- Email: [email protected]
Strong-coupling calculation of fluctuation pressure of a membrane between walls
11 October 1999
Physics Letters A 261 Ž1999. 127–133 www.elsevier.nlrlocaterphysleta
Strong-coupling calculation of fluctuation pressure of a membrane between walls Michael Bachmann ) , Hagen Kleinert 1, Axel Pelster
2
Institut fur ¨ Theoretische Physik, Freie UniÕersitat ¨ Berlin, Arnimallee 14, 14195 Berlin, Germany Received 2 June 1999; received in revised form 26 June 1999; accepted 26 June 1999 Communicated by P.R. Holland
Abstract We calculate analytically the proportionality constant in the pressure law of a membrane between parallel walls from the strong-coupling limit of variational perturbation theory up to third order. Extrapolating the zeroth to third approximations to infinity yields the pressure constant a s 0.0797149. This result lies well within the error bounds of the most accurate available Monte Carlo result a MC s 0.0798 " 0.0003. q 1999 Elsevier Science B.V. All rights reserved.
tions by Helfrich w1x and by Janke and Kleinert w2x which yielded
1. Membrane between walls The violent thermal out-of-plane fluctuations of a membrane between parallel walls generate a pressure p following the law psa
k B2 T 2
k Ž dr2 .
3
,
Ž 1.1 .
whose form was first derived by Helfrich w1x. Here, k denotes the elasticity constant of the membrane, and d the distance between the walls. The exact value of the prefactor a is unknown, but estimates have been derived from crude theoretical approxima-
) Supported by the Studienstiftung des deutschen Volkes, Email: [email protected]. 1 E-mail: [email protected] 2 E-mail: [email protected]
a Hth f 0.0242,
th a JK f 0.0625.
Ž 1.2 .
More precise values were found from Monte Carlo simulations by Janke and Kleinert w2x and by Gompper and Kroll w3x which gave MC a JK f 0.079 " 0.002,
MC a GK f 0.0798 " 0.0003. Ž 1.3 .
In previous work w4x, a systematic method was developed for calculating a with any desired high accuracy. Basis for this method is the strong-coupling version of variational perturbation theory w5x. In that theory, the free energy of the membrane is expanded into a sum of connected loop diagrams, which is eventually taken to infinite coupling strength to account for the hard walls. As a first approximation, an infinite set of diagrams was calculated, others were estimated by invoking a mathematical analogy with a
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 2 7 - 2
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
128
similar one-dimensional system of a quantum mechanical particle between walls. The result of this procedure was a pressure constant
a Kth s
p2 128
s 0.0771063 . . . ,
Ž 1.4 .
very close to Ž1.3.. It is the purpose of this Letter to go beyond this estimate by calculating all diagrams up to four loops exactly. In this way, we improve the analytic approximation Ž1.4. and obtain a value
tions. This problem is solved by the strong-coupling theory of Ref. w4x as follows. If the area of the membrane is denoted by A, the partition function Ž2.6. determines the free energy per area as fsy
1 A
ln Z.
Ž 2.3 .
By differentiating f with respect to the distance d of the walls, we obtain the pressure p s yE frE d. 2.1. Smooth potential adapting walls
a th f 0.0797149,
Ž 1.5 .
which is in excellent agreement with the precise MC MC value a GK in Eq. Ž1.3..
We introduce some smooth potential restricting the fluctuations w Ž x . to the interval Žydr2,dr2., for instance V Ž w Ž x. . sm
2. The smooth potential model of the membrane between walls
4
d2
p
2
tan2
p d
' m4w 2 Ž x . q To set up the theory, we let the membrane lie in the x-plane and fluctuate in the z-direction with vertical displacements w Ž x .. The walls at z s "dr2 restrict the displacements to the interval w g Žydr2,dr2.. Near zero temperature, the thermal fluctuations are small, w Ž x . f 0. The curvature energy E of the membrane has the harmonic approximation w1x 2
E s 12 k dx 2 E 2w Ž x . .
H
Ž 2.1 .
The thermodynamic partition function Z of the membrane is given by the sum over all Boltzmann factors of field configurations w Ž x . dw Ž x .
qdr2
Zs Ł x
Hydr2 (2p k
½
=exp y
k 2 k BT
Hd
2
x E wŽ x.
2
5
.
Ž 2.2 .
This simple harmonic functional integral poses the problem of dealing with a finite range of fluctua-
d2
Vint Ž w Ž x . . ,
Vint Ž w Ž x . . s m4 ´4 w 4 Ž x . q ´6
p q´ 8
Ž 2.4 .
2
p
6
ž /wŽ d
x.
4 8
ž /wŽ d
x. q. . . ,
Ž 2.5 .
with ´4 s 2r3, ´6 s 17r45, ´ 8 s 62r315, . . . . Thus we are left with the functional integral Zs
E Dw Ž x . exp
ž
qm w Ž x . q
2
p2
where we have split the potential into harmonic and interacting part
4 2
B Trk
wŽ x.
