ZHEN YE: Green’s Function Method in Thermo-Field Dynamics 117

phys. stat. sol. (b) 179, 117 (1993)

Subject classification: 71.10 and 71.45

Department of Physics, University of Ottuwa ’)

Green’s Function Method in Thermo-Field Dynamics

Applications to Plasma and Laser Heated Matter

BY ZHEN YE^)

This paper, on one hand, is an introduction to thermo-field dynamics (TFD), on the other hand a derivation of some interesting features of Green’s functions in the frame of thermo-field dynamics. It is proved that in the language of TFD, Green’s functions at finite temperature take a very simple form. Through the applications of TFD to many-body systems it is shown that TFD is a straightforward method in dealing with finite temperature many-body systems. As an application of Green’s function method in TFD, we derive a Kogan-type formula for the non-equilibrium dynamic conductivity or laser-pulse heated condensed matters, where electrons and ions are held at different temperatures.

1. Introduction

As a basic theory for our future work, the present paper has twofold purposes. On one hand, it studies the general structure of the Green’s function at finite temperature in the frame of thermo-field dynamics (TFD). O n the other hand, it discusses the applications of Green’s function method in TFD, especially we study the dynamical conductivity of the laser-pulse heated matter which is of interest not only for basic physics but also in many other fields such as optics, laser or ion driven fusion, and microstructured materials [l]. We shall derive a Kogan formula for the relaxation time of an electron-phonon system which is heated by short laser pulses. The applications, I hope, can convince readers of the simplicity of thermo-field dynamics. Our future work will be fully based on the theory presented here.

Thermo-field dynamics (TFD) [2] is a useful method for the study of macy-body systems at finite temperature. It was applied to high-T, superconductivity [ 3 ] . In our previous paper [4], the Green’s function method in T F D was investigated in detail for the bosonic system and we applied this method in surface infrared absorption. In the present paper, we will extend our discussion to the fermionic systems, particularly electron systems. Using Lehmann’s spectral representation formalism, we shall prove that the Green’s functions in these systems are having simple structures. The paper will be structured as follows. In Section 2 we review some basic concepts in TFD. Green’s functions in thermo-field dynamics for fermion systems are studied in Section 3. In this section, we shall prove that Green’s function at finite temperature takes a very simple form even with the presence of interaction. Some applications of T F D Green’s function method are presented in Section 4. In Section 4, we first use T F D Green’s function method to study the well-known plasma problem. Then we study the relaxation process in the laser heated condensed matter. A final remark concludes the paper.

I ) Ottawa, Ontario, Canada KlN6N5. ‘) Present address: Institute of Ocean Sciences, P.O. Box 6000, Sidney, B.C., Canada V8L 492.

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2. Basic Concepts in TFD

We briefly review some basic concepts in TFD [2]. In the language of thermo-field dynamics, a tilde field denoted by A" is introduced to associate with any Heisenberg field A in calculating thermal average. Physically, the tilde operator of the creation operator for the energy hw is equivalent to the creation operator of a negative energy - hw. The tilde operation changes the sign of the imaginary part of a complex number. Any field and its tilde field obey the same commutation or anti-communation relations. The properties of the tilde operation are defined as follows:

N

A,A, = &A", ; A" = & A ,

c,A, + C'A, = c:& + c;A",, N

where eF is 1 for the boson-like operator and -1 for the fermion-like operator. The star means the complex-conjugate.

If the vacuum at zero temperature is 10) with annihilation operator a(k) , and its tilde conjugate E(k ) , we have

a(k), E(k) 10) = 0 .

