Abstract
Full Text
UDC 533.95.2
PHYSICS
V. N. MEL’NIKOV
GREEN’S FUNCTIONS FOR QUANTUM FERMI SYSTEMS OF CHARGED PARTICLES WITH COLLISIONS
(Presented by Academician N. N. Bogolyubov, 4 V 1967)
Charged Fermi particles are considered, interacting according to Coulomb’s law and situated in a compensating field of charge of the opposite sign. We shall start from equations for quantum distribution functions of the form
\[ D_1(t,x_1,x'_1)=\langle \Psi^+(t,x'_1);\ \Psi(t,x_1)\rangle, \]
where \(\langle\ldots\rangle = Q^{-1}\operatorname{Sp}(\ldots e^{-(H-\lambda N)/\theta});\)
\(Q=\operatorname{Sp}e^{-(H-\lambda N)/\theta}\), \(\lambda\) is the chemical potential;
\(x=\{\bar r,\sigma\}\),
\[ \Psi(t,x)=\frac{1}{(2\pi)^{3/2}}\int e^{ipx}a_{p\sigma}\,dp,\qquad \Psi^+(t,x)=\frac{1}{(2\pi)^{3/2}}\int e^{-ipx}a^+_{p\sigma}\,dp . \]
Then
\[ D_1(t,x_1,x'_1)=\frac{1}{(2\pi)^3}\iint e^{i(p_1x_1-p'_1x'_1)} \langle a^+_{p'_1\sigma'_1},a_{p_1\sigma_1}\rangle\,dp_1\,dp'_1 = \]
\[ =\frac{1}{(2\pi)^3}\iint e^{i(p_1x_1-p'_1x'_1)} F(p_1,\sigma_1,p'_1\sigma'_1)\,dp_1\,dp'_1 . \]
The equilibrium value is
\[ F^0(p_1,p'_1)=n_{p_1}\delta(p_1-p'_1)\delta(\sigma_1-\sigma'_1),\qquad n_p=2/(e^{(\varepsilon-\lambda)/\theta}+1), \]
\[ \varepsilon=\hbar^2p^2/2m,\qquad D^0_1(x_1,x'_1)=\frac{1}{(2\pi)^3}\frac{\delta(\sigma_1-\sigma'_1)}{2} \int e^{ip_1(x_1-x'_1)}n_{p_1}\,dp_1 . \]
The functions \(D\), in contrast to Bogolyubov distribution functions \((^1)\), are normalized to \(n\), and not to 1. Let us take the kinetic equation with a self-consistent term in the Hartree approximation
\[ i\hbar\frac{\partial D_1(t,x_1,x'_1)}{\partial t} +\frac{\hbar^2}{2m}(\Delta_{r_1}-\Delta_{r'_1})D_1(t,x_1,x'_1)- \]
\[ -D_1(t,x_1,x'_1)\int [\Phi(r_1-r'_2)-\Phi(r'_1-r'_2)]D_1(t,x'_2,x'_2)\,dx'_2 =I_{\mathrm{st}}. \tag{1} \]
The case \(I_{\mathrm{st}}=0\) was considered in work \((^2)\). We linearize equation (1) and perform a symmetric Fourier transformation with respect to spatial variables of the type
\[ F\{\ldots\}=\frac{1}{(2\pi)^3}\int\{\ldots\}e^{-ikr+ik'r'}\,dr\,dr' . \tag{2} \]
As a result we obtain the equation in \(k\)-space
\[ \left[i\hbar \frac{\partial}{\partial t}-\frac{\hbar^2}{2m}\left(k_1^2-k_1^{\prime 2}\right)\right]\delta F(k_1 k_1')- \]
\[ -\frac{\Delta(\sigma_1-\sigma_1')}{2(2\pi)^{3/2}}\rho(k_1-k_1')\Phi(k_1-k_1')\left(n_{k_1'}-n_{k_1}\right)=I_{\mathrm{st}}^{\mathrm{lin}} \tag{3} \]
where
\[ \rho(q)=\sum_{\sigma_2}\int \delta F(k,\sigma_2,k-q,\sigma_2)\,dk;\qquad p_1-p_1'=q,\quad p_1=k. \]
Let us imagine that the system is placed in an infinitesimally small external field and that the variation \(\delta F\) is due precisely to this field. Let
\[ \delta H=e^{-iEt}\delta\Omega(x_1,x_2')/2\pi, \]
then (3) becomes