1 y
Hd 2
2
x
½
E 2w Ž x .
p2 d2
Vint Ž w Ž x . .
5/
,
2
Ž 2.6 .
where we have set k s k B T s 1. After truncating the Taylor expansion around the origin, the periodicity of the trigonometric function is lost and the integrals over w Ž x . in Ž2.2. can be taken from y` to q`. The interacting part is treated perturbatively. Then, the harmonic part of V Ž w Ž x ))leads to an exactly
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
129
integrable partition function Zm 2 . The mass parameter m is arbitrary at the moment, but will eventually taken to zero, in which case the potential V Ž w Ž x ))describes two hard walls at w s "dr2. We shall now calculate a perturbation expansion for Z up to four loops. This will serve as a basis for the limit m ™ 0, which will require the strong-coupling theory of Ref. w4x.
where the sum runs over all pair contractions, and P denotes the associated index permutations. The harmonic correlation function Ž2.11. is in momentum space
2.2. Perturbation expansion for free energy
thus being proportional to the difference of two ordinary correlation functions Ž p 2 y m 2 .y1 with an imaginary square mass m2 s "im2 . From their known x-space form we have immediately
The perturbation expansion proceeds from the harmonic part of Eq. Ž2.20.: Zm 2 s
A m 2 w w xr2 yA
E Dw Ž x . e
s eyA f m 2 ,
Ž 2.7 .
Gm 2 Ž k . s
1
i
4
k qm
4
s 2m
i s
H
½
2
E 2w Ž x . q m4w 2 Ž x . .
5
f m 2 s 18 m 2 .
Ž 2.9 .
The harmonic correlation functions associated with Ž2.7. are ² O 1 Ž w Ž x 1 . . O 2 Ž w Ž x 2 . . PPP :m 2 1 s Zm 2
E Dw Ž x . O Ž w Ž x . . 1
1
=O 2 Ž w Ž x 2 . . PPP eyAAm 2 w w xr2 ,
Gm 2 Ž x 1 , x 2 . s ² w Ž x 1 . w Ž x 2 . :m 2
Ž 2.10 .
Ž 2.11 .
determines, by Wick’s rule, all correlation functions Ž2.10. as sums of products of Ž2.11.:
s
Ý
g
Hd
2A
g2 1 y 2! 4 A
2
Gm 2 Ž x P Ž1. , x P Ž2. . PPP Gm 2 Ž x P Ž ny1. , x P Ž n. . ,
Ž 2.12 .
Ž 2.14 .
Hd
x ² Vint Ž w Ž x . . :m 2 ,c
2
x 1 d 2 x 2 ² Vint Ž w Ž x 1 . .
=Vint Ž w Ž x . 2 . :m 2 ,c q . . . ,
Ž 2.15 .
where the subscript c indicates the cumulants. Inserting the expansion Ž2.5. and using Ž2.10. and Ž2.12., the series can be written as f s m2 a0 q
pairs
, k y im2 Ž 2.13 . 2
K 0 Ž 'i m < x 1 y x 2 < .
4p m 2
`
² w Ž x 1 . PPP w Ž x n . :m 2
y
where K 0 Ž z . is a modified Bessel function w6x. At zero distance, the ordinary harmonic correlations are logarithmically divergent, but the difference is finite yielding Gm 2 Ž0. s 1r8m2 . We now expand the partition function Ž2.6. in powers of gVint Ž f Ž x .., where g ' p 2rd 2 , and use Ž2.10. to obtain a perturbation series for Z. Going over to the cumulants, we find the free energy per unit area f s fm2 q
where the functions Oi Ž w Ž x j .. may be arbitrary polynomials of w Ž x j .. The basic harmonic correlation function
k q im
1
2
yK 0 Ž 'y i m < x 1 y x 2 < . ,
Ž 2.8 .
From Refs. w2,4x, the harmonic free energy per unit area f m 2 is known as
2
Gm 2 Ž x 1 , x 2 . s Gm 2 Ž x 1 y x 2 .
with Am 2 w w x s d 2 x
1 2
Ý ns1
an
g
ž / m2
n
,
Ž 2.16 .
where the coefficients a n are dimensionless real numbers, starting with a 0 s 1r8. The higher expan-
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
130
sion coefficients a n are combinations of integrals over the connected correlation functions: a1 s ´ 4 a2 s ´6
m2
x ² w 4 Ž x . :m 2 ,c ,
Ž 2.17 .
2
Hd
2A
m10
x ² w 6 Ž x . :m 2 ,c
Hd
8A
2
x 1 d 2 x 2 ² w 4 Ž x 1 . w 4 Ž x 2 . :m 2 ,c ,
Ž 2.18 . a3 s ´ 8
m8 2A
Hd
2
x ² Õ 8 Ž x . :m 2 ,c
12
y ´4 ´6 q ´43
m
Hd 4A
m16 48 A
Hd
2
2
x 1 d 2 x 2 ² w 6 Ž x 1 . w 4 Ž x 2 . :m 2 ,c 2
2
x1 d x 2 d x 3² w Ž x1 .
Ž 2.19 .
