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Regularization of Feynman diagrams in the high density expansion method for spin systems near critical points
Volume 116, number 7
PHYSICS LETTERS A
30 June 1986
R E G U L A R I Z A T I O N OF F E Y N M A N D I A G R A M S IN THE HIGH DENSITY EXPANSION METHOD FOR SPIN S Y S T E M S NEAR C R I T I C A L P O I N T S Z. O N Y S Z K I E W I C Z 1
Department of Physics, Universityof Florida, Gainesville,FL 32611, USA and A. W I E R Z B I C K I 2
Department of Chemistry, Universityof Florida, Gainesville,FL 32611, USA Received 28 March 1986; accepted for publication 29 April 1986
A regularization of the high density expansion for spin systems is proposed. This regularization is used to remove imaginary parts of observableswhich appear near critical points. This regularization was applied to a first approximation for the simple Ising model.
One of the most successful tools to investigate thermodynamic properties of spin models is the high density expansion method ( H D E M ) [1-12]. As a starting point to H D E M in the theory of magnetism we choose the following decomposition of the hamiltonian:
H = ( H - H1) + Ht f Ho + H 1,
(1)
where the perturbative part H 1 is defined by the transformation H ~ H 1 = H ( S z --) 8S~),
(2)
where
*S~=S~-(SZ),
(S')=Tr{S~exp[fl(F-H)]},
f l = ( k a T ) -a,
and S ~ is the operator of the z-component of the spin. In the theory of magnetism H D E M is based on the classification of Feymnan diagrams with respect to powers of the parameter l / z , where z is the effective number of spins interacting with any given spin [13]. The approach is valid only when z >> 1 and when H 1 is small. In general, for the average value ( A ) of any observable A of the spin system given, the high density expansion in the thermodynamic limit can be expressed as follows:
=
=
=
lso:
zn Vn r
n n r ( k l ' k2 . . . . . kLnr) df~,,~,
(3)
1 Permanent address: Institute of Physics, A. Mickiewicz University, Poznah, Poland. 2 Permanent address: Department of Chemistry, A. Miekiewicz University, Poznah, Poland. 0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
335
Volume 116, number 7
PHYSICS LETYERS A
30 June 1986
where
~ r = fJ~nrd~nr, R~ is the number of Feynman diagrams of the nth order, Lnr is the number of loops in the Feynman diagram corresponding to A~ and f~,~ is the region of the momentum space volume Vnr o v e r which the integration is performed. In general near critical points in any order of approximation for n > 0 we have (A> 4= (A)*.
(4)
This relation is physically unacceptable and is the consequence of the used approximations or divergences of the expansion (3) near critical points. If the expansion (3) were exact (provided that the function was analytical) all imaginary parts of this expansion should compensate one another, however, when we approximate (3) by taking only a limited class of diagrams the imaginary part remains. These imaginary parts cannot be simply neglected because this leads to nonphysical results [14]. Taking this into regard we propose to remove the imaginary part in each Feynman diagram independently by means of the following "regularization" procedure. The idea of regularization consists in replacing the expansion (3) by the following one: n" z" 1 V.'~ 1 f./.~(I,~, E ,
(A) -- rega,
k~. . . . . kL~)d['~nr,
(5)
r=l
where
f. The set ~2'= {~2',~} c f~ = { ~ r } and ~2"~ is the maximal subregion of the region ~-~nr for which the following equation Imfa a n r ( k l , k 2 . . . . . kL.r) d~nr = Pnr
0
(6)
holds. From (5) it follows that lim rega,(A) = (A),
(A) = rega,(A )
f~, --, f~
outside the vicinity of the critical points. As an example of the proposed regularization we perform the regularization of Feynman diagrams for the first approximation of HDEM for a simple cubic one-sublattice Ising model (S = 1/2) described by the hamiltonian
= - ~ E ~,S/S;,.
(7)
f~f'
In the first approximation of HDEM we obtained the following expression [5-7]: = ~ = ; 336
+ ~ ! +(-~"!+
+O(1/z2), !E~-~][~
(S) [ + ....
(9)
Volume 116, number 7
PHYSICS LETTERSA
30 June 1986
where I'" "''
"] = 2 - " d " L ( y ) / d y m ,
b(y)=ln2coshy,
y = ½flr(o)(s z) = o/t,
(10)
m
g
I
',
1
!=-~ ~_,flT(k),
o = 2(SZ),
t = 4[fiT(0)]-'
k
Substituting (9) and (10) into (8) we obtain in the thermodynamic limit a [1 -- w ( ~')] t h ( o / t ) ,
(11)
where w(r) is the so-called generalized Watson integral
1 w( ~ ) = - -
g
,ff3,,O ~0 ~0 "l'--y
dk x dky dkz,
(12)
which is the complex number for r ~ (0,1) [15], where
v = ~ (cos kx + cos ky + cos ks)
(13)
(here the lattice constant a = 1) and r = t [ 1 - t h 2 ( o / t ) ] -1 In accordance with proposed regularization we are looking for f~l for which Im w ( r ) = 0.
