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Influence of the Coulomb interaction on the Fermi hole in an electron liquid
Physica A 255 (1998) 48–64
In uence of the Coulomb interaction on the Fermi hole in an electron liquid Masafumi Horiuchi ∗ , Takanori Endo, Hiroshi Yasuhara Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Received 10 July 1997
Abstract From the diagrammatic analysis, it is proved that the spin-parallel part of the static structure factor of an electron liquid, S ↑↑ (q), has the following exact asymptotic form for q/pf : S ↑↑ (q) − 1 = 4( rs =)(pf =q)6 d2 g↑↑ (r)=d(pf r)2 |pf r=0 + · · ·, = (4=9)1=3 , where pf is the Fermi wave number and g↑↑ (r) the spin-parallel pair correlation function. The second derivative of c 1998 g↑↑ (r) at zero separation is evaluated in the particle–particle ladder approximation. Elsevier Science B.V. All rights reserved PACS: 71.10.Ca Keywords: Electron gas; Pair correlation function; Short-range correlation; Cusp conditions
1. Introduction In the early study of the electron gas model, Hubbard [1,2] and Nozieres–Pines [3] interpolated between long- and short-range correlations in an attempt to construct a theory valid for intermediate densities. They treated long-range correlation associated with small momentum transfers in the RPA, while in their treatment of short-range correlation associated with large momentum transfers they resorted to second-order perturbation theory. Their treatment of short-range correlation between spin-antiparallel electrons, however, still remained inappropriate for such a strongly correlated region. Singwi and his co-workers [4] were the rst who have succeeded in a good description of short-range correlation at metallic densities as well as long-range correlation. The pair correlation function g(r) calculated from their theory remains positive at short distances throughout almost the whole region of metallic densities. We shall hereafter refer to their work as the STLS theory. They have begun with a classical electron gas in the presence of an external eld and calculated the dielectric function (q; !). ∗
Corresponding author. Fax: +8122 217 7754; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 7 0 - 3
M. Horiuchi et al. / Physica A 255 (1998) 48–64
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The basic assumption in the STLS theory is that the two-particle distribution function f(p; r; p0 ; r0 ; t) in the Liouville equation can be replaced by f(p; r; t)f(p0 ; r0 ; t)g(r − r0 ) where f(p; r; t) is one particle distribution function and g(r − r0 ) the pair correlation function in the equilibrium state. The local eld factor G(q) entering their expression for the dielectric function can then be given as a functional of the static structure factor S(q), which is the Fourier transform of g(r). They have obtained a set of equations to determine S(q) or G(q) self-consistently with the help of the uctuation–dissipation theorem, which connects S(q) with the frequency integral of the imaginary part of the inverse dielectric function. Finally, they have made the quantum version of theory by replacing the classical non-interacting polarization function and the static structure factor S(q) by their quantum analogues. A careful examination of the STLS theory reveals that it satis es the following relation: r g(0) + · · · g(r) = g(0) + a0 1=3 4 = 0:5210 · · · ; (1) = g(0) + rs pf r g(0) + · · · ; = 9 where rs is the usual density parameter, given by the radius of a sphere with volume per electron, measured in units of the Bohr radius a0 ; (4=3)(a0 rs )3 = n−1 , n being the average electron density. According to Fourier transformation, (1) is equivalent to the following self-consistent requirement imposed upon the asymptotic form of S(q) for large q, 4 8 rs pf S(q) = 1 − g(0) + · · · ; 3 q Z 1 dq g(r) = 1 + (S(q) − 1)ei q · r : (2) n (2 )3 In fact, Eq. (1), or Eq. (2) is exact and often referred to as the cusp condition. This contains the following physical meaning: In the short-distance limit of their interspace, the motion of any two spin-antiparallel electrons of many-body systems is essentially reduced to the two-body problem in the medium and a close relation between the value of the wave function at zero separation and its derivative is established under the in uence of the short-distance singularity of the Coulomb interaction. The cusp condition can readily be derived by examining the analyticity of two-electron interacting parts of the many-body wave function, but the value of the pair correlation function at zero separation, g(0) itself cannot possibly be obtained without any practical calculations. The satisfaction of the cusp condition is, therefore, necessary but not enough for the description of short-range correlation. A clear understanding of short-range correlation is now acquired in the diagrammatic language of many-body perturbation theory [ 5 –7], instead of resorting to the STLS theory. Consider an in nite series of particle–particle ladder interactions and all the ladder interacting parts in the Goldstone perturbation expansion. This type of interactions
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
between spin-antiparallel electrons exclusively contribute to the predominant asymptotic form of S(q) − 1, or S ↑↓ (q) of order q−4 for large q and their total contributions can be written exactly as 4 4 rs pf S ↑↓ (q) = − g↑↓ (0) + · · · ; (3) 3 q or equivalently the following relation holds for small r: r ↑↓ g (0) + · · · : g↑↓ (r) = g↑↓ (0) + a0
(4)
That is, particle–particle ladder interactions have the intrinsic property of satisfying the cusp condition. Furthermore, the particle–particle ladder approximation which involves a single ladder series enables one to evaluate g↑↓ (0) itself. For a paramagnetic electron liquid g(r) is the arithmetic average of the spin-parallel and the spin-antiparallel pair correlation functions, g↑↑ (r) and g↑↓ (r). The value of g(0) then equals g↑↓ (0)=2 because g↑↑ (0) = 0 due to the Pauli principle. One of the authors (H. Y.) and his co-workers [8–10] have numerically showed that the ladder approximation gives a reasonable evaluation of g↑↓ (r) and g↑↑ (r) at short distances (06pf r . 3:0) for any rs . The reason for the success of the STLS theory probably may be that it includes ladder interactions in a sense through their self-consistent scheme. In fact, there is a close resemblance between the equation of G(q) in the STLS theory and the integral equation of the particle–particle ladder interaction for large q. The present paper is a straightforward extension of a previous paper [7]. From the diagrammatic analysis we shall show that the spin-parallel part of the static structure factor S ↑↑ (q) has the following exact asymptotic form of order q−6 for q/pf : r p 6 d2 g↑↑ (r) s f ↑↑ S (q) − 1 = 4 + ··· ; (5) q d(pf r)2 pf r=0 where the spin-parallel pair correlation function g↑↑ (r) is the Fourier transform of S ↑↑ (q), Z 2 dq ↑↑ (S ↑↑ (q) − 1)ei q · r : (6) g (r) = 1 + n (2 )3 The asymptotic form above occurs as a consequence of the complete cancellation between the contribution of all the direct particle–particle ladder interacting parts and that of their exchange counterparts; each contribution is of order q−4 in the same way as S ↑↓ (q). That is, a direct and exchange pair of ladder interactions between two particle states with the same spin have the property of satisfying the above self-consistent requirement imposed upon the asymptotic form of S ↑↑ (q) for large q. The Fourier transformation of Eq. (5) means that g↑↑ (r) can rigorously be written for pf r.1 in the form 2 ↑↑ r r2 d g (r) 1+ g↑↑ (r) = + ··· : (7) 2 2 a0 dr 2 r=0 This is another cusp condition on g↑↑ (r).
M. Horiuchi et al. / Physica A 255 (1998) 48–64
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These cusp conditions on g↑↓ (r) and g↑↑ (r) have originally been noticed by chemists and for a long time applied to their study of electron correlation in molecules, without being communicated to physicists who study electron correlation in condensed matter. Kato [11] has established the mathematical foundation on the cusp behavior of manyelectron wave functions. Kimball [12,13] has rederived the cusp condition on g↑↓ (r). Rajagopal et al. [14] have stressed the importance of the cusp condition on g↑↑ (r) in the fully ferromagnetic electron liquid. A concise derivation of these cusp conditions is as follows: In the short-distance limit of their interspace, the motion of two electrons in many-body systems is dominated only by their mutual Coulomb interaction. The radial part of the wave function of the two electrons with relative angular momentum l then satis es the following equation: −
˜2 1 d 2 r 2 dr
r2
dRl (r) dr
+
e2 l (l + 1) ˜2 Rl (r) + Rl (r) = E Rl (r) ; r 2 r2
(8)
where is the e ective mass and E denotes a many-body operator which is bounded as r → 0. Substituting into the equation an assumed form of the wave function at short distances, Rl (r) = a + b r + · · · (or c r + d r 2 + · · ·) for l = 0 (or 1), we can immediately obtain b = a=(2 a0 ) and d = c=(4 a0 ) from the analyticity of E. The two equalities above are nothing but the cusp conditions. An alternative derivation of the cusp conditions in the present paper will be helpful to the thorough understanding of their physical implications. Furthermore, it enables us to evaluate the second derivative of g↑↑ (r) at zero separation in the particle–particle ladder approximation. The ratio of the second derivative of g↑↑ (r) at zero separation to its value in the Hartree–Fock approximation can be regarded as a measure of the in uence of short-range Coulomb repulsion upon the Fermi hole. It should be compared with the value of g↑↓ (0). From the diagrammatic analysis, we shall also show that, in the fully ferromagnetic electron liquid, the asymptotic form of S ↑↑ (q) for large q is closely related to the asymptotic form of the momentum distribution function n(p) for large p. As has been well known, the term of order ln rs leads the high-density expansion of correlation energy per electron, in units of Rydberg. However, this does not seem to be understood in terms of g↑↑ (r) and g↑↓ (r). We shall then comment on what features of these two functions are responsible for the logarithmic term.
