Optimized perturbation methods for the free energy of the anharmonic oscillator

28 March 1994 PHYSICS LETTERS A

PhysicsLetters A 186 (1994) 375-381

ELSEVIER

Optimized perturbation methods for the free energy of the anharmonic oscillator Kostas Vlachos Department of Physws, Patras Umverszty, GR 26110Patras, Greece

Anna Okopifiska 1 Instztute of Physws, WarsawUmverslty, BzatystokBranch, Lzpowa41, 15-424Btatystok, Poland Received 3 December 1993;revised manuscript received 19 January 1994;acceptedfor pubhcatlon 20 January 1994 Communicatedby J.P. Vlgier

Abstract

Two possible apphcations of the optimized expansion for the free energy of thc quantum-mechanical anharmonic oscillator are discussed. The first method is for the fimte temperature effective potential; the second for the classical effective potential. The results of both methods show a quick convergenceand agree well w~ththe exact free energyin the whole range of temperature. 1. The description of the quantum-mechanical anharmonic oscillator (AO) with a Hamiltonian H-- ½p2+ ½m2x2 d_ ~x4

( 1)

is equivalent [ 1 ] to that of a scalar field in the Euclidean space-time of one dimension (time) with a classical action given by S [ x ] = f [ ½ x ( t ) ( - O 2 + m 2 ) x ( t ) + 2 x 4 ( t ) ] dr.

(2)

After rescaling all quantities in terms of 2, only one dimensionless parameter z = m2/,~ 2/3 remains; therefore, when discussing numerical results, we put 2 = 1 without loss of generality. The energy spectrum of the AO can be calculated numerically and provides a simplest test for approximation methods in quantum field theory (QFT) (e.g., the loop expansion [ 2 ], the 1/N expansion, the loop expansion for composite operators [ 3 ], the optimized expansion (OE) [ 4 ] ). The conventional loop expansion gives the Rayleigh-Schr~dinger series which is asymptotic for the AO [ 1 ]; hence, the numerical results for energy levels are good only for large values of the parameter m2/)~ 2/3. In the 1/N expansion the energy levels can be calculated from the spectral properties of Green's functions, derived from the given order effective action. The results show a quick convergence and agree well with the exact spectrum of the AO, the quality of approximation decreases for increasing excitation level [ 5 ]. This is similar in the OE; however, in this case the approximation becomes worse as the parameter m2/ 2 2/3 decreases, becoming inapplicable in the double-well case (m 2< 0) [ 5 ]. E-mall: [email protected]. 0375-9601/94/$07.00 © 1994ElsevierSoence B.V All rights reserved SSDI0375-9601 (94)00073-X

K Vlachos,A Okopthska/ PhyswsLettersA 186(1994)375-381

376

It is also interesting to discuss the partition function and the free energy which contain information on the whole spectrum of the quantum system. The field-theoretical methods for systematic approximations of the effective action can be extended to finite temperature T in the Euclidean formalism by compactifylng the "imaginary time" dimension with a period fl= 1/T. The conventional loop expansion coincides with termodynamic perturbation theory for the free energy [ 6 ]. Here we study the approximations for the free energy, obtained by two applications of the OE: (1) for the finite temperature effective potential, (ii) for the classical effective potential. 2. The first method is an extension of the OE for the effective action in QFT [4] to a finite temperature effective potential and has been studied in space-time of arbitrary dimension [7 ]. Here we discuss the onedimensional case when the vacuum persistence amplitude is defined by the integral

Z[J]=fDxexp(-S[x]+fJ(t)x(t)dt),

(3)

over functions x(t) which vanish at infinity. The effective action is defined by

Z[J] - j ~(t)J(t) dt,

/'[~] =ln

(4)

where the expectation value of x is given by ~(t) = fi In Z/~J(t). The extension to finite temperature T consists in replacing the set of functions in the integral (3) by periodic functions with a period fl= 1/ T. The infinite interval in all integrals over t has to be replaced by the interval [0, fl], we denote the corresponding quantities by Zp,/'B. The finite temperature effective potential defined by l

vp(O= ~ rB[~] I~/>=~

(5)

has the meaning of the free energy of a quantum system interacting with a constant electric field J, when the expectation value of the coordinate x is given by (-- cost. The free energy F of the AO is determined by the value of the finite temperature effective potential at minimum which corresponds to J = 0. The OE for the effective action is generated by the application of the steepest-descent method to the generating functional Z[J] with the classical action modified to the form

s tx] = j ½x(t)G-l(t, t')x(t') dtdt'

+~(f½x(t)[D-'(t,t')-G-'(t,t')]x(t')dtdt'+2fx4(t)dt),

(6,

where D-~(t, t ' ) = ( - 0 2 + m 2 ) j ( t - t ') and G(t, t') is an arbitrary propagator. After shifting x(t) by Xo(t), chosen to satisfy the classical equation of motion 8S~/8Xo(t)= -J(t), and expanding the exponential into a Taylor series we obtain

Z[J] = e x p ( - X ~ [ x o ]

+

f J(t)Xo(t) dt) f Dx' e x p ( - f ½ x ' ( t ) G - ' ( t ,

X [ 1 - - e ( f ½ x ' ( t ) [3 -1 (t,

t')-G-l(t, t')]x'(t')dt

dt'+2

t')x'(t) dt dt')

fxo(t)x'3(t)dt+2 fx'4(t)dt)+...

