Nonperturbative Quantum Physics from Low-Order Perturbation Theory
The Stark effect in hydrogen and the cubic anharmonic oscillator furnish examples of quantum systems where the perturbation results in a certain ionization probability by tunneling processes. Accordingly, the perturbed ground-state energy is shifted and broadened, thus acquiring an imaginary part which is considered to be a paradigm of nonperturbative behavior. Here we demonstrate how the low order coefficients of a divergent perturbation series can be used to obtain excellent approximations to both real and imaginary parts of the perturbed ground state eigenenergy. The key is to use analytic continuation functions with a built in analytic structure within the complex plane of the coupling constant, which is tailored by means of Bender-Wu dispersion relations. In the examples discussed the analytic continuation functions are Gauss hypergeometric functions, which take as input fourth order perturbation theory and return excellent approximations to the complex perturbed eigenvalue. These functions are Borel-consistent and dramatically outperform widely used Padé and Borel-Padé approaches, even for rather large values of the coupling constant.
pacs:11.15.Bt, 11.10.Jj, 32.60.+i
Since the pioneering work of Dyson Dyson1952 (), who provided an argument as to why perturbation series in quantum electrodynamics (QED) are divergent, the fundamental problem of how to reconstruct physical observables from power-series expansions with zero convergence radius has remained an active area of research Caliceti2007 (); Marino2014 (). This problem has been encountered in virtually all areas of quantum physics, such as statistical Brezin1977 (); Kleinert2001 (); Yukalov1990 (), string Marino2014 () and quantum field theories Lipatov1977 (); Dunne2002 (), as well as many-body problems of condensed matter physics Pollet2010 ().
The simplest examples can be found in single-particle quantum mechanics Caliceti2007 (); Marino2014 (); Brezin1977 (); Dunne2002 (); Bender1969 (); Janke1995 (); Jentschura2001 (). For instance, the perturbation expansion for the Stark Hamiltonian has zero radius of convergence Jentschura2001 (); Privman1980 (). When a hydrogen atom is placed in a homogeneous electric field, the electronic ground state energy is shifted and broadened as the electric field intensity, , increases. As a function of , the perturbed ground state energy then acquires both a real part and an imaginary part , . The latter reflects the tunneling rate in and out of the Coulomb potential, which is very difficult to obtain perturbatively. To see this, let and consider the perturbative expansion Privman1980 () for the ground state energy of hydrogen in powers of around ,
where atomic units (a.u.) are used throughout the paper. The same-sign expansion coefficients are real and grow factorially with . Therefore, the series in Eq. (Nonperturbative Quantum Physics from Low-Order Perturbation Theory) has zero radius of convergence. No matter how small is, the series in Eq.(Nonperturbative Quantum Physics from Low-Order Perturbation Theory) will never converge to , and so “” is used in Eq. (Nonperturbative Quantum Physics from Low-Order Perturbation Theory) to indicate that the RHS is an asymptotic expansion of the LHS.
Accurate calculations of have been achieved by a combination of perturbation theory with Borel-Padé Reinhardt1982 (); Jentschura2001 () resummation techniques. Unfortunately, these techniques are ultimately impractical since they require too many coefficients. In most quantum-mechanical problems of interest, one has available only low-order coefficients, obtained by calculating Feynman diagrams up to a given order and/or by summing a few classes of diagrams (out of infinitely many possible) to “infinite order” Stefanucci2013 (); Mera2013 (). In contrast, the recent development of the diagrammatic Monte Carlo (DiagMC) method Pollet2010 () has made it possible to sample Feynman diagrams up to large orders. However, DiagMC is computationally demanding and relies on resummation and regularization techniques Pollet2010 () in order to treat divergent or slowly convergent series. Other resummation approaches Kleinert2001 () rely on large-order information Bender1969 (); Brezin1977 () and knowledge of for large values of to design rapidly converging strong-coupling expansions from divergent weak-coupling expansions. While promising, these approaches have remained largely unexplored.
