Recursive calculation of effective potential and variational resummation

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS 46, Recursive calculation of effective potential and variational resummation Sebastian F. Brandt and Hagen Kleinert Freie Universität Berlin, Institut für Theoretische Physik, Arnimallee 14, Berlin, Germany Axel Pelster Universität Duisburg-Essen, Campus Essen, Fachbereich Physik, Universitätsstraße 5, Essen, Germany Received 13 July 2004; accepted 7 September 2004; published online 4 February 2005 We set up a method for a recursive calculation of the effective potential which is applied to a cubic potential with imaginary coupling. The result is resummed using variational perturbation theory, yielding an exponentially fast convergence American Institute of Physics. DOI: / I. INTRODUCTION Perturbation theory is the most commonly used technique for an approximate description of nonexactly solvable systems. However, most perturbation series are divergent and yield acceptable results only after resummation. In recent years, based on a variational approach due to Feynman and Kleinert, 1 a systematic and uniformly convergent variational perturbation theory VPT has been developed. 2 5 It permits the conversion of divergent weak-coupling into convergent strongcoupling expansions and has been applied successfully in quantum mechanics, quantum statistics, condensed matter physics, and the theory of critical phenomena. The convergence of VPT has been proved to be exponentially fast, 3,4 and this has been verified for the ground-state energy of different quantum mechanical model systems. If the underlying potential is mirror symmetric, one introduces a trial oscillator whose frequency is regarded as a variational parameter and whose influence is minimized according to the principle of minimal sensitivity. 6 In this way, the ground-state energy of the quartic anharmonic oscillator was analyzed up to very high orders in Refs. 7 and 8. If the potential is not mirror symmetric, the center of fluctuations no longer lies at the origin but at some nonzero place X. In VPT, this situation is accounted for by regarding the nonvanishing center of fluctuations X as a second variational parameter. An extreme example is a complete antisymmetric potential, such as Vx=Ax 3, which for real A does not correspond to a stable system. Interestingly, if the parameter A is chosen to be imaginary, so that there does not exist a classical system at all, the quantum-mechanical system turns out to be well defined, and the spectrum of the Hamilton operator 2 H = 1 2 x 2 + ix3 1 is real and positive. This remarkable property of the non-hermitian Hamilton operator, found in Refs. 9 13, can be attributed to the fact that it possesses a different symmetry: it is invariant under the combined application of the parity and the time-reversal operation. In this paper, we apply VPT to the PT-symmetric Hamilton operator 1. In a first naive approach, we ignore the necessary shift X of the center of fluctuations and resum the weakcoupling series of the ground-state energy for the anharmonic oscillator /2005/463/032101/11/$ , American Institute of Physics

2 Brandt, Kleinert, and Pelster J. Math. Phys. 46, Vx = 2 2 x2 + igx 3 2 in the strong-coupling limit. In this limit, the potential 2 reduces to the purely cubic potential of 1. It turns out that the VPT results approach the corresponding numerical value for the groundstate energy of 1 with increasing order, but the rate of convergence is not satisfactory. Afterwards, we allow for a nonvanishing center of fluctuations X by using the effective potential, whose calculation is accomplished by an efficient recursion scheme. This refined approach improves the convergence of the results drastically. In Sec. II, we derive the weak-coupling series for the ground-state energy of 2 by evaluating connected vacuum diagrams. In Sec. III, we show how this perturbation series can be obtained more efficiently by means of the Bender Wu recursion method. 14 In Sec. IV, we resum the weak-coupling series for the ground-state energy of 2 by applying VPT and examine the resulting convergence. In Sec. V, we determine the effective potential with the background method 15,16 from one-particle irreducible vacuum diagrams. In Sec. VI, we set up new recursion relations for a more efficient calculation of the effective potential. In Sec. VII, we finally treat the resulting expansion with VPT and examine the improved convergence. II. PERTURBATION THEORY The perturbation series for the ground-state energy of the anharmonic oscillator 2 can be calculated from connected vacuum diagrams. Up to the fourth order in the coupling constant g, the ground-state energy is given by the Feynman diagrams with the propagator 3 and the vertices 4 Evaluating the Feynman diagrams 3 leads to the following analytical expression for the groundstate energy: 5 E = g g Og 6. 6 Since evaluating Feynman diagrams of higher orders is cumbersome, only low perturbation orders are feasible by this procedure. If we want to study higher orders, we better use the Bender Wu recursion relations. 14