1, (7)

where 3 - ~(t,

t') =

[ - 0 2 + mE+ 122X02(t) ] J ( t - t'). After a Legendre transformation, the effective action (4) is

K. Vlachos, A. Okopl~ka / Physzcs Letters A 186 (1994) 3 75- 381

377

Fig. 1. The effective action to third order of the optimized expansion The full hne is an arbitrary propagator G(t, t'), the rule denotes the two-particle vertex A-2 (t, t') - G -1 (t, t').

obtained as a series in e. The nth order term can be represented diagrammatically as a sum of n-vertex vacuum one-particle irreducible diagrams with the Feynman rules of the modified theory (6). The third order result is shown in Fig. 1. 3. The finite temperature effective potential in the OE is given by the same set of diagrams as the effective action, only the Feynman rules are replaced by those for finite temperature. Since Vp is a function of the constant ~, the propagator can be chosen in the form

G~(t,t')=

1 oo ~m~_

1 cosh[½£2(lt-t'[-fl)] toE +£2 2 -£2 sinh(½P£2) '

exp[-ito,,(t-t')]

with an arbitrary parameter £2. The Matsubara frequencies are given by O~m= propagator becomes equal to ]

1

Goo(t,t')= ~ ~nnexp[-ip(t-t') p2.~.£22--

e x p ( - ½£2lt-t'l ) 2£2

(8)

2~rn/fl. At zero temperature the (9)

The parameter ~ is a formal parameter of the expansion and is set equal to one at the end. The exact Vp(~), obtained as a sum of an infinite series, does not depend on £2, but a finite order truncation does. We can make the nth order approximant V~") (£2, ~) as insensitive as possible to a small variation of the unphysical parameter by choosing £2 equal to ~ ( n , ~, fl) satisfying the gap equation 8V~")/5~=0.

(10)

In the approximate expression for the finite temperature effective potential V~") (~) = V~") (~, ~(n, ~, fl)) the optimal value of £2 changes from order to order, assuring the convergence of the expansion [ 8 ]. The first order of the OE coincides with the finite temperature Hartree approximation, but the variational interpretation cannot be maintained beyond the first order. We calculated the finite temperature effective potential to third order in the OE. In second order the gap equation has no real solution in some range of temperature, in this case the real part of the result has been taken. For the single-well AO ( m 2 > 0) the symmetric minimum of the finite temperature effective potential at ~= 0 (OES) gives a very good description of the free energy. The quality of approximation becomes worse for decreasing m 2/22/3. In the most unfavourable case of the quartic oscillator (m 2= 0) the results of the three lowest orders are shown in Fig. 2 in comparison with the exact free energy, calculated by the numerical procedure based on the modification of the linear variational method [ 9 ]. In the double-well ease (m 2< 0), there is a critical value zcr (n, fl) of the parameter m 2/22/3, above which the only minimum of V~") (~) is at ~= 0, in agreement with the exact result. However, for m2/22/3 below zcr a lower

K Vlachos,A Okopthska/ Phystcs Letters A 186 (1994) 375-381

378

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Fig. 2. The free energy F of the quartlc oscillator (m 2= 0), obtained as the value of the fimte temperature effectwe potential at ~= 0 m the first three orders (OES1, OES2, OES3) of the OE, plotted versus the reverse temperature 1/7". m i n i m u m at ~ # 0 appears. As can be seen m Fig. 3, for m 2= _ 20, the value of the finite temperature effective potential in the non-symmetric m i n i m u m ( O E N ) gives a much better approximation of the free energy than the value at ~ = 0 ( O E S ) . 4. Different approximation schemes for the free energy can be obtained [ 10] approximating the classical effective potential Vcl ( x ) , defined by a simple integral Za[J=0] =e-BF=

f

dx°

exp[ -flVcl(Xo) ]

.

(11)

The local partition function Z x° can be written as Z ~ ° = e x p [ -flVcl(xo) ] = f Dx x / ~ f ( X o

- ~ ) exp( - S [ x ] ) ,

(12)

where g = fdz x (z)~ft. For T = 0 the classtcal effective potential coincides with the constraint effective potential

[11]. The OE for the classical effective potential is generated by the application o f the steepest-descent method to Z x° ( 1 2 ) with the modified classical action ( 6 ) . After shifting x ( t ) by a constant Xo and expanding the exponential into a Taylor series we have

379

K. Vlachos, A. Okoptftska / Physcs Letters A 186 (1994) 375-381

-5 OESI

-i0

/ / I'

-15

i l

-20

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OENI

OEN3

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"~ k \\'\" OES2

-30

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-35

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F*g. 3. The free energy F of the double-well oscillator ( m e = (OES), and at ~ 0 (OEN).

