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On the exact renormalization of the two-dimensional coulomb gas of point particles
Volume 105A, number 4,5
PHYSICS LETTERS
15 October 1984
ON THE EXACT RENORMALIZATION OF THE TWO-DIMENSIONAL COULOMB GAS OF POINT PARTICLES
E.C. MARINO
Departamento di Fisica, Universidade Federal de Sdo Carlos, Cx.P. 676, 13.560 Sdo Carlos, SP, Brazil Received 17 July 1984 An exact renormalization of the two-dimensional Coulomb gas of point particles is presented in the grand-canonical ensemble. This renormalization eliminates all short distance divergences and a sensible equation of state is thereby obtained.
The exact equation of state for a two-dimensional plasma of point charges -+q interacting through the Coulomb potential (Coulomb gas) is well known since a long time [1,2] (eq. (6) below). Although it predicts the right temperature, T c = q2/4k (k is the Boltzmann constant) for the Kosterlitz-Thouless phase transition [3] it does not seem to make sense for temperatures below 2T c [1,2]. This happens due to the electrostatic energy infinities which appear when charges o f different sign occupy the same place. These infinities make the integrals appearing in the partition function to diverge and the usual procedure leading to the equation of state no longer makes sense. The standard solution to this problem is to consider particles with a hard core 8 and to make the integrals start from 5 instead o f zero [3,4]. This is a bad procedure if one wants to describe a gas of point particles. In this letter, we show how to eliminate these divergences without resorting to a hard core, but by an exact renormalization o f the zero point energy o f the system. This is made possible by the use of a grand-canonical ensemble. After renormalization we are led to the same thermodynamic equation of state which now makes sense down to the critical temperature. Further divergences which affect this system are the classical self-energies o f the point particles. These can be eliminated by an exact renormalization o f the chemical potential, as is shown below. This last renormalization is straightforward [5]. The first one is not. The Coulomb gas w i t h N point particles is characterized by a charge density
N p(x) = q ~ Xi6(x - x i ) , i=1
(1)
where k i = -+1 and x is a two-dimensional variable. Using the short distance regularized Green function o f the two-dimensional laplacian operator [6], G(x)---limlel_, 0 ln(Ixl + lel), we obtain the hamiltonian
H= ~
q
~ Xik/ln(Ix i - x i l + lel) i,]--#i
i=1 2m
-
-
1Nq2lnlel
=H 0 +H I .
(2)
At the end we will take the limit Ie I --" 0. The last term above is the classical self-energy of the charges. Let us consider a neutral plasma (Z/N=1Xi = 0; N even), first in a canonical ensemble. Due to neutrality, we can make
N Hl=-lq2
~
i,j,i
[Ix i - x / I + lel) Xihiln( ' a !
-- ~Nq 2 In (lel/a) ,
(3)
where a is an arbitrary, finite constant with dimension of length. The partition function is N
Z(T,V,i)=f
N
H d%f iH d2pi e - " H
ixil
"=
'
(4) 0.375-9601/84/$ 03.00 ©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
215
Volume 105A, number 4,5
PHYSICS LETTERS
where R is the radius of the system (V = rrR 2) and fl l/kT. In order to make Z dimensionless, we multiply it by (1/aK) 2N, where r is an arbitrary finite constant with dimension of momentum and a is the same as above. Multiplying Z by this arbitrary finite constant does not affect the thermodynamic properties of the system. Introducing H in (4), we obtain straightforwardly =
Z(T, V , N ) =
Xexp
N d2xi ZN f I-I lel~0 Ixil
positive charge and n negative. From (7) it is easy to see that we can eliminate the divergences associated to the classical self-energies by introducing a renormalized fugacity & = aZ 0. To this renormalized fugacity there corresponds a renormalized chemical potential
2 ~ XiX/.In i,]>i
with ~ = e fl~ and Z 0 given by (5c). In terms offi, we write o~
zo(r, v, (50
a
(5b)
with (5c)
From (5), upon rescaling x i = Rfci, one is immediately led to the equation of state [1] p V = A r k T [ 1 - i1~ 2 l .