(14)
From the numerical solution of eq. (14) we obtained that f~ia is the cubic in momentum space of the volume which is equal to [~r - b(p)] 3, where b(z) is the smallest value (for a given z) of the lower limit of the integral (12) (see fig. 1). Hence, according to the above considerations and in agreement with (5) we can replace eq. (11) by the following one: o = [1 - w ( r , b ( z ) ) ] t h ( o / t ) ,
(15)
where w(r, b(r))=
f,n" f,n" f,n" "~ 3' dkx dky dkz, [ ~ r - b(~')] _3 Jb(r)Jb(~)Jb(~)z--
(16)
and
w(r, b ( r ) ) = w*('r, b(r)). The magnetization o, integrals w(r) and w(r, b(r)) were computed numerically. The results obtained are presented graphically in figs. 1 and 2. Using eq. (15) we obtained a Curie temperature t c = 0.926 (in relative units). The results obtained allow us to state that the proposed regularization of Feynman diagrams in H D E M leads to good physical results. In previous works [5-7] the first approximation of H D E M was considered to be a sufficient one for the description of the simple Ising model in the whole range of temperatures this assumption was based on the values of the Curie temperature which were obtained taking into account only the real part of w(r). Our results do not support those assumptions because the function a(t) obtained this way has a jump, contrary to the continuous behavior of o(t) obtained by means of another 337
Volume 116, number 7
PHYSICS LETTERS A I
30 June 1986
J
1.0
05
\
\ (7-
'/ /
1
\
/
/0
/ / / / /
/ -0.5
-
/
i/ /i / i 0.5
m
/ \ \
\
/
\ \
-I.0
0
I 0.5
I 1.0
Fig. 1. The dependence of bOO (solid line), the integral w(r, b(T)) (dashed line), and Re w(z) (dotted line) on the parameter r.
0
I 05
I {.0
Fig. 2. Magnetization o for the model (7) versus temperature t obtained numerically from eq. (11) (solid line) (only real part), from eq. (15) (dashed line), and in the molecular field approximation (dotted line).
m e t h o d [16]. The regularization proposed in this paper removes the imaginary part of o ( t ) and considerably decreases the j u m p of o(t). W e were not able to remove the j u m p of o ( t ) completely because it is not the consequence of the imaginary part of o ( t ) but the consequence of an approximation [17,18]. W e believe that the regularization proposed m a y be very useful in higher approximations of H D E M and for more complicated spin models. This research was supported by National Science F o u n d a t i o n grant C H E 8411932.
References
[1] R. Brout, Phys. Rev. 118 (1960) 1009; 122 (1961) 469. [2] G. Horwitz and H.B. Callen, Phys. Rev. 124 (1961) 1757. 338
Volume 116, number 7
PHYSICS LETTERS A
30 June 1986
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
F. Englert, Phys. Rex,. 129 (1963) 567. R.B. Stinchcombe, G. Horwitz, F. Englert and R. Brout, Phys. Rev. 130 (1963) 155. V.G. Vaks, A.D. Larkin and S.A. Pikin, Soy. Phys. JETP 26 (1968) 647. Y.A. Izyumov and F.A. Kassan-Ogly, Phys. Met. MetaUogr. 30 (1970) 1. D. Hsing-Yen Yang and Yung-Li Wang, Phys. Rev. B 10 (1974) 4714; 12 (1975) 1057. K. Skrobi~ and B. Westwahski, Physica A 83 (1976) 257. Z. Onyszkiewicz, Phys. Lett, A 57 (1976) 480. B. Westwaflski, Physica A 92 (1978) 501. Z. Onyszkiewicz and H. Cotta, Acta Phys. Polon. A 55 (1979) 189. G.S. Psaltakis and M.G. Cottam, J. Phys. C 15 (1982) 4847. M.G. Cottam and R.B. Stinchcombe, J. Phys. C 3 (1970) 2283. B. Westwanski, private communication. T. Horiguchi, J. Phys. Soc. Japan A 5 (1972) 67. C. Domb, Ising model, in: Phase transitions and critical phenomena, Vol. 3, eds. C. Domb and M.S. Green (Academic Press, London, 1974). [17] Z. Onyszkiewicz, Phys. Lett. A 76 (1980) 411. [18] Z. Onyszldewicz, Physica A 103 (1980) 257.
