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On determining excitation energies from the poles of green functions
Volume 1, numlzn" 5
PHYSICS
ON
DETERMINING THE POLES
LETTERS
EXCITATION OF GREEN
D. T E R I ~ R
1 Jmm 1969.
ENERGIES
FROM
FUNCTIONS
and W. E. P A R R Y
The Clarer~n Laboratory, O~ord Rec~ ~ ' ~ d 27 April 1962 F r o m (I), (2), (4), and (5) one finds R is commonly stated that the poles of the r, ~urler transform of the single-particle Green f,/m,1 tion determine the energies of the low-lying .~zc'tations of a system 1-4)] Recently Bolsterli ~) h~ts drawn attention to the fact that the single-p~o~,dc.1~. -1 < = . . . = <<:; ~+>> = . . . tremely simple, soluble model to shew that unde,~ c~rtaln circumstances it m a y even easily led to = <(~+~; ~a+>>= . . . -- 0 . wrong conclusions, especially in a perturbation Consider now the HRmiltonian theory approach *. Consider a system described by the (generalis~ ~I) H' = (c - ~) (I - 28) 0+O + (, - ~) 4Wi---'U) (s + ~+) Hamiltonian 3) =
(~ -
~) ~+,~,
(~')
where ~ and ~+ satisfy the fermion commutation relations, ~=~+~+-0,
~+~+~+=I
,
(2)
and where ~ is an adjustable parameter. Consider now the retarded Green functions <>defined by t h e relation
< = - i ([A(t), B(t')] +> ,
t> t ' ;
,
t< t~ ,
w h e r e [ , ] + i s an a n t i c o m m u t a t o r , w h e r e the single pointed b r a c k e t s i n d i c a t e an e n s e m b l e a v e r a g e (~ = 1 t ~ ,
(..> = T r e ' 0 H . .
/ Tr e - 0 H ,
(4)
which for z e r o t e m p e r a t u r e (~ -- ~) r e d u c e s to a g r o u n d - s t a t e a v e r a g e , and w h e r e t h e A(t) a n d / 3 ( t ' ) a r e ( g e n e r a l i s e d ) H e i s e n b e r g o p e r a t o r s . T h e Four i e r t r a n s f o r m <(A; B>>E (=- ((A; B)>: we d r o p the indox E in the followin~,) of ((A(t): Btt')~ s a t i s f i e s the equation of m o t i o n 2 ,'4") 1 E <> = ~-~-<[A, B] +> + <<[A,HI .; B>> ,
(7)
w h e r e S and 0+ s a t i s f y t h e f e r m i o n c o m m u t a t i o n r e l a t i o n s . To find t h e e x c i t a t i o n s p e c t r u m , one would consider <(0; 0+>>. One can again use (5) to evaluate this Green function as w e shall do presently. One can, however, also evaluate it through a suitable canonical transformation. If one applies the canonical transformation
0 = (I - 5) a - 5a+ +jS(l - 5) (~R+ - ~+~) ,
i3) =0
+ (~ - ~ ) ~ ,
(5)
where [ , ] _ is the commutator. * After this note was written we received a preprint of a paper by R. Balian, L.H. Nosanow and N. R. Werthamer who have aJ~o found spurious poles.
(8)
H' reduces to H of (1). Using (8) w e can express <>in terms of <<:L;~+>>, <>, <>, . . . , and we get then, using (6),
<>=
[
_~)9. + ~+~ E + , -
,
;]"
(g)
W e note here the occurrence of two excitation energies: s - ~ and - s + ~. W e m u s t now discuss the physical nature of the quantity ~. The term ~a+R occurs in the generalised Hamiltonian (1) as a consequence of the lact that the temperaturedependent Green Junctions used in statistical manybody problems involve averages over grand ensembles. In that case ~ is the chemical potential. It is oRen convenient to reckon all energies from the c h e m i c a l p o t e n t i a l a s z e r o , a n d c - ~ - the pole o c c u r r i n g in the firs't of e q u a t i o n s (6) - i s t h e n , i n d e e d , the e x c i t a t i o n e n e r g y . In f a c t , the p o l e s of the Green f u n c t i o n s give u s e ~ e r g y differences, a s is weU-known. If, f o r a m o m e n t , we put = 0, we get f r o m (9) the e x c i t a t i o n e n e r g i e s , and
145
Volume 1, number 5 .