(6)
This equation predicts a phase transition at flcq 2 = 4. Observe, however, that for/~/2 >/2, the integrals in (5a) diverge and we can no longer say that the equation of state (6) corresponds to the partition function Z. The divergences occur whenever x i ~ xi, with hi 4: X/, that is, when charges of different sign occupy the same position. Let us turn now to the grand-canonical ensemble. Only neutral configurations contribute in the thermodynamic limit. We must have, therefore, N = 2n, that is n positive and n negative charges. The grand-partition function is given by
z (r, v, u)=
** (c~Z0)2n
E
lel~On=O
(n[) 2
2n
d2xi
ixil
a2
f
H
2n
[flq2 ~ Xik, l n ( l X i - X ] l + [ e l . ) ] X exp_ i,]>i a
'
(7)
where ot = eOU is the fugacity and/a, the chemical potential. l[(n!) 2 is a combinatorial factor corresponding to the fact that there are 2n particles, n of which have
.-.2n
E
no= (n!)"
v),
(9)
withHzn defined by (5b) (H 0 = I). Again, making a rescaling xi = RYei = (Vfir)l/2 Yci and observing that p = - O f l / b V)T' # with fl = - k T In ZG, we obtain right away the equation of state p V = (iV) k T [ 1 - ~ a
Z 0 = (27rm/flr 2) (lel/a)Oq 2 ]2.
(8)
fi = I.t + k T l n Z 0 ,
lim
-= lim Z f f H N ( T , V ) , lel~0
216
15 October 1984
2],
(10)
which coincides with (6) in the thermodynamic limit. In eq. (10), oo
(N)=Z~I
~ (t~)2n 2nH2n(T, V) n=O (n!) 2
(N=En).(ll)
Eq. (10) again does not make sense for/k/2/> 2 because the integrals in H2n diverge in this case. Let us analyze in detail the divergences of the functions H2n. In order to do that, let us separate the integration region in each of the H2n , in two distinct parts. The first one is such that the integrands are completely regular, that is Ix i - X / l > ~ for h i ¢ X/and some small 6. The rest of the integration region always contains parts in which Ix i - x~ I < 6 for one or more pairs (i, ]), with Xi 4: X/. Upon doing this, we obtain the finite part ofH2n, which we call F2n , and the various divergent part in terms of Ie I. In the usual treatments, where a hard core 8 is used, only the first part of the integration region (Ix i - x/I > 5) is considered. The details of the calculation are given in ref. [6] and the result is H 0 =1, H 2 = I(e) + F 2 ,
H 4 = 2/2(e) + 4 I ( e ) F 2 + F 4 ,
(12)
responds a renormalized hamiltonian/7, through
H 6 = 6/3(e) +
1812(e)F2 + 91(e)F4 + F 6 ,
H 8 = 4!I4(e) +
4(4!)13(e)F2 + 3(4!)12(e)F4
ZG(T, V,/2) = Tr exp [---~(/7 - / i N ) ]
(12 cont'd)
+ 16/(e)F 6 + F 8 , . . . . In the expressions above,
3q 2 = 2 ,
lel~O
(a/lel) ~q2- 2
lel
3q2>2 (13)
and the various combinatoric, numerical coefficients represent the number of different ways in which n charges of a given sign and n of the opposite one can form pairs of different charge occupying the same place. The 8 dependence of the terms containing I(e) cancels that of F2n in the limit 8 + 0 [6]. Let us introduce now expression (12) in (9). Multiplying each H2n by the appropriate factor (6) 2n/(n !)2 and factorizing all F2n appearing in the expansion, we can rearrange the sum (9) in the following way
(602n_H2n(T, 10 = D ( e ) ~ (s)2n F2n(T, V) , n=0 (n!) 2
n=0 (n!) 2
(14)
where D (e) = exp [(S) 2I(e) ] •
(15)
The remarkable fact that we can absorb all divergences in the multiplicative factor D(e) allows us to introduce a renormalized grand-partition function ZG = D - I (e)ZG, ZG(T, V,/2) = ~ (6)2n n=0 (n[) 2
= D - l(e) Tr exp whence /7 = H + H 0 ,
I(e)=2QrR/a) 2 lim l n ( a l e l ) , = 2(rrR/a) lim
15 October 1984
PHYSICS LETTERS
Volume 105A, number 4,5
F2n(T, V),
(16)
[-3(H -/2N)] ,
H 0 = kr(~)2X(e).
