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Many-time green functions and the resolvent operator
Volume 27A, number 10
MANY-TIME
PHYSICS LETTERS
GREEN
FUNCTIONS
AND
THE
7 October 1968
RESOLVENT
OPERATOR*
J. C. H E R Z E L Physics Department, University of Melbourne, Parkville, Victoria 3052, Australia Received 15 July 1968
The resolvent operator is used to define many-time thermodynamic Green functions, the equations of motion of which are thus shown equivalent to identities based on that operator. An alternative method for calculating the many-time Green functions is suggested.
In quantum s t a t i s t i c a l m e c h a n i c s the d o u b l e - t i m e t h e r m o d y n a m i c G r e e n function has been defined in t e r m s of the o r d i n a r y G r e e n o p e r a t o r (resolvent). An identity b a s e d on the r e s o l v e n t then l e a d s to the b a s i c equation of motion for the t h e r m o d y n a m i c functions [1]. The p r e s e n t l e t t e r g e n e r a l i z e s t h e s e i d e a s within c l a s s i c a l s t a t i s t i c a l m e c h a n i c s . One defines the c l a s s i c a l t h e r m o d y n a m i c G r e e n functions a r i s i n g in n o n l i n e a r r e s p o n s e t h e o r y , cons i s t e n t l y with both the c o r r e s p o n d i n g quantum s t a t i s t i c a l definition [3], and the o r i g i n a l c l a s s i c a l t w o t i m e definition [4], to be
G N , ~ + I ( B , A ) ~-[~ (1 --
G#+I(B,A)¢o = (½~)~
~-I S d ~ j . ~ ( T j ) e x p [ i w ( { ~ + l - j } T j ) ] x 1..
I n t e g r a t i n g out the {~j}, a s s u m i n g I m w > 0, and w r i t i n g (L - w) -1 ~- g(w), one obtains
G ~ + I ( B , A ) = ( 1 / - 2 ~ ) ~t ( [ g ( - w ) [ g ( - 2 w ) [ . . . ~ ( - ~ w ) S ; A ] . . .
];A];A]) .
(1)
A l t e r n a t i v e l y , s t a r t i n g f r o m the c l a s s i c a l f o r m of eq. (9) of r e f . 3, one can w r i t e
6 U +I ( B , A )~ = (1/2~i)U f d r B • g(uw)[A;g({ u - 1 } w )[n; . . . . ;g(w)[A;D~]. . . ]] .
(25
R e p l a c i n g the g ( - ~ w ) of eq. (1) and the g(~w) of eq. (2) by the i d e n t i c a l l y - z e r o e x p r e s s i o n s {g(-pw)" (L + ~ 0 5 - 1} and {g(~w)(L - Uw) - 1} r e s p e c t i v e l y , one obtains i d e n t i t i e s each of which a r e equivalent to t h e equation of motion: -i~.wGu+I ( B , A ) = (1/2~)G([B;A],A) w + G~+I ([B;HN],A)w , which: i s the c l a s s i c a l l i m i t of eq. (155 of r e f . 3. F o r m u l a e (1) and (2) can be u s e d to d e t e r m i n e m a n y - t i m e G r e e n functions by using f a m i l i a r e x p a n s ions of the r e s o l v e n t . C o m p a r e d with the a p p r o a c h s u g g e s t e d in r e f . 3, which i s b a s e d on the equations of motion, (15) of r e f . 3, the p r e s e n t m e t h o d h a s t h e a d v a n t a g e that i t a v a i l s of t h e a l r e a d y w e l l - d e v e l oped r e s o l v e n t o p e r a t o r techniques. It i s to be noted that r e c e n t l y , [5], a r e s o l v e n t o p e r a t o r a p p r o a c h has been d e v e l o p e d f r o m equations (15) of r e f . 3. Eq. (2) h a s , in p r i n c i p l e , been u s e d to c a l c u l a t e the l i n e a r d i e l e c t r i c function [6], u s i n g t h e d i a g r a m * This work has been financed with a Melbourne University Research grant. :~ The notation used here is as in ref. 2 except that here W is used instead of E. 654
Volume 27A, number 10
PHYSICS LETTERS
7 October 1968
t e c h n i q u e d e v e l o p e d by the B r u s s e l s School. O t h e r r e l e v a n t work in this d i r e c t i o n i s d i s c u s s e d and r e f e r e n c e d in [7] and [8]. The p r e s e n t viewpoint can be i l l u s t r a t e d by studying the e l e m e n t a r y e x a m p l e of sect i o n 3B of r e f . 3, in which it is found a p p r o p r i a t e to employ f o r m u l a (1). One now finds that the f i r s t - o r d e r - i n - ~ c a l c u l a tion of G3(x , x) w can be i n t e r p r e t e d in f a m i l i a r t e r m s , n a m e l y :
G3(x, x)¢o =- (1/21r) 2 ([go(-W)" [ g o ( - 2 w ) ( - L i )go(-2co)x;x];x]) , w h e r e go(-W) --- ( L o + W ) - i , and H = H o + HI, with H o e p 2 / 2 m + ½mw2x 2 and H I ~ _ ~ 3. Evaluation of this e x p r e s s i o n f o r G3(x , x) w i n v o l v e s b i n o m i a l expansion of go, r e s u l t i n g in p r o d u c t s of s i m p l e s e r i e s which can be e a s i l y w r i t t e n in c l o s e d f o r m [3, eq. (34)]. T he author i s indebted to K. C. Hines f o r h is i n t e r e s t in this work.