.
.
.
.
.
PHYSICS
'
.
- ~, c o r r e s p o n d i n g c o r r e c t l y to t~he two p o s s i b l e d i f f e r e n c e s between 0 a r ~ ¢. If, h o w e v e r , we i n t r o d u c e p, we do n o t g e t ¢ - ~ and - ¢ - ~, a s s h o u l d follow f ~ ) m a s h i f t of the origin. O u r f i r s t c o n c l u s i o n i s thus t h a t a l t h o u g h the t e m p e r a t u r e d e p e n d e n t Green f u n c t i o n s involving a v e r a g e s o v e r grand ensembles are the most suitable ones for s t a t i s t i c a l p u r p o s e s , we s h o u l d u s e c ~ m n t c a l e n s e m b l e a v e r a g e s , t h a t i s , a v e r a g e s (4) with H e q u a l to t h e t r u e Hamiltoni~m H " , H" = H + ~
= H + pa+a = ~a+a ,
(10)
if we w i s h to d e t e r m i n e t h e e n e r g y s p e c t r t t m of a system. If t h e a and a + a r e t h e r e a l p a r t i c l e a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s f o r o u r s y s t e m , ~ i s t h e c h e m i c a l potential. T h e o c c u r r e n c e of t h e p o l e s at - ~ a n d - ¢ + ~ i s t h e n h a r d l y s u r p r i s i n g , but ff one ~ r t s f r o m t h e H a m i l t o n i a n (7) it is i m p o s s i b l e to f i n d t h e value c~ ~, a s t h i s Hamlltonian would be w r i t t e n in the f o r m H ' = A e+e + s(~ + e+) + C .
(79
If, on the other hand, the ~ and ~+ are the r e a l p a r t i t l e o p e r ~ ' o r s , we would w r i t e H' in the f o r m a " = ( , ' - ~ ' ) S+~ + s ( S + ~+) + c ,
by p'(1 - 2e) -1 w h e r e 6 i s not known a prio.~. L e t u s now e v a l u a t e ~\$, ~' " ¢+)) u s i n g the equation of m o t i o n (5), and let u s p u t ~ = 0 to s i m p l i f y t h e d 2 s c u s s i o n , F r o m (5) we have the follovdng e q u a tions ~ ( i - 2~)]
J5(1
=~i+
~ .
5) <
-
.
S+S
e*}>
,
(lla).
[~ + ~(i - ,',,'~)] <> --, EBB + -
~+8;
,~g(i-z-~ <<~B+- s+~; B+>), (,~lb)
{~+}>
2~ ~
:
[<>- <(~+; s+>>] , ( l l c )
f r o m which we get = _~,.rvl
..
- - ~
+ ....
-- .
.
.
.
.
.
.a.
L' . . . . .
,L%.
_
E - - - ~ - -
~ .~ .
.
.
.
(12) .
,.~At
s p u r i o u s pole at the orig~n (which we sh~ll not dis-. c u s s h e r e ) , the 3ame a_ equation (8), T h i s i s not s u r p r i s i n g , as o u r H a m f l t o n i a a is suf!.~cienfly s i m ple to lead to a c l o s e d ~et of equations f r o m which we can solve for <
i46
. . . . .
: Ijmm1962
' ,
,,
:
. . . . . . . . . . . . . . . . . . . . .
: . , ,
_
.