2. Particle–particle ladder approximation As is evident from the Goldstone perturbation formula, the ground-state energy shift of an electron liquid due to the Coulomb interaction, E can be regarded as a functional of the Coulomb interaction v(q) and the free-electron energy spectrum p . Then, 0 the following expressions for the static structure factor S (q) and the momentum distribution function n (p) result if one proceeds along the procedure of deriving the
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
coupling constant integration algorithm that can be seen in most textbooks on manybody theory [15]: 0 n E = (S (q) − 0 ); 0 v (q) 2
0
S (q) =
1 D 0 E q −q ; N
(E0 + E) = n (p) = a†p ap ; p
(9) (10)
where q is the density uctuation operator of electrons with spin and ap the annihilation operator of an electron with wavenumber p and spin ; E0 is the ground-state energy of an non-interacting Fermi gas; N is the total electron number in the volume
of the system and n the average electron density; h· · ·i denotes the expectation value with respect to the exact ground state. The expressions above are convenient especially for evaluating the asymptotic be0 havior of S (q) and n (p) for large q and p, respectively. The Coulomb interaction in an electron liquid, of course, does not depend on the spin orientation, but we introduce 0 here the notation v (q) for the convenience of specifying the interaction between two electrons with spins and 0 . The static structure factor S(q) is de ned as 1 X D 0 E S(q) = q −q (11) N 0
and it can be divided into the spin-parallel and spin-antiparallel parts as follows: S(q) = S ↑↑ (q) + S ↑↓ (q) ; S ↑↑ (q) =
1 X q −q ; N
(12) S ↑↓ (q) =
1 X − q −q : N
(13)
The spin-parallel and spin-antiparallel pair correlation functions, g↑↑ (r) and g↑↓ (r) are the Fourier transforms of S ↑↑ (q) and S ↑↓ (q), respectively, Z 2 dq g↑↑ (r) = 1 + (S ↑↑ (q) − 1)ei q · r ; (14) n (2 )3 Z 2 dq ↑↓ g↑↓ (r) = 1 + S (q)ei q · r : (15) n (2 )3 In the previous paper [7], we have proved that only the functional derivatives of particle–particle ladder parts in the Goldstone perturbation expansion with respect to v(q) make a contribution to the predominant asymptotic form of S ↑↓ (q) of order q−4 for large q and their total contributions amount exactly to 4 rs − 3
pf q
4
g↑↓ (0) :
In this paper we shall also proceed along the lines of the previous paper.
(16)
M. Horiuchi et al. / Physica A 255 (1998) 48–64
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Fig. 1. An in nite series of particle–particle ladder energy diagrams and their exchange counter parts.
We shall begin with the lowest-order perturbation term of the ground-state energy that is given in the Hartree–Fock approximation. EHF = −
1 1 X 0 f(p) f(p0 ) v(p − p0 ); 2 pp0
v(q) =
4 e2 ; q2
(17)
0
where f(p) is the Fermi distribution function at 0 K. The functional di erentiation of Eq. (17) with respect to v(q) gives S ↑↓ (q) in this approximation. ↑↑ (q) = 1 − SHF
21 X f(p) f(p − q) : n p
The corresponding spin-parallel pair correlation function can be written as 2 Z Z 0 1 dp dp0 2 ↑↑ gHF (r) = f(p) f(p0 ) | 1 − ei(p−p ) · r |2 : 3 3 n (2) (2) 2
(18)
(19)
Its second derivative at zero separation is ( 25 ) pf2 . In this approximation there is no ↑↓ correlation between any two electrons with antiparallel spins and gHF (r) = 1. Next, we shall consider those energy diagrams which are constructed from an in nite series of particle–particle ladder interactions (see Fig. 1). Note that no self-energy corrections are attached to the particle or hole lines in the diagrams. The ground-state energy shift due to these ladder diagrams can formally be written as follows: ↑↓ ↑↑ + Eladd ; Eladd = Eladd
↑↓ = Eladd
1 1 XX (1 − 0 ) v(q) 2 2 q pp0 0
(20)
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
×
↑↑ = Eladd
f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) I (p; p0 ; q) ; p − p+q + p0 − p0 −q
(21)
f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) 1 1 XX 0 v(q) 2 2 q pp0 p − p+q + p0 − p0 −q 0
× {I (p; p0 ; q) − I (p; p0 ; −p + p0 − q)} ;
(22)
↑↓ ↑↑ and Eladd are spin-antiparallel and spin-parallel contributions, respecwhere Eladd tively and the particle–particle ladder interaction I (p; p0 ; q) is the solution of the following integral equation:
I (p; p0 ; q) = v(q) +
1 X (1 − f(p + k))(1 − f(p0 − k)) v(q − k) I (p; p0 ; k) :
p − p+k + p0 − p0 −k k
(23) The interaction I (p; p0 ; q) has the symmetry property I (p; p0 ; −q) = I (p; p0 ; q). By taking ↑↓ ↑↑ and Eladd with respect to v(q) we can readily obtain the functional derivative of Eladd ↑↓ ↑↑ Sladd (q) and Sladd (q) in this approximation, ↑↓ (q) = Sladd
2 1 X f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) 0) (1 − I (p; p0 ; q) n 2 pp0 p − p+q + p0 − p0 −q 0
+
X (1 − f(p + k))(1 − f(p0 − k)) 1 1 X 0) (1 − I (p; p0 ; k) n 3 pp0 p − p+k + p0 − p0 −k k
0
×
↑↑ (q) = Sladd
(1 − f(p + q + k))(1 − f(p0 − q − k)) I (p; p0 ; q + k) : p − p+q+k + p0 − p0 −q−k
(24)
2 1 X f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) 0 2 n pp0 p − p+q + p0 − p0 −q 0
× {I (p; p0 ; q) − I (p; p0 ; −p + p0 − q)} +
X (1 − f(p + k))(1 − f(p0 − k)) 1 1 X 0 I (p; p0 ; k) 3 n pp0 p − p+k + p0 − p0 −k 0
×
k
(1 − f(p + q + k))(1 − f(p0 − q − k)) p − p+q+k + p0 − p0 −q−k
× {I (p; p0 ; q + k) − I (p; p0 ; −p + p0 − q − k)} :
(25)
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The spin-antiparallel pair correlation function corresponding to Eq. (24) can be written as ↑↓ gladd (r) =
2 Z Z dp dp0 2 f(p)f(p0 ) n (2)3 (2)3 2 Z 0 (1 − f(p + q))(1 − f(p − q)) dq 0 iq · r I (p; p ; q) e × 1+ : 3 0 0 (2) − + − p p+q p p −q (26)
The spin-parallel pair correlation function, when incorporated with the Hartree–Fock expression, can be written in a closed form ↑↑ ↑↑ ↑↑ (r) = gHF (r) + gladd (r) gHF+ladd 2 Z Z 1 dp dp0 2 f(p) f(p0 ) = 3 3 n (2) (2) 2 Z 0 dq (1 − f(p + q))(1 − f(p0 − q)) × (1 − ei(p−p ) · r ) + (2)3 p − p+q + p0 − p0 −q 2 0 × I (p; p0 ; q)e−iq · r (1 − ei(p−p +2q) · r ) : (27)
Eq. (27) vanishes at zero separation as it should. For the convenience of the forthcoming discussions we shall here give the expression for the second derivative of g(r) at zero separation: ↑↑ (r) d2 gHF+ladd dr 2 r=0 Z 1 2 dq 2 ↑↑ =− q (S (q)HF+ladd − 1) 3 n (2)3 =
2 Z Z 2 dp dp0 f(p) f(p0 ) 3 n (2) (2)3 Z dq (1 − f(p + q))(1 − f(p0 − q)) 0 × (p − p ) + (2)3 p − p+q + p0 − p0 −q 1 3
2 × I (p; p0 ; q) (p − p0 + 2q) :
(28)
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
Now, let us consider the predominant asymptotic form of S ↑↑ (q) for large q. Obviously, S ↑↑ (q) consists of two parts: direct and exchange; it includes the expression for S ↑↓ (q) as the direct part. We shall rst consider the asymptotic form of the direct part, namely S ↑↓ (q). As q becomes large enough, the interaction I (p; p0 ; q) tends to Z 0 dk (1 − f(p + k))(1 − f(p − k)) 0 I (p; p ; k) : (29) v(q) 1 + (2)3 p − p+k + p0 − p0 −k Note that the second term on the right-hand side of Eq. (24) is written in a convolution form with respect to the integral over k. In this case, one must pay special care in taking its asymptotic form for large q. Eventually, we are led to the following expression for the asymptotic form of S ↑↓ (q) for large q: Z v(q) n dk ↑↓ ↑↓ + (q) → − S (k) + · · · Sladd q 2 (2)3 ladd 4 rs =− 3
pf q
4
↑↓ gladd (0) + · · · :
(30)
The exchange part of S ↑↑ (q) has quite the same asymptotic form of order q−4 with the opposite sign and as a consequence of the cancellation S ↑↑ (q) has the asymptotic form of order q−6 for large q. The asymptotic form of S ↑↑ (q) of order q−6 cannot be obtained from anywhere except a direct and exchange pair of ladder interactions. A straightforward but tedious calculation leads us to the following expression for the ↑↑ ↑↑ (q) + Sladd (q) for large q: asymptotic form of SHF ↑↑ SHF+ladd (q) − 1 Z Z v(q) 1 2 dp0 dp → f(p) f(p0 ) q q 2 n (2)3 (2)3 Z dq (1 − f(p + q))(1 − f(p0 − q)) × (p − p0 ) + (2)3 p − p+q + p0 − p0 −q
2 × I (p; p0 ; q) (p − p0 + 2q) v(q) 1 q q 2
Z
dk 2 ↑↑ k (SHF+ladd (k) − 1) + · · · (2)3 r p 6 d2 g↑↑ s f HF+ladd (r) +··· : =4 q d(pf r)2 =−
(31)
pf r=0
That is, the asymptotic form of S ↑↑ (q) − 1 of order q−6 for large q occurs only as a consequence of the cancellation between a direct and exchange pair of ladder
M. Horiuchi et al. / Physica A 255 (1998) 48–64
57