(17)
(18)
From ZG(T, V,/2) we can derive a renorrnalized grand-canonical potential ~ = - k T In ZG, with = ~ 2 + H 0. We may now take the limit l el -+ 0, obtaining a finite 2 G, given by (16). F r o m p = - ( 0 ~ / 0 V) T, ~ and the rescaling trick, we arrive at the same equation of state (10), but with a completely well defined car):
(a)2n ( N ) = Z d 1 n=~0
"~2nF2n(T,
If).
(19)
We conclude, therefore, that the equation of state (10) is perfectly sensible for a Coulomb gas of point particles. It is not surprising that it predicts the right critical temperature.
References [1] A.M. Salzberg and S. Prager, J. Chem. Phys. 38 (1963) 2587; R.M. May, Phys. Lett. 25A (1967) 282; G. Knorr, Phys. Lett. 28A (1968) 166. [2] C. Deutsch and M. Lavaud, Phys. Rev. A9 (1974) 2598. [3] J.M. Kosterlitz and D.J. Thouless, J, Phys. C6 (1973) 1181; J.M. Kosterlitz, J. Phys. C7 (1974) 1046. [4] J.V. Jos6, L.P. Kadanoff, S. Kirkpatrick and D. Nelson, Phys. Rev. B16 (1978) 1217. [5] S. Samuel, Phys. Rev. D18 (1978) 1916. [6] E.C. Marino, Dynamical mass generation in the Thirring model: a functional integral approach, U.F.S. Carlos preprint (1984).
which is completely finite. To this function, there cor-
217
PHYSICS LETTERS
15 October 1984
ON THE EXACT RENORMALIZATION OF THE TWO-DIMENSIONAL COULOMB GAS OF POINT PARTICLES
E.C. MARINO
Departamento di Fisica, Universidade Federal de Sdo Carlos, Cx.P. 676, 13.560 Sdo Carlos, SP, Brazil Received 17 July 1984 An exact renormalization of the two-dimensional Coulomb gas of point particles is presented in the grand-canonical ensemble. This renormalization eliminates all short distance divergences and a sensible equation of state is thereby obtained.
The exact equation of state for a two-dimensional plasma of point charges -+q interacting through the Coulomb potential (Coulomb gas) is well known since a long time [1,2] (eq. (6) below). Although it predicts the right temperature, T c = q2/4k (k is the Boltzmann constant) for the Kosterlitz-Thouless phase transition [3] it does not seem to make sense for temperatures below 2T c [1,2]. This happens due to the electrostatic energy infinities which appear when charges o f different sign occupy the same place. These infinities make the integrals appearing in the partition function to diverge and the usual procedure leading to the equation of state no longer makes sense. The standard solution to this problem is to consider particles with a hard core 8 and to make the integrals start from 5 instead o f zero [3,4]. This is a bad procedure if one wants to describe a gas of point particles. In this letter, we show how to eliminate these divergences without resorting to a hard core, but by an exact renormalization o f the zero point energy o f the system. This is made possible by the use of a grand-canonical ensemble. After renormalization we are led to the same thermodynamic equation of state which now makes sense down to the critical temperature. Further divergences which affect this system are the classical self-energies o f the point particles. These can be eliminated by an exact renormalization o f the chemical potential, as is shown below. This last renormalization is straightforward [5]. The first one is not. The Coulomb gas w i t h N point particles is characterized by a charge density
N p(x) = q ~ Xi6(x - x i ) , i=1
(1)
where k i = -+1 and x is a two-dimensional variable. Using the short distance regularized Green function o f the two-dimensional laplacian operator [6], G(x)---limlel_, 0 ln(Ixl + lel), we obtain the hamiltonian
H= ~
q
~ Xik/ln(Ix i - x i l + lel) i,]--#i
i=1 2m
-
-
1Nq2lnlel
=H 0 +H I .