have r e c o u r s e to d e c o u p ~ p r o c e d u r e s . The f i r s t o r d e r d e c o u p l i n g would c o r r e s p o n d to w r i t i n g
<(Be+ - ~+~; ~+)) ~ (1 - 2 ~ <(1; S+)) ~ 0 ,
(13)
= <~+e),
a n d we would g e t f r o m (I l a )
((~; ~+~>= 2 ~ [ £
1 - ,(i"
20]
'
(14)
which leads to an incorrect excitation energy equal to ¢ ( 1 - 2 0 . We w~.y a s k i n how f a r t h e d e c o u p l i n g (13) i s j u s t i f i e d . T h e IL~.~ltonian ( 7 ) i s s u c h that we can t r e a t 6 o r ! r a t h e r 6a a s a s m a l l pw--ameter. In t h e l i m i t a s 8a -- 0, we get ~ - a , and H' - / / . We would e x p e c t t h a t t h e d.ecoupling (13) would be exact in t h e l i m i t a s 8~ - O , H we r e t a i n only t e r m s of o r d e r 5½, we can n e g l e c t t h e s e c o n d t e r m on t h e r i g h t - h a n d s i d e of ( l l a ) ,
but i n t h a t c a s e w e m u s t
n e g l e c t t h e t e r m 28¢ in t h e d e n o m i n a t o r in ( 1 4 ) , and w e h a v e o n l y the ( c o r r e c t ) p o l e at e. If we r e tain t e r m s up to o r d e r ~, we m u s t u s e (11c) to write I
((e~ + - ~ + e ;
~+))~~
2¢~
((e; ~+>>,
(1~)
~here we have used (11b). CombinLng ( l l a ) and (15) we g e t
<> ~
[E-
2¢26 `
¢(1 .- 2 5 ) - - - ~ - - ]
-]
,
or
((~; S+))~ ~ [ E
2 - E e ( 1 - 25) - 2e25] " l , ( ~ 6 )
<<~; ,~+>>
I
E
,
(7")
and then the energies are not displaced by p', but
[_~ -
LETTERS . . . . . .
,,
~
f r o m which, again up to t e r m s of o r d e r 5, we get f o r the e x c i t a t i o n e n e r g y . We get, h o w e v e r , a l s o a s p u r i o u s e x t r a pole at - 26 ¢. We h a v e noticed the o c c u r r e n c e of s p u r i o u s r e s u l t s in s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y a l s o in o t h e r p r o b l e m s , and we hope to d i s c u s s at a l a t e r s t a g e hbe r e s u l t s of s o m e i n v e s t i g a t i o n s of m o r e complicated Hamiltonians. I) V.M. Galltski~ ~nd A. B. Migdal, J. Exptl. Theoret. Phys. (USSR) 34 (1958) 139; translation: Soviet Phys. JETP 7 (1958) 96. 2) D.N. Zubarev° Usp. Fiz. Nauk 71 (1960) 71; translation; Soviet Phys. Uspekhi 3 (1960) 320. 3) V.L. Bonch-Bruevich and S. V. T~-abifKov, "r--heGreen function mothod in statistical mechanics (Ffzmatglz, Moscow, 1961: translation: North-HoPand Publ. Co., Amsterdam, 1962). 4) D.ter Haar, in Fluctuation, relaxation and resonance in magnetic systems (Oliver aad Boyd, Ed~aburgh, 196~, p. 119. M. P~lsterlt, Phy~,. ~cv. Letters 4 (1960) £2.
PHYSICS
ON
DETERMINING THE POLES
LETTERS
EXCITATION OF GREEN
D. T E R I ~ R
1 Jmm 1969.
ENERGIES
FROM
FUNCTIONS
and W. E. P A R R Y
The Clarer~n Laboratory, O~ord Rec~ ~ ' ~ d 27 April 1962 F r o m (I), (2), (4), and (5) one finds R is commonly stated that the poles of the r, ~urler transform of the single-particle Green f,/m,1 tion determine the energies of the low-lying .~zc'tations of a system 1-4)] Recently Bolsterli ~) h~ts drawn attention to the fact that the single-p~o~,dc.1~. -1 <
(~ -
~) ~+,~,
(~')
where ~ and ~+ satisfy the fermion commutation relations, ~=~+~+-0,
~+~+~+=I
,
(2)
and where ~ is an adjustable parameter. Consider now the retarded Green functions <
<
t> t ' ;
,
t< t~ ,
w h e r e [ , ] + i s an a n t i c o m m u t a t o r , w h e r e the single pointed b r a c k e t s i n d i c a t e an e n s e m b l e a v e r a g e (~ = 1 t ~ ,
(..> = T r e ' 0 H . .