interactions and its coecient is closely related to the second derivative of the corresponding spin-parallel pair correlation function at zero separation. As is evident from the derivation above, the self-consistency imposed upon the asymptotic form above is intrinsic to an in nite series of ladder interactions between two particle states with the same spin. ↑↓ ↑↑ ↑↑ and EHF + Eladd It is to be noted that there is an alternative expression for Eladd using the q → 0 limit of the ladder interaction I (p; p0 ; q). 11 X ↑↓ Eladd = (1 − 0 )f(p)f(p0 ) lim (I (p; p0 ; q) − v(q)) ; (32) q→0 2 pp0 0
↑↑ ↑↑ + Eladd = EHF
11 X 0 f(p)f(p0 ) 2 pp0 0
× lim (I (p; p0 ; q) − v(q) − I (p; p0 ; −p + p0 − q)) : q→0
(33)
Here it is interesting that the lowest order of the exchange ladder interaction I (p; p0 ; −p + p0 − q) gives rise to the Hartree–Fock energy in the limit q → 0. With the expression above one can easily understand that the asymptotic forms of S ↑↓ (q) and S ↑↑ (q) for large q are ascribed to the following properties of the ladder interaction between two electrons with anti-parallel and parallel spins, respectively. As q becomes large enough limk→0 (I (p; p0 ; k0 ) − v(k)) v(q) ) ( X lim 0 (I (p; p0 ; k) − v(k0 )) 1 v(q) k →0 ; 1+ →−
q v(k)
(34)
k
limk→0 (I (p; p0 ; k) − v(k) − I (p; p0 ; −p + p0 − k)) v(q) 1 v(q) 1 X 2 limk0 →0 (I (p; p0 ; k0 ) − v(k0 ) − I (p; p0 ; −p + p0 − k0 )) →− : k
q q2 v(k) k
(35) 3. Particle–particle ladder parts of larger diagrams In the previous section, we have proved that an in nite series of ladder interactions between two particle states with the same spin contribute to the predominant asymptotic form of S ↑↑ (q) of order q−6 for large q and its coecient is closely related to the second derivative of the corresponding spin-parallel pair correlation function at zero separation. In this section we shall prove that the relation above holds generally. That is, any direct and exchange pair of particle–particle ladder parts in larger Goldstone
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
Fig. 2. Direct and exchange particle–particle ladder parts to be substituted independently for all the interactions in the irreducible diagrams.
diagrams also contribute to the asymptotic form of S ↑↑ (q) of order q−6 for large q and its coecient is rigorously given by the second derivative of the spin-parallel pair correlation function at zero separation. We shall consider every Goldstone diagram for E and reduce all the ladder parts in it to the simple interaction v(q). By ladder part of Goldstone diagrams we mean that there are no interactions elsewhere while the ladder interactions occur. All these diagrams thus obtained are called irreducible. Therefore, any irreducible diagram no longer contributes to the predominant asymptotic form of S ↑↓ (q) or S ↑↑ (q). All diagrams can be restored by substituting ladder parts independently for all the interactions v(k) in these irreducible diagrams (see Fig. 2). A ladder part of larger diagrams satis es the following integral equation: I (p; p0 ; k; ER ) = v(k) +
1 X (1 − f(p + k0 ))(1 − f(p0 − k0 )) v(k − k0 )
0 −p+k0 − p0 −k0 − ER k
0
0
× I (p; p ; k ; ER ) ;
(36)
where p; p0 are the momenta of two electrons to be engaged in the interaction and k is the total momentum transferred by the ladder part. ER is the excitation energy of the other electrons and holes present while the ladder interactions occur. It takes di erent values according as which interaction in irreducible diagrams the ladder part is to be substituted for. ER is related to the excitation energy of the intermediate state at the beginning of the interaction, E, as −ER = p + p0 − E and does not involve the Coulomb interaction at all. A ladder part between two electrons with anti-parallel spins is represented by I (p; p0 ; k; ER ). A ladder part between electrons with parallel spins, on the other hand, is represented by I (p; p0 ; k; ER ) − I (p; p0 ; −p + p0 − k; ER ) since it is always accompanied by the exchange counterpart.
M. Horiuchi et al. / Physica A 255 (1998) 48–64
59
Let us take the functional derivatives of I (p; p0 ; k; ER ) and I (p; p0 ; k; ER )−I (p; p0 ; −p + p0 − k; ER ) with respect to v(q) and examine how the two derivatives behave in the limit of large q. The result can be written in a compact form. As q becomes large enough, 1 v(q) X I (p; p0 ; k; ER ) I (p; p0 ; k; ER ) →− ; v(q)
q v(k0 ) 0
(37)
k
(I (p; p0 ; k; ER ) − I (p; p0 ; −p + p0 − k; ER )) v(q) →−
1 v(q) 1 X 0 2 (I (p; p0 ; k; ER ) − I (p; p0 ; −p + p0 − k; ER )) : k
q q2 0 v(k0 )
(38)
k
Now, it is evident that the functional derivative of all the ladder parts appearing in larger Goldstone diagrams, together with the derivative of a single ladder part treated in the previous section, gives the exact predominant asymptotic form of S ↑↓ (q) or S ↑↑ (q) for large q. The contributions from larger Goldstone diagrams can be written as −
1 v(q)
q
X
XXX
sum over all the diagrams with ladder parts
ER
k
···
X I (p; p0 ; k; ER )
pp0
k0
v(k0 )
··· ;
(39) −
1 v(q) 1
q q 2 X
k0
X
XXX
sum over all the diagrams with ladder parts
2 (I (p; p
k0
0
ER
k
···
pp0
; k; ER ) − I (p; p0 ; −p + p0 − k; ER )) ··· ; v(k0 )
(40)
P where ER denotes the sum over all the positions of ladder parts in a larger Goldstone diagram and by · · · · · · we have represented the factors coming from the other parts of the diagram than the ladder part of which we have just taken the functional derivative with respect to v(q); it is therefore possible that the other ladder parts may be involved in the part denoted by · · · · · · . To sum up we have the following results: v(q) S (q) → − q ↑↓
S ↑↑ (q) − 1 → −
! 1 X ↑↓ n + S (k) + · · · ; 2
(41)
k
v(q) 1 1 X 2 ↑↑ k (S (k) − 1) + · · · : q q 2
k
(42)
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
Next, we shall the predominant asymptotic form of the momentum distribuD discuss E † tion function ak ak for large k. From quite a similar analysis we have, in a previous paper [16], proved that as k becomes large enough, D
a†k ak
E
→ =
v(k) 2 k
! n 1 X ↑↓ 0 n2 + S (k ) + · · · ; 4 2 0
2
4 r 2 p 8 s
9
f
k
k
g↑↓ (0) + · · · :
(43)
That is, the momentum distribution function ha†k ak i for k /pf is closely related to the value of g↑↓ (r) at zero separation. The point is that in the evaluation of E= k only the functional derivatives of particle–particle ladder parts contribute to the predominant asymptotic form of order k −8 for large k. This is because, when one di erentiates an energy denominator in the expression with respect to k , the very denominator and its neighboring two Coulomb interactions alone are connected with the variable k and give rise to order k −8 for large k; k is not involved in the other parts at all. The above relation is a direct consequence of the following properties of the ladder interaction. As k becomes large enough, limq→0 (I (p; p0 ; q) − v(q)) k 1 →
v(k) 2 k
2 ( 1+
X limq→0 (I (p; p0 ; q) − v(q)) v(k0 )
k0
1 I (p; p0 ; q; ER ) → k
v(k) 2 k
2 X k0
) ;
I (p; p0 ; q; ER ) : v(k0 )
(44)
(45)
So far we have treated a paramagnetic electron liquid. In the fully ferromagnetic electron liquid, on the other hand, the above asymptotic form of order k −8 for large k completely vanishes since there are no ladder interactions between two electrons with anti-parallel spins at all. Instead, the following predominant asymptotic form of order k −10 for large k appears as an immediate consequence of the cancellation between a direct and exchange pair of ladder interactions between two electrons with parallel spins: D
a†k ak
E
→−
=
v(k) 2 k
2
1 n 1 X 2 ↑↑ q (S (q) − 1) + · · · k2 2 q
4 rs 2 pf 10 3 k
d2 g↑↑ (r) d(pf r)2
pf r = 0
+ ··· :
(46)
M. Horiuchi et al. / Physica A 255 (1998) 48–64
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The basic properties of the ladder interaction underlying Eq. (46) are given as follows. As k becomes large enough, limq→0 (I (p; p0 ; q) − v(q) − I (p; p0 ; −p + p0 − q)) k →−
2 1 v(k) 1 X 02 limq→0 (I (p; p0 ; q) − v(q) − I (p; p0 ; −p+p0 −q)) ; k
2 k k2 0 v(k0 ) k
(47) (I (p; p0 ; q; ER ) − I (p; p0 ; −p + p0 − q; ER )) k →−
1
v(k) 2 k
2
1 X 02 (I (p; p0 ; q; ER ) − I (p; p0 ; −p + p0 − q; ER )) : k k2 0 v(k0 ) k