(2)
At the end we will take the limit Ie I --" 0. The last term above is the classical self-energy of the charges. Let us consider a neutral plasma (Z/N=1Xi = 0; N even), first in a canonical ensemble. Due to neutrality, we can make
N Hl=-lq2
~
i,j,i
[Ix i - x / I + lel) Xihiln( ' a !
-- ~Nq 2 In (lel/a) ,
(3)
where a is an arbitrary, finite constant with dimension of length. The partition function is N
Z(T,V,i)=f
N
H d%f iH d2pi e - " H
ixil
"=
'
(4) 0.375-9601/84/$ 03.00 ©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
215
Volume 105A, number 4,5
PHYSICS LETTERS
where R is the radius of the system (V = rrR 2) and fl l/kT. In order to make Z dimensionless, we multiply it by (1/aK) 2N, where r is an arbitrary finite constant with dimension of momentum and a is the same as above. Multiplying Z by this arbitrary finite constant does not affect the thermodynamic properties of the system. Introducing H in (4), we obtain straightforwardly =
Z(T, V , N ) =
Xexp
N d2xi ZN f I-I lel~0 Ixil
positive charge and n negative. From (7) it is easy to see that we can eliminate the divergences associated to the classical self-energies by introducing a renormalized fugacity & = aZ 0. To this renormalized fugacity there corresponds a renormalized chemical potential
2 ~ XiX/.In i,]>i
with ~ = e fl~ and Z 0 given by (5c). In terms offi, we write o~
zo(r, v, (50
a
(5b)
with (5c)
From (5), upon rescaling x i = Rfci, one is immediately led to the equation of state [1] p V = A r k T [ 1 - i1~ 2 l .
(6)
This equation predicts a phase transition at flcq 2 = 4. Observe, however, that for/~/2 >/2, the integrals in (5a) diverge and we can no longer say that the equation of state (6) corresponds to the partition function Z. The divergences occur whenever x i ~ xi, with hi 4: X/, that is, when charges of different sign occupy the same position. Let us turn now to the grand-canonical ensemble. Only neutral configurations contribute in the thermodynamic limit. We must have, therefore, N = 2n, that is n positive and n negative charges. The grand-partition function is given by
z (r, v, u)=
** (c~Z0)2n
E
lel~On=O
(n[) 2
2n
d2xi
ixil
a2
f
H
2n
[flq2 ~ Xik, l n ( l X i - X ] l + [ e l . ) ] X exp_ i,]>i a
'
(7)
where ot = eOU is the fugacity and/a, the chemical potential. l[(n!) 2 is a combinatorial factor corresponding to the fact that there are 2n particles, n of which have
.-.2n
E
no= (n!)"
v),
(9)
withHzn defined by (5b) (H 0 = I). Again, making a rescaling xi = RYei = (Vfir)l/2 Yci and observing that p = - O f l / b V)T' # with fl = - k T In ZG, we obtain right away the equation of state p V = (iV) k T [ 1 - ~ a
Z 0 = (27rm/flr 2) (lel/a)Oq 2 ]2.