R e f e~F~c e s
1. 2. 3. 4. 5. 6. 7. 8.
E.R. Pike, Proc. Phys. Soe. (London) 84 (1964) 83. J.C. Herzel, J. Math. Phys. 8 (1967) 1650. T. Tanaka, K. Moorjani and T. Morita, Phys. Rev. 155 0967) 388. N.N. Bogolyubov Jr. and B.I. Sadovnikov, Zh. Eksperim. i Teor. Fiz. 43 (1962) 677; Soviet Phys. JETP 16 (1963) 482. T.Kawasaki, Prog. Theor. Phys. (Kyoto) 39 (1968} 331. A.Ron, J. Math. Phys. 4 (1963} 811. S.A. Rice and P. Gray, The statistical mechanics of simple liquids (Interscience Publishers, New York, 1965} especially section 7.3.E. P. R~sibois, in Physics of many-particle systems, Vol. 1. Ed. E. Meeron (Gordon and Breach, Science Publishers, Inc., New York, 1966} Chap. 6 section 8.
DE
HAAS-VAN
ALPHEN
EFFECT
AND
FERMI
SURFACE
IN NIOBIUM
t
G. B. SCOTT, M. SPRINGFORD and J . R. STOCKTON
School of Mathematical and Physical Sciences, University of Sussex, Brighton, UK Received 26 August 1968
De Haas-van Alphen oscillations have been investigated in niobium and are discussed in terms of a model for the Fermi surface proposed by Mattheiss.
We have studied de H a a s - v a n Alphen (dHvA) o s c i l l a t i o n s in niobium $ f o r m a g n e t i c f i e l d d i r e c tions in the { 110} p l a n e s . T h e r e s u l t s a r e i n t e r p r e t e d in t e r m s of a m o d e l f o r the F e r m i s u r f a c e p r o p o s e d by M a t t h e i s s [1] f o r the Group VB t r a n sition m e t a l s but s o m e f e a t u r e s of this m o d e l a r e not o b s e r v e d in our e x p e r i m e n t s . T h e dHvA o s c i l lations w e r e i n v e s t i g a t e d in a 100 kG s u p e r c o n t This work is supported by the Science Research Council. We are indebted to Professor W. F. Vinen, Department of Physics, University of Birmingham, and to Dr. G. Taylor, Department of Metallurgy, University of Oxford, for the niobium crystals.
ducting m a g n e t using the f i el d modulation t e c h nique [2]. Fig. 1 shows the o b s e r v e d a n g u l a r v a r i a t i o n of e x t r e m a l a r e a in the ( l I 0 ) plane. T h e high o _ f r e q u e n c y b r a n c h has a m i n i m u m a r e a of 1.85A -~ with m * / m o = 3.2 along [111] and extends f o r about 20 ° on e i t h e r s i d e of the [111] o r i e n t a t i o n . This is in a g r e e m e n t with m a g n e t o t h e r m a l e x p e r i m e n t s of G r a e b n e r et al. [3] who have s u g g e s t e d that this b r a n c h d e r i v e s f r o m e x t r e m a l o r b i t s on t h e open hole sh eet about the H point in the Bril]ouin zone, and r e c e n t c a l c u l a t i o n s by M a t t h e i s s [4] a g r e e quantitively with the p r o p o s a l . Fig. 1 a l s o shows a set of dHvA f r e q u e n c i e s which
655
MANY-TIME
PHYSICS LETTERS
GREEN
FUNCTIONS
AND
THE
7 October 1968
RESOLVENT
OPERATOR*
J. C. H E R Z E L Physics Department, University of Melbourne, Parkville, Victoria 3052, Australia Received 15 July 1968
The resolvent operator is used to define many-time thermodynamic Green functions, the equations of motion of which are thus shown equivalent to identities based on that operator. An alternative method for calculating the many-time Green functions is suggested.