To find the free energy Ž2.16. between walls, we must go to the limit m 2 ™ 0. Following w4,5x, we substitute m2 by the variational parameter M 2 , which is introduced via the trivial identity m 2 ' M 4 y gr with r s Ž M 4 y m 4 .rg, and expand this in powers of g up to the order g N. In the limit m 2 ™ 0, this expansion reads
(
m2 Ž M 2 . s M 2 y 1
1 r 2 M r
gy 2
1 r2 8 M
6
g2
N
Ý
a n g n M 2Ž1yn. bn ,
ns1
Ž 2.21 . with Nyn
Ž 1 y n . r2 Ý Ž y1. k k
ž
Ž 2.23 .
pN s a N
d
ž / 2
y3
.
Ž 2.24 .
3. Evaluation of the fluctuation pressure up to four-loop order The correlation functions appearing in Ž2.17. – Ž2.19. are conveniently represented by Feynman graphs. Green functions are pictured as solid lines and local interactions as dots, whose coordinates are integrated over: x 1 –- x 2 ' Gm 2 Ž x 1 , x 2 . ,
Ž 3.1 .
Ø ' d 2 x.
Ž 3.2 .
3
f N Ž M 2 ,d . s M 2 a0 b 0 q
ks0
!
s 0,
whose solution gives the optimal value MN2 Ž d .. Resubstituting this result into Eq. Ž2.21. produces the optimized free energy f N Ž d . s f N Ž MN2 Ž d .,d ., which only depends on the distance as f N Ž d . s 4a N rd 2 . Its derivative with respect to d yields the desired pressure law with the Nth order approximation for the constant a N :
H
3
g y... . Ž 2.20 . 16 M 10 Inserting this into Ž2.16., reexpanding in powers of g, and truncating after the Nth term, we arrive at the free energy per unit area y
2
We must now calculate the cumulants occuring in the expansion Ž2.16..
4
=w 4 Ž x 2 . w 4 Ž x 3 . :m 2 ,c .
bn s
E f N Ž M 2 ,d . EM
m6
y ´42
2
Hd
2A
of Ž2.21. is done as usual w5x by determining the minimum of f N Ž M 2 ,d . with respect to the variational parameter M 2 , i.e. by the condition
/
Ž 2.22 .
being the binomial expansion of Ž1 y 1.Ž1yn.r2 truncated after the Ž N y n.th term w4x. The optimization
These rules can be taken over to momentum space in the usual way. One easily verifies that the integrals over the connected correlation functions in Ž2.17. – Ž2.19. have a dimension Arm2Ž nqVy1., where V is the number of the vertices of the associated Feynman diagrams. Thus we parametrize each Feynman diagram by ÕArm2Ž nqVy1., with a dimensionless number Õ, which includes the multiplicity. In Table 1, we have listed the values Õ for all diagrams up to four loops. No divergences are encountered. Exact results are stated as fractional numbers. The other numbers are obtained by numerical integration, which are reliable up to the last written digit. The right-hand column shows numbers Õ K obtained by the earlier approximation w4x, where all the Feynman diagrams
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133 Table 1 Feynman diagrams with loops L, multiplicities s, and their dimensionless values Õ. The last column shows the values Õ K s Õ PB r4 L used in Ref. w4x.
131
in M 2 . To see how the results evolve from order to order, we start with the first order 1
f 1 Ž M 2 ,d . s
2
a0 M 2 q a1
p2 d2
,
Ž 3.3 .
with a 0 s 1r8 and a1 s 1r64. Here, an optimal value of M 2 does not exist. Thus we simply use the perturbative result for m s 0 which is equal to Ž3.3. for M s 0. Differentiating f 1Ž0,d . with respect to d yields the pressure constant in Ž2.24.: 1 p2 p2 a 1 s a1 2 s f 0.038553. 4 d 256
Ž 3.4 .
This value is about half as big as the Monte Carlo estimates Ž1.3. and agrees with the value found in w4x. To second order, the reexpansion Ž2.21. reads 3
f 2 Ž M 2 ,d . s
a0 M 2 q a1
8
p2 d2
q a2
p4 1 d4 M 2
Ž 3.5 .
with a 2 f 1.0882 P 10y3 from Table 1. Minimizing this energy in M 2 yields an optimal value M22 Ž d . s
(
8 a2 p 2 3 a0 d 2
p2 f 0.152362
d2
,
Ž 3.6 .
and
p2 f2 Ž d . s
d2
3 2
ž a q( a a / . 1
0
Ž 3.7 .