339
PHYSICS LETTERS A
30 June 1986
R E G U L A R I Z A T I O N OF F E Y N M A N D I A G R A M S IN THE HIGH DENSITY EXPANSION METHOD FOR SPIN S Y S T E M S NEAR C R I T I C A L P O I N T S Z. O N Y S Z K I E W I C Z 1
Department of Physics, Universityof Florida, Gainesville,FL 32611, USA and A. W I E R Z B I C K I 2
Department of Chemistry, Universityof Florida, Gainesville,FL 32611, USA Received 28 March 1986; accepted for publication 29 April 1986
A regularization of the high density expansion for spin systems is proposed. This regularization is used to remove imaginary parts of observableswhich appear near critical points. This regularization was applied to a first approximation for the simple Ising model.
One of the most successful tools to investigate thermodynamic properties of spin models is the high density expansion method ( H D E M ) [1-12]. As a starting point to H D E M in the theory of magnetism we choose the following decomposition of the hamiltonian:
H = ( H - H1) + Ht f Ho + H 1,
(1)
where the perturbative part H 1 is defined by the transformation H ~ H 1 = H ( S z --) 8S~),
(2)
where
*S~=S~-(SZ),
(S')=Tr{S~exp[fl(F-H)]},
f l = ( k a T ) -a,
and S ~ is the operator of the z-component of the spin. In the theory of magnetism H D E M is based on the classification of Feymnan diagrams with respect to powers of the parameter l / z , where z is the effective number of spins interacting with any given spin [13]. The approach is valid only when z >> 1 and when H 1 is small. In general, for the average value ( A ) of any observable A of the spin system given, the high density expansion in the thermodynamic limit can be expressed as follows:
=
=
=
lso:
zn Vn r
n n r ( k l ' k2 . . . . . kLnr) df~,,~,
(3)
1 Permanent address: Institute of Physics, A. Mickiewicz University, Poznah, Poland. 2 Permanent address: Department of Chemistry, A. Miekiewicz University, Poznah, Poland. 0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
335
Volume 116, number 7
PHYSICS LETYERS A
30 June 1986
where
~ r = fJ~nrd~nr, R~ is the number of Feynman diagrams of the nth order, Lnr is the number of loops in the Feynman diagram corresponding to A~ and f~,~ is the region of the momentum space volume Vnr o v e r which the integration is performed. In general near critical points in any order of approximation for n > 0 we have (A> 4= (A)*.
(4)
This relation is physically unacceptable and is the consequence of the used approximations or divergences of the expansion (3) near critical points. If the expansion (3) were exact (provided that the function was analytical) all imaginary parts of this expansion should compensate one another, however, when we approximate (3) by taking only a limited class of diagrams the imaginary part remains. These imaginary parts cannot be simply neglected because this leads to nonphysical results [14]. Taking this into regard we propose to remove the imaginary part in each Feynman diagram independently by means of the following "regularization" procedure. The idea of regularization consists in replacing the expansion (3) by the following one: n" z" 1 V.'~ 1 f./.~(I,~, E ,
(A) -- rega,
k~. . . . . kL~)d['~nr,
(5)
r=l
where
f. The set ~2'= {~2',~} c f~ = { ~ r } and ~2"~ is the maximal subregion of the region ~-~nr for which the following equation Imfa a n r ( k l , k 2 . . . . . kL.r) d~nr = Pnr
0
(6)
holds. From (5) it follows that lim rega,(A) = (A),
(A) = rega,(A )
f~, --, f~
outside the vicinity of the critical points. As an example of the proposed regularization we perform the regularization of Feynman diagrams for the first approximation of HDEM for a simple cubic one-sublattice Ising model (S = 1/2) described by the hamiltonian
= - ~ E ~,S/S;,.
(7)
f~f'
In the first approximation of HDEM we obtained the following expression [5-7]:
+ ~ ! +(-~"!+
+O(1/z2), !E~-~][~
(S) [ + ....
(9)
Volume 116, number 7
PHYSICS LETTERSA
30 June 1986
where I'" "''
"] = 2 - " d " L ( y ) / d y m ,
b(y)=ln2coshy,
y = ½flr(o)(s z) = o/t,
(10)
m
g
I
',
1
!=-~ ~_,flT(k),
o = 2(SZ),
t = 4[fiT(0)]-'
k
Substituting (9) and (10) into (8) we obtain in the thermodynamic limit a [1 -- w ( ~')] t h ( o / t ) ,
(11)
where w(r) is the so-called generalized Watson integral
1 w( ~ ) = - -
g
,ff3,,O ~0 ~0 "l'--y
dk x dky dkz,
(12)
which is the complex number for r ~ (0,1) [15], where
v = ~ (cos kx + cos ky + cos ks)
(13)
(here the lattice constant a = 1) and r = t [ 1 - t h 2 ( o / t ) ] -1 In accordance with proposed regularization we are looking for f~l for which Im w ( r ) = 0.