At finite temperature, the thermal vacuum is no longer the vacuum of operators a(k ) and E(k) , i.e. a(k), E(k) I@) + 0. Instead the annihilation operators z(k, P), E(k, 8) of thermal vacuum l O ( p ) ) at finite temperature are obtained through a so-called thermal Bogolyubov transformation,

a(k, P) = a(k ) cosh @k - Et(k ) sinh 0, , d(k , P) = E ( k ) cosh 0, - crt(k) sinh 0, (2.1)

for bosons, and

a(k, j~') = a(k) cos dk - Et(k ) sin 0, , E(k, 0) = d(k) cos 0, + at(k) sin 0, (2.2)

for fermions. These two transformations, coupling the field and its tilde conjugate field, bear similarity to the spin canonical transformation in superconductivity theory where a spin-up electron is coupled with a spin-down one (or vice versa), forming a Cooper pair [6]. Thermal vacuum denotes as lO(P)) satisfies

a(k, f l), P) lO(P)> = 0 '

In the above we have

for bosons,

for the fermions with ,8 = l/k,Tand E , being the energy spectrum.

resulting in the thermal average ( A ) , is replaced by the thermal expectation In the scheme described above, the conventional thermal trace Tr (epPHA)/Tr e-OH [5] ,

( A ) = (O(P)I A lO(P)>. (2.3)

Green's Function Method in Thermo-Field Dynamics 119

The operators A or A can always be expanded in terms of annihilation operators a ( k ) or 5(k ) . Using the Bogolyubov transformations in (2.1) and (2.2), the operator A(A) can thus be expressed in terms of a(k, j3), E(k, p). Then the thermal average given in (2.3) is easy to be calculated. Considering the property of the tilde operation, the total Hamiltonian of the system is written as

H = H - E ? , (2.4)

because the tilde field obeys the Heisenberg equation

a at

ih - A(t) = [A"@), -d] .

This can be easily checked by performing a tilde operation on the Heisenberg equation of non-tilde operator A. We call H the system dynamical Hamiltonian.

3. Green's Functions in TFD

In this section we study the Green's function method in TFD for an electronic system. For the bosonic system, readers should refer to [4].

3.1 Green's functions and spectral representation

The dynamical Hamiltonian of a many-electron system can be usually written as

H = C J d3x d(x) '('1 Y ~ ( x ) + Hint > (3.1) S

where the kinetic energy operator T(V)eik" = ckeikx - - ((h2k2/2m) - p) eikx with m being the electron mass and p the chemical potential determined from N = x f F ( Q . Hint may

be the Coulomb interaction or electron-ion interaction, etc. For example, the dynamical Hamiltonian of an electron gas is usually writen as

P

S

+ $ 1 f d3x d3x' y . i (x) w:(x') V(x , x')ss,.OO, yU,(x') V,,(X).

H = H - E ? , (3.3)

(3.2) IS'. 00'

The total Hamiltonian is then given by

which, in thermal doublet notation, can be rewriten as

H = 2 C C, J d3.x w:~(x) T(V) I/I,"(x) s z = 1 , 2

+ C F, 2 J d3x d3x' W:~(X) w",(x') V(x, x'),,,,,,, 1/7",.(x') y:.(x) (3.4) I ss', nu'

with thermal doublets

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-CF(cp) d F ( E p ) CF(Ep) dF(Ep) ’ + d 3 E p ) p o - G, + i6 p o - E, - 16 p o - c, + i6 po - cp - i6

G 3 P ) = -CF(cp) d F ( E p ) + C F ( E p ) d F ( E p ) &Ep) + C&,)

Throughout the paper, we reserve index s, o . . . (= 7 , I) for the spin index, CI, /I, y.. . (= 1,2) for the thermal doublet index. However, if the interaction does not depend on the spin, we will simply ignore the spin index.

