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On an objection to the Patashinskii-Pokrovskii theory of the λ-transition in liquid 4He
PHYSICS
Volume 25A. number 7
LETTERS
presents a coherent state defined by the new coordinates (x0 - k(t), Yo - h(8).
9 October 1967
1. P.Carruthere and N.M. Nieto. Am. J, Phys. 33 (1965) 537. 2. k. J.klauber. Phys. Letters 21 (1966) 650. 3. C. L. Mehta and E. C.G.Sudarshan, Phys. Letters
I would like to express my appreciation to Professor Edward L. O’Neill and Professor
22 (1966) 574. 4. W. H. Louisell,
Jerald Weiss of the Worcester Polytechnic Institute for their steady interest and many stimulat-
Radiation
and noise in quantum
electronics (McGraw-Hill, 1964) p. 103. (eq. 3.28).
5. M. Lax and W. H. Louisell. (1967) 47.
ing discussions.
IEEE.
Vol. QE-3
*****
ON AN OBJECTION TO THE PATASHINSKII-POKROVSKII OF THE X-TRANSITION IN LIQUID 4He
THEORY
M. A. MOORE Department
of Theoretical Received
Physics.
Oxford, UK
25 August 1967
It is shown that the often-made criticism of the Patashinskii-Pokrovskii the Ising model implies no criticism of its validity for the A -transition.
The theory of the X-transition proposed by Patashinskii and Pokrovskli [l] * while in excellent agreement with experiment [2,3] has come under attack on several accounts. We wish to point out that at least one of these objections is without foundation. The particular objection is one raised by Kadanoff et al. [4] who observed that when the ideas of PP are extended directly to the Ising model, as was done by Abe [5], results are obtained in clear disagreement with the known Ising model answers. Now the reason why the PP theory necessarily fails for the Ising model has been given by Stillinger [6]. We shall show that Stillinger’s argument does not work in the case of the h-transition, allowing the possibility that the PP theory is really valid, but that it is just the attempt to apply it to the Ising model which is incorrect. Stillinger’s argument goes as follows. Let F(k,P) denote the Fourier transform of the pair correlation function (I_r1p2) for p < PC, where PC = (kBTC)-1, and TC is the Curie temperature. A self-energy operator W(k,B) can be defined via the equation F(k, p) = [l + W(k, /?)]-I. Because pi = f 1, we have the relation * We shall refer to this paper as PP.
theory that it fails when applied to
[l + W(k,p)]-l,
1 = ;Jd3k 7
where T is the volume of the Brillouin zone. The energy of the system can be written as
E(P) __ = & Jd3k N
v(k) F(k,P),
7
where v(k) is the Fourier transform of the interspin potential. The specific heat is given by
alap
C(P) = -kBp2
E(P)
c(p) _ O2 j-& -_kgN 277
i.e.
v(k) [l+W(k, P)12
awe?, PI*
(3)
ap
This can be cast into a more revealing form by multiplying (1) by iv(O), applying P2 a/ap to it and adding to (3) c(p)
= @
2T ;
,-$k
[v(k)-v(O)]aw(k,P) [1+W(k,/3)12
ap
(4)
’
In the theory of Abe and PP, aW(k,B)/ap remains finite a\ p = PC, and for small k,F(k,P) behaves as k-i. For a short-range potential [v(k) - v(O)] will vary as k2, so that C(&-+) calculated from (4) should be finite, the apparent 499
Volume 25A. n u m b e r 7
PHYSICS
s i n g u l a r i t y in the d e n o m i n a t o r at k = 0, fl = tiC being c o m p l e t e l y quenched. But C(~3C) is a l m o s t c e r t a i n l y infinite. In addition to this f a i l u r e , the A b e - P P t h e o r y g i v e s a C u r i e - W e i s s dependence f o r the s t a t i c s u s c e p t i b i l i t y , i.e. F(0, fl) = = p W ( 0 , fi)/afi i/3=fl~ (flr,-fl)] -1, which is known to be i n c o r r e c t , and a v a r i a t i o n of F(k, tiC) as k -1"94 is thought m o r e l i k e l y than k-~ as k - - 0 [6]. F o r liquid 4He the situation is quite different. H e r e Cv(~), the s p e c i f i c heat at constant v o l u m e does r e m a i n finite [7]. Infinite s i n g u l a r i t i e s do a r i s e in c e r t a i n q u a n t i t i e s like Cp, the s p e c i f i c heat at constant p r e s s u r e and K T, the i s o t h e r m a l c o m p r e s s i b i l i t y . Since t h e s e two quantities a r e r e l a t e d by Cp = (Cv/Ks)K T w h e r e K S is the a d i a batic c o m p r e s s i b i l i t y which r e m a i n s finite at the A-line, we need only c o n s i d e r how the infinite s i n g u l a r i t y in KT d ev e l o p s . KT is e a s i l y e x p r e s sed in t e r m s of the o n e - p a r t i c l e G r e e n function G(k, corn) [8], which p la y s the analogous r o l e to F ( k , fl) in the t h e o r y of the A - t r a n s i t i o n .