\[ i\hbar\frac{\partial \delta F(t,k_1,\sigma_1,k_1',\sigma_1')}{\partial t} -\frac{\hbar^2}{2m}(k_1^2-k_1^{\prime 2})\delta F(t,k_1,\sigma_1,k_1',\sigma_1')- \]
\[ -\frac{\delta\Omega(t,k_1,\sigma_1,k_1',\sigma_1')}{4\pi} \left(n_{k_1'}-n_{k_1}\right)- \]
\[ -\frac{\Delta(\sigma_1-\sigma_1')}{2(2\pi)^{3/2}}\rho(k_1-k_1')\Phi(k_1-k_1')\left(n_{k_1'}-n_{k_1}\right) =I_{\mathrm{st}}^{\mathrm{lin}}. \tag{4} \]
We apply the theorem on variations of the mean value of a dynamical quantity (see \((^3)\), and for the classical case \((^{2,4})\))
\[ \delta F(t,k_1,k_1',\sigma_1,\sigma_1')/\delta\Omega(t,p,p',\sigma,\sigma') = G^{\mathrm{ret}}(E;k_1,k_1',\sigma_1,\sigma_1';p,p',\sigma,\sigma'), \tag{5} \]
\[ \delta\Omega(t,k_1,k_1',\sigma_1,\sigma_1')/\delta\Omega(t,p,p',\sigma,\sigma') = \delta(k_1-p)\delta(k_1'-p')\Delta(\sigma-\sigma_1)\Delta(\sigma_1'-\sigma'). \tag{6} \]
Then for the Green’s function we obtain the equation
\[ \hbar E G_E(k_1,k_1',\sigma_1,\sigma_1';k,k',\sigma,\sigma') -\frac{\hbar^2}{2m}(k_1^2-k_1^{\prime 2})G_E- \]
\[ -\frac{\Delta(\sigma_1-\sigma_1')}{2(2\pi)^{3/2}}\Phi(k_1-k_1')\rho(k_1-k_1';k,k',\sigma,\sigma') \left(n_{k_1'}-n_{k_1}\right)- \]
\[ -\frac{\Delta(\sigma-\sigma_1)\Delta(\sigma'-\sigma_1')}{4\pi} \delta(k-k_1)\delta(k'-k_1') \left(n_{k_1'}-n_{k_1}\right) =I_{\mathrm{st}}^G, \]
\[ \rho(q;k,k',\sigma,\sigma') = \sum_{\sigma_1}\int G_E(k_1,k_1-q,\sigma_1,\sigma_1;k,k',\sigma,\sigma')\,dk_1. \tag{7} \]
It remains to choose the collision integral and to write it in terms of Green’s functions. The quantum collision integral was obtained in works \((^{5,6})\), etc. Let us consider the expression obtained by Wyld and Fried \((^7)\) for the spatially homogeneous case and for particles without spin,
\[ I_{\mathrm{st}} = -\frac{i}{(2\pi)^2}\iint dp'\,dq' \left| \frac{\Phi(q')} {K\left\{q',\frac{1}{\hbar}(E_{p+q'}-E_p)\right\}} \right|^2 \times \]
\[ \times \delta(E_{p+q'}+E_{p'-q'}-E_p-E_{p'}) \left\{ F(p)F(p')[1-F(p+q')] \right\} \times \]
\[ \times [1-F(p'-q')] - F(p+q')F(p'-q')[1-F(p)][1-F(p')] \Big\}, \tag{8} \]
where
\[ K(q,\Omega) = 1+ \frac{\Phi(q)}{(2\pi)^{3/2}} \int \frac{F(p''+q/2)-F(p''-q/2)} {\hbar\Omega+E_{p''-q/2}-E_{p''+q/2}+i\eta} \,dp'', \tag{9} \]
\[ E_p=\hbar^2p^2/2m,\qquad \eta>0,\quad \eta\to 0. \]
Let us recall that this integral was obtained under the assumption that the higher distribution functions depend on time only through the first func-
i.e., the equation is valid for \(t>\tau_{\mathrm{coll}}\). Moreover, \(I_{\mathrm{st}}\) is proportional to the small parameter \(\varepsilon=8\pi e^2m/r_dp_F^2\),
\[
r_d=(8\pi e^2mn/p_F^2)^{-1/2},\qquad n=N/V,\qquad p_F=(3\pi^2n)^{1/3}\hbar,
\]
\[
\varepsilon=\sqrt{\pi}/3(8e^2m/\hbar p_F)^{3/2}=\sqrt{\pi}/3\sigma^{3/2}\ll1.
