Ground State of a Semiconductor-like System with Transverse s-d Interaction and Perturbation Method†)

Progress of Theoretical Physics, May 1970

Application of perturbation method to the ground state of a semiconductor-like system with transverse s-d interaction is accompanied by a serious difficulty of logarithmic divergence similar to that of the ordinary s-d problem, as the width of the forbidden band, Δ, goes to zero. As a preliminary study to improving Abrikosov approximation by considering less divergent terms, it is here attempted to remove just the logarithmic divergence of the second order term of the ground state expectation value of the longitudinal component of a localized spin, ≪ Sz >. For this object, an approximate perturbational ground state wave function is constructed from the Rayleigh-Schrödinger perturbational wave function, by expecting that the weight of many-electron excited configurations should increase with decreasing Δ. The obtained ≪ Sz > is markedly different from the result of the second order perturbation theory and gradually tends to zero as Δ→0, statisfying the physical condition 0 ≤≪ Sz > ≤S all over the region of Δ. On the other hand, the obtained expectation values of the energy ≪H> and the number of the excited electrons ≪n>> coincide essentially with those of the second order perturbation theory. The present approximation may be regarded as the furthest extreme very opposite to Abrikosov approximation in which only the diagrams consisting of just one closed loop of particle line are summed up. (See Fig. 3.) The truth is expected to lie between the two approximations.

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Ground State of a Semiconductor-like System with Transverse s-d Interaction and Perturbation Method†)

Ground State of a Semiconductor-like System with Transverse s-d Interaction and Perturbation Methodt) Itsuo OHNARI 0 Koji ISHIKAWA 0 1 Yukio MIZUNO 0 0 Institute of Physics, College of General Education University of Tokyo , Meguro-ku, Tokyo 1 > His present address is Department of Physics, Faculty of Science, University of Nagoya Application of perturbation method to the ground state of a semiconductor-like system with transverse s-d interaction is accompanied by a serious difficulty of logarithmic divergence similar to that of the ordinary s-d problem, as the width of the forbidden band, J, goes to zero. As a preliminary study to improving Abrikosov approximation by considering less divergent terms, it is here attempted to remove just the logarithmic divergence of the second order term of the ground state expectation value of the longitudinal component of a localized spin, <Sz>· For this object, an approximate perturbational ground state wave function is constructed from the Rayleigh-Schrodinger perturbational wave function, by expecting that the weight of many-electron excited configurations should increase with decreasing J. The obtained <Sz) is markedly different from the result of the second order perturbation theory and gradually tends to zero as J~O, satisfying the physical condition O<<Sz)<S all over the region of J. On the other hand, the obtained expectation values of the energy (H) and the number of the excited electrons <n>) coincide essentially with those of the second order perturbation theory. The present approximation may be regarded as the furthest extreme very opposite to Abrikosov approximation in which only the diagrams consisting of just one closed loop of particle line are summed up. (See Fig. 3.) The truth is expected to lie between the two approximations. - Model and perturbation series H=Ho+ V, (2 · 2) a i 1 -o-t Fig. 1. State density of semiconductor-like model. <n>)= (2 · 7a) +4(ln 2-S)ln ~ + const} + ···J. (2· 7b) An effect of many-electron excited configurations lff=i:,{ p n=O W -H0 (JE = <(f) IV!lff) . (V-&E)}n(f), (3 ·1a) (3·1b) approximate wave function: ....., 2N 7J! ~ 7Jf =@ + ~ (jjn • ?t=l Now we introduce several notations: Ap=ek-sl, ap+=-a~ci"tal~, p= (k, l),*) (3 ·10) (3 ·11) (3·12) Then, (3 ·10) may be rewritten as (3 ·13) (3 ·14) (3 ·15) (3 ·16) (3 ·17) Namely Hence =1+~2Nl( -J)2n n=l \N d e l1- - } ·ek- el 2 =exp{(J) (-ln-LI-+ln D+LI )}· D D+LI 2D+LI i 2 = - 1(for S= ~ ). 2 D D+LI 2D+LI Energy expectation value <H) First we shall calculate <fJJIH0 jfi!). Using the relation we obtain <fJJ IHoI fJJ) = ~ <(f)n IHoI (f)1~> n By means of the relations <fJJ IVI fJJ) IS calculated as follows, <fJJ lV[ fJJ) = 2 ~ <(f)n+ll VI (f)n) n <(f)nl VI (f)n) = 0 , Thus the right-hand side reduces to We obtain, therefore, the energy expectation value <H> = (<iJf IHoI iJf >+ <iJf IVI iJf >)/in= W- ( J) ~ l_ 2 N P J..11 (3·18) Then, we get ~ 1 P1>P2>··>Pn (A'p1 A'p2 · · · l''.pn) 2 Hence (3 ·19) § 4. Discussion and conclusions Fig. 2. Diagrams summed up in the treatment of § 3. These diagrams consist of plural closed loops of particle line except for the first two. =_!_ex { 2 (J/DY In (11/D) } 2 p 1-4/3(J/D)2ln2 (11/D) . Acknowledgement Appendix 1 4__,0 In ( -D ) (s + s')2 ~ 2L1 ' 4->o _!_ ln3 (!2_) -ln2 (!2_) + (?) ln(!!_) + (?) ~ 3 ! \ 2J 2J 2J d-->0 1 1 nn( D-}\_,.. n! 2LI (n-12)! 1 nn-1(-2DL·f) ·+ · · inA= .ooE1- (J-)2n Inn (-D) =exp [(J-\;2 In (-D)]. n=o n!\D 2LI D 2LI 1 ) ]. Kondo , Frog. Theor. Phys. 32 ( 1964 ), 37 . 2) A. A. Abrikosov , Physics 2 ( 1965 ), 5 . H. Suhl , Phys. Rev . 138 ( 1965 ) A515; 141 ( 1966 ), 483 ; Physics 2 ( 1965 ), 39 . 3) K. Yosida and A. Okiji , Frog. Theor. Phys. 34 ( 1965 ), 505 . 4) H. Miwa , Frog. Theor. Phys. 34 ( 1965 ), 375 . 5 ) ]. Kondo , Frog. Theor. Phys. 40 ( 1968 ), 683 . 6 ) ]. Kondo , "Theory of Dilute Magnetic Alloys" (Researches of the Electrotechnical Laboratory , No. 688 ). 7) S. D. Silverstein and C. B. Duke , Phys. Rev . 161 ( 1967 ), 456 . 8) P. W. Anderson , Phys. Rev. Letters 18 ( 1967 ), 1049 . 9) K. Yosida , Phys. Rev . 147 ( 1966 ) 223; Frog . Theor. Phys. 36 ( 1966 ), 875 .


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Itsuo Ohnari, Koji Ishikawa, Yukio Mizuno. Ground State of a Semiconductor-like System with Transverse s-d Interaction and Perturbation Method†), Progress of Theoretical Physics, 1970, 1186-1198, DOI: 10.1143/PTP.43.1186