(48) The derivation of the relations above is quite parallel to that of Eqs. (34) and (35) in the preceding paragraph. A virial-theorem-like relation holds between two functional derivatives of the ladder part for suciently large k, 4 k lim (I (p; p0 ; q) − v(q)) = 0 ; + v(k) k v(k) q→0 4 k I (p; p0 ; q; ER ) = 0 : + v(k) (49) k v(k) Another relation holds in the fully ferromagnetic electron liquid. For suciently large k, 4 k lim (I (p; p0 ; q) − v(q) − I (p; p0 ; −p + p0 − q)) = 0 ; − v(k) k v(k) q→0 4 k (I (p; p0 ; q; ER ) − I (p; p0 ; −p + p0 − q; ER )) = 0 : − v(k) (50) k v(k) Note that in Eq. (50) the minus sign is attached to the functional derivative with respect to the Coulomb interation, in contrast to Eq. (49). We shall nish this section with an approximate evaluation of the second derivative of g↑↑ (r) at zero separation, which appears in Eqs. (5) and (46). The second derivative of g↑↑ (r) at zero separation can be written as Z 2 ↑↑ 1 2 dq 2 ↑↑ d g (r) = − q (S (q) − 1) : (51) 2 dr 3 n (2)3 r=0 The lowest-order correction to the Hartree–Fock value comes from a direct and exchange pair of second-order perturbation terms which are included in the
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
particle–particle ladder series and is given as 2 ↑↑ rs 2 2 d g (r) p : 1 − = f dr 2 5 3:45 r=0
(52)
We have estimated the above lowest-order contribution from the corresponding value in the fully ferromagnetic case which has been calculated numerically in a paper by Rajagopal et al. [14]. Their value is ! ↑↑ d2 gfully rs 2 2 ferro (r) p : (53) 1 − = f dr 2 5 2:9 r=0
A di erence between the two corrections relative to each Hartree–Fock value arises 1=3 from a factor of ( 32 ) · ( 12 ). The in uence of the Coulomb interaction on the second derivative in the fully ferromagnetic case is enhanced compared with the paramagnetic case, as it should be. Next, we shall make an approximate estimate of higher-order corrections. The particle–particle ladder approximation is essential for the evaluation of the second derivative of g↑↑ (r) at zero separation as well as g↑↓ (0). This can easily be seen from the fact that it satis es a couple of cusp conditions. Let us start with an analytic expression for g↑↓ (0) in the particle–particle ladder approximation, which has been given by one of the authors [5,6]: 2 rs 21=2 ↑↓ g (0) = ; (54) ; = I1 (41=2 ) where I1 (x) is the rst-order modi ed Bessel function. Then we shall adjust the variable in the above formula so as to reproduce exactly the high-density expansion of the ratio of the second derivative to its Hartree–Fock value. By replacing by 0:437 in the formula we can thus obtain an approximate formula for the second derivative of g↑↑ (r) at zero separation, which may be expected to be valid throughout the whole region of metallic densities. 2 ↑↑ 2 2 2 d g (r) 2(0:437)1=2 = pf : (55) dr 2 5 I1 (4(0:437)1=2 ) r=0 4. Concluding remarks Because of the Pauli principle, the spin-parallel pair correlation function g↑↑ (r) is xed to zero at zero separation. Under the in uence of the Coulomb interaction g↑↑ (r) is depressed on the short distance side and enhanced beyond unity on the long distance side so as to conserve the sum rule Z n dr(g↑↑ (r) − 1) = − 1 : (56) 2 The above in uence of the Coulomb interaction on the Fermi hole is minor compared with the Coulomb hole, but it gives rise to signi cant lowering in the potential energy.
M. Horiuchi et al. / Physica A 255 (1998) 48–64
63
In the present paper we have clari ed the physical implications of cusp conditions on g↑↓ (r) and g↑↑ (r) from the diagrammatic analysis of many-body perturbation theory. That is, particle–particle ladder interactions are essential for the description of cusp conditions. In addition, we have shown that the particle–particle ladder approximation not only satis es cusp conditions but also gives a reasonable evaluation of the second derivative of g↑↑ (r) at zero separation as well as g↑↓ (0), throughout the whole region of metallic densities. The magnitude of g↑↓ (0) can be regarded as a measure of the in uence of shortrange Coulomb repulsion on the Coulomb hole. In a quite analogous way, the ratio of the second derivative of g↑↑ (r) at zero separation to its Hartree–Fock value can be regarded as a measure of the in uence of short-range Coulomb repulsion upon the Fermi hole. Both of the two show a similar trend as a function of rs . The particle–particle ladder approximation is essential for the proper description of the short-distance behavior of g↑↓ (r) and g↑↑ (r), but it is quite inappropriate for the purpose of describing their long-distance behaviors. This can readily be understood from the fact that in the evaluation of correlation energy it includes the second-order direct term with no screening e ect involved, thus leading to divergence. What longdistance behavior of g↑↑ (r) and g↑↓ (r) in the ladder approximation is responsible for the divergence? This question is closely related to the origin of the logarithmic term, ln rs appearing in the high-density expansion of correlation energy. The spin-parallel and spin-antiparallel parts of the static structure factor derived from the ladder series both behave like −rs pf =q over a range of 0¡q=pf . 1 for very small rs . According to Fourier transformation, this means that both of g↑↓ (r) − 1 and g↑↑ (r)(= g↑↑ (r) − ↑↑ (r)) behave like −rs =(pf r)2 over a range of pf r¿1, thus leading to divergence gHF of the potential energy. The RPA considers the most divergent series of perturbation terms. In the high-density limit it amounts to cutting o the above term of −rs pf =q derived from the second-order direct energy such that rs1=2 ¡q=pf . 1. In real space, the above behavior of g↑↑ (r) and g↑↓ (r) in the ladder approximation is cut o such that 1¡pf r¡rs−1=2 , thus giving the term of order ln rs . We note that so far as the leading term of order ln rs is concerned, spin-parallel and spin-antiparallel correlations both make an equivalent contribution to the correlation energy. So far, we have treated the homogeneous electron liquid. The cusp conditions, of course, hold for any inhomogeneous electron systems and can be generally written as r ↑↓ g (0; R) + · · · ; a0 2 ↑↑ r r2 d g (r; R) 1+ g↑↑ (r; R) = + ··· ; 2 2a0 dr 2 r=0
g↑↓ (r; R) = g↑↓ (0; R) +
0
(57) (58)
where the pair correlation function g (r1 ; r2 ) are rewritten in terms of the inter particle separation, r(= r1 − r2 ) and the center of gravity, R(= (r1 + r2 )=2). In principle, these conditions can also be derived from the diagrammatic analysis of perturbation theory. It should be emphasized that the particle–particle ladder series of perturbation terms are
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
essential for the satisfaction of these conditions and the proper evaluation of g↑↓ (0; R) and d2 g↑↑ (r; R)=dr 2 |r=0 . As has been widely recognized, the theoretical formulation that is capable of dealing with divergence caused by long-range parts of the Coulomb interaction is indispensable for the description of long-range correlation such as the existence of plasmon excitations and its counterpart e ect, screening. Almost all approaches to the study of electron correlation have therefore been advanced from the long-range side. The cusp conditions, on the other hand, are a characteristic feature caused by short-range parts of the Coulomb interaction. Their importance seems to have been overlooked in condensed matter physics. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J. Hubbard, Proc. Roy. Soc. A 240 (1957) 530. J. Hubbard, Proc. Roy. Soc. A 243 (1958) 336. P. Nozieres, D. Pines, Phys. Rev. 111 (1958) 442. K.S. Singwi, M.P. Tosi, R.H. Land, A. Sjolander, Phys. Rev. 176 (1968) 589. H. Yasuhara, Solid State Commun. 11 (1972) 1481. H. Yasuhara, J. Phys. Soc. Japan 36 (1974) 361. H. Yasuhara, Physica 78 (1974) 420. Y. Ousaka, H. Suehiro, H. Yasuhara, J. Phys. C 19 (1986) 4247. Y. Ousaka, H. Suehiro, H. Yasuhara, J. Phys. C 19 (1986) 4263. H. Yasuhara, H. Suehiro, Y. Ousaka, J. Phys. C 21 (1988) 4045. T. Kato, Comm. Pure Appl. Math. 10 (1957) 151. J.C. Kimball, Phys. Rev. A 7 (1973) 1648. J.C. Kimball, J. Phys. A 8 (1975) 1513. A.K. Rajagopal, J.C. Kimball, M. Banerjee, Phys. Rev. B 18 (1978) 2339. See, for example, in: A.L. Fetter and J.D. Walecka (Eds.), Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. [16] H. Yasuhara, Y. Kawazoe, Physica A 85 (1976) 416.