KT=
1
/ an \
1 1 (2~) 3 ~n 2
~mjd3k 6 2 ( k ' win) [1 - ~bZ- ( k ,
corn)];
n is the density of the s y s t e m , Wm = 2~mfl-1 w h e r e rn = 0 , 1, . . . , a n d Z ( k , wm) is the s e l f e n e r g y o p e r a t o r in the n o r m a l Bose s y s t e m . P P c l a i m that on the ~ - l i n e (1 - aZ(k, 0 ) / a ~ ) and hE(k, 0)/a~ a r e n o n - z e r o and finite and that
ZUR
IMPULSVERTEILUNG
LETTERS
9 O c t o b e r 1967
G(k, 0) ~ - (AkJ) -1. Using t h e s e r e s u l t s it is e a s y to see how an infinite l o g a r i t h m i c s i n g u l a r i t y in Cp and K T can develop. F o r Cp and KT, but not f o r Cv, t h e r e is no equation analogous to (4) which can r e m o v e an apparent infinite s i n g u l a r ity, and t h e r e f o r e no r e a s o n why the t h e o r y of P P is obviously w o r m g for the A - t r a n s i t i o n .
References 1. A. Z. P a t a s h i n s k i i and V. L. P o k r o v s k i i . Soy. P h y s . J E T P 19 (1964) 677•
2. E.G. Batyev, A. Z. Patashinskii and V. L. Pokrovskii. Sov. Phys. JETP 20 (1965) 398. 3. M.A. Moore and R.B.Stinchcombe, Phys. Letters 24A (1967) 619. 4. L.P. Kadanoff et al. Rev. Mod. Phys. 39 (1967) 395. 5. R.Abe, Prog. Theoret. Phys. (Kyoto) 35 (1965) 209. 6. F.H.Stillinger, Phys. Rev. 146 (1966) 209. 7. M.J. Buckingham and W. M. Fairbank, Progress in low temperature physics III, ed. C . J . G o r t e r (North-Holland Publishing Company. Amsterdam. 1961) ch. 3. 8. A.A.Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods of quantum field theory in statistical physics (Prentice-Hall, New Jersey, 1963) ch. 3.
ENDLICHER
FERMI-GASE
E. J A H N I G * und P. Z I E S C H E
Piidiigogisches Institut Dresden, Abteilung f~r Theoretische Physik, Dresden, Germany Eingegangen am 5 September 1967
In the one-dimensional case we discuss, how the ~sharpness ~ of the Fermi-surface depends on the partisle-number, or on the size of the system.
In der Metallphysik spielt bekanntlich die Er* Jetzige Adresse: D D R 123 Beeskow, Erweiterte Oberschule. 500
f o r s c h u n g d e r F e r m i - F l t L c h e n wegen i h r e r Inf o r m a t i o n e n tiber wichtige M e t a l l e i g e n s c h a [ t e n s i n e g r o s z e R o l l e . D ah er ist es von I n t e r e s s e nach d e r ~Scharfe ~ d i e s e r F e r m i - F 1 R c h e n zu
Volume 25A. number 7
LETTERS
presents a coherent state defined by the new coordinates (x0 - k(t), Yo - h(8).