For the free electron field, we have

exp (ikx - i ~ , t ) a t ( k ) . d3k

The bare one-body Green’s function is defined as

iG.?“”(x) = (O(P)I T[WPW lu,OBt(O)I lW)> ‘ (3.7)

Using the Bogolyubov transformation in (2.2), we obtain the bare Green’s function in momentum space,

G,P(P) = U(Ep) [ P o - E p + i6z I - l U+(EP) or

(3.10)

with cF(cP) = [1 - fF(c,)]”2, dF(c,) = f ’ k ’ 2 ( ~ p ) . It is easy to check that

u-’ = U t . In the presence of any interaction, the Green’s function is defined as

i G W = (O(P)I T[v,Z(x) v t t (011 lO(P)> ’ (3.1 1)

The (1, 1) component of the Green’s function corresponds to the conventional Green’s function. One can prove that the spectral representation of the perturbed Green’s function in momentum space is given by

G , ( p ) = d o o,(w,p) U ( W ) [po - w + ids]-’ U - ’ ( o ) , (3.12)

where gS(w, p ) is the spectral function. Considering the commutation relations, we have thc sum rule

(3.13) J d u o , ( w , ~ ) = 1 .

O , ( W > P ) = 6(w - w p ) .

Obviously, for the free electron Green’s function, we have

Green’s Function Method in Thermo-Field Dynamics 121

The above spectral representation of the Green’s function has a nice form and can be regarded as an exact form of the Green’s function. However, in practical calculations we need to consider the Dyson equation in certain approximation from which we can obtain the spectral function.

As usual, the Dyson equation is written as

(3.14)

From now on we ignore the spin indices unless we need them. Comparing (3.14) and (3.12) we conclude that the self-energy matrix C ( p ) must have the following structure:

C ( P ) = ReC(P)I + i Y ( P ) f l ( P o ) (3.15) or

(3.16)

So we see that the real part of the self-energy matrix is always diagonal while the imaginary part of the self-energy matrix has off-diagonal elements. Here we note Re C’l = Re Z22 = Re Z, I is a unitary matrix, y ( p ) is a c-number function, and the matrix D(p0) is given by

n ( P 0 ) = U(P0) TUt(P0). (3.17)

Using the properties Ut = Up’ , z- ’ - - z, we have

Y(P) = Im Z(P) U(P0) 7Ut(PO) ’ (3.18)

The imaginary part of the Green’s function in (3.12) reads as

Im G(P) = - 7 . 4 4 R ( P 0 ) . (3.19)

When we compare this with that obtained from taking the imaginary part of the Dyson equation in (3.14), we find the spectral function,

(3.20)

This is an import result. Now we expand the real part of the self-energy near the bare energy shell cp ,

Re C(p) = Re Z ( c p , p ) + + ...

Then to the first derivative order, the spectral function reads

(3.21)

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where E;, Z@) are the renormalized energy and the wave function renormalization factor given by

The renormalization of the wave function gives the relation between Heisenberg and free fields,

(3.22) lp(x) = Z"2(,) lp"(X) + ... with the definition Z(8) eipx = Z ( p ) eipx.

written in a nice form, In the Tamm-Dancoff (TD) approximation, i.e. Z ( p ) z 1, the Green's function can be

G ( p ) = U ( F J [ p o - CZ + iy(p) TI-' Ut(t$,), (3.23)

which bears similar form to the bare Green's function. From the above equation we see clearly the physical meaning of y - ' ( p ) : it represents the lifetime of the physical electron due to interaction. Without interaction, we have y ( p ) = 6 = 0'.

3.2 Feynman rules and examples

In condensed matter physics, there are two basic interactions of interest: electron-electron Coulomb interaction and electron-phonon interaction. All other interactions can more or less be reduced into a form similar to one of these two interactions. Therefore we start with the electron-electron Coulomb interaction Hamiltonian which has been given ealier,

Hin, = $ C E, C J d3x d3x' &(x) v",(x') V ( X , x ' ) , ,~ , , ,~~~~(x ' ) I&,(x) . (3.24) a ss'.uu'

In the T F D scheme, all the zero-temperature Feynman diagram techniques are reserved except that we need to add thermal doublet index and vertex index represented by M.. . and E,. . ., respectively. We summarize the Feynman rules for this interaction as follows:

1. We use the broken line for the interaction V ( x - x'), the solid line for the electron propagator iG(x - x'). All the quantities have 2 x 2 matrix structure unless otherwise noted. The thermal doublet index is put on the shoulder of the propagator and the spin index at the leg.