In quantum s t a t i s t i c a l m e c h a n i c s the d o u b l e - t i m e t h e r m o d y n a m i c G r e e n function has been defined in t e r m s of the o r d i n a r y G r e e n o p e r a t o r (resolvent). An identity b a s e d on the r e s o l v e n t then l e a d s to the b a s i c equation of motion for the t h e r m o d y n a m i c functions [1]. The p r e s e n t l e t t e r g e n e r a l i z e s t h e s e i d e a s within c l a s s i c a l s t a t i s t i c a l m e c h a n i c s . One defines the c l a s s i c a l t h e r m o d y n a m i c G r e e n functions a r i s i n g in n o n l i n e a r r e s p o n s e t h e o r y , cons i s t e n t l y with both the c o r r e s p o n d i n g quantum s t a t i s t i c a l definition [3], and the o r i g i n a l c l a s s i c a l t w o t i m e definition [4], to be
G N , ~ + I ( B , A ) ~-[~ (1 --
G#+I(B,A)¢o = (½~)~
~-I S d ~ j . ~ ( T j ) e x p [ i w ( { ~ + l - j } T j ) ] x 1..
I n t e g r a t i n g out the {~j}, a s s u m i n g I m w > 0, and w r i t i n g (L - w) -1 ~- g(w), one obtains
G ~ + I ( B , A ) = ( 1 / - 2 ~ ) ~t ( [ g ( - w ) [ g ( - 2 w ) [ . . . ~ ( - ~ w ) S ; A ] . . .
];A];A]) .
(1)
A l t e r n a t i v e l y , s t a r t i n g f r o m the c l a s s i c a l f o r m of eq. (9) of r e f . 3, one can w r i t e
6 U +I ( B , A )~ = (1/2~i)U f d r B • g(uw)[A;g({ u - 1 } w )[n; . . . . ;g(w)[A;D~]. . . ]] .
(25
R e p l a c i n g the g ( - ~ w ) of eq. (1) and the g(~w) of eq. (2) by the i d e n t i c a l l y - z e r o e x p r e s s i o n s {g(-pw)" (L + ~ 0 5 - 1} and {g(~w)(L - Uw) - 1} r e s p e c t i v e l y , one obtains i d e n t i t i e s each of which a r e equivalent to t h e equation of motion: -i~.wGu+I ( B , A ) = (1/2~)G([B;A],A) w + G~+I ([B;HN],A)w , which: i s the c l a s s i c a l l i m i t of eq. (155 of r e f . 3. F o r m u l a e (1) and (2) can be u s e d to d e t e r m i n e m a n y - t i m e G r e e n functions by using f a m i l i a r e x p a n s ions of the r e s o l v e n t . C o m p a r e d with the a p p r o a c h s u g g e s t e d in r e f . 3, which i s b a s e d on the equations of motion, (15) of r e f . 3, the p r e s e n t m e t h o d h a s t h e a d v a n t a g e that i t a v a i l s of t h e a l r e a d y w e l l - d e v e l oped r e s o l v e n t o p e r a t o r techniques. It i s to be noted that r e c e n t l y , [5], a r e s o l v e n t o p e r a t o r a p p r o a c h has been d e v e l o p e d f r o m equations (15) of r e f . 3. Eq. (2) h a s , in p r i n c i p l e , been u s e d to c a l c u l a t e the l i n e a r d i e l e c t r i c function [6], u s i n g t h e d i a g r a m * This work has been financed with a Melbourne University Research grant. :~ The notation used here is as in ref. 2 except that here W is used instead of E. 654
Volume 27A, number 10
PHYSICS LETTERS
7 October 1968
t e c h n i q u e d e v e l o p e d by the B r u s s e l s School. O t h e r r e l e v a n t work in this d i r e c t i o n i s d i s c u s s e d and r e f e r e n c e d in [7] and [8]. The p r e s e n t viewpoint can be i l l u s t r a t e d by studying the e l e m e n t a r y e x a m p l e of sect i o n 3B of r e f . 3, in which it is found a p p r o p r i a t e to employ f o r m u l a (1). One now finds that the f i r s t - o r d e r - i n - ~ c a l c u l a tion of G3(x , x) w can be i n t e r p r e t e d in f a m i l i a r t e r m s , n a m e l y :
G3(x, x)¢o =- (1/21r) 2 ([go(-W)" [ g o ( - 2 w ) ( - L i )go(-2co)x;x];x]) , w h e r e go(-W) --- ( L o + W ) - i , and H = H o + HI, with H o e p 2 / 2 m + ½mw2x 2 and H I ~ _ ~ 3. Evaluation of this e x p r e s s i o n f o r G3(x , x) w i n v o l v e s b i n o m i a l expansion of go, r e s u l t i n g in p r o d u c t s of s i m p l e s e r i e s which can be e a s i l y w r i t t e n in c l o s e d f o r m [3, eq. (34)]. T he author i s indebted to K. C. Hines f o r h is i n t e r e s t in this work.