In uence of the Coulomb interaction on the Fermi hole in an electron liquid Masafumi Horiuchi ∗ , Takanori Endo, Hiroshi Yasuhara Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Received 10 July 1997
Abstract From the diagrammatic analysis, it is proved that the spin-parallel part of the static structure factor of an electron liquid, S ↑↑ (q), has the following exact asymptotic form for q/pf : S ↑↑ (q) − 1 = 4( rs =)(pf =q)6 d2 g↑↑ (r)=d(pf r)2 |pf r=0 + · · ·, = (4=9)1=3 , where pf is the Fermi wave number and g↑↑ (r) the spin-parallel pair correlation function. The second derivative of c 1998 g↑↑ (r) at zero separation is evaluated in the particle–particle ladder approximation. Elsevier Science B.V. All rights reserved PACS: 71.10.Ca Keywords: Electron gas; Pair correlation function; Short-range correlation; Cusp conditions
1. Introduction In the early study of the electron gas model, Hubbard [1,2] and Nozieres–Pines [3] interpolated between long- and short-range correlations in an attempt to construct a theory valid for intermediate densities. They treated long-range correlation associated with small momentum transfers in the RPA, while in their treatment of short-range correlation associated with large momentum transfers they resorted to second-order perturbation theory. Their treatment of short-range correlation between spin-antiparallel electrons, however, still remained inappropriate for such a strongly correlated region. Singwi and his co-workers [4] were the rst who have succeeded in a good description of short-range correlation at metallic densities as well as long-range correlation. The pair correlation function g(r) calculated from their theory remains positive at short distances throughout almost the whole region of metallic densities. We shall hereafter refer to their work as the STLS theory. They have begun with a classical electron gas in the presence of an external eld and calculated the dielectric function (q; !). ∗
Corresponding author. Fax: +8122 217 7754; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 0 7 0 - 3
M. Horiuchi et al. / Physica A 255 (1998) 48–64
49
The basic assumption in the STLS theory is that the two-particle distribution function f(p; r; p0 ; r0 ; t) in the Liouville equation can be replaced by f(p; r; t)f(p0 ; r0 ; t)g(r − r0 ) where f(p; r; t) is one particle distribution function and g(r − r0 ) the pair correlation function in the equilibrium state. The local eld factor G(q) entering their expression for the dielectric function can then be given as a functional of the static structure factor S(q), which is the Fourier transform of g(r). They have obtained a set of equations to determine S(q) or G(q) self-consistently with the help of the uctuation–dissipation theorem, which connects S(q) with the frequency integral of the imaginary part of the inverse dielectric function. Finally, they have made the quantum version of theory by replacing the classical non-interacting polarization function and the static structure factor S(q) by their quantum analogues. A careful examination of the STLS theory reveals that it satis es the following relation: r g(0) + · · · g(r) = g(0) + a0 1=3 4 = 0:5210 · · · ; (1) = g(0) + rs pf r g(0) + · · · ; = 9 where rs is the usual density parameter, given by the radius of a sphere with volume per electron, measured in units of the Bohr radius a0 ; (4=3)(a0 rs )3 = n−1 , n being the average electron density. According to Fourier transformation, (1) is equivalent to the following self-consistent requirement imposed upon the asymptotic form of S(q) for large q, 4 8 rs pf S(q) = 1 − g(0) + · · · ; 3 q Z 1 dq g(r) = 1 + (S(q) − 1)ei q · r : (2) n (2 )3 In fact, Eq. (1), or Eq. (2) is exact and often referred to as the cusp condition. This contains the following physical meaning: In the short-distance limit of their interspace, the motion of any two spin-antiparallel electrons of many-body systems is essentially reduced to the two-body problem in the medium and a close relation between the value of the wave function at zero separation and its derivative is established under the in uence of the short-distance singularity of the Coulomb interaction. The cusp condition can readily be derived by examining the analyticity of two-electron interacting parts of the many-body wave function, but the value of the pair correlation function at zero separation, g(0) itself cannot possibly be obtained without any practical calculations. The satisfaction of the cusp condition is, therefore, necessary but not enough for the description of short-range correlation. A clear understanding of short-range correlation is now acquired in the diagrammatic language of many-body perturbation theory [ 5 –7], instead of resorting to the STLS theory. Consider an in nite series of particle–particle ladder interactions and all the ladder interacting parts in the Goldstone perturbation expansion. This type of interactions
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
between spin-antiparallel electrons exclusively contribute to the predominant asymptotic form of S(q) − 1, or S ↑↓ (q) of order q−4 for large q and their total contributions can be written exactly as 4 4 rs pf S ↑↓ (q) = − g↑↓ (0) + · · · ; (3) 3 q or equivalently the following relation holds for small r: r ↑↓ g (0) + · · · : g↑↓ (r) = g↑↓ (0) + a0
(4)
That is, particle–particle ladder interactions have the intrinsic property of satisfying the cusp condition. Furthermore, the particle–particle ladder approximation which involves a single ladder series enables one to evaluate g↑↓ (0) itself. For a paramagnetic electron liquid g(r) is the arithmetic average of the spin-parallel and the spin-antiparallel pair correlation functions, g↑↑ (r) and g↑↓ (r). The value of g(0) then equals g↑↓ (0)=2 because g↑↑ (0) = 0 due to the Pauli principle. One of the authors (H. Y.) and his co-workers [8–10] have numerically showed that the ladder approximation gives a reasonable evaluation of g↑↓ (r) and g↑↑ (r) at short distances (06pf r . 3:0) for any rs . The reason for the success of the STLS theory probably may be that it includes ladder interactions in a sense through their self-consistent scheme. In fact, there is a close resemblance between the equation of G(q) in the STLS theory and the integral equation of the particle–particle ladder interaction for large q. The present paper is a straightforward extension of a previous paper [7]. From the diagrammatic analysis we shall show that the spin-parallel part of the static structure factor S ↑↑ (q) has the following exact asymptotic form of order q−6 for q/pf : r p 6 d2 g↑↑ (r) s f ↑↑ S (q) − 1 = 4 + ··· ; (5) q d(pf r)2 pf r=0 where the spin-parallel pair correlation function g↑↑ (r) is the Fourier transform of S ↑↑ (q), Z 2 dq ↑↑ (S ↑↑ (q) − 1)ei q · r : (6) g (r) = 1 + n (2 )3 The asymptotic form above occurs as a consequence of the complete cancellation between the contribution of all the direct particle–particle ladder interacting parts and that of their exchange counterparts; each contribution is of order q−4 in the same way as S ↑↓ (q). That is, a direct and exchange pair of ladder interactions between two particle states with the same spin have the property of satisfying the above self-consistent requirement imposed upon the asymptotic form of S ↑↑ (q) for large q. The Fourier transformation of Eq. (5) means that g↑↑ (r) can rigorously be written for pf r.1 in the form 2 ↑↑ r r2 d g (r) 1+ g↑↑ (r) = + ··· : (7) 2 2 a0 dr 2 r=0 This is another cusp condition on g↑↑ (r).
M. Horiuchi et al. / Physica A 255 (1998) 48–64
51
These cusp conditions on g↑↓ (r) and g↑↑ (r) have originally been noticed by chemists and for a long time applied to their study of electron correlation in molecules, without being communicated to physicists who study electron correlation in condensed matter. Kato [11] has established the mathematical foundation on the cusp behavior of manyelectron wave functions. Kimball [12,13] has rederived the cusp condition on g↑↓ (r). Rajagopal et al. [14] have stressed the importance of the cusp condition on g↑↑ (r) in the fully ferromagnetic electron liquid. A concise derivation of these cusp conditions is as follows: In the short-distance limit of their interspace, the motion of two electrons in many-body systems is dominated only by their mutual Coulomb interaction. The radial part of the wave function of the two electrons with relative angular momentum l then satis es the following equation: −
˜2 1 d 2 r 2 dr
r2
dRl (r) dr
+
e2 l (l + 1) ˜2 Rl (r) + Rl (r) = E Rl (r) ; r 2 r2
(8)
where is the e ective mass and E denotes a many-body operator which is bounded as r → 0. Substituting into the equation an assumed form of the wave function at short distances, Rl (r) = a + b r + · · · (or c r + d r 2 + · · ·) for l = 0 (or 1), we can immediately obtain b = a=(2 a0 ) and d = c=(4 a0 ) from the analyticity of E. The two equalities above are nothing but the cusp conditions. An alternative derivation of the cusp conditions in the present paper will be helpful to the thorough understanding of their physical implications. Furthermore, it enables us to evaluate the second derivative of g↑↑ (r) at zero separation in the particle–particle ladder approximation. The ratio of the second derivative of g↑↑ (r) at zero separation to its value in the Hartree–Fock approximation can be regarded as a measure of the in uence of short-range Coulomb repulsion upon the Fermi hole. It should be compared with the value of g↑↓ (0). From the diagrammatic analysis, we shall also show that, in the fully ferromagnetic electron liquid, the asymptotic form of S ↑↑ (q) for large q is closely related to the asymptotic form of the momentum distribution function n(p) for large p. As has been well known, the term of order ln rs leads the high-density expansion of correlation energy per electron, in units of Rydberg. However, this does not seem to be understood in terms of g↑↑ (r) and g↑↓ (r). We shall then comment on what features of these two functions are responsible for the logarithmic term.