2. For the n-th-order perturbation, draw all topologically distinct diagrams with n interaction lines and 2n + 1 directed electron Green's function, i.e. G$(x - y) running from y to x. Label each vertex with a four-dimensional space-time point xi and associate each end of the solid line with a spin index si and a thermal doublet index cli. However, the two ends of a solid line will share the same thermal doublet index if they are also connected by an interaction line, and add a factor E ~ , to these two ends.

3. Add a sign factor ( - l)', F is the number of closed fermion loops in the diagram. A factor ( - i/h)" is added to n-th-order perturbation.

4. Equal time Green's function interpreted as G(x, t ; x', t'). 5. Integrate over all internal variables over space and time. Sum up the internal spin

and thermal doublet indices.

Green’s Function Method in Thermo-Field Dynamics 123

6. If we work in momentum space, we should add a factor eisukotl to G““(k) which is either in a closed loop or connected by a broken line, when the interaction potential has the form

I/ss’,aa’(x> x’) = Vss.,aa.(X - -4. 7. In many cases, using the following notation will reduce a lot of confusions as we will

see later in the study of plasma system:

where we also assume the thermal doublet for the interaction line. For the spin-independent interaction, we have I/ss.,ou,(x, x’) = V(x , x’) dssrduu,. Now we emphasize that the definition of the self-energy C is given by: iG = iGo + iGo( - iC) iG.

To make an example of using above Feynman rules, we consider the vertex correction as shown in Fig. 1. Here the solid line represents the free electron propagator and the dashed line is for the Coulomb interaction. We have,

T I Js Vax’(q) iGo“’p’(p - k ) iGoy’“’(p + q - k) Vy’O’(k) iGoyy‘(p + q) iGoPP’(p) a’y B

= 1 V‘“’(q) Ta’B’y‘(p, p + q) iGoyy’(p + q) iGopO’(p) a’y’p’

(3.25)

with vertex correction,

Goa’o’(p - k) Goy”’(p + q - k ) Vy‘o’(k) . d4k

r a ’ o ’ y ’ ( p , p + 4) = i 2 Jm In the above we defined VaO(k) = (- i/h) Sao&,V(k) with V(k) = 4ne2/k2.

Now we consider the electron-phonon interaction H e - p ,

f i e - p = c j d3X [VBW V S ( 4 - 4 g(a) Cp(4 - c s d3X [ Z ( 4 G S ( 4 - 4 g(a) @(x) S S

(3.26)

where n = (O(P)l ~ t t p lO(P)) and g(a) eikx = g, eikx which is the coupling constant. The thermal doublet arrange operator Pa is defined as

P1(A’B’ ... C’) = A’B’ .. c’ , P’(A’l3’ ... C’) = c’ ... B’A’

Fig. 1. Lowest-order vertex correction in the electron-elec- tron Coulomb interaction

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The Feynman rules for this interaction are almost the same as the electron-electron interaction except the following aspects: the interaction line V(x,x‘) is replaced by the phonon propagator iD@(x - x’), each vertex is attached with a factor c,.

Now we provide some examples of calculating Feynman diagrams in thermo-field dynamics. Since we will discuss electron-electron and electron-phonon interactions in the next section, here we would like to study the phonon-phonon interaction as a brief example. We consider anharmonic phonon interaction ( V3/3! ) ‘p3 + (V4/4!) ‘p4, which produces some Feynman diagrams as depicted in Fig. 2.

For Fig. 2a, we have

(3.27)

Taking into account the spectral representation of phonon Green’s function [4], we calculate the bubble loop as

x a,(k - k , , 0”) [U,(O”) z(w - w1 - W” + i6z)- UB(W”)]

= J dwp L,(k, u p ) [UB(Wp) Z(W - up + i6z) - UB(Wp)l

with L,(k, up) = (-i) C J dw’ dw“ ac(k, , w’) o(k - k , , w” )

ki

x [l + fR (W‘ ) + fR(O”)] S(WP - 0’ - a”).