2
Inserting a 0 s 1r8 and a1 ,a 2 from Table 1, we obtain
a 2 f 0.073797, were estimated by an analogy to the the problem of a particle in a box. In Ref. w4x, it was shown that the value Õ of a large class of diagrams of the membrane problem can be obtained by simply dividing the value of the corresponding particle-in-a-box-diagram Õ PB by a factor 1r4 L , where L is the number of loops in the diagrams. Inserting the numbers in Table 1 into Ž2.17. – Ž2.19., we obtain the coefficients a1 ,a2 ,a 3 of the free energy per area Ž2.21., which is then extremized
Ž 3.8 .
thus improving drastically the first-order estimate Ž3.4. This value is by a factor 1.026 larger than that obtained in the approximation of Ref. w4x. Continuing this proceeding to third order, we must minimize 2
f 3 Ž M ,d . s
5 16
p2
2
a 0 M q a1
q a3
p6 1 d6 M 4
,
d2
3 q 2
a2
p4 1 d4 M 2
Ž 3.9 .
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
132
with a 3 f 2.7631 P 10y5 . The optimal value of M 2 is M32 Ž d . s
(
32 a2 5 a0
cos
1 3
)
arccos
5 a0 a32
p2
2 a 23
d2
p2 f 0.219608
. Ž 3.10 . d2 Inserted into Ž3.9., we find the four-loop approximation for the proportionality constant a :
a 3 f 0.079472.
Ž 3.11 .
This result is in very good agreement of the Monte Carlo results in Ž1.3.. It differs from the approximate value of the method presented in Ref. w4x by a factor 1.047. An even better result will now be obtained by extrapolating the sequence a 1 , a 2 , a 3 to infinite order. 4. Extrapolation towards the exact constant Variational perturbation theory exhibits typically an exponentially fast convergence. This was exactly proven for the anharmonic oscillator w5x. Other systems treated by variational perturbation theory show a similar behavior w7x. Assuming that an exponential convergence exists also here, we may extrapolate the sequence of values a 1 , a 2 , a 3 calculated above to infinite order. It is useful to extend this sequence by one more value at the lower end, a 0 s 0, which follows from the one-loop energy Ž2.9. at m 2 s 0. This sequence is now extrapolated towards a hypothetical exact value a ex by parametrizing the approach as
a ex y a N s exp Ž yh y j N e . .
Ž 4.1 .
The parameters h , j , ´ , and the unknown value of a ex are determined from the four values a 0 , . . . , a 3 , with the result
h s 2.529298,
j s 0.660946,
e s 1.976207,
Ž 4.2 . and the extrapolated value for the exact constant:
a ex s 0.0797149.
Ž 4.3 .
This is now in perfect agreement with the Monte Carlo values Ž1.3..
Fig. 1. Difference between the extrapolated pressure constant a ex and the optimized N-th order value a N obtained from variational perturbation theory for the method presented in this paper Žsolid line. and the first four values of the approximation scheme introduced in Ref. w4x Ždashed line.. Dots represent the values to order N in these approximations.
The approach is graphically shown in Fig. 1 where the optimized values a 0 , . . . , a 3 all lie on a straight line Žsolid line.. For comparison, we have also extrapolated the first four values a K0 , . . . , a K3 in the approach of Ref. w4x yielding a value a ex K f 0.0759786, which is 4.9% smaller than Ž4.3.. 5. Summary We have calculated the universal constant a occuring in the pressure law Ž1.1. of a membrane fluctuating between two walls. This has been done by replacing the walls by a smooth potential with a parameter m2 . This potential approaches the wall potential in the limit m2 ™ 0. The anharmonic part of the smooth potential was treated perturbatively. The limit m2 ™ 0 corresponds to a strong-coupling limit of the power series, and was calculated by variational perturbation theory. Extrapolating the lowest four approximations to infinity yields a pressure constant a , which is in very good agreement with Monte Carlo values. References w1x W. Helfrich, Z. Naturforsch. A 33 Ž1978. 305; W. Helfrich, R.M. Servuss, Nuovo Cimento D 3 Ž1984. 137. w2x W. Janke, H. Kleinert, Phys. Lett. A 117 Ž1986. 353. Žhttp: r r www.physik.fu y berlin.de r ; kleinert r kleiner_ e133. w3x W. Janke, H. Kleinert, M. Meinhart, Phys. Lett. B 217 Ž1989.
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133 525 Ž http:rrwww.physik.fu y berlin.der ; kleinertr kleiner_re184.; G. Gompper, D.M. Kroll, Europhys. Lett. 9 Ž1989. 59; F. David, J. de Phys. 51 Ž1990. C7-115; R.R. Netz, R. Lipowsky, Europhys. Lett. 29 Ž1995. 345. w4x H. Kleinert, cond-matr9811308 Ž1998.. Žhttp:rrwww. physik.fuyberlin.der ; kleinertrkleiner_re277. w5x H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, World Scientific, Singapore 1995, sec-
133
ond edition, Chap. 5. Žthird edition: http:rrwww.physik.fuy berlin.der ; kleinertrkleiner_reb3r3rded.html. w6x I.S. Gradstein, I.M. Ryshik, Tables of Series, Products, and Integrals vol. 2, Verlag Harry Deutsch, Thun 1981, first edition. w7x H. Kleinert, W. Kurzinger, A. Pelster, J. Phys. A: Math. ¨ Gen. Ž31., 8307 Ž1998. Žquant-phr980616.. Žhttp:rrwww. physik.fuyberlin.der ; kleinertrkleiner_re267.