2. Particle–particle ladder approximation As is evident from the Goldstone perturbation formula, the ground-state energy shift of an electron liquid due to the Coulomb interaction, E can be regarded as a functional of the Coulomb interaction v(q) and the free-electron energy spectrum p . Then, 0 the following expressions for the static structure factor S (q) and the momentum distribution function n (p) result if one proceeds along the procedure of deriving the
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
coupling constant integration algorithm that can be seen in most textbooks on manybody theory [15]: 0 n E = (S (q) − 0 ); 0 v (q) 2
0
S (q) =
1 D 0 E q −q ; N
(E0 + E) = n (p) = a†p ap ; p
(9) (10)
where q is the density uctuation operator of electrons with spin and ap the annihilation operator of an electron with wavenumber p and spin ; E0 is the ground-state energy of an non-interacting Fermi gas; N is the total electron number in the volume
of the system and n the average electron density; h· · ·i denotes the expectation value with respect to the exact ground state. The expressions above are convenient especially for evaluating the asymptotic be0 havior of S (q) and n (p) for large q and p, respectively. The Coulomb interaction in an electron liquid, of course, does not depend on the spin orientation, but we introduce 0 here the notation v (q) for the convenience of specifying the interaction between two electrons with spins and 0 . The static structure factor S(q) is de ned as 1 X D 0 E S(q) = q −q (11) N 0
and it can be divided into the spin-parallel and spin-antiparallel parts as follows: S(q) = S ↑↑ (q) + S ↑↓ (q) ; S ↑↑ (q) =
1 X q −q ; N
(12) S ↑↓ (q) =
1 X − q −q : N
(13)
The spin-parallel and spin-antiparallel pair correlation functions, g↑↑ (r) and g↑↓ (r) are the Fourier transforms of S ↑↑ (q) and S ↑↓ (q), respectively, Z 2 dq g↑↑ (r) = 1 + (S ↑↑ (q) − 1)ei q · r ; (14) n (2 )3 Z 2 dq ↑↓ g↑↓ (r) = 1 + S (q)ei q · r : (15) n (2 )3 In the previous paper [7], we have proved that only the functional derivatives of particle–particle ladder parts in the Goldstone perturbation expansion with respect to v(q) make a contribution to the predominant asymptotic form of S ↑↓ (q) of order q−4 for large q and their total contributions amount exactly to 4 rs − 3
pf q
4
g↑↓ (0) :
In this paper we shall also proceed along the lines of the previous paper.
(16)
M. Horiuchi et al. / Physica A 255 (1998) 48–64
53
Fig. 1. An in nite series of particle–particle ladder energy diagrams and their exchange counter parts.
We shall begin with the lowest-order perturbation term of the ground-state energy that is given in the Hartree–Fock approximation. EHF = −
1 1 X 0 f(p) f(p0 ) v(p − p0 ); 2 pp0
v(q) =
4 e2 ; q2
(17)
0
where f(p) is the Fermi distribution function at 0 K. The functional di erentiation of Eq. (17) with respect to v(q) gives S ↑↓ (q) in this approximation. ↑↑ (q) = 1 − SHF
21 X f(p) f(p − q) : n p
The corresponding spin-parallel pair correlation function can be written as 2 Z Z 0 1 dp dp0 2 ↑↑ gHF (r) = f(p) f(p0 ) | 1 − ei(p−p ) · r |2 : 3 3 n (2) (2) 2
(18)
(19)
Its second derivative at zero separation is ( 25 ) pf2 . In this approximation there is no ↑↓ correlation between any two electrons with antiparallel spins and gHF (r) = 1. Next, we shall consider those energy diagrams which are constructed from an in nite series of particle–particle ladder interactions (see Fig. 1). Note that no self-energy corrections are attached to the particle or hole lines in the diagrams. The ground-state energy shift due to these ladder diagrams can formally be written as follows: ↑↓ ↑↑ + Eladd ; Eladd = Eladd
↑↓ = Eladd
1 1 XX (1 − 0 ) v(q) 2 2 q pp0 0
(20)
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
×
↑↑ = Eladd
f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) I (p; p0 ; q) ; p − p+q + p0 − p0 −q
(21)
f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) 1 1 XX 0 v(q) 2 2 q pp0 p − p+q + p0 − p0 −q 0
× {I (p; p0 ; q) − I (p; p0 ; −p + p0 − q)} ;
(22)
↑↓ ↑↑ and Eladd are spin-antiparallel and spin-parallel contributions, respecwhere Eladd tively and the particle–particle ladder interaction I (p; p0 ; q) is the solution of the following integral equation:
I (p; p0 ; q) = v(q) +
1 X (1 − f(p + k))(1 − f(p0 − k)) v(q − k) I (p; p0 ; k) :
p − p+k + p0 − p0 −k k
(23) The interaction I (p; p0 ; q) has the symmetry property I (p; p0 ; −q) = I (p; p0 ; q). By taking ↑↓ ↑↑ and Eladd with respect to v(q) we can readily obtain the functional derivative of Eladd ↑↓ ↑↑ Sladd (q) and Sladd (q) in this approximation, ↑↓ (q) = Sladd
2 1 X f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) 0) (1 − I (p; p0 ; q) n 2 pp0 p − p+q + p0 − p0 −q 0
+
X (1 − f(p + k))(1 − f(p0 − k)) 1 1 X 0) (1 − I (p; p0 ; k) n 3 pp0 p − p+k + p0 − p0 −k k
0
×
↑↑ (q) = Sladd
(1 − f(p + q + k))(1 − f(p0 − q − k)) I (p; p0 ; q + k) : p − p+q+k + p0 − p0 −q−k
(24)
2 1 X f(p)(1 − f(p + q))f(p0 )(1 − f(p0 − q)) 0 2 n pp0 p − p+q + p0 − p0 −q 0
× {I (p; p0 ; q) − I (p; p0 ; −p + p0 − q)} +
X (1 − f(p + k))(1 − f(p0 − k)) 1 1 X 0 I (p; p0 ; k) 3 n pp0 p − p+k + p0 − p0 −k 0
×
k
(1 − f(p + q + k))(1 − f(p0 − q − k)) p − p+q+k + p0 − p0 −q−k
× {I (p; p0 ; q + k) − I (p; p0 ; −p + p0 − q − k)} :
(25)
M. Horiuchi et al. / Physica A 255 (1998) 48–64
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The spin-antiparallel pair correlation function corresponding to Eq. (24) can be written as ↑↓ gladd (r) =
2 Z Z dp dp0 2 f(p)f(p0 ) n (2)3 (2)3 2 Z 0 (1 − f(p + q))(1 − f(p − q)) dq 0 iq · r I (p; p ; q) e × 1+ : 3 0 0 (2) − + − p p+q p p −q (26)
The spin-parallel pair correlation function, when incorporated with the Hartree–Fock expression, can be written in a closed form ↑↑ ↑↑ ↑↑ (r) = gHF (r) + gladd (r) gHF+ladd 2 Z Z 1 dp dp0 2 f(p) f(p0 ) = 3 3 n (2) (2) 2 Z 0 dq (1 − f(p + q))(1 − f(p0 − q)) × (1 − ei(p−p ) · r ) + (2)3 p − p+q + p0 − p0 −q 2 0 × I (p; p0 ; q)e−iq · r (1 − ei(p−p +2q) · r ) : (27)
Eq. (27) vanishes at zero separation as it should. For the convenience of the forthcoming discussions we shall here give the expression for the second derivative of g(r) at zero separation: ↑↑ (r) d2 gHF+ladd dr 2 r=0 Z 1 2 dq 2 ↑↑ =− q (S (q)HF+ladd − 1) 3 n (2)3 =
2 Z Z 2 dp dp0 f(p) f(p0 ) 3 n (2) (2)3 Z dq (1 − f(p + q))(1 − f(p0 − q)) 0 × (p − p ) + (2)3 p − p+q + p0 − p0 −q 1 3
2 × I (p; p0 ; q) (p − p0 + 2q) :
(28)
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
Now, let us consider the predominant asymptotic form of S ↑↑ (q) for large q. Obviously, S ↑↑ (q) consists of two parts: direct and exchange; it includes the expression for S ↑↓ (q) as the direct part. We shall rst consider the asymptotic form of the direct part, namely S ↑↓ (q). As q becomes large enough, the interaction I (p; p0 ; q) tends to Z 0 dk (1 − f(p + k))(1 − f(p − k)) 0 I (p; p ; k) : (29) v(q) 1 + (2)3 p − p+k + p0 − p0 −k Note that the second term on the right-hand side of Eq. (24) is written in a convolution form with respect to the integral over k. In this case, one must pay special care in taking its asymptotic form for large q. Eventually, we are led to the following expression for the asymptotic form of S ↑↓ (q) for large q: Z v(q) n dk ↑↓ ↑↓ + (q) → − S (k) + · · · Sladd q 2 (2)3 ladd 4 rs =− 3
pf q
4
↑↓ gladd (0) + · · · :
(30)
The exchange part of S ↑↑ (q) has quite the same asymptotic form of order q−4 with the opposite sign and as a consequence of the cancellation S ↑↑ (q) has the asymptotic form of order q−6 for large q. The asymptotic form of S ↑↑ (q) of order q−6 cannot be obtained from anywhere except a direct and exchange pair of ladder interactions. A straightforward but tedious calculation leads us to the following expression for the ↑↑ ↑↑ (q) + Sladd (q) for large q: asymptotic form of SHF ↑↑ SHF+ladd (q) − 1 Z Z v(q) 1 2 dp0 dp → f(p) f(p0 ) q q 2 n (2)3 (2)3 Z dq (1 − f(p + q))(1 − f(p0 − q)) × (p − p0 ) + (2)3 p − p+q + p0 − p0 −q
2 × I (p; p0 ; q) (p − p0 + 2q) v(q) 1 q q 2
Z
dk 2 ↑↑ k (SHF+ladd (k) − 1) + · · · (2)3 r p 6 d2 g↑↑ s f HF+ladd (r) +··· : =4 q d(pf r)2 =−
(31)
pf r=0
That is, the asymptotic form of S ↑↑ (q) − 1 of order q−6 for large q occurs only as a consequence of the cancellation between a direct and exchange pair of ladder
M. Horiuchi et al. / Physica A 255 (1998) 48–64
57