These considerations lead us to the following questions: Do we really need a large number of coefficients to predict quantities like ? Or can it be done with just a few coefficients? In this Letter, we demonstrate that low-order approximations can, paradoxically, reproduce nonperturbative quantities like with excellent accuracy even for rather large values of the perturbation strength. This is achieved by using the so-called Bender-Wu dispersion relations Bender1969 () to guess the branch cut structure of , for complex . Then we design approximants with the property of having the desired branch-cut structure “built-in”, with flexible branch points. This flexibility is essential to approximate in a wide variety of problems. For the Stark problem the branch cut structure is indeed known from Bender-Wu dispersion relations Bender1969 (): possesses branch points at and .
Let us then start by trying to calculate using a divergent series such as Eq. (Nonperturbative Quantum Physics from Low-Order Perturbation Theory). Traditionally, the first choice is to calculate Padé approximants Baker1996 (). These are parameterized rational approximations, , of the form
where the parameters and are determined by equating each order up to in the Taylor and asymptotic series of and , respectively, around , so that . Padé approximants and other similarly simple sequence transformations are valuable tools for analytic continuation (AC), and can work well in many cases of practical interest Mera2013 (). They provide a family of rational functions that are easily built order by order: first-order perturbation theory gives and ; second-order perturbation theory yields , and ; etc. By studying the resulting Padé table, one can in many cases extract good approximations to the expectation value of interest. However, by approximating with a rational function of , one is imposing an asymptotic behavior for large values of which is in general not physical. Approximating can be difficult because the denominator in can vanish for specific values of the interaction strength. More importantly, is a real number for real , and therefore . This means that the standard Padé approximants cannot work for our problem as they fail to give Reinhardt1982 (); Jentschura2001 ().
Nevertheless, the idea behind Padé approximants is very general and it can be used to propose new approximations. For example, one can choose a parameterized analytic function to approximate , fixing the parameters so that the Taylor series for is equal to the asymptotic series of up to the desired order. An example is provided in Ref. Bender1996 (), which considers continued-exponential approximations of the form , where the parameters are fixed from the perturbation expansion of , as is the case with Padé approximants. Since here we are concerned with the determination of , we initially aimed for a function with the following desirable properties: (i) it is a complex function of real , with the ability to mimic the branch cut structure discussed above; (ii) it can be built from low-order perturbation theory, as Padé approximants and continued-functions are built; (iii) it is amenable to generalization by being a member of a more general family of “higher order” functions; and (iv) it is sufficiently general and flexible in order to include many possible functions as particular cases.
A possible candidate satisfying all desirable properties (i)-(iv) is the Gauss hypergeometric function . It satisfies (i) and (ii) as it is complex (and has a branch cut) for , and it contains at most four parameters so it can be built from the coefficients . It also satisfies condition (iii) because can be obtained by approximating the ratio between consecutive expansion coefficients, , by a 1/1 Padé approximant that reproduces exactly for . If higher-order Padé approximants could be used as well, one would obtain hypergeometric functions of higher order, , which are actually instances of an even “higher-order” function—the so-called Meijer- function nistlibrary (). Finally, satisfies condition (iv) as it is well known that many functions are particular cases of .
The Taylor series for is given by:
where is a so-called Pochhammer symbol. To obtain the , one equates each order in the asymptotic series for with the corresponding order in the Taylor series for to obtain a system of four equations with four unknowns
Once the coefficients are determined, a hypergeometric approximation for, e.g., the Stark case can be constructed as:
We apply this scheme to three Hamiltonians from single-particle quantum mechanics, offering illustrative examples of divergent perturbation series as well as nonperturbative behavior: the Stark Hamiltonian, , with asymptotic series expansion described Privman1980 (); Jentschura2001 () by Eq. (Nonperturbative Quantum Physics from Low-Order Perturbation Theory); the cubic one-dimensional anharmonic oscillator with real perturbation Alvarez1988 (), ; and the cubic one-dimensional anharmonic oscillator with imaginary perturbation Bender1999 (), . Here, is the force constant taken as in the numerical analysis below. Furthermore, the perturbed ground-state eigenvalue has in the first two cases, while in the third case one has a non-Hermitian, but -symmetric Bender1999 (), Hamiltonian with real eigenvalues. For simplicity, all equations are written assuming the Stark Hamiltonian problem.
Figure 1 shows [top panels] and [bottom panels] as a function of for these three Hamiltonians. In Fig. 1(a), values of and are shown for the Stark Hamiltonian. Exact results taken from Ref. Kolosov1987 () are compared with those calculated using the simple hypergeometric approximant and the same-order 2/2 Padé approximant. The simple approximant introduced here provides excellent approximations to both and , while the 2/2 Padé approximant fails to approximate either quantity. In Fig. 1(b) a similar comparison is made for the cubic anharmonic oscillator with real perturbation, taking the exact numerical values from Ref. Alvarez1988 (). Once again the approximant dramatically outperforms the 2/2 Padé approximant. Finally, in Fig. 1(c) we see the results obtained from the cubic anharmonic oscillator with imaginary perturbation. In this case, both Padé and exact results are taken from Ref. Bender1999 (). The Padé results have been obtained in Ref. Bender1999 () by means of a Cesaro sum of the energies obtained from the 22/22 and 22/23 Padé approximants. Figure 1(c) shows that outperforms large-order Padé () for the calculation of , and they both reproduce the exact value of . Therefore the hypergeometric approximant offers an excellent fourth-order approximation, likely to outperform Padé approximants of much higher order. Note that a single hypergeometric approximant yields the results shown in Fig. 1(b) and 1(c), just by replacing by .
A comparison between the hypergeometric approximant and Padé approximants is admittedly not very fair. To obtain from Padé approximants, the standard procedure Jentschura2001 () thus far has been to employ the Borel-Padé method Caliceti2007 (). In this method, one starts from a large number of coefficients and evaluates the Borel-transformed coefficients , which are then employed to calculate Padé approximants and integrals
This yields the Borel-Padé approximation, . The Borel method removes from the coefficients, sums the transformed series, and puts back into the series by means of the Laplace transform in Eq. (6). The essence of the Borel-Padé method Caliceti2007 () is to perform AC on the Borel transformed coefficients and use the resulting analytic function to evaluate the integral in Eq. (6). While the Borel-Padé method allows accurate calculations of from the perturbation series, it also requires Jentschura2001 () very large orders of perturbation theory that are unavailable in practice and lead to accuracy issues, impeding the calculation at high values of the perturbation strength.
In the Borel-Padé method, the analytic function is a Padé approximant. In the same spirit, we use hypergeometric functions as analytic functions to construct the Borel-hypergeometric method, by performing hypergeometric AC of the Borel-transformed series, calculating the coefficients that define the hypergeometric function from . The Borel-hypergeometric approximation, , is then
and specifies the integration contour Jentschura2001 (). An expression somewhat similar to Eq. (7) was used in Ref. Kleinert2001 () as the starting point to construct convergent strong-coupling expansions, while requiring the knowledge of both and .