(8)
fi = I.t + k T l n Z 0 ,
lim
-= lim Z f f H N ( T , V ) , lel~0
216
15 October 1984
2],
(10)
which coincides with (6) in the thermodynamic limit. In eq. (10), oo
(N)=Z~I
~ (t~)2n 2nH2n(T, V) n=O (n!) 2
(N=En).(ll)
Eq. (10) again does not make sense for/k/2/> 2 because the integrals in H2n diverge in this case. Let us analyze in detail the divergences of the functions H2n. In order to do that, let us separate the integration region in each of the H2n , in two distinct parts. The first one is such that the integrands are completely regular, that is Ix i - X / l > ~ for h i ¢ X/and some small 6. The rest of the integration region always contains parts in which Ix i - x~ I < 6 for one or more pairs (i, ]), with Xi 4: X/. Upon doing this, we obtain the finite part ofH2n, which we call F2n , and the various divergent part in terms of Ie I. In the usual treatments, where a hard core 8 is used, only the first part of the integration region (Ix i - x/I > 5) is considered. The details of the calculation are given in ref. [6] and the result is H 0 =1, H 2 = I(e) + F 2 ,
H 4 = 2/2(e) + 4 I ( e ) F 2 + F 4 ,
(12)
responds a renormalized hamiltonian/7, through
H 6 = 6/3(e) +
1812(e)F2 + 91(e)F4 + F 6 ,
H 8 = 4!I4(e) +
4(4!)13(e)F2 + 3(4!)12(e)F4
ZG(T, V,/2) = Tr exp [---~(/7 - / i N ) ]
(12 cont'd)
+ 16/(e)F 6 + F 8 , . . . . In the expressions above,
3q 2 = 2 ,
lel~O
(a/lel) ~q2- 2
lel
3q2>2 (13)
and the various combinatoric, numerical coefficients represent the number of different ways in which n charges of a given sign and n of the opposite one can form pairs of different charge occupying the same place. The 8 dependence of the terms containing I(e) cancels that of F2n in the limit 8 + 0 [6]. Let us introduce now expression (12) in (9). Multiplying each H2n by the appropriate factor (6) 2n/(n !)2 and factorizing all F2n appearing in the expansion, we can rearrange the sum (9) in the following way
(602n_H2n(T, 10 = D ( e ) ~ (s)2n F2n(T, V) , n=0 (n!) 2
n=0 (n!) 2
(14)
where D (e) = exp [(S) 2I(e) ] •
(15)
The remarkable fact that we can absorb all divergences in the multiplicative factor D(e) allows us to introduce a renormalized grand-partition function ZG = D - I (e)ZG, ZG(T, V,/2) = ~ (6)2n n=0 (n[) 2
= D - l(e) Tr exp whence /7 = H + H 0 ,
I(e)=2QrR/a) 2 lim l n ( a l e l ) , = 2(rrR/a) lim
15 October 1984
PHYSICS LETTERS
Volume 105A, number 4,5
F2n(T, V),
(16)
[-3(H -/2N)] ,
H 0 = kr(~)2X(e).
(17)
(18)
From ZG(T, V,/2) we can derive a renorrnalized grand-canonical potential ~ = - k T In ZG, with = ~ 2 + H 0. We may now take the limit l el -+ 0, obtaining a finite 2 G, given by (16). F r o m p = - ( 0 ~ / 0 V) T, ~ and the rescaling trick, we arrive at the same equation of state (10), but with a completely well defined car):
(a)2n ( N ) = Z d 1 n=~0
"~2nF2n(T,
If).
(19)
We conclude, therefore, that the equation of state (10) is perfectly sensible for a Coulomb gas of point particles. It is not surprising that it predicts the right critical temperature.
References [1] A.M. Salzberg and S. Prager, J. Chem. Phys. 38 (1963) 2587; R.M. May, Phys. Lett. 25A (1967) 282; G. Knorr, Phys. Lett. 28A (1968) 166. [2] C. Deutsch and M. Lavaud, Phys. Rev. A9 (1974) 2598. [3] J.M. Kosterlitz and D.J. Thouless, J, Phys. C6 (1973) 1181; J.M. Kosterlitz, J. Phys. C7 (1974) 1046. [4] J.V. Jos6, L.P. Kadanoff, S. Kirkpatrick and D. Nelson, Phys. Rev. B16 (1978) 1217. [5] S. Samuel, Phys. Rev. D18 (1978) 1916. [6] E.C. Marino, Dynamical mass generation in the Thirring model: a functional integral approach, U.F.S. Carlos preprint (1984).
which is completely finite. To this function, there cor-
217