\tag{10}
\]
Consequently, in the present theory \(\varepsilon\), or \(\sigma=8e^2m/p_F\hbar\), are small parameters. Since \(\varepsilon=(8\pi e^2m/\hbar^2)^{3/2}/3\pi^2n^{1/2}\), it is clear that \(\varepsilon\ll1\) for sufficiently dense systems at low temperatures. Let us calculate the collision integral (8) and assume that the linearized equation obtained is also valid for a spatially inhomogeneous system.
Applying the variational theorem to (5), (6), we find
\[
I_{\mathrm{st}}^G=-\frac{i}{(2\pi)^2}\iint dp'\,dq\,Q^0(q,k_1)\,
\delta(E_{k_1+q}+E_{p'-q}-E_{k_1}-E_{p'})
\times
\]
\[
\times\{G_E(k_1,k_1',\sigma_1,\sigma_1';k,k',\sigma,\sigma')s_1+
G_E(p',k_1',\sigma_1,\sigma_1';k,k',\sigma,\sigma')s_2-
\]
\[
-G_E(k_1+q,k_1',\sigma_1,\sigma_1';k,k',\sigma,\sigma')s_3-
\]
\[
-G_E(p'-q,k_1',\sigma_1,\sigma_1';k,k',\sigma,\sigma')s_4\},
\tag{11}
\]
where
\[ Q^0(q,k_1)=\left|\frac{\Phi(q)} {K^0(q,(1/\hbar)(E_{k_1+q}-E_{k_1}))}\right|^2; \]
\[
s_1=n_{p'}-n_{p'}n_{k_1+q}-n_{p'}n_{p'-q}+n_{k_1+q}n_{p'-q};
\]
\[
s_2=n_{k_1}-n_{k_1}n_{k_1+q}-n_{k_1}n_{p'-q}+n_{k_1+q}n_{p'-q};
\]
\[
s_3=n_{p'-q}+n_{k_1}n_{p'}-n_{p'-q}n_{k_1}-n_{p'-q}n_{p'};
\]
\[
s_4=n_{k_1+q}+n_{k_1}n_{p'}-n_{k_1+q}n_{k_1}-n_{k_1+q}n_{p'},
\]
and \(K^0\) means that \(n_p\) has been substituted in place of \(F\).
The solution of equation (7), which for simplicity we consider for spinless particles, with \(I_{\mathrm{st}}^G\) in the form (11), will symbolically be
\[ G=\frac{L+\mathcal{L}}{\hbar E-(\hbar^2/2m)(k_1^2-k_1'^2)+iM}, \tag{12} \]
where
\[
M=\frac{1}{(2\pi)^2}\iint dp'\,dq\,Q^0(q,k_1)s_1
\delta(E_{k_1+q}+E_{p'-q}-F_{k_1}-E_{p'}),
\]
\[
L=\frac{1}{(2\pi)^{3/2}}\rho(k_1-k_1';k,k')\Phi(k_1-k_1')
(n_{k_1'}-n_{k_1})+
\]
\[
+\frac{1}{2\pi}\delta(k-k_1)\delta(k_1'-k')(n_{k_1'}-n_{k_1}),
\tag{13}
\]
\[
\mathcal{L}=I_{\mathrm{st}}^G-iMG_E.
\]
The structure of expression (12) is analogous to the classical expression for the Green’s function. Therefore here, too, one may neglect \(\mathcal{L}\) with high accuracy for values of the wave numbers much smaller than \(k_F\). Let us show this by passing to the dimensionless \(k/k_F\) and \(q/k_F\). \(L\sim e^2/q^2\). \(Q^0\) will be \(\sim me^4/k_F^3\). In order not to take \(\mathcal{L}\) into account, the condition \(\mathcal{L}\ll L\) must be fulfilled; hence \(q^2\ll k_F^2/\sigma\), i.e., for \(q<k_F\) and even \(q\sim k_F\), since \(\sigma\ll1\).
Therefore, for \(q<k_F\) we have
\[ G_E=L/[\hbar E-(\hbar^2/2m)(k_1^2-k_1'^2)+iM]. \tag{14} \]
Introduce the substitution \(k_1'+q=k_1,\quad k_1=k_1\). Then (14) becomes
\[
[\hbar E-(\hbar^2/2m)(2k_1-q)q+iM(k_1)]G_E(k_1,k_1-q;k,k')-
\]
\[
-(2\pi)^{-3/2}\rho(q;k,k')\Phi(q)(n_{k_1-q}-n_{k_1})=
\]
\[
=(2\pi)^{-1}\delta(k-k_1)\delta(k'+q-k_1)(n_{k_1-q}-n_{k_1}).