Physics Letters A 261 Ž1999. 127–133 www.elsevier.nlrlocaterphysleta
Strong-coupling calculation of fluctuation pressure of a membrane between walls Michael Bachmann ) , Hagen Kleinert 1, Axel Pelster
2
Institut fur ¨ Theoretische Physik, Freie UniÕersitat ¨ Berlin, Arnimallee 14, 14195 Berlin, Germany Received 2 June 1999; received in revised form 26 June 1999; accepted 26 June 1999 Communicated by P.R. Holland
Abstract We calculate analytically the proportionality constant in the pressure law of a membrane between parallel walls from the strong-coupling limit of variational perturbation theory up to third order. Extrapolating the zeroth to third approximations to infinity yields the pressure constant a s 0.0797149. This result lies well within the error bounds of the most accurate available Monte Carlo result a MC s 0.0798 " 0.0003. q 1999 Elsevier Science B.V. All rights reserved.
tions by Helfrich w1x and by Janke and Kleinert w2x which yielded
1. Membrane between walls The violent thermal out-of-plane fluctuations of a membrane between parallel walls generate a pressure p following the law psa
k B2 T 2
k Ž dr2 .
3
,
Ž 1.1 .
whose form was first derived by Helfrich w1x. Here, k denotes the elasticity constant of the membrane, and d the distance between the walls. The exact value of the prefactor a is unknown, but estimates have been derived from crude theoretical approxima-
) Supported by the Studienstiftung des deutschen Volkes, Email: [email protected]. 1 E-mail: [email protected] 2 E-mail: [email protected]
a Hth f 0.0242,
th a JK f 0.0625.
Ž 1.2 .
More precise values were found from Monte Carlo simulations by Janke and Kleinert w2x and by Gompper and Kroll w3x which gave MC a JK f 0.079 " 0.002,
MC a GK f 0.0798 " 0.0003. Ž 1.3 .
In previous work w4x, a systematic method was developed for calculating a with any desired high accuracy. Basis for this method is the strong-coupling version of variational perturbation theory w5x. In that theory, the free energy of the membrane is expanded into a sum of connected loop diagrams, which is eventually taken to infinite coupling strength to account for the hard walls. As a first approximation, an infinite set of diagrams was calculated, others were estimated by invoking a mathematical analogy with a
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 4 2 7 - 2
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
128
similar one-dimensional system of a quantum mechanical particle between walls. The result of this procedure was a pressure constant
a Kth s
p2 128
s 0.0771063 . . . ,
Ž 1.4 .
very close to Ž1.3.. It is the purpose of this Letter to go beyond this estimate by calculating all diagrams up to four loops exactly. In this way, we improve the analytic approximation Ž1.4. and obtain a value
tions. This problem is solved by the strong-coupling theory of Ref. w4x as follows. If the area of the membrane is denoted by A, the partition function Ž2.6. determines the free energy per area as fsy
1 A
ln Z.
Ž 2.3 .
By differentiating f with respect to the distance d of the walls, we obtain the pressure p s yE frE d. 2.1. Smooth potential adapting walls
a th f 0.0797149,
Ž 1.5 .
which is in excellent agreement with the precise MC MC value a GK in Eq. Ž1.3..
We introduce some smooth potential restricting the fluctuations w Ž x . to the interval Žydr2,dr2., for instance V Ž w Ž x. . sm
2. The smooth potential model of the membrane between walls
4
d2
p
2
tan2
p d
' m4w 2 Ž x . q To set up the theory, we let the membrane lie in the x-plane and fluctuate in the z-direction with vertical displacements w Ž x .. The walls at z s "dr2 restrict the displacements to the interval w g Žydr2,dr2.. Near zero temperature, the thermal fluctuations are small, w Ž x . f 0. The curvature energy E of the membrane has the harmonic approximation w1x 2
E s 12 k dx 2 E 2w Ž x . .
H
Ž 2.1 .
The thermodynamic partition function Z of the membrane is given by the sum over all Boltzmann factors of field configurations w Ž x . dw Ž x .
qdr2
Zs Ł x
Hydr2 (2p k
½
=exp y
k 2 k BT
Hd
2
x E wŽ x.
2
5
.
Ž 2.2 .
This simple harmonic functional integral poses the problem of dealing with a finite range of fluctua-
d2
Vint Ž w Ž x . . ,
Vint Ž w Ž x . . s m4 ´4 w 4 Ž x . q ´6
p q´ 8
Ž 2.4 .
2
p
6
ž /wŽ d
x.
4 8
ž /wŽ d
x. q. . . ,
Ž 2.5 .
with ´4 s 2r3, ´6 s 17r45, ´ 8 s 62r315, . . . . Thus we are left with the functional integral Zs
E Dw Ž x . exp
ž
qm w Ž x . q
2
p2
where we have split the potential into harmonic and interacting part
4 2
B Trk
wŽ x.
1 y
Hd 2
2
x
½
E 2w Ž x .
p2 d2
Vint Ž w Ž x . .
5/
,
2
Ž 2.6 .
where we have set k s k B T s 1. After truncating the Taylor expansion around the origin, the periodicity of the trigonometric function is lost and the integrals over w Ž x . in Ž2.2. can be taken from y` to q`. The interacting part is treated perturbatively. Then, the harmonic part of V Ž w Ž x ))leads to an exactly
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
129
integrable partition function Zm 2 . The mass parameter m is arbitrary at the moment, but will eventually taken to zero, in which case the potential V Ž w Ž x ))describes two hard walls at w s "dr2. We shall now calculate a perturbation expansion for Z up to four loops. This will serve as a basis for the limit m ™ 0, which will require the strong-coupling theory of Ref. w4x.
where the sum runs over all pair contractions, and P denotes the associated index permutations. The harmonic correlation function Ž2.11. is in momentum space
2.2. Perturbation expansion for free energy
thus being proportional to the difference of two ordinary correlation functions Ž p 2 y m 2 .y1 with an imaginary square mass m2 s "im2 . From their known x-space form we have immediately
The perturbation expansion proceeds from the harmonic part of Eq. Ž2.20.: Zm 2 s
A m 2 w w xr2 yA
E Dw Ž x . e
s eyA f m 2 ,
Ž 2.7 .
Gm 2 Ž k . s
1
i
4
k qm
4
s 2m
i s
H
½
2
E 2w Ž x . q m4w 2 Ž x . .
5
f m 2 s 18 m 2 .
Ž 2.9 .
The harmonic correlation functions associated with Ž2.7. are ² O 1 Ž w Ž x 1 . . O 2 Ž w Ž x 2 . . PPP :m 2 1 s Zm 2
E Dw Ž x . O Ž w Ž x . . 1
1
=O 2 Ž w Ž x 2 . . PPP eyAAm 2 w w xr2 ,
Gm 2 Ž x 1 , x 2 . s ² w Ž x 1 . w Ž x 2 . :m 2
Ž 2.10 .
Ž 2.11 .
determines, by Wick’s rule, all correlation functions Ž2.10. as sums of products of Ž2.11.:
s
Ý
g
Hd
2A
g2 1 y 2! 4 A
2
Gm 2 Ž x P Ž1. , x P Ž2. . PPP Gm 2 Ž x P Ž ny1. , x P Ž n. . ,
Ž 2.12 .
Ž 2.14 .
Hd
x ² Vint Ž w Ž x . . :m 2 ,c
2
x 1 d 2 x 2 ² Vint Ž w Ž x 1 . .
=Vint Ž w Ž x . 2 . :m 2 ,c q . . . ,
Ž 2.15 .
where the subscript c indicates the cumulants. Inserting the expansion Ž2.5. and using Ž2.10. and Ž2.12., the series can be written as f s m2 a0 q
pairs
, k y im2 Ž 2.13 . 2
K 0 Ž 'i m < x 1 y x 2 < .
4p m 2
`
² w Ž x 1 . PPP w Ž x n . :m 2
y
where K 0 Ž z . is a modified Bessel function w6x. At zero distance, the ordinary harmonic correlations are logarithmically divergent, but the difference is finite yielding Gm 2 Ž0. s 1r8m2 . We now expand the partition function Ž2.6. in powers of gVint Ž f Ž x .., where g ' p 2rd 2 , and use Ž2.10. to obtain a perturbation series for Z. Going over to the cumulants, we find the free energy per unit area f s fm2 q
where the functions Oi Ž w Ž x j .. may be arbitrary polynomials of w Ž x j .. The basic harmonic correlation function
k q im
1
2
yK 0 Ž 'y i m < x 1 y x 2 < . ,
Ž 2.8 .
From Refs. w2,4x, the harmonic free energy per unit area f m 2 is known as
2
Gm 2 Ž x 1 , x 2 . s Gm 2 Ž x 1 y x 2 .
with Am 2 w w x s d 2 x
1 2
Ý ns1
an
g
ž / m2
n
,
Ž 2.16 .
where the coefficients a n are dimensionless real numbers, starting with a 0 s 1r8. The higher expan-
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
130
sion coefficients a n are combinations of integrals over the connected correlation functions: a1 s ´ 4 a2 s ´6
m2
x ² w 4 Ž x . :m 2 ,c ,
Ž 2.17 .
2
Hd
2A
m10
x ² w 6 Ž x . :m 2 ,c
Hd
8A
2
x 1 d 2 x 2 ² w 4 Ž x 1 . w 4 Ž x 2 . :m 2 ,c ,
Ž 2.18 . a3 s ´ 8
m8 2A
Hd
2
x ² Õ 8 Ž x . :m 2 ,c
12
y ´4 ´6 q ´43
m
Hd 4A
m16 48 A
Hd
2
2
x 1 d 2 x 2 ² w 6 Ž x 1 . w 4 Ž x 2 . :m 2 ,c 2
2
x1 d x 2 d x 3² w Ž x1 .
Ž 2.19 .
To find the free energy Ž2.16. between walls, we must go to the limit m 2 ™ 0. Following w4,5x, we substitute m2 by the variational parameter M 2 , which is introduced via the trivial identity m 2 ' M 4 y gr with r s Ž M 4 y m 4 .rg, and expand this in powers of g up to the order g N. In the limit m 2 ™ 0, this expansion reads
(
m2 Ž M 2 . s M 2 y 1
1 r 2 M r
gy 2
1 r2 8 M
6
g2
N
Ý
a n g n M 2Ž1yn. bn ,
ns1
Ž 2.21 . with Nyn
Ž 1 y n . r2 Ý Ž y1. k k
ž
Ž 2.23 .
pN s a N
d
ž / 2
y3
.
Ž 2.24 .
3. Evaluation of the fluctuation pressure up to four-loop order The correlation functions appearing in Ž2.17. – Ž2.19. are conveniently represented by Feynman graphs. Green functions are pictured as solid lines and local interactions as dots, whose coordinates are integrated over: x 1 –- x 2 ' Gm 2 Ž x 1 , x 2 . ,
Ž 3.1 .
Ø ' d 2 x.
Ž 3.2 .
3
f N Ž M 2 ,d . s M 2 a0 b 0 q
ks0
!
s 0,
whose solution gives the optimal value MN2 Ž d .. Resubstituting this result into Eq. Ž2.21. produces the optimized free energy f N Ž d . s f N Ž MN2 Ž d .,d ., which only depends on the distance as f N Ž d . s 4a N rd 2 . Its derivative with respect to d yields the desired pressure law with the Nth order approximation for the constant a N :
H
3
g y... . Ž 2.20 . 16 M 10 Inserting this into Ž2.16., reexpanding in powers of g, and truncating after the Nth term, we arrive at the free energy per unit area y
2
We must now calculate the cumulants occuring in the expansion Ž2.16..
4
=w 4 Ž x 2 . w 4 Ž x 3 . :m 2 ,c .
bn s
E f N Ž M 2 ,d . EM
m6
y ´42
2
Hd
2A
of Ž2.21. is done as usual w5x by determining the minimum of f N Ž M 2 ,d . with respect to the variational parameter M 2 , i.e. by the condition
/
Ž 2.22 .
being the binomial expansion of Ž1 y 1.Ž1yn.r2 truncated after the Ž N y n.th term w4x. The optimization
These rules can be taken over to momentum space in the usual way. One easily verifies that the integrals over the connected correlation functions in Ž2.17. – Ž2.19. have a dimension Arm2Ž nqVy1., where V is the number of the vertices of the associated Feynman diagrams. Thus we parametrize each Feynman diagram by ÕArm2Ž nqVy1., with a dimensionless number Õ, which includes the multiplicity. In Table 1, we have listed the values Õ for all diagrams up to four loops. No divergences are encountered. Exact results are stated as fractional numbers. The other numbers are obtained by numerical integration, which are reliable up to the last written digit. The right-hand column shows numbers Õ K obtained by the earlier approximation w4x, where all the Feynman diagrams
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133 Table 1 Feynman diagrams with loops L, multiplicities s, and their dimensionless values Õ. The last column shows the values Õ K s Õ PB r4 L used in Ref. w4x.
131
in M 2 . To see how the results evolve from order to order, we start with the first order 1
f 1 Ž M 2 ,d . s
2
a0 M 2 q a1
p2 d2
,
Ž 3.3 .
with a 0 s 1r8 and a1 s 1r64. Here, an optimal value of M 2 does not exist. Thus we simply use the perturbative result for m s 0 which is equal to Ž3.3. for M s 0. Differentiating f 1Ž0,d . with respect to d yields the pressure constant in Ž2.24.: 1 p2 p2 a 1 s a1 2 s f 0.038553. 4 d 256
Ž 3.4 .
This value is about half as big as the Monte Carlo estimates Ž1.3. and agrees with the value found in w4x. To second order, the reexpansion Ž2.21. reads 3
f 2 Ž M 2 ,d . s
a0 M 2 q a1
8
p2 d2
q a2
p4 1 d4 M 2
Ž 3.5 .
with a 2 f 1.0882 P 10y3 from Table 1. Minimizing this energy in M 2 yields an optimal value M22 Ž d . s
(
8 a2 p 2 3 a0 d 2
p2 f 0.152362
d2
,
Ž 3.6 .
and
p2 f2 Ž d . s
d2
3 2
ž a q( a a / . 1
0
Ž 3.7 .
2
Inserting a 0 s 1r8 and a1 ,a 2 from Table 1, we obtain
a 2 f 0.073797, were estimated by an analogy to the the problem of a particle in a box. In Ref. w4x, it was shown that the value Õ of a large class of diagrams of the membrane problem can be obtained by simply dividing the value of the corresponding particle-in-a-box-diagram Õ PB by a factor 1r4 L , where L is the number of loops in the diagrams. Inserting the numbers in Table 1 into Ž2.17. – Ž2.19., we obtain the coefficients a1 ,a2 ,a 3 of the free energy per area Ž2.21., which is then extremized
Ž 3.8 .
thus improving drastically the first-order estimate Ž3.4. This value is by a factor 1.026 larger than that obtained in the approximation of Ref. w4x. Continuing this proceeding to third order, we must minimize 2
f 3 Ž M ,d . s
5 16
p2
2
a 0 M q a1
q a3
p6 1 d6 M 4
,
d2
3 q 2
a2
p4 1 d4 M 2
Ž 3.9 .
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133
132
with a 3 f 2.7631 P 10y5 . The optimal value of M 2 is M32 Ž d . s
(
32 a2 5 a0
cos
1 3
)
arccos
5 a0 a32
p2
2 a 23
d2
p2 f 0.219608
. Ž 3.10 . d2 Inserted into Ž3.9., we find the four-loop approximation for the proportionality constant a :
a 3 f 0.079472.
Ž 3.11 .
This result is in very good agreement of the Monte Carlo results in Ž1.3.. It differs from the approximate value of the method presented in Ref. w4x by a factor 1.047. An even better result will now be obtained by extrapolating the sequence a 1 , a 2 , a 3 to infinite order. 4. Extrapolation towards the exact constant Variational perturbation theory exhibits typically an exponentially fast convergence. This was exactly proven for the anharmonic oscillator w5x. Other systems treated by variational perturbation theory show a similar behavior w7x. Assuming that an exponential convergence exists also here, we may extrapolate the sequence of values a 1 , a 2 , a 3 calculated above to infinite order. It is useful to extend this sequence by one more value at the lower end, a 0 s 0, which follows from the one-loop energy Ž2.9. at m 2 s 0. This sequence is now extrapolated towards a hypothetical exact value a ex by parametrizing the approach as
a ex y a N s exp Ž yh y j N e . .
Ž 4.1 .
The parameters h , j , ´ , and the unknown value of a ex are determined from the four values a 0 , . . . , a 3 , with the result
h s 2.529298,
j s 0.660946,
e s 1.976207,
Ž 4.2 . and the extrapolated value for the exact constant:
a ex s 0.0797149.
Ž 4.3 .
This is now in perfect agreement with the Monte Carlo values Ž1.3..
Fig. 1. Difference between the extrapolated pressure constant a ex and the optimized N-th order value a N obtained from variational perturbation theory for the method presented in this paper Žsolid line. and the first four values of the approximation scheme introduced in Ref. w4x Ždashed line.. Dots represent the values to order N in these approximations.
The approach is graphically shown in Fig. 1 where the optimized values a 0 , . . . , a 3 all lie on a straight line Žsolid line.. For comparison, we have also extrapolated the first four values a K0 , . . . , a K3 in the approach of Ref. w4x yielding a value a ex K f 0.0759786, which is 4.9% smaller than Ž4.3.. 5. Summary We have calculated the universal constant a occuring in the pressure law Ž1.1. of a membrane fluctuating between two walls. This has been done by replacing the walls by a smooth potential with a parameter m2 . This potential approaches the wall potential in the limit m2 ™ 0. The anharmonic part of the smooth potential was treated perturbatively. The limit m2 ™ 0 corresponds to a strong-coupling limit of the power series, and was calculated by variational perturbation theory. Extrapolating the lowest four approximations to infinity yields a pressure constant a , which is in very good agreement with Monte Carlo values. References w1x W. Helfrich, Z. Naturforsch. A 33 Ž1978. 305; W. Helfrich, R.M. Servuss, Nuovo Cimento D 3 Ž1984. 137. w2x W. Janke, H. Kleinert, Phys. Lett. A 117 Ž1986. 353. Žhttp: r r www.physik.fu y berlin.de r ; kleinert r kleiner_ e133. w3x W. Janke, H. Kleinert, M. Meinhart, Phys. Lett. B 217 Ž1989.
M. Bachmann et al.r Physics Letters A 261 (1999) 127–133 525 Ž http:rrwww.physik.fu y berlin.der ; kleinertr kleiner_re184.; G. Gompper, D.M. Kroll, Europhys. Lett. 9 Ž1989. 59; F. David, J. de Phys. 51 Ž1990. C7-115; R.R. Netz, R. Lipowsky, Europhys. Lett. 29 Ž1995. 345. w4x H. Kleinert, cond-matr9811308 Ž1998.. Žhttp:rrwww. physik.fuyberlin.der ; kleinertrkleiner_re277. w5x H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, World Scientific, Singapore 1995, sec-
133
ond edition, Chap. 5. Žthird edition: http:rrwww.physik.fuy berlin.der ; kleinertrkleiner_reb3r3rded.html. w6x I.S. Gradstein, I.M. Ryshik, Tables of Series, Products, and Integrals vol. 2, Verlag Harry Deutsch, Thun 1981, first edition. w7x H. Kleinert, W. Kurzinger, A. Pelster, J. Phys. A: Math. ¨ Gen. Ž31., 8307 Ž1998. Žquant-phr980616.. Žhttp:rrwww. physik.fuyberlin.der ; kleinertrkleiner_re267.