(3.28)

/--- \ / \ I \ \ I \ /

I --I--

‘ y y

k’- b a

k C K - 6 - G d

Fig. 2. Some Fynman diagrams due to phonon-phonon anharmonic interactions

Green’s Function Method in Thermo-Field Dynamics

For Fig. 2b, we have

iCT$(k, w ) = s,,,v; c CyEJY(k, w = 0) DyX, t ; x, t’)

iC;;(k, w ) = 6,,.vk&,D?y’(x, t ; x, t’).

iCZ$(k, 0) = Ey&)>.v:L,,(k, 0)

OL

Fig. 2c leads to

The dephasing diagram in Fig. 2d gives,

125

(3.29)

(3.30)

x DYY’(k - k , - k,, o - w1 - 0,). (3.31)

Simple calculation gives

L,,(k, w ) = J dw‘ L,(k, 0’) [uB(w’) z(w - o’ + i 6 ~ ) - ’ uB(w’)],

&(k, w) = -

(3.32) where

1 J d o , dw2 dw, a,(ki, 0 1 ) oc(k2, w2) a,(k - k , - k, , wj) k i . k z

x [1 + f B ( 4 + f B ( W 2 ) l U + f R ( W 3 ) + f B ( 0 - 4 1 x6 (w - w , - 0 2 - w 3 ) . (3.33)

We can extend the above recipe to consider, for example, the n-bubble diagram, in which case the loop calculation will be simply

L”(k ,w) = ( - i Y - l C J d w , ... dw, k i ... k ,

(3.34) In the above calculation we did not specify whether the dashed line in Fig. 2 represents the

bare or corrected propagator for a phonon. When the dashed line is for the bare phonon Green’s function, we simply substitute o,(k, w ) = (1/2) sign (0) [6(w - wk) + 6(w + wk)].

4. Applications

In the previous section, we studied the Green’s functions in thermo-field dynamics and derived that the Green’s functions at finite temperature take very simple form. In this section, we shall apply the above results to many-body systems, particularly the electron gas system and electron-phonon systems.

4.1 Plasmon in electron gas

The dynamical Hamiltonian for the electron gas was given in (3.2). Since the plasmon is the collective mode excitation due to the fluctuation of the electron density, we define the density correlation function

ZX“’(X - x’) = (O(p)I T[AtY(x) Ari’(x’)] lO(f l)) , (4.1)

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where An represents the density fluctuation,

Afi"(x) = ($(;)) '

with Afi = vitv - (O(P)l ytv lo(/?)). Please note here that we ignore the spin index as long as we add a spin factor at the end of calculation.

The bare density correlation function is given by

ix0"'(x - x') = (O(P) / TIAfioa(x) AfiO'(~')l (O(P)) , (4.2)

where Afro = yotvo - (O(/?)I yotv0 lo(/?)) with yo the free electron field. Simple calculation leads to

(4.3) ixoaP(x - x') = &,cS[2S + 11 GOaP(, - x') GoPa(x' - x),

where S is the electron spin, and the free electron propagator Go(x - x') is given previously. In the momentum space, we easily have

where k = (k, ko). Simple calculation leads to

in the above, we use ck for the electron energy spectrum. In the presence of Coulomb interaction, the Dyson equation for the density correlation

function is, under chain approximation (Fig. 3), according to the Feynman rules described in the last section,

ix"'(x - x') = i x y x - x')

+ 1 dx, dx, ixoay(x - xl) ( - i /h ) V(xl - x2) cy ( - 1) 2 tx . Y O (x2 - x') , (4.8) Y

which can be arranged into a compact form

with p(k) = ( l /h) V ( k ) z and V(k) = 4ne2/k2. To find the energy spectrum of plasmon excitation, we need find the denominator of xo/(l - xoa). It is easy to check that the matrix of the density correlation function has the following form:

