variables ~/(x), ~'(x) and ~'+(x) exp (iSint(ff +, ~b)) : const f DO f Dr D~"+ x exp [~ r~Ullr/+ i~+U;l~ + i~?(~+~) + i~+(~/+a~) + i~(~+a+~) I ,
(16)
where U~.~ are the operations inversed with respect 1 to U~,~. To define these operations as one-valued ones, it is convenient to determine the integration space in equation (16) by imposition of the periodicity conditions on the border of the system: r/( L ) =
r/(+L);
~.(L)
= ~'(+2)"
(17)
551
The field r~(x) may be taken as real, whereas the field ~'(x) must be complex to ensure the true structure of the functional (13). That the field ~'(x) is one of the Bose type follows from the fact that the two terms in equation (14) containing/-]2 have the same sign. After substituting the expression (16)into equation (13) we get in the exponent the functional to be bilinear with respect to the Fermi fields: G(A)=
constf D¢ D¢+f DnD~D~+exp {i(¢+K6) i + i~'(ff+a+~k) + i~b+A + iA+~bI .
(18)
In this formula the direct dissipative interaction between electrons is represented as a result of their interaction with the new "quantized" Bose-fields 77 and ~'. The excitations of these fields bear the "quanta" of dissipative characteristics of the system rather than those of its dynamic values, as do the usual quantized fields. The fact that the generating functional for Green functions of the electronic system, which defines all the features of its dynamics, has been expressed in the form (1 8), being so much close to the generating functionals in the quantum fields theory, has far going consequences. Namely, the conclusion is to be drawn that fluctuations are intrinsic to the nature of relaxation processes as much as they are intrinsic to the nature of quantum motion. In other words, the values characterizing the irreversible process (the relaxation time etc.) are not something rigidly set for the system under consideration. On the contrary, those values are produced in the very course of the relaxation process under the influence of fluctuations of a special kind, i.e. dissipative fluctuations, which have an origin to be common with the genesis of the irreversibility itself. The effect of dissipative fluctuations on the formation of dissipative values is reflected in the fact that mathematical expressions for such values always include an averaging over configurations of a random field. It should be mentioned that the dissipative fluctuations have no direct connection with the dimension of the system. Such fluctuations exist in one-dimensional systems as well as in three-dimensional ones. The distinction of one-dimensional systems lies in the fact that the dissipative fluctuations in a three-dimensional conductor are mostly small, provided (pol) >> 1. It may be shown that the formation of the relaxation time in this case is affected especially by the binary correlations of a random field, whereas the higher order correlations give corrections to be proportional to (pol) -1. This fact
552
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
manifests itself in the diagram techniques as smallness of all diagrams which include intersections of impurity lines [l ]. On the contrary, in a one-dimensional system the dissipative fluctuations become much increased. The anomalous growth of fluctuations in general is a typical feature of one-dimensional systems. In this case, not only the binary correlations of a random field contribute to the formation of the relaxation time, but so do also the higher order correlations to the effect of vanishing of the small parameter (pol) -1. Correspondingly, increases the number of important diagrams with intersections of impurity lines, which are responsible for those fluctuations. 3. THE DIELECTRIC RESPONSE FUNCTION FOR A ONE-DIMENSIONAL CONDUCTOR IN A STEADY FIELD Now we shall show that the concept of the fluctuations of the relaxation time is by no means a quite formal one. To shed light upon this fact, let us consider a one-dimensional conductor in an applied electric field varying in space and time. The dissipative fluctuations are found to be large in the low-frequency limit oo72 ~ 1, where r2 is the mean free time of electron with respect to the back scattering. In this case Anderson's localization takes place and a one-dimensional electronic system is an insulator rather than a conductor. Taking advantage of the generating functional (18), we can obtain the following expression for the dielectric response function of the system:
e(co, K) = 1 + (%,r2 cosot)Z[Q(00, K) + Q ( 0 0 , - K)]. (19) Here 00p is the plasma frequency of electrons, a is the angle between the high conductivity direction (z) and the wavevector q of an applied field, 00 and K are, respectively, the dimensionless frequency and dimensionless component of q along the direction z. In order to obtain dimensional units one has to put 00 ~ 00r2, K + qzl2. Only the conduction electrons have been taken into account to obtain the formula (1 9). Neglecting a small delocalization of electronic wavefunctions, the function Q(00, K) is found in the white-noise approximation to be quite the same as it was defined by Berezinsky [3]:
Q(00, ~) = 2 J dxW(x)Ko(ZV~).