\tag{15}
\]
Next, after simple operations of division and integration, we arrive at the following solution for the Green’s functions:
\[ \rho(q;k,k')= \frac{1}{2\pi} \frac{(n_{k'}-n_k)\delta(k'+q-k)} {\hbar E-(\hbar^2/2m)(2k-q)q+iM(k)} \times \left[ 1+\frac{\Phi(q)}{(2\pi)^{3/2}} \int \frac{(n_k-n_{k-q})\,dk} {\hbar E-(\hbar^2/2m)(2k-q)q+iM(k)} \right]^{-1}. \tag{16} \]
Thus, in contrast to work \((^2)\), here the damping of the system due to collisions is taken into account. Damping without allowance for polarization by the method of kinetic equations was considered in work \((^8)\).
Knowing the function \(\rho\), with the aid of the spectral theorems \((^3)\) it is not difficult to calculate the thermodynamic characteristics of the system, as well as the correlation functions. For this it is necessary to find the spectral density \(I\) from the formula
\[ I(E)=-2\operatorname{Im}G_E/(e^{E/\theta}+1). \]
After simple calculations we obtain
\[ \begin{aligned} I(E)=&-\delta(k-k'-q)(n_{k'}-n_k) \{M(k)\operatorname{Re}K^0+\\ &+[\hbar E-(\hbar^2/2m)(2k-q)q]\operatorname{Im}K^0\}\times\\ &\times\{2\pi(e^{E/\theta}+1)(\operatorname{Re}^2K^0+\operatorname{Im}^2K^0) [[\hbar E-\\ &-(\hbar^2/2m)(2k-q)q]^2+M^2(k)]\}^{-1}. \end{aligned} \]
The damping and frequencies of the system are determined as the poles of the Green’s function (16). For this purpose, the values \(K^0\) and \(M\) are first calculated from formulas (9) and (13). In the case of small values of the wave numbers at \(\theta=0\),
\[ Q^0(q)=\frac{2e^4}{\pi(q^2+4ck_F)^2},\qquad c=\frac{me^4}{2\pi\hbar^2}\sim\sigma. \]
For \(\theta=0\), and also for small values of the wave numbers, the frequencies coincide with the results of the works of V. P. Silin; the collisional damping is absent for \(k\ll k_F\). It appears only in a term \(\sim(k/k_F)^2\), which corresponds to the result of Dubois \((^9)\).
Analogous results are also obtained for \(\theta\ll\varepsilon_F=\mu_0\), with the values \(Q^0(q)\) differing from the case \(\theta=0\) only by a change of the factor
\[ c^*\to c:\qquad c^*=c\left\{1-\frac{\pi^2}{12}\left(\frac{\theta}{\mu_0}\right)^2\right\}. \]
It should be noted that the exact limits of applicability of equation (11) can be estimated only on the basis of a more general inhomogeneous quantum kinetic equation with collisions. For the present one may say that it is applicable for times in the interval \(\tau_{\mathrm{int}}<t<\tau_{\mathrm{rel}}\), where \(\tau_{\mathrm{rel}}\) is determined by the damping of the system, and also in the interval of wave vectors \(q<k_F\).
In conclusion I express my deep gratitude to my scientific adviser, Academician N. N. Bogolyubov, for his constant attention and numerous consultations in the course of the work.
Moscow State University
named after M. V. Lomonosov
Received
8 IV 1967
CITED LITERATURE
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- N. N. Bogolyubov, Jr., B. I. Sadovnikov, Vestn. Mosk. Univ., 2 (1963).
- D. N. Zubarev, UFN, 71, 1 (1960).
- V. N. Melnikov, DAN, 171, No. 5 (1966).
- R. Balescu, H. Taylor, Phys. Fluids, 4, 85 (1961).
- V. P. Silin, ZhETF, 40, 1768 (1961).
- H. W. Wyld Jr., B. D. Fried, Ann. Phys., 23, 3 (1963).
- I. B. Aleksandrov et al., Vestn. Mosk. Univ., 2 (1963).
- D. F. Dubois et al., Phys. Rev. Lett., 8, 11 (1962).