Green’s Function Method in Thermo-Field Dynamics 127

Fig. 3. Chain approximation for the electron density-density correlation function. The solid line is the bare electron propagator and the dashed one is the Coulomb interaction

with the real part of xo being calculated as

(4.11)

which is exactly the same result as obtained from other methods such as Matsubara’s method in RPA approximation [5 ] . The matrix Q takes the following form:

whose matrix elements are calculated as

4ne2 4ne2 1 - Re xoil(q) - i ~ [Qf(q) - Q:(q)]1’2 = 0 ,

4 q2 (4.12)

which determines the plasmon energy spectrum opl(q) and the lifetime zpl,

w,”,(q) = o,”l(o) + $ q 2 u $ + ... , (4.13)

(4.14)

where w,”,(O) = 4ne2n/m. It is easy to check: 7;’ (0) = 0. Please note above that it is important to keep the off-diagonal matrix elements, such as Q2(q), in thermo-field dynamics. To compare our lifetime result with that obtained by other methods [6], we simply need to notice that [Qf(q) - Q:(q)]”2 = Im xg”(q), where xR is the retarded density-density correlation function. How to obtain the retarded Green’s function from the casual correlation function will be discussed in the Appendix.

4.2 Relaxation time of electron-phonon interaction

The theory of the relaxation time of the electron-phonon system measured in electrical resistivity has been studied extensively by many authors for the case that the electrons and ions are at the same temperature, i.e. in equilibrium state. One of the excellent work was

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done by Mahan [7]. Considering a force-force correlation function, Mahan proposed a new expression of electronic relaxation time. This expression is valid over the entire range of frequency.

The recent short-pulse laser experiments [l, 8,9] enable us to study the condensed states under extreme conditions. Our particular interest lies in the transport properties of these states. Due to the short-pulse laser interaction, electrons in the matter can be heated up to lo6 K, in such as aluminum, while the ions are held at "room temperature", 7; [I]. This is possible because the laser impulse is so short that the ions do not change their equilibrium state in that short period of time [lo]. Obviously, in order to study the transport properties in such non-equilibrium system we need to consider the interaction between two subsystems which are held at different temperatures. Here please note that the word "non-equilibrium" is not accurate, because we have assumed that electrons and phonons are kept at different thermal equilibrium states and we ignored the temperature change due to the phonon-elec- tron collision. As matter of fact, the electron-phonon collision time is very short compared to the time of changing temperature so that the assumption is valid that electrons and phonons are at different thermal equilibrium states (denoted by temperatures f lc , pi, respectively). The interaction between electrons and phonons is weak in most materials [5] , which means that the linear response theory is applicable. In this section, we will derive a formula in T F D theory for the electron-phonon relaxation time z(w), following closely the definition given by Mahan [7]. Once the relaxation time is obtained, the dynamical resistivity can be calculated as

(4.15)

where n, is the electron density. The electron-phonon interaction is given by

In the laser heated experiment, the electrons and phonons are held at two different temperatures. However, electrons and phonons are assumed to be kept at two different thermal equilibrium states and the electron-phonon interaction does not change their equilibrium states. Define the current-current correlation function n(t) [7],

or, in momentum space,

in" (w) = J dt eiwf (O(P)I T[iil ( t ) jf'(0)l lO(P)> , where j i is the z-component of the current operator

(4.1 8)

The superscript 11 means the (1, 1) component in the thermal doublet notation. The relaxation time z ( u ) is related to the retarded current-current correlation function nkl (w) as

(4.19)

Green's Function Method in Thermo-Field Dynamics 129

where the Drude formula [5] is assumed to be valid, This equation can be simplified, under the weak electron-phonon interaction, to

(4.20)

y1, is the density of electrons.

to the imaginary part of the causal Green's function by two ways: one is by According to the Appendix, the imaginary part of the retarded Green's function is related

Im nA'(w) = tanh (P0/2) Im .n"(w). (4.21)

This relation can be also obtained by using the Kubo-Martin-Schwinger [ l l ] identity which says, for any two operators at temperature T, that

(O(P)I A( t ) Wt') lO(P)> = (O(P)I B(t') A(t + iP) lO(P)> > (4.22)

where /3 = l/k,T. Another way, much simpler, is to replace f i 6 in the denominator by i6. We will use the second way. Therefore we simply need to calculate the imaginary part of the casual current-current Green's function to obtain the relaxation time of the electron- phonon system.