We shall try the solution of this equation in the form of the Kontorovich-Lebedev transformation: 1 W(x) =
f dlaKiu(2X/x)q~(t~),
(22)
0
which satisfies proper boundary conditions, W(x) being integrable at x ~ + 0 and decreasing at x ~ + oo. After substituting this expression into equation (21), we get the equation oo
f dtaKiu(Zx/x)[v(la) o
i(00 -- .K)]~b(/a) = 2x/~Ko(2X/~), (23)
where u(g) = (1 +/a2)/4. The solution of this integral equation is
2_
e~
U Sh (7r/1)
( dx
Kiu( 2X/x)Ko(2X/~). (24)
Making use of the formulas (20), (22) and (24), we obtain n2 oo
Sh n/.t2
1
Q(00, K) = -~ f dlala 7rlav(#)--i(00--K) . o ch a - 2
(25)
Hence, v(kt) is in fact the collision frequency of electrons and ~ is the corresponding "quantum number". The expression (25) means averaging the typical energy denominator (v -- i00 + iK)-1 over the fluctuations of v, which has the probability distribution function (going back to dimensional units)
w(v) = r?r=O(vr= - ~) Sh rr(vr= - ~[)1/2 ch 3 rr(vr2 ~)i/2 , -
-
(26)
where O(x) = 0 at x < 0 and O(x) = 1 at x > 0. It is easily seen that w(v) is non-negative and duly normalized. It follows now from equations (19) and (25) that the expression for the dielectric response function in a steady field (w = 0) is oo
e(K) = 1 + (copr2 coso02 f dv 00(v) v2 2v+~:2.
(27)
In a uniform field K = 0, and we obtain e(0) = 1 + 4~'(3)(00pr2 cos a) 2,
(28)
(20)
0
Here Ko(z) is the Macdonald function, and W(x) is to be found by solving the differential equation I-- x 2 -d- ~ -- 2x d + ] W(x) -- i(00 -- K)W(x) dx 2 G = 2Ko(2x/-x).
Vol. 47, No. 7
(21)
where ~'(x) is the ~'-function of Riemann. This expression is in full agreement with the known result [2, 6]. It is worthwhile to make note of the fact that the operator in the brackets in the 1.h.s. of equation (21) may be called "collision frequency operator" in a sense to be very close to the quantum mechanical one. Indeed, the eigenvalues of the operator give possible values of
FLUCTUATIONS OF THE RELAXATION TIME IN 1-D SYSTEMS
Vol( 47, No. 7
the collision frequency u, while the formula (22) corresponds to an expansion of IV(x) into the complete set of orthogonal and normalized eigenfunctions of this operator: Cu(x) =
f
Sh n#]
Kiu(2V%),
'
o
,
in compliance with conventional expansions in the quantum mechanics as well. So we are led once more to the fact that the dissipative fluctuations show resemblance to the quantum ones [8].
(29)
REFERENCES 1.
d~t~u(x)~u(x ) = 6 ( x - - x ); (30)
2.
dx ¢u(x)Ov(x) = 8(la -- v).
3.
0
The first relation (30) has been proved in [7], and the second one is to be found in physical literature and may be proved as well using the Nicholson formula known in the theory of Bessel's functions. The Green function of the equation (21) proves to be: G ( x , x ' ) = _( du 0
gPt~(x)cku(x') v(la) -- i(¢o -- K) '
(31)
553
4. 5. 6. 7. 8.
A.A. Abrikosov, L.P, Gor'kov & I.E. Dzjaloshinsky, Methods o f the Quantum FieM Theory in the Statistical Physics, Nauka, Moscow (1961). A.A. Abrikosov & I.A. Ryzhkin, Adv. Phys. 24, 148 (1978). V.L. Berezinsky, Soy. Phys. - JETP 65, 1251 (1973). S.F. Edwards, Phil. Mag. 3, 1020 (1958). F. Yonezawa & T. Matsubara, Progr. Theor. Phys. 35,759 (1966). A.A. Gogolin, V.I. Melnikov & E.I. Rashba, Soy. Phys. - JETP 69,327 (1975). N.N. Lebedev, Soy. Phys. Appl. Math. Mech. 13, 465 (1949). L.V. Chebotarev, Solid State Commun. 45,553 (1983).