3 Recursive calculation of effective potential J. Math. Phys. 46, TABLE I. Expansion coefficients for the ground-state energy of the anharmonic oscillator 2 up to the 10th order. k k k k III. BENDER WU RECURSION RELATIONS The Schrödinger eigenvalue equation for the anharmonic oscillator 2, 2 2 x x2 + igx 3x = Ex, 7 is solved as follows: We write the wave function x as x = 1/4 xˆ2 exp 2 + xˆ, 8 with the abbreviation xˆ =x /, and expand the exponent in powers of the dimensionless coupling constant ĝ=g / 5 by using xˆ = ĝ k k xˆ. 9 The k xˆ are expanded in powers of the rescaled coordinate xˆ, k=1 For the ground-state energy we make the ansatz k+2 k xˆ = c k m xˆm. m=1 E = Inserting 8 11 into 7, we obtain to first order k=1 ĝ k k c 1 1 = i, c 2 1 =0, c 3 1 = i 3, 1 =0. 12 For k2, we find the following recursion relation for the expansion coefficients in 10: c k m +2m +1 m = 2m k c m k 1 m+1 2m l=1 n=1 nm +2 nc l k l n c m+2 n, 13 with c m k 0 for mk+2. The expansion coefficients of the ground-state energy follow from k 1 k = c k 2 1 c l 2 1 c k l 1. l=1 14 Table I shows the coefficients k up to the 10th order. We observe that all odd orders vanish for

4 Brandt, Kleinert, and Pelster J. Math. Phys. 46, symmetry reasons and that the first two even orders agree, indeed, with the earlier result 6. IV. RESUMMATION OF GROUND-STATE ENERGY In this section, we consider the strong-coupling limit of the perturbation series 11. Rescaling the coordinate according to x xg 1/5, the Schrödinger equation 7 becomes x 2x 2 g 4/5 2 x 2 + ix 3x = g 2/5 Ex. Expanding the wave function and the energy in powers of the coupling constant yields 15 and x = 0 xˆ + ĝ 4/5 1 xˆ + ĝ 8/5 2 xˆ + E = ĝ 2/5 b 0 + ĝ 4/5 b 1 + ĝ 8/5 b By considering 15 in the limit g, we find that the leading strong-coupling coefficient b 0 equals the ground-state energy associated with the Hamilton operator 1. A precise numerical value for this ground-state energy was given by Bender, 9,18 The weak-coupling series 11 is of the form b 0 = E N, = k=1 N 5 k 2k, with the abbreviation =g 2. Table I suggests that 19 represents a divergent Borel series which is resummable by applying VPT. 2 5 To this end, an artificial parameter is introduced in the perturbation series, which is most easily obtained by Kleinert s square-root trick, with 1+r, 20 r = Thus, one replaces the frequency in the weak-coupling series 19 according to 20 and reexpands the resulting expression in powers of up to the order N. Afterwards, the auxiliary parameter r is replaced according to 21. The ground-state energy thus becomes dependent on the variational parameter, E N, E N,,. The influence of is then optimized according to the principle of minimal sensitivity, 6 i.e., one approximates the ground-state energy to Nth order by E N = E N,, N, where N denotes that value of the variational parameter for which E N,, has an extremum or a turning point. Consider, as an example, the weak-coupling series 19 to first order, 22 E 1, = Inserting 20, re-expanding in to first order, and taking into account 21, we obtain

5 Recursive calculation of effective potential J. Math. Phys. 46, FIG. 1. Convergence of the results for the strong-coupling coefficient b 0 obtained from resummation of the weak-coupling series of the ground-state energy circles. The resummation involving a variational path average X converges much faster triangles. The lines represent fits of the respective data to straight lines. E 1,, = Extremizing this and going to large coupling constants, we obtain the strong-coupling behavior of the variational parameter, with the abbreviation ˆ =ĝ 2 and the coefficients 1 = ˆ 1/ ˆ 2/ ˆ 4/5 +, = 5 22, 1 1 = , 2 1 = , Inserting the result 25 and 26 into 24, we obtain the strong-coupling series 17 with the first-order coefficients b 0 1 = , b 1 1 = , b 2 1 = , The numerical value of the leading strong-coupling coefficient is b Thus, to first order, the relative deviation of the result from the precise value 18 is b 1 0 b 0 24%. 28 b 0 Despite this relatively poor agreement, it turns out that the VPT results for b N 0 in higher orders converge towards the exact value 18. In Refs. 3 and 4 it is proved that VPT in general yields approximations whose relative deviation from the exact value vanishes exponentially. In our case we have b N 0 b 0 exp CN 3/5, 29 b 0 where the exponent 3/5 is determined by the structure of the strong-coupling series 17. In Fig. 1 the exponential convergence of our variational results is shown up to the 20th order. Fitting the logarithm of the relative deviation to a straight line yields