9 October 1967
1. P.Carruthere and N.M. Nieto. Am. J, Phys. 33 (1965) 537. 2. k. J.klauber. Phys. Letters 21 (1966) 650. 3. C. L. Mehta and E. C.G.Sudarshan, Phys. Letters
I would like to express my appreciation to Professor Edward L. O’Neill and Professor
22 (1966) 574. 4. W. H. Louisell,
Jerald Weiss of the Worcester Polytechnic Institute for their steady interest and many stimulat-
Radiation
and noise in quantum
electronics (McGraw-Hill, 1964) p. 103. (eq. 3.28).
5. M. Lax and W. H. Louisell. (1967) 47.
ing discussions.
IEEE.
Vol. QE-3
*****
ON AN OBJECTION TO THE PATASHINSKII-POKROVSKII OF THE X-TRANSITION IN LIQUID 4He
THEORY
M. A. MOORE Department
of Theoretical Received
Physics.
Oxford, UK
25 August 1967
It is shown that the often-made criticism of the Patashinskii-Pokrovskii the Ising model implies no criticism of its validity for the A -transition.
The theory of the X-transition proposed by Patashinskii and Pokrovskli [l] * while in excellent agreement with experiment [2,3] has come under attack on several accounts. We wish to point out that at least one of these objections is without foundation. The particular objection is one raised by Kadanoff et al. [4] who observed that when the ideas of PP are extended directly to the Ising model, as was done by Abe [5], results are obtained in clear disagreement with the known Ising model answers. Now the reason why the PP theory necessarily fails for the Ising model has been given by Stillinger [6]. We shall show that Stillinger’s argument does not work in the case of the h-transition, allowing the possibility that the PP theory is really valid, but that it is just the attempt to apply it to the Ising model which is incorrect. Stillinger’s argument goes as follows. Let F(k,P) denote the Fourier transform of the pair correlation function (I_r1p2) for p < PC, where PC = (kBTC)-1, and TC is the Curie temperature. A self-energy operator W(k,B) can be defined via the equation F(k, p) = [l + W(k, /?)]-I. Because pi = f 1, we have the relation * We shall refer to this paper as PP.
theory that it fails when applied to
[l + W(k,p)]-l,
1 = ;Jd3k 7
where T is the volume of the Brillouin zone. The energy of the system can be written as
E(P) __ = & Jd3k N
v(k) F(k,P),
7
where v(k) is the Fourier transform of the interspin potential. The specific heat is given by
alap
C(P) = -kBp2
E(P)
c(p) _ O2 j-& -_kgN 277
i.e.
v(k) [l+W(k, P)12
awe?, PI*
(3)
ap
This can be cast into a more revealing form by multiplying (1) by iv(O), applying P2 a/ap to it and adding to (3) c(p)
= @
2T ;
,-$k
[v(k)-v(O)]aw(k,P) [1+W(k,/3)12
ap
(4)
’
In the theory of Abe and PP, aW(k,B)/ap remains finite a\ p = PC, and for small k,F(k,P) behaves as k-i. For a short-range potential [v(k) - v(O)] will vary as k2, so that C(&-+) calculated from (4) should be finite, the apparent 499
Volume 25A. n u m b e r 7
PHYSICS
s i n g u l a r i t y in the d e n o m i n a t o r at k = 0, fl = tiC being c o m p l e t e l y quenched. But C(~3C) is a l m o s t c e r t a i n l y infinite. In addition to this f a i l u r e , the A b e - P P t h e o r y g i v e s a C u r i e - W e i s s dependence f o r the s t a t i c s u s c e p t i b i l i t y , i.e. F(0, fl) = = p W ( 0 , fi)/afi i/3=fl~ (flr,-fl)] -1, which is known to be i n c o r r e c t , and a v a r i a t i o n of F(k, tiC) as k -1"94 is thought m o r e l i k e l y than k-~ as k - - 0 [6]. F o r liquid 4He the situation is quite different. H e r e Cv(~), the s p e c i f i c heat at constant v o l u m e does r e m a i n finite [7]. Infinite s i n g u l a r i t i e s do a r i s e in c e r t a i n q u a n t i t i e s like Cp, the s p e c i f i c heat at constant p r e s s u r e and K T, the i s o t h e r m a l c o m p r e s s i b i l i t y . Since t h e s e two quantities a r e r e l a t e d by Cp = (Cv/Ks)K T w h e r e K S is the a d i a batic c o m p r e s s i b i l i t y which r e m a i n s finite at the A-line, we need only c o n s i d e r how the infinite s i n g u l a r i t y in KT d ev e l o p s . KT is e a s i l y e x p r e s sed in t e r m s of the o n e - p a r t i c l e G r e e n function G(k, corn) [8], which p la y s the analogous r o l e to F ( k , fl) in the t h e o r y of the A - t r a n s i t i o n .
