Properties of quantum systems via diagonalization of transition amplitudes. I. Discretization effects

Abstract

We analyze the method for calculation of properties of nonrelativistic quantum systems based on exact diagonalization of space-discretized short-time evolution operators. In this paper we present a detailed analysis of the errors associated with space discretization. Approaches using direct diagonalization of real-space discretized Hamiltonians lead to polynomial errors in discretization spacing Delta . Here we show that the method based on the diagonalization of the short-time evolution operators leads to substantially smaller discretization errors, vanishing exponentially with 1/Delta(2). As a result, the presented calculation scheme is particularly well suited for numerical studies of few-body quantum systems. The analytically derived discretization errors estimates are numerically shown