1/T

3

4

20), obtained in the OE as the value of the effective potent,al at ¢ = 0

z~°= exp ( -p{½~x8 +E[ ½(m~-~)x8 +XXo~ ] }) P

o P P P X[1-e(½(m2+122x2-.Q2) ~ x'2(t) dt+2Xo ~ X'a(t) dt+). ~ x'4(t) dt)+...] . o

o

(13)

0

After performing the Gaussian integrals, the local partition function and the classical effective potential can be calculated as series in e. The given order classical effective potential Vctr) (Xo) in the OE can be represented diagrammatically, only one-particle irreducible diagrams are present, because by definition x'(t) has zero vacuum expectation value. Therefore, the set of diagrams is the same as in the case of the OE for the finite temperature effective potential, the only difference is in the propagator. In the case of the OE for the classical effective potential the propagator does not contain zero modes and equals

G'~m(t, t') =

~2 = ,~o exp[--itOm(t--t') ] to 2 +2~12

=Gp(t, t ' ) -

1

(14)

where Gp(t, t') is the finite temperature propagator (8). Performing the integration over x0 in ( 1 1 ) and calculating the free energy to the given order in e we would obtain a result coincident with the OE for the finite temperature effective potential. However, a better approximation for the partition function can be obtained by reversing the order of operation: first to optimize the given order classical effective potential

5Vg')/8~=O,

(15)

380

K Vlachos, A Okopthska / Physws Letters A 186 (1994) 375-381

and then to p e r f o r m the integration over Xo in ( 11 ) numerically. The first order of this a p p r o x i m a t i o n coincides with the F e y n m a n - K l e l n e r t ( F K ) variational d e t e r m i n a t i o n o f the classical effective potential [ 12,13 ] The OE gives a possibility to calculate corrections to the variational classical effective potential, i m p r o v i n g the F K app r o x i m a t i o n for the p a r t i t i o n function in a systematic way. F o r alternative methods o f calculating corrections to the F K a p p r o x i m a t i o n , see Refs. [ 10,14 ]. 5. We have studied the numerical results o f the OE for the classical effective potential in the first and third order (to a v o i d c o m p l i c a t i o n s with complex solutions o f the second o r d e r ) . F o r m2<~ 0 the quality of the app r o x i m a t i o n becomes worse for decreasing m 2 / ) . 2/3, the results for the quartlc oscillator (m 2 = 0) are c o m p a r e d with the exact free energy in Fig. 4. The first order results ( F K 1 ) are better than those obtained from the value o f the finite temperature effective potential at ~ = 0 (OES1), however, the third order results ( F K 3 ) and (OES3) are very similar a n d agree well with the exact free energy calculated numerically The differences between the studied m e t h o d s decrease to zero for T = 0. In the double-well case, the results o f the OE for the classical effective potential are better than those o b t a i n e d from the lowest m i n i m u m (at ~ 0 ) o f the effective potential ( O E N ) . The quality o f the a p p r o x i m a t i o n improves for increasing Ira21/22/3. F o r m E-- - 2 0 the first order results ( F K 1 ) are indistinguishable from the exact results on the scale o f Fig. 3. In Fig. 5 we c o m p a r e the results o f both m e t h o d s for a larger range o f temperature, the convergence in the F K a p p r o a c h is better than in the case o f the OEN. However, one has to notice that the numerical calculations are m u c h m o r e c o m p l i c a t e d in the OE for the classical effective potential. Moreover, a generalisatlon o f the F K a p p r o a c h to the true Q F T (with a non-trivial space d i m e n s i o n ) is difficult, even in the lowest o r d e r [ 12,15 ] I

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Fig. 4 The free energy Fofthe quart~c oscdlator (m2=0), calculated w~th the classical effecnve potentml m the first (FKI) and third (FK3) orders of the OE

381

K Vlachos, A. Okopthska/Physms LettersA 186 (1994) 375-381 I

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Fig. 5. The free energy Fofthe double-well oscillator (m2= -20), calculated with the classical effective potential in the first (FK1) and third (FK3) orders of the OE, compared with the value of the fimte temperature effective potentml at ~# 0 (OEN 1, OEN3 ).

This work has been s u p p o r t e d partially by the C o m m i t t e e for Scientific Research u n d e r G r a n t PB-2-0956-91-01. One o f the authors (K.V.) is grateful to the m e m b e r s o f the Institute o f Physics (Bia{ystok Branch o f W a r s a w U n i v e r s i t y ) a n d the S o h a n Institute for Nuclear Studies for the kind hospitality.

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