/ Tr e - 0 H ,
(4)
which for z e r o t e m p e r a t u r e (~ -- ~) r e d u c e s to a g r o u n d - s t a t e a v e r a g e , and w h e r e t h e A(t) a n d / 3 ( t ' ) a r e ( g e n e r a l i s e d ) H e i s e n b e r g o p e r a t o r s . T h e Four i e r t r a n s f o r m <(A; B>>E (=- ((A; B)>: we d r o p the indox E in the followin~,) of ((A(t): Btt')~ s a t i s f i e s the equation of m o t i o n 2 ,'4") 1 E <
(7)
w h e r e S and 0+ s a t i s f y t h e f e r m i o n c o m m u t a t i o n r e l a t i o n s . To find t h e e x c i t a t i o n s p e c t r u m , one would consider <(0; 0+>>. One can again use (5) to evaluate this Green function as w e shall do presently. One can, however, also evaluate it through a suitable canonical transformation. If one applies the canonical transformation
0 = (I - 5) a - 5a+ +jS(l - 5) (~R+ - ~+~) ,
i3) =0
+ (~ - ~ ) ~ ,
(5)
where [ , ] _ is the commutator. * After this note was written we received a preprint of a paper by R. Balian, L.H. Nosanow and N. R. Werthamer who have aJ~o found spurious poles.
(8)
H' reduces to H of (1). Using (8) w e can express <
<
[
_~)9. + ~+~ E + , -
,
;]"
(g)
W e note here the occurrence of two excitation energies: s - ~ and - s + ~. W e m u s t now discuss the physical nature of the quantity ~. The term ~a+R occurs in the generalised Hamiltonian (1) as a consequence of the lact that the temperaturedependent Green Junctions used in statistical manybody problems involve averages over grand ensembles. In that case ~ is the chemical potential. It is oRen convenient to reckon all energies from the c h e m i c a l p o t e n t i a l a s z e r o , a n d c - ~ - the pole o c c u r r i n g in the firs't of e q u a t i o n s (6) - i s t h e n , i n d e e d , the e x c i t a t i o n e n e r g y . In f a c t , the p o l e s of the Green f u n c t i o n s give u s e ~ e r g y differences, a s is weU-known. If, f o r a m o m e n t , we put = 0, we get f r o m (9) the e x c i t a t i o n e n e r g i e s , and
145
Volume 1, number 5 .
.
.
.
.
.
PHYSICS
'
.
- ~, c o r r e s p o n d i n g c o r r e c t l y to t~he two p o s s i b l e d i f f e r e n c e s between 0 a r ~ ¢. If, h o w e v e r , we i n t r o d u c e p, we do n o t g e t ¢ - ~ and - ¢ - ~, a s s h o u l d follow f ~ ) m a s h i f t of the origin. O u r f i r s t c o n c l u s i o n i s thus t h a t a l t h o u g h the t e m p e r a t u r e d e p e n d e n t Green f u n c t i o n s involving a v e r a g e s o v e r grand ensembles are the most suitable ones for s t a t i s t i c a l p u r p o s e s , we s h o u l d u s e c ~ m n t c a l e n s e m b l e a v e r a g e s , t h a t i s , a v e r a g e s (4) with H e q u a l to t h e t r u e Hamiltoni~m H " , H" = H + ~
= H + pa+a = ~a+a ,
(10)
if we w i s h to d e t e r m i n e t h e e n e r g y s p e c t r t t m of a system. If t h e a and a + a r e t h e r e a l p a r t i c l e a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s f o r o u r s y s t e m , ~ i s t h e c h e m i c a l potential. T h e o c c u r r e n c e of t h e p o l e s at - ~ a n d - ¢ + ~ i s t h e n h a r d l y s u r p r i s i n g , but ff one ~ r t s f r o m t h e H a m i l t o n i a n (7) it is i m p o s s i b l e to f i n d t h e value c~ ~, a s t h i s Hamlltonian would be w r i t t e n in the f o r m H ' = A e+e + s(~ + e+) + C .
(79
If, on the other hand, the ~ and ~+ are the r e a l p a r t i t l e o p e r ~ ' o r s , we would w r i t e H' in the f o r m a " = ( , ' - ~ ' ) S+~ + s ( S + ~+) + c ,
by p'(1 - 2e) -1 w h e r e 6 i s not known a prio.~. L e t u s now e v a l u a t e ~\$, ~' " ¢+)) u s i n g the equation of m o t i o n (5), and let u s p u t ~ = 0 to s i m p l i f y t h e d 2 s c u s s i o n , F r o m (5) we have the follovdng e q u a tions ~ ( i - 2~)]
J5(1
=~i+
~ .
5) <
-
.
S+S
e*}>
,
(lla).
[~ + ~(i - ,',,'~)] <
~+8;
,~g(i-z-~ <<~B+- s+~; B+>), (,~lb)
{~+}>
2~ ~
:
[<
f r o m which we get = _~,.rvl
..