interactions and its coecient is closely related to the second derivative of the corresponding spin-parallel pair correlation function at zero separation. As is evident from the derivation above, the self-consistency imposed upon the asymptotic form above is intrinsic to an in nite series of ladder interactions between two particle states with the same spin. ↑↓ ↑↑ ↑↑ and EHF + Eladd It is to be noted that there is an alternative expression for Eladd using the q → 0 limit of the ladder interaction I (p; p0 ; q). 11 X ↑↓ Eladd = (1 − 0 )f(p)f(p0 ) lim (I (p; p0 ; q) − v(q)) ; (32) q→0 2 pp0 0
↑↑ ↑↑ + Eladd = EHF
11 X 0 f(p)f(p0 ) 2 pp0 0
× lim (I (p; p0 ; q) − v(q) − I (p; p0 ; −p + p0 − q)) : q→0
(33)
Here it is interesting that the lowest order of the exchange ladder interaction I (p; p0 ; −p + p0 − q) gives rise to the Hartree–Fock energy in the limit q → 0. With the expression above one can easily understand that the asymptotic forms of S ↑↓ (q) and S ↑↑ (q) for large q are ascribed to the following properties of the ladder interaction between two electrons with anti-parallel and parallel spins, respectively. As q becomes large enough limk→0 (I (p; p0 ; k0 ) − v(k)) v(q) ) ( X lim 0 (I (p; p0 ; k) − v(k0 )) 1 v(q) k →0 ; 1+ →−
q v(k)
(34)
k
limk→0 (I (p; p0 ; k) − v(k) − I (p; p0 ; −p + p0 − k)) v(q) 1 v(q) 1 X 2 limk0 →0 (I (p; p0 ; k0 ) − v(k0 ) − I (p; p0 ; −p + p0 − k0 )) →− : k
q q2 v(k) k
(35) 3. Particle–particle ladder parts of larger diagrams In the previous section, we have proved that an in nite series of ladder interactions between two particle states with the same spin contribute to the predominant asymptotic form of S ↑↑ (q) of order q−6 for large q and its coecient is closely related to the second derivative of the corresponding spin-parallel pair correlation function at zero separation. In this section we shall prove that the relation above holds generally. That is, any direct and exchange pair of particle–particle ladder parts in larger Goldstone
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
Fig. 2. Direct and exchange particle–particle ladder parts to be substituted independently for all the interactions in the irreducible diagrams.
diagrams also contribute to the asymptotic form of S ↑↑ (q) of order q−6 for large q and its coecient is rigorously given by the second derivative of the spin-parallel pair correlation function at zero separation. We shall consider every Goldstone diagram for E and reduce all the ladder parts in it to the simple interaction v(q). By ladder part of Goldstone diagrams we mean that there are no interactions elsewhere while the ladder interactions occur. All these diagrams thus obtained are called irreducible. Therefore, any irreducible diagram no longer contributes to the predominant asymptotic form of S ↑↓ (q) or S ↑↑ (q). All diagrams can be restored by substituting ladder parts independently for all the interactions v(k) in these irreducible diagrams (see Fig. 2). A ladder part of larger diagrams satis es the following integral equation: I (p; p0 ; k; ER ) = v(k) +
1 X (1 − f(p + k0 ))(1 − f(p0 − k0 )) v(k − k0 )
0 −p+k0 − p0 −k0 − ER k
0
0
× I (p; p ; k ; ER ) ;
(36)
where p; p0 are the momenta of two electrons to be engaged in the interaction and k is the total momentum transferred by the ladder part. ER is the excitation energy of the other electrons and holes present while the ladder interactions occur. It takes di erent values according as which interaction in irreducible diagrams the ladder part is to be substituted for. ER is related to the excitation energy of the intermediate state at the beginning of the interaction, E, as −ER = p + p0 − E and does not involve the Coulomb interaction at all. A ladder part between two electrons with anti-parallel spins is represented by I (p; p0 ; k; ER ). A ladder part between electrons with parallel spins, on the other hand, is represented by I (p; p0 ; k; ER ) − I (p; p0 ; −p + p0 − k; ER ) since it is always accompanied by the exchange counterpart.
M. Horiuchi et al. / Physica A 255 (1998) 48–64
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Let us take the functional derivatives of I (p; p0 ; k; ER ) and I (p; p0 ; k; ER )−I (p; p0 ; −p + p0 − k; ER ) with respect to v(q) and examine how the two derivatives behave in the limit of large q. The result can be written in a compact form. As q becomes large enough, 1 v(q) X I (p; p0 ; k; ER ) I (p; p0 ; k; ER ) →− ; v(q)
q v(k0 ) 0
(37)
k
(I (p; p0 ; k; ER ) − I (p; p0 ; −p + p0 − k; ER )) v(q) →−
1 v(q) 1 X 0 2 (I (p; p0 ; k; ER ) − I (p; p0 ; −p + p0 − k; ER )) : k
q q2 0 v(k0 )
(38)
k
Now, it is evident that the functional derivative of all the ladder parts appearing in larger Goldstone diagrams, together with the derivative of a single ladder part treated in the previous section, gives the exact predominant asymptotic form of S ↑↓ (q) or S ↑↑ (q) for large q. The contributions from larger Goldstone diagrams can be written as −
1 v(q)
q
X
XXX
sum over all the diagrams with ladder parts
ER
k
···
X I (p; p0 ; k; ER )
pp0
k0
v(k0 )
··· ;
(39) −
1 v(q) 1
q q 2 X
k0
X
XXX
sum over all the diagrams with ladder parts
2 (I (p; p
k0
0
ER
k
···
pp0
; k; ER ) − I (p; p0 ; −p + p0 − k; ER )) ··· ; v(k0 )
(40)
P where ER denotes the sum over all the positions of ladder parts in a larger Goldstone diagram and by · · · · · · we have represented the factors coming from the other parts of the diagram than the ladder part of which we have just taken the functional derivative with respect to v(q); it is therefore possible that the other ladder parts may be involved in the part denoted by · · · · · · . To sum up we have the following results: v(q) S (q) → − q ↑↓
S ↑↑ (q) − 1 → −
! 1 X ↑↓ n + S (k) + · · · ; 2
(41)
k
v(q) 1 1 X 2 ↑↑ k (S (k) − 1) + · · · : q q 2
k
(42)
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Next, we shall the predominant asymptotic form of the momentum distribuD discuss E † tion function ak ak for large k. From quite a similar analysis we have, in a previous paper [16], proved that as k becomes large enough, D
a†k ak
E
→ =
v(k) 2 k
! n 1 X ↑↓ 0 n2 + S (k ) + · · · ; 4 2 0
2
4 r 2 p 8 s
9
f
k
k
g↑↓ (0) + · · · :
(43)
That is, the momentum distribution function ha†k ak i for k /pf is closely related to the value of g↑↓ (r) at zero separation. The point is that in the evaluation of E= k only the functional derivatives of particle–particle ladder parts contribute to the predominant asymptotic form of order k −8 for large k. This is because, when one di erentiates an energy denominator in the expression with respect to k , the very denominator and its neighboring two Coulomb interactions alone are connected with the variable k and give rise to order k −8 for large k; k is not involved in the other parts at all. The above relation is a direct consequence of the following properties of the ladder interaction. As k becomes large enough, limq→0 (I (p; p0 ; q) − v(q)) k 1 →
v(k) 2 k
2 ( 1+
X limq→0 (I (p; p0 ; q) − v(q)) v(k0 )
k0
1 I (p; p0 ; q; ER ) → k
v(k) 2 k
2 X k0
) ;
I (p; p0 ; q; ER ) : v(k0 )
(44)
(45)
So far we have treated a paramagnetic electron liquid. In the fully ferromagnetic electron liquid, on the other hand, the above asymptotic form of order k −8 for large k completely vanishes since there are no ladder interactions between two electrons with anti-parallel spins at all. Instead, the following predominant asymptotic form of order k −10 for large k appears as an immediate consequence of the cancellation between a direct and exchange pair of ladder interactions between two electrons with parallel spins: D
a†k ak
E
→−
=
v(k) 2 k
2
1 n 1 X 2 ↑↑ q (S (q) − 1) + · · · k2 2 q
4 rs 2 pf 10 3 k
d2 g↑↑ (r) d(pf r)2
pf r = 0
+ ··· :
(46)
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The basic properties of the ladder interaction underlying Eq. (46) are given as follows. As k becomes large enough, limq→0 (I (p; p0 ; q) − v(q) − I (p; p0 ; −p + p0 − q)) k →−
2 1 v(k) 1 X 02 limq→0 (I (p; p0 ; q) − v(q) − I (p; p0 ; −p+p0 −q)) ; k
2 k k2 0 v(k0 ) k
(47) (I (p; p0 ; q; ER ) − I (p; p0 ; −p + p0 − q; ER )) k →−
1
v(k) 2 k
2
1 X 02 (I (p; p0 ; q; ER ) − I (p; p0 ; −p + p0 − q; ER )) : k k2 0 v(k0 ) k