6 Brandt, Kleinert, and Pelster J. Math. Phys. 46, ln b 0 N b 0 = N 3/ b 0 In the following, we show how this exponential convergence is improved drastically by allowing for a shift of the center of fluctuations. V. DIAGRAMMATIC APPROACH TO EFFECTIVE POTENTIAL In the presence of a constant external current j, the quantum statistical partition function reads Zj ª Dx exp 1 Axj, 31 where Axj is the Euclidean action, Axj =0 d 1 2 ẋ2 + Vx jx. 32 The free energy thus becomes a function of the external current, Fj = 1 ln Zj. 33 The path average, X = Zj 1 d Dx0 x exp 1 Axj, 34 then follows from the first derivative of the free energy with respect to the external current: X = Fj. 35 j Assuming that the last identity can be inverted to yield the current j as a function of the average X, one defines the effective potential V eff X as the Legendre transform of the free energy with respect to the external current: V eff X = FjX + jxx. Furthermore, the first derivative of the effective potential gives back the external current j: 36 V eff X = jx. 37 X Thus, the free energy FFj=0 can be obtained by extremizing the effective potential, with F = V eff X e, 38 V effx X X=X e =0. 39 In the zero-temperature limit, extremizing the effective potential then yields the ground-state energy.

7 Recursive calculation of effective potential J. Math. Phys. 46, The effective potential is usually not calculated by performing explicitly the Legendre transformation 36 but by a diagrammatic technique derived via the so-called background method. 15,16 There, the effective potential is expanded in powers of the Planck constant, and the expansion terms are one-particle irreducible vacuum diagrams. The result is V eff X = VX + 2 Tr ln G 1 + V int X, 40 where the trace-log term is given by the ground-state energy of a harmonic oscillator of X-dependent frequency =VX, 2 Tr ln G 1 = The interaction term V int X contains the sum of all one-particle irreducible vacuum diagrams. For the anharmonic oscillator 2, the relevant subset of the diagrams in 3 is 42 These one-particle irreducible vacuum diagrams are derived most easily by an efficient graphical recursion method. 17 The frequency of the propagators is now given by By evaluating the diagrams 42 we obtain = 2 +6igX. 43 V eff X = 2 2 X2 + igx g 2 +6igX igX g igX 9/2 + O4. 44 The ground-state energy of the anharmonic oscillator 2 is found by extremizing the effective potential 44. To this end, we expand the extremal background according to X e = ix 0 + X X X 3 + O 4. Inserting 45 into the vanishing first derivative of 44 and re-expanding in, we obtain a system of equations which are solved by X 0 =0, X 1 = 3g 2 3, X 2 = 33g Inserting 45 and 46 into 44 and re-expanding in yields again the ground-state energy 6. In order to go to higher orders, we shall now develop a recursion relation for the effective potential VI. RECURSIVE APPROACH TO EFFECTIVE POTENTIAL In the presence of a constant external current j, the Schrödinger eigenvalue equation for the anharmonic oscillator 2 reads 2 2 x x2 + igx 3 jxx = Ex. 47 Taking into account the Legendre identities 36 and 37, Eq. 47 becomes

8 Brandt, Kleinert, and Pelster J. Math. Phys. 46, x x2 + igx 3 V eff Xxx = V eff X V eff XXx. 48 If the coupling constant g vanishes, Eq. 48 is solved by x = N expxˆ xˆ2 xˆ 2, V eff X = Xˆ 2, where the path average has been rescaled by the oscillator length, Xˆ =X /. For a nonvanishing coupling constant g, we solve the differential equation 48 by the expansions x = N expxˆ xˆ xˆ2 V eff X = Xˆ xˆ, ĝ k V k Xˆ. For the correction to the wave function, xˆ, we make again the ansatz 9 and 10. Thus, we obtain from 48 for k=1, c 1 1 = i +2iXˆ ixˆ 2, c = 2, c 3 1 = i 3, V 1Xˆ 3iXˆ = 2 + ixˆ For k2 one finds for m2 the following recursion relation for the expansion coefficients of the wave function: c k m +2m +1 m = 2m k c m+2 + Xˆ m +1 m with c m k 0 for mk+2. For m=1, we have k=1 k c m k 1 m+1 2m l=1 n=1 52 nm +2 nc l k l n c m+2 n, 54 k 1 c k 1 =3c k 3 +2Xˆ c k 2 + V k Xˆ + c k l 2 c l 1 + c k l 1 c l 2. l=1 55 The expansion coefficients of the effective potential follow from k 1 V k Xˆ = c k 2 3Xˆ c k 3 2Xˆ 2 c k 2 Xˆ c k l 2 c l 1 + c k l 1 c l 2 1 k 1 c l l=1 2 1 c k l 1. l=1 56 Using these results, the effective potential can be determined recursively, yielding an expansion in the coupling constant g, V eff X = X2 + ig X3 3X g 2 +9X2 4 4 ig 33X4 +9X2 4 6 g X X Og This result is in agreement with the expansion of 44 in powers of g and can be carried to higher orders without effort. The expansion coefficients for the -expansion