- - ~
+ ....
-- .
.
.
.
.
.
.a.
L' . . . . .
,L%.
_
E - - - ~ - -
~ .~ .
.
.
.
(12) .
,.~At
s p u r i o u s pole at the orig~n (which we sh~ll not dis-. c u s s h e r e ) , the 3ame a_ equation (8), T h i s i s not s u r p r i s i n g , as o u r H a m f l t o n i a a is suf!.~cienfly s i m ple to lead to a c l o s e d ~et of equations f r o m which we can solve for <
i46
. . . . .
: Ijmm1962
' ,
,,
:
. . . . . . . . . . . . . . . . . . . . .
: . , ,
_
.
have r e c o u r s e to d e c o u p ~ p r o c e d u r e s . The f i r s t o r d e r d e c o u p l i n g would c o r r e s p o n d to w r i t i n g
<(Be+ - ~+~; ~+)) ~ (1 - 2 ~ <(1; S+)) ~ 0 ,
(13)
= <~+e),
a n d we would g e t f r o m (I l a )
((~; ~+~>= 2 ~ [ £
1 - ,(i"
20]
'
(14)
which leads to an incorrect excitation energy equal to ¢ ( 1 - 2 0 . We w~.y a s k i n how f a r t h e d e c o u p l i n g (13) i s j u s t i f i e d . T h e IL~.~ltonian ( 7 ) i s s u c h that we can t r e a t 6 o r ! r a t h e r 6a a s a s m a l l pw--ameter. In t h e l i m i t a s 8a -- 0, we get ~ - a , and H' - / / . We would e x p e c t t h a t t h e d.ecoupling (13) would be exact in t h e l i m i t a s 8~ - O , H we r e t a i n only t e r m s of o r d e r 5½, we can n e g l e c t t h e s e c o n d t e r m on t h e r i g h t - h a n d s i d e of ( l l a ) ,
but i n t h a t c a s e w e m u s t
n e g l e c t t h e t e r m 28¢ in t h e d e n o m i n a t o r in ( 1 4 ) , and w e h a v e o n l y the ( c o r r e c t ) p o l e at e. If we r e tain t e r m s up to o r d e r ~, we m u s t u s e (11c) to write I
((e~ + - ~ + e ;
~+))~~
2¢~
((e; ~+>>,
(1~)
~here we have used (11b). CombinLng ( l l a ) and (15) we g e t
<
[E-
2¢26 `
¢(1 .- 2 5 ) - - - ~ - - ]
-]
,
or
((~; S+))~ ~ [ E
2 - E e ( 1 - 25) - 2e25] " l , ( ~ 6 )
<<~; ,~+>>
I
E
,
(7")
and then the energies are not displaced by p', but
[_~ -
LETTERS . . . . . .
,,
~
f r o m which, again up to t e r m s of o r d e r 5, we get f o r the e x c i t a t i o n e n e r g y . We get, h o w e v e r , a l s o a s p u r i o u s e x t r a pole at - 26 ¢. We h a v e noticed the o c c u r r e n c e of s p u r i o u s r e s u l t s in s e c o n d - o r d e r p e r t u r b a t i o n t h e o r y a l s o in o t h e r p r o b l e m s , and we hope to d i s c u s s at a l a t e r s t a g e hbe r e s u l t s of s o m e i n v e s t i g a t i o n s of m o r e complicated Hamiltonians. I) V.M. Galltski~ ~nd A. B. Migdal, J. Exptl. Theoret. Phys. (USSR) 34 (1958) 139; translation: Soviet Phys. JETP 7 (1958) 96. 2) D.N. Zubarev° Usp. Fiz. Nauk 71 (1960) 71; translation; Soviet Phys. Uspekhi 3 (1960) 320. 3) V.L. Bonch-Bruevich and S. V. T~-abifKov, "r--heGreen function mothod in statistical mechanics (Ffzmatglz, Moscow, 1961: translation: North-HoPand Publ. Co., Amsterdam, 1962). 4) D.ter Haar, in Fluctuation, relaxation and resonance in magnetic systems (Oliver aad Boyd, Ed~aburgh, 196~, p. 119. M. P~lsterlt, Phy~,. ~cv. Letters 4 (1960) £2.