Using the conservation law of charge, we can prove that [7]

x (O(Pw Pill Ttq(t, 4) q(0, 4') Afi(t, d Afi(O, 4'11 lO(Pe, Pi)> . (4.23)

Here g, is the electron-phonon interaction constant. AA(t, q) = d3x e-iq.x Aiz(t, x) with the quantity Aiz being defined before. The bracket can be approximately calculated to the order g i as

iD''(4 q) i x ' ' (4 4) >

where D ( t , q) and ~ ( t , q) are the q-phonon Green's function and the electron density-density correlation function, respectively. Further we assume [7] that x contains no electron-phonon interaction and take x z xo, D as the exact phonon Green's function, then we have

(4.24)

where we made use of the property D(q, w ) = D(q , -w). The calculation of the loop

leads to

9 physica (b) 179/1

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where n,(o, bi) = l/(ePi" - I), nF(Ek, 8,) = l/(e"'"k + l), and E~ is the energy spectrum of the electron, oc(q, w) is the spectral function for the phonon discussed in last section.

Considering the Appendix, the retarded loop is given by,

[ (l + n B ( w ' , Pi)) - nF(Ek3 P c ) ) n F ( E k + q , P e ) - nB(o', Pi) (l - n F ( E k + q , P e ) ) nF(Ekr f i e )

0 f 0' + &k - & k + q f i6 0 + w' + &k - &k+,, + id (4.26)

Then we have the relaxation time, after considering the summation over the electron spin,

by using the following identity:

we can simplify (4.27) to

This is a Kogan-type formula for the relaxation time of the electron-phonon system at non-equilibrium states. Its derivation from Green's function method has not yet been reported. This formula was used in [lo] to study the dynamical resistivity in a hot electron system, yielding a good result compared to the existing experimental result. The original Kogan formula [12] is derived for the energy transfer from the hot electron to the lattice.

When T, = T, i.e. electron and phonon are at the same temperature, (4.27) becomes

1 x Im [nB(w' ) - nB(O f w')l > (4.29)

E(4,W + 0') which is exactly the same results as obtained by Mahan [7] for the relaxation time of the equilibrium electron-phonon system. Here c(q, 0) is the dielectrical function given by

(4.30)

Green’s Function Method in Thermo-Field Dynamics 131

From the above discussion, we see that TFD Green’s function method is not only useful in the study of thermal equilibrium many-body systems, but it can also be easily used to study a many-body system that has several subsystems which are held at different thermal equilibria. The essence in the TFD method is that all the quantities are defined in a real time-space and the thermal effects are represented clearly by a so-called thermal Bogolyubov matrix. The sophisticated temperature-frequency summation in the Matubara method is replaced by a simple real frequency integration in TFD, in which it is not required that all the propagators in a Feynman diagram be at the same temperature.

5. Conclusion

In the paper, we first briefly discussed some basic concepts of thermo-field dynamics. Then we proved that the Green’s function at finite temperature has a very simple structure which bears the similarity to the Green’s function at zero temperature. We provided Feynman rules and some examples to show how TFD works. It is very important to keep off-diagonal matrix elements in TFD. As matter of fact, these off-diagonal matrix elements, always imaginary, are associated with thermal dissipation because they become zero at zero temperature. To convince the readers of the simplicity of the T F D treatment, we applied TFD to the well-known plasma problems and found the same results as obtained by using other methods. Then we went to study the transport properties of the electron-phonon system in which electrons and ions are held at different temperatures by short-pulse lasers. We derived the Kogan formula for the relaxation time of such electron-phonon systems in a much simpler way. This relaxation time is of interest to experimentalists because it relates to the resistivity measured in the experiments.