(14)
From the numerical solution of eq. (14) we obtained that f~ia is the cubic in momentum space of the volume which is equal to [~r - b(p)] 3, where b(z) is the smallest value (for a given z) of the lower limit of the integral (12) (see fig. 1). Hence, according to the above considerations and in agreement with (5) we can replace eq. (11) by the following one: o = [1 - w ( r , b ( z ) ) ] t h ( o / t ) ,
(15)
where w(r, b(r))=
f,n" f,n" f,n" "~ 3' dkx dky dkz, [ ~ r - b(~')] _3 Jb(r)Jb(~)Jb(~)z--
(16)
and
w(r, b ( r ) ) = w*('r, b(r)). The magnetization o, integrals w(r) and w(r, b(r)) were computed numerically. The results obtained are presented graphically in figs. 1 and 2. Using eq. (15) we obtained a Curie temperature t c = 0.926 (in relative units). The results obtained allow us to state that the proposed regularization of Feynman diagrams in H D E M leads to good physical results. In previous works [5-7] the first approximation of H D E M was considered to be a sufficient one for the description of the simple Ising model in the whole range of temperatures this assumption was based on the values of the Curie temperature which were obtained taking into account only the real part of w(r). Our results do not support those assumptions because the function a(t) obtained this way has a jump, contrary to the continuous behavior of o(t) obtained by means of another 337
Volume 116, number 7
PHYSICS LETTERS A I
30 June 1986
J
1.0
05
\
\ (7-
'/ /
1
\
/
/0
/ / / / /
/ -0.5
-
/
i/ /i / i 0.5
m
/ \ \
\
/
\ \
-I.0
0
I 0.5
I 1.0
Fig. 1. The dependence of bOO (solid line), the integral w(r, b(T)) (dashed line), and Re w(z) (dotted line) on the parameter r.
0
I 05
I {.0
Fig. 2. Magnetization o for the model (7) versus temperature t obtained numerically from eq. (11) (solid line) (only real part), from eq. (15) (dashed line), and in the molecular field approximation (dotted line).
m e t h o d [16]. The regularization proposed in this paper removes the imaginary part of o ( t ) and considerably decreases the j u m p of o(t). W e were not able to remove the j u m p of o ( t ) completely because it is not the consequence of the imaginary part of o ( t ) but the consequence of an approximation [17,18]. W e believe that the regularization proposed m a y be very useful in higher approximations of H D E M and for more complicated spin models. This research was supported by National Science F o u n d a t i o n grant C H E 8411932.
References
[1] R. Brout, Phys. Rev. 118 (1960) 1009; 122 (1961) 469. [2] G. Horwitz and H.B. Callen, Phys. Rev. 124 (1961) 1757. 338
Volume 116, number 7
PHYSICS LETTERS A
30 June 1986
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
F. Englert, Phys. Rex,. 129 (1963) 567. R.B. Stinchcombe, G. Horwitz, F. Englert and R. Brout, Phys. Rev. 130 (1963) 155. V.G. Vaks, A.D. Larkin and S.A. Pikin, Soy. Phys. JETP 26 (1968) 647. Y.A. Izyumov and F.A. Kassan-Ogly, Phys. Met. MetaUogr. 30 (1970) 1. D. Hsing-Yen Yang and Yung-Li Wang, Phys. Rev. B 10 (1974) 4714; 12 (1975) 1057. K. Skrobi~ and B. Westwahski, Physica A 83 (1976) 257. Z. Onyszkiewicz, Phys. Lett, A 57 (1976) 480. B. Westwaflski, Physica A 92 (1978) 501. Z. Onyszkiewicz and H. Cotta, Acta Phys. Polon. A 55 (1979) 189. G.S. Psaltakis and M.G. Cottam, J. Phys. C 15 (1982) 4847. M.G. Cottam and R.B. Stinchcombe, J. Phys. C 3 (1970) 2283. B. Westwanski, private communication. T. Horiguchi, J. Phys. Soc. Japan A 5 (1972) 67. C. Domb, Ising model, in: Phase transitions and critical phenomena, Vol. 3, eds. C. Domb and M.S. Green (Academic Press, London, 1974). [17] Z. Onyszkiewicz, Phys. Lett. A 76 (1980) 411. [18] Z. Onyszldewicz, Physica A 103 (1980) 257.
339