(48) The derivation of the relations above is quite parallel to that of Eqs. (34) and (35) in the preceding paragraph. A virial-theorem-like relation holds between two functional derivatives of the ladder part for suciently large k, 4 k lim (I (p; p0 ; q) − v(q)) = 0 ; + v(k) k v(k) q→0 4 k I (p; p0 ; q; ER ) = 0 : + v(k) (49) k v(k) Another relation holds in the fully ferromagnetic electron liquid. For suciently large k, 4 k lim (I (p; p0 ; q) − v(q) − I (p; p0 ; −p + p0 − q)) = 0 ; − v(k) k v(k) q→0 4 k (I (p; p0 ; q; ER ) − I (p; p0 ; −p + p0 − q; ER )) = 0 : − v(k) (50) k v(k) Note that in Eq. (50) the minus sign is attached to the functional derivative with respect to the Coulomb interation, in contrast to Eq. (49). We shall nish this section with an approximate evaluation of the second derivative of g↑↑ (r) at zero separation, which appears in Eqs. (5) and (46). The second derivative of g↑↑ (r) at zero separation can be written as Z 2 ↑↑ 1 2 dq 2 ↑↑ d g (r) = − q (S (q) − 1) : (51) 2 dr 3 n (2)3 r=0 The lowest-order correction to the Hartree–Fock value comes from a direct and exchange pair of second-order perturbation terms which are included in the
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particle–particle ladder series and is given as 2 ↑↑ rs 2 2 d g (r) p : 1 − = f dr 2 5 3:45 r=0
(52)
We have estimated the above lowest-order contribution from the corresponding value in the fully ferromagnetic case which has been calculated numerically in a paper by Rajagopal et al. [14]. Their value is ! ↑↑ d2 gfully rs 2 2 ferro (r) p : (53) 1 − = f dr 2 5 2:9 r=0
A di erence between the two corrections relative to each Hartree–Fock value arises 1=3 from a factor of ( 32 ) · ( 12 ). The in uence of the Coulomb interaction on the second derivative in the fully ferromagnetic case is enhanced compared with the paramagnetic case, as it should be. Next, we shall make an approximate estimate of higher-order corrections. The particle–particle ladder approximation is essential for the evaluation of the second derivative of g↑↑ (r) at zero separation as well as g↑↓ (0). This can easily be seen from the fact that it satis es a couple of cusp conditions. Let us start with an analytic expression for g↑↓ (0) in the particle–particle ladder approximation, which has been given by one of the authors [5,6]: 2 rs 21=2 ↑↓ g (0) = ; (54) ; = I1 (41=2 ) where I1 (x) is the rst-order modi ed Bessel function. Then we shall adjust the variable in the above formula so as to reproduce exactly the high-density expansion of the ratio of the second derivative to its Hartree–Fock value. By replacing by 0:437 in the formula we can thus obtain an approximate formula for the second derivative of g↑↑ (r) at zero separation, which may be expected to be valid throughout the whole region of metallic densities. 2 ↑↑ 2 2 2 d g (r) 2(0:437)1=2 = pf : (55) dr 2 5 I1 (4(0:437)1=2 ) r=0 4. Concluding remarks Because of the Pauli principle, the spin-parallel pair correlation function g↑↑ (r) is xed to zero at zero separation. Under the in uence of the Coulomb interaction g↑↑ (r) is depressed on the short distance side and enhanced beyond unity on the long distance side so as to conserve the sum rule Z n dr(g↑↑ (r) − 1) = − 1 : (56) 2 The above in uence of the Coulomb interaction on the Fermi hole is minor compared with the Coulomb hole, but it gives rise to signi cant lowering in the potential energy.
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In the present paper we have clari ed the physical implications of cusp conditions on g↑↓ (r) and g↑↑ (r) from the diagrammatic analysis of many-body perturbation theory. That is, particle–particle ladder interactions are essential for the description of cusp conditions. In addition, we have shown that the particle–particle ladder approximation not only satis es cusp conditions but also gives a reasonable evaluation of the second derivative of g↑↑ (r) at zero separation as well as g↑↓ (0), throughout the whole region of metallic densities. The magnitude of g↑↓ (0) can be regarded as a measure of the in uence of shortrange Coulomb repulsion on the Coulomb hole. In a quite analogous way, the ratio of the second derivative of g↑↑ (r) at zero separation to its Hartree–Fock value can be regarded as a measure of the in uence of short-range Coulomb repulsion upon the Fermi hole. Both of the two show a similar trend as a function of rs . The particle–particle ladder approximation is essential for the proper description of the short-distance behavior of g↑↓ (r) and g↑↑ (r), but it is quite inappropriate for the purpose of describing their long-distance behaviors. This can readily be understood from the fact that in the evaluation of correlation energy it includes the second-order direct term with no screening e ect involved, thus leading to divergence. What longdistance behavior of g↑↑ (r) and g↑↓ (r) in the ladder approximation is responsible for the divergence? This question is closely related to the origin of the logarithmic term, ln rs appearing in the high-density expansion of correlation energy. The spin-parallel and spin-antiparallel parts of the static structure factor derived from the ladder series both behave like −rs pf =q over a range of 0¡q=pf . 1 for very small rs . According to Fourier transformation, this means that both of g↑↓ (r) − 1 and g↑↑ (r)(= g↑↑ (r) − ↑↑ (r)) behave like −rs =(pf r)2 over a range of pf r¿1, thus leading to divergence gHF of the potential energy. The RPA considers the most divergent series of perturbation terms. In the high-density limit it amounts to cutting o the above term of −rs pf =q derived from the second-order direct energy such that rs1=2 ¡q=pf . 1. In real space, the above behavior of g↑↑ (r) and g↑↓ (r) in the ladder approximation is cut o such that 1¡pf r¡rs−1=2 , thus giving the term of order ln rs . We note that so far as the leading term of order ln rs is concerned, spin-parallel and spin-antiparallel correlations both make an equivalent contribution to the correlation energy. So far, we have treated the homogeneous electron liquid. The cusp conditions, of course, hold for any inhomogeneous electron systems and can be generally written as r ↑↓ g (0; R) + · · · ; a0 2 ↑↑ r r2 d g (r; R) 1+ g↑↑ (r; R) = + ··· ; 2 2a0 dr 2 r=0
g↑↓ (r; R) = g↑↓ (0; R) +
0
(57) (58)
where the pair correlation function g (r1 ; r2 ) are rewritten in terms of the inter particle separation, r(= r1 − r2 ) and the center of gravity, R(= (r1 + r2 )=2). In principle, these conditions can also be derived from the diagrammatic analysis of perturbation theory. It should be emphasized that the particle–particle ladder series of perturbation terms are
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M. Horiuchi et al. / Physica A 255 (1998) 48–64
essential for the satisfaction of these conditions and the proper evaluation of g↑↓ (0; R) and d2 g↑↑ (r; R)=dr 2 |r=0 . As has been widely recognized, the theoretical formulation that is capable of dealing with divergence caused by long-range parts of the Coulomb interaction is indispensable for the description of long-range correlation such as the existence of plasmon excitations and its counterpart e ect, screening. Almost all approaches to the study of electron correlation have therefore been advanced from the long-range side. The cusp conditions, on the other hand, are a characteristic feature caused by short-range parts of the Coulomb interaction. Their importance seems to have been overlooked in condensed matter physics. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
J. Hubbard, Proc. Roy. Soc. A 240 (1957) 530. J. Hubbard, Proc. Roy. Soc. A 243 (1958) 336. P. Nozieres, D. Pines, Phys. Rev. 111 (1958) 442. K.S. Singwi, M.P. Tosi, R.H. Land, A. Sjolander, Phys. Rev. 176 (1968) 589. H. Yasuhara, Solid State Commun. 11 (1972) 1481. H. Yasuhara, J. Phys. Soc. Japan 36 (1974) 361. H. Yasuhara, Physica 78 (1974) 420. Y. Ousaka, H. Suehiro, H. Yasuhara, J. Phys. C 19 (1986) 4247. Y. Ousaka, H. Suehiro, H. Yasuhara, J. Phys. C 19 (1986) 4263. H. Yasuhara, H. Suehiro, Y. Ousaka, J. Phys. C 21 (1988) 4045. T. Kato, Comm. Pure Appl. Math. 10 (1957) 151. J.C. Kimball, Phys. Rev. A 7 (1973) 1648. J.C. Kimball, J. Phys. A 8 (1975) 1513. A.K. Rajagopal, J.C. Kimball, M. Banerjee, Phys. Rev. B 18 (1978) 2339. See, for example, in: A.L. Fetter and J.D. Walecka (Eds.), Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. [16] H. Yasuhara, Y. Kawazoe, Physica A 85 (1976) 416.