Appendix

In this Appendix we will discuss the relations between the causal and retarded Green’s functions.

The retarded Green’s function of two operators A(x) , B(y) is defined as

where [ , I + means commutation or anti-commutation operation. The spectral representation for the retarded Green’s function is easily obtained as [2]

po - o + iS for the case that both operators A and B are fermion-like operators. I is a unit matrix. While the retarded Green’s function reads

p o - o + iS for the case that both A and B are boson-like operators.

The causal Green’s function reads

132 ZHEN YE: Green's Function Method in Thermo-Field Dynamics

for A, B being fermion-like, or ,l

for boson-like A, B. From these equations, we see clearly that to obtain the retarded Green's function from

the causal one, we simply replace the z matrix in the denominator by the unit matrix. We can also easily get

Re GR(p) = Re G ( p ) > Im GR(p) = v(PO) zUt(pO) lm G ( P ) for fermion-like A, B, and

Re GR(P) = Re G ( d , Im G,(P) = u i 2 ( p d Im G ( P ) for boson-like A, B. Since the current operator is boson-like, we easily have (4.21). However, for fermion-like operators, we have

Re GR(p) = Re G ( p ) , Im Gii(p) = tanh-' ( f lpo /2) Im"G(p). We should use these relations to obtain the retarded Green's function from the causal Green's function, especially for the many-body systems with subsystems being in different thermal equilibrium states.

The advanced Green's function can be obtained as follows:

GA(p) = G & ( p ) .

Acknowledgements

We would like to thank Dr. M. W. C. Dharma-wardana at NRC, Canada for his stimulating discussion and bringing to our attention the problem of non-equilibrium electron-phonon relaxation. The author also thanks Prof. P.Piercy and H.Y. Chu for their constant encouragement and discussion.

References H. M. MILCHBERG, R. R. FREEMAN, S. C. DAVEY, and R. M. MORE, Phys. Rev. Letters 61. 2364 (1988). H. UMEZAWA, H. MATSUMOTO, and M. TACHIKI, Thermo Field Dynamics and Condensed States. North-Holland Publ. Co., Amsterdam 1982. Z. YE, H. UMEZAWA, and R. TESHIMA, Solid State Commun. 65, 1563 (1990). Z. YE, phys. stat. sol. (b) 171, 193 (1992). Z. YE and H. UMEZAWA, Phys. Letters A 162, 63 (1992). Z. YE. H. UMEZAWA and R. TESHIMA, Phys. Rev. B 44, 352 (1991). Z. YE, Phys. Rev. B 46, 2628 (1992). G. D. MAHAN, Many-Particle Physics, Plenum Press, New York 1981. J . R. SCHRIEFFER, Theory of Superconductivity, W. A. Benjamin, Inc., Reading (Mass.) 1964. G. D. MAHAN, J . Phys. Chem. Solids 31, 1477 (1970).

D. MEYERHOFER, J . DELETTREZ, D. STRICKLAND, P. BADO, and G. MOUROU, Phys. Rev. Lellers 62, 760 (1989). R. L. SHEPHERD, D. R. KANIA, and L. A. JONES, Phys. Rev. Letters 61, 1278 (1988). M. W. C. DHARMA-WARDANA and F. PERROT, Phys. Letters A 163, 223 (1992). R. KUBO, J . Phys. Soc. Japan 12, 57 (1957). P. MARTIN and J. SCHWINGER, Phys. Rev. 115, 1342 (1959). SH. M. KOGAN, Soviet Phys. - Solid State 4, 1813 (1963).

J. C. KIEFFER, P. AUDEBERT, M. CHAKER, J. P. MATTE, H. PEPIN, T. W. JOHNSTON, P. MAINE,

(Received December 8, 1992; in reuisedform June 24, 1993)