R e f e~F~c e s
1. 2. 3. 4. 5. 6. 7. 8.
E.R. Pike, Proc. Phys. Soe. (London) 84 (1964) 83. J.C. Herzel, J. Math. Phys. 8 (1967) 1650. T. Tanaka, K. Moorjani and T. Morita, Phys. Rev. 155 0967) 388. N.N. Bogolyubov Jr. and B.I. Sadovnikov, Zh. Eksperim. i Teor. Fiz. 43 (1962) 677; Soviet Phys. JETP 16 (1963) 482. T.Kawasaki, Prog. Theor. Phys. (Kyoto) 39 (1968} 331. A.Ron, J. Math. Phys. 4 (1963} 811. S.A. Rice and P. Gray, The statistical mechanics of simple liquids (Interscience Publishers, New York, 1965} especially section 7.3.E. P. R~sibois, in Physics of many-particle systems, Vol. 1. Ed. E. Meeron (Gordon and Breach, Science Publishers, Inc., New York, 1966} Chap. 6 section 8.
DE
HAAS-VAN
ALPHEN
EFFECT
AND
FERMI
SURFACE
IN NIOBIUM
t
G. B. SCOTT, M. SPRINGFORD and J . R. STOCKTON
School of Mathematical and Physical Sciences, University of Sussex, Brighton, UK Received 26 August 1968
De Haas-van Alphen oscillations have been investigated in niobium and are discussed in terms of a model for the Fermi surface proposed by Mattheiss.
We have studied de H a a s - v a n Alphen (dHvA) o s c i l l a t i o n s in niobium $ f o r m a g n e t i c f i e l d d i r e c tions in the { 110} p l a n e s . T h e r e s u l t s a r e i n t e r p r e t e d in t e r m s of a m o d e l f o r the F e r m i s u r f a c e p r o p o s e d by M a t t h e i s s [1] f o r the Group VB t r a n sition m e t a l s but s o m e f e a t u r e s of this m o d e l a r e not o b s e r v e d in our e x p e r i m e n t s . T h e dHvA o s c i l lations w e r e i n v e s t i g a t e d in a 100 kG s u p e r c o n t This work is supported by the Science Research Council. We are indebted to Professor W. F. Vinen, Department of Physics, University of Birmingham, and to Dr. G. Taylor, Department of Metallurgy, University of Oxford, for the niobium crystals.
ducting m a g n e t using the f i el d modulation t e c h nique [2]. Fig. 1 shows the o b s e r v e d a n g u l a r v a r i a t i o n of e x t r e m a l a r e a in the ( l I 0 ) plane. T h e high o _ f r e q u e n c y b r a n c h has a m i n i m u m a r e a of 1.85A -~ with m * / m o = 3.2 along [111] and extends f o r about 20 ° on e i t h e r s i d e of the [111] o r i e n t a t i o n . This is in a g r e e m e n t with m a g n e t o t h e r m a l e x p e r i m e n t s of G r a e b n e r et al. [3] who have s u g g e s t e d that this b r a n c h d e r i v e s f r o m e x t r e m a l o r b i t s on t h e open hole sh eet about the H point in the Bril]ouin zone, and r e c e n t c a l c u l a t i o n s by M a t t h e i s s [4] a g r e e quantitively with the p r o p o s a l . Fig. 1 a l s o shows a set of dHvA f r e q u e n c i e s which
655