KT=
1
/ an \
1 1 (2~) 3 ~n 2
~mjd3k 6 2 ( k ' win) [1 - ~bZ- ( k ,
corn)];
n is the density of the s y s t e m , Wm = 2~mfl-1 w h e r e rn = 0 , 1, . . . , a n d Z ( k , wm) is the s e l f e n e r g y o p e r a t o r in the n o r m a l Bose s y s t e m . P P c l a i m that on the ~ - l i n e (1 - aZ(k, 0 ) / a ~ ) and hE(k, 0)/a~ a r e n o n - z e r o and finite and that
ZUR
IMPULSVERTEILUNG
LETTERS
9 O c t o b e r 1967
G(k, 0) ~ - (AkJ) -1. Using t h e s e r e s u l t s it is e a s y to see how an infinite l o g a r i t h m i c s i n g u l a r i t y in Cp and K T can develop. F o r Cp and KT, but not f o r Cv, t h e r e is no equation analogous to (4) which can r e m o v e an apparent infinite s i n g u l a r ity, and t h e r e f o r e no r e a s o n why the t h e o r y of P P is obviously w o r m g for the A - t r a n s i t i o n .
References 1. A. Z. P a t a s h i n s k i i and V. L. P o k r o v s k i i . Soy. P h y s . J E T P 19 (1964) 677•
2. E.G. Batyev, A. Z. Patashinskii and V. L. Pokrovskii. Sov. Phys. JETP 20 (1965) 398. 3. M.A. Moore and R.B.Stinchcombe, Phys. Letters 24A (1967) 619. 4. L.P. Kadanoff et al. Rev. Mod. Phys. 39 (1967) 395. 5. R.Abe, Prog. Theoret. Phys. (Kyoto) 35 (1965) 209. 6. F.H.Stillinger, Phys. Rev. 146 (1966) 209. 7. M.J. Buckingham and W. M. Fairbank, Progress in low temperature physics III, ed. C . J . G o r t e r (North-Holland Publishing Company. Amsterdam. 1961) ch. 3. 8. A.A.Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods of quantum field theory in statistical physics (Prentice-Hall, New Jersey, 1963) ch. 3.
ENDLICHER
FERMI-GASE
E. J A H N I G * und P. Z I E S C H E
Piidiigogisches Institut Dresden, Abteilung f~r Theoretische Physik, Dresden, Germany Eingegangen am 5 September 1967
In the one-dimensional case we discuss, how the ~sharpness ~ of the Fermi-surface depends on the partisle-number, or on the size of the system.
In der Metallphysik spielt bekanntlich die Er* Jetzige Adresse: D D R 123 Beeskow, Erweiterte Oberschule. 500
f o r s c h u n g d e r F e r m i - F l t L c h e n wegen i h r e r Inf o r m a t i o n e n tiber wichtige M e t a l l e i g e n s c h a [ t e n s i n e g r o s z e R o l l e . D ah er ist es von I n t e r e s s e nach d e r ~Scharfe ~ d i e s e r F e r m i - F 1 R c h e n zu