9 Recursive calculation of effective potential J. Math. Phys. 46, TABLE II. Expansion coefficients for the effective potential of 2 up to five loops. l V l X 2 X 2 /2+igX 3 2 g l V l X 51g g g N V eff X = l V l X + O N+1 l=0 can then be obtained easily. 19 Iterating the recursion relations up to the order g 8,we obtain the effective potential up to five loops as shown in Table II. VII. RESUMMATION OF EFFECTIVE POTENTIAL We now apply VPT to the loop expansion of the effective potential 58. Since the Planck constant is now the expansion parameter rather than the coupling constant g, Kleinert s squareroot trick will be modified accordingly, 58 with 1+r, 59 r = As an example, we consider again the first order V 1 eff X = 2 2 X2 + igx igX After substituting according to 59, re-expanding in, and taking into account 60, we obtain V 1 eff X, = 2 2 X2 + igx igX In order to calculate an approximation for the ground-state energy, we now optimize in and extremize in X, yielding V eff 1 =0, X,X=X 1,= 1 63 X V eff 1 =0. X,X=X 1,= 1 64 Equation 63 is solved by Afterwards, we obtain from 64, 1 =0. 65

10 Brandt, Kleinert, and Pelster J. Math. Phys. 46, TABLE III. Variational results for the ground-state energy of 1 compared to the numerical result 18 of Refs. 9 and loop VPT loop VPT loop VPT loop VPT loop VPT Numerical X ig + =0. 2 6igX 1 3/2 66 This equation allows us to determine the strong-coupling behavior of X, X 1 = iĝ 1/5 X X 1 1 ĝ 4/5 + X 2 1 ĝ 8/5 +, 67 where the coefficients read X 1 0 = , X 1 1 = 2 15, X 2 1 = 5 24, Inserting the results 65, 67, and 68 into 62 yields the strong-coupling behavior of the ground-state energy 17, with the new coefficients b 0 1 = , b 1 1 = , b 2 1 = , The new numerical value of the leading strong-coupling coefficient is b , which is in much better agreement with 18 than the previous value of 27, b 1 0 b 0 3%. 70 b 0 Thus, the variational path average has led to a significant improvement of the first-order result. Table III summarizes our results for b N 0 up to the fifth order. Figure 1 shows the much faster exponential convergence up to the fifth order. A least square fit of the data yields ln b 0 N b 0 = N 3/ b 0 which is to be compared with 30. VIII. CONCLUSION AND OUTLOOK We have developed a recursive technique to determine the effective potential, which is far more efficient than diagrammatic methods. In combination with VPT, this leads to a fast converging determination of the ground-state energy of quantum-mechanical systems with nonmirror symmetric potentials. It will be interesting to analyze in a similar way systems with a coordinate dependent mass term, where only a lowest order effective potential has been calculated so far. 20 Interesting future applications will address the effective potential of 4 theories in 4 dimensions to obtain equations of state near to a critical point. A first attempt in this direction is Ref. 21.

11 Recursive calculation of effective potential J. Math. Phys. 46, ACKNOWLEDGMENTS The authors thank C.M. Bender for fruitful discussions. S.F.B. thanks R. Graham for hospitality during a stay at the Universität Duisburg-Essen in R.P. Feynman and H. Kleinert, Phys. Rev. A 34, H. Kleinert, Phys. Lett. A 173, H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. World Scientific, Singapore, H. Kleinert and V. Schulte-Frohlinde, Critical Properties of 4 -Theories World Scientific, Singapore, Fluctuating Paths and Fields Dedicated to Hagen Kleinert on the Occasion of his 60th Birthday, edited by W. Janke, A. Pelster, H.-J. Schmidt, and M. Bachmann World Scientific, Singapore, P.M. Stevenson, Phys. Rev. D 23, W. Janke and H. Kleinert, Phys. Rev. Lett. 75, H. Kleinert and W. Janke, Phys. Lett. A 206, C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, C.M. Bender, S. Boettcher, and P.N. Meisinger, J. Math. Phys. 40, See Ref. 5, p C.M. Bender, D.C. Brody, and H.F. Jones, Phys. Rev. Lett. 89, C.M. Bender, D.C. Brody, and H.F. Jones, Am. J. Phys. 71, C.M. Bender and T.T. Wu, Phys. Rev. 184, ; Phys. Rev. D 7, B. De Witt, Dynamical Theory of Groups and Fields Gordon and Breach, New York, R. Jackiw, Phys. Rev. D 9, A. Pelster and H. Kleinert, Physica A 323, C.M. Bender private communication. 19 More details can be found in S.F. Brandt, Diploma thesis, Freie Universität Berlin, 2004, 20 H. Kleinert and A. Chervyakov, Phys. Lett. A 299, C.M.L. de Aragão and C.E.I. Carneiro, Phys. Rev. D 68,