We now apply the Borel-hypergeometric method to approximate and for the same three Hamiltonians studied in Fig. 1. Figure 2 demonstrates that in all three problems considered the Borel-hypergeometric method gives excellent approximations to both and , and reproduces the results given by the hypergeometric approximant. Comparing Borel-hypergeometric and hypergeometric approximants reveals that the hypergeometric approximant is Borel consistent to a very good approximation. It is well known that a convergent sum and its Borel resummation give the same result. The hypergeometric approximations discussed here clearly satisfy this highly desirable property. The careful reader will surely notice that in Fig. 2(b) the Borel-hypergeometric sum diverges for very small . This is not a problem since the simple hypergeometric approximant is already well-behaved for . Alternatively, one can calculate one extra order of perturbation theory and build the Borel-hypergeometric approximant from the coefficients of the once-subtracted series, . As shown in Fig. 2(b) that procedure mitigates this minor problem, while leading to similarly accurate overall results. We emphasize that the hypergeometric and Borel-hypergeometric approaches are fourth-order approximations and thus much simpler and less expensive than the widely used Borel-Padé method Caliceti2007 (), while being of comparable accuracy. For instance, in the case of the Stark Hamiltonian with very large , approximately 70 orders of perturbation theory were required in Ref. Jentschura2001 () as the input for the Borel-Padé scheme to produce , which can be contrasted with our result , and with numerically exact data Kolosov1987 () . For the Stark Hamiltonian, a.u. MV cm corresponds to a rather large electric field.
It is easy to understand why the hypergeometric and Borel-hypergeometric method dramatically outperform the traditional Borel-Padé method Caliceti2007 (). To obtain one needs approximants with a branch cut in the complex plane with branch points at and Bender1969 (); Reinhardt1982 (). Padé approximants typically have both poles and zeroes on the real axis, thereby lacking the correct analytic structure of –essential for rapid convergence. The function has a branch cut running from to . When calculating from the low-order coefficients , we typically obtain a large value for , thus mimicking the correct branch cut structure in , as illustrated in Fig. 3.
Our findings have interesting physical implications. At the end of Ref. Dyson1952 (), Dyson wonders about the possibility that a series with zero radius of convergence might contain all that there is to know in a quantum system, arguing that if this were the case then an extension of QED would be needed. In the three examples considered in our study, the “extension” needed consists in supplementing the low-order information with an AC function able to mimic the branch cut structure required to obtain Reinhardt1982 (); Dunne2002 () .
We speculate that the approximations developed here have potential applications in nonequilibrium many-body perturbation theory Stefanucci2013 () for condensed matter systems. For example, there is a habit of partially resumming classes of Feynman diagrams for such problems, as exemplified by the self-consistent Born approximation (SCBA) Frederiksen2007 (); Mahfouzi2013 () for electron-boson interacting systems or the self-consistent approximation Spataru2009 () for electron-electron interacting systems out of equilibrium. These are both self-consistent in order to conserve charge current Stefanucci2013 (), and can therefore be viewed as “infinite order” approximations. However, they are exact only to first order because, to second order, SCBA misses the polarization bubble diagram Mahfouzi2013 () and the first-order vertex correction Dash2011 () to the Fock diagram, while the self-consistent approximation accounts for the former but also misses the latter. Thus, such resummation schemes are uncontrolled because an error is “summed” to all orders (starting from second order) Mera2013 (). An alternative could be to build the exact diagrammatic series—including vertex corrections—at low orders, and then use a suitable AC technique. The hypergeometric and Borel-hypergeometric techniques proposed here look promising in this respect.
In conclusion, by analogy with traditional Padé and Borel-Padé techniques we have developed a fourth-order hypergeometric approximant and its natural Borel extension. We demonstrate the effectiveness of this technique by summing the perturbative expansions with zero radius of convergence for three well-known examples from single-particle quantum mechanics, obtaining excellent approximations to their perturbed ground state eigenenergies, even for rather large values of the perturbation strength. The imaginary part of the perturbed eigenvalue is obtained with good precision using only fourth-order perturbation theory, thereby evading the calculation of a large number of coefficients in perturbation theory. Our results show that nonperturbative physics can be obtained from the low-order coefficients of a divergent perturbation series, as long as a carefully tailored analytic continuation technique is implemented.
Acknowledgements.H. M. and B. K. N. were supported by NSF under Grant No. ECCS 1202069. T. G. P. acknowledges funding for the QUSCOPE center by the Villum foundation.
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