Elsevier

Physics Letters A

Volume 360, Issue 2, 25 December 2006, Pages 205-209
Physics Letters A

Efficient calculation of energy spectra using path integrals

https://doi.org/10.1016/j.physleta.2006.08.035Get rights and content

Abstract

A newly developed method for systematically improving the convergence of path integrals for transition amplitudes [A. Bogojević, A. Balaž, A. Belić, Phys. Rev. Lett. 94 (2005) 180403, A. Bogojević, A. Balaž, A. Belić, Phys. Rev. B 72 (2005) 064302, A. Bogojević, A. Balaž, A. Belić, Phys. Lett. A 344 (2005) 84] and expectation values [J. Grujić, A. Bogojević, A. Balaž, Phys. Lett. A 360 (2006) 217] is here applied to the efficient calculation of energy spectra. We show how the derived hierarchies of effective actions lead to substantial speedup of the standard path integral Monte Carlo evaluation of energy levels. The general results and the ensuing increase in efficiency of several orders of magnitude are shown using explicit Monte Carlo simulations of several distinct models.

Introduction

Feynman's path integrals [5], [6] provide the general mathematical framework for dealing with quantum and statistical systems. The formalism has been successfully applied in generalizing the quantization procedure from the archetypical quantum mechanical problem of the dynamics of a single particle moving in one dimension, to more particles, more dimensions, as well as to more complicated objects such as fields, strings [7], etc. Symmetries of physical systems can be more easily treated and applied in this formalism, since it gives a simple and natural setup for their use [8]. Various approximation techniques are more easily derived within the framework of this formalism, and it has been successfully used for deriving non-perturbative results. The parallel application of this formalism in both high energy and condensed matter physics makes it an important general tool [9], [10]. The analytical and numerical approaches to path integrals have by now become central to the development of many other areas of physics, chemistry and materials science, as well as to the mathematics and finance [11], [12], [13], [14]. In particular, general numerical approaches such as the path integral Monte Carlo method have made possible the treatment of a wealth of non-trivial and previously inaccessible models.

The key impediment to the development of the path integral formalism is a lack of complete understanding of the general mathematical properties of these objects. In numerical approaches limited analytical input generally translates into lower efficiency of employed algorithms. The best path generating algorithms, for example, are efficient precisely because they have built into them the kinematic consequences of the stochastic self-similarity of paths [15]. A recent series of papers [1], [2], [3] has for this reason focused on the dynamical implications of stochastic self-similarity by studying the relation between path integral discretizations of different coarseness. This has resulted in a systematic analytical construction of a hierarchy of N-fold discretized effective actions SN(p) labeled by a whole number p and built up from the naively discretized action in the mid-point ordering prescription (corresponding to p=1). The level p effective actions lead to discretized transition amplitudes and expectation values differing from the continuum limit by a term of order 1/Np.

In this Letter we extend the applicability of the above method for improving the efficiency of path integral calculations to the evaluation of energy spectra. We show how the increased convergence of path integrals translates into the speedup in the numerical calculation of energy levels. Throughout the Letter we present and comment on the Monte Carlo simulations conducted using the hierarchy of effective actions for the case of several different models including anharmonic oscillator, Pöschl–Teller potential, and Morse potential. All the numerical simulations presented were done using Grid-adapted Monte Carlo code and were run on EGEE-II and SEE-GRID-2 infrastructure [16], [17]. The effective actions and the codes used can be found on our web site [18].

Section snippets

Partition function and energy spectra

The partition function is the central object in statistical mechanics. The path integral formalism gives us an elegant framework for calculating partition functions which can be used either for deriving analytical approximation techniques or for carrying out numerical evaluation. The starting point is the expression for the partition function in the coordinate basis,Z(β)=daA(a,a;β), where A(a,b;β)=b|eβHˆ|a is the quantum mechanical transition amplitude for going from a to b in

Numerical results

As we have seen in the previous section, F(β) can be evaluated with arbitrary precision on any interval of inverse temperatures [0,βmax] for any given potential by appropriately increasing and adjusting N, p, and NMC. Let us now numerically compare the quality of different discretizations of the free energy FN(p) with F, the most accurate one that may be calculated on a given set {βi}. To do this we use the standard χ2 function,χ2(N,p)=1M{βi}(FN(p)(βi)F(βi))2(ΔFN(p)(βi))2+(ΔF(βi))2, where M

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Supported by the Ministry of Science and Environmental Protection of the Republic of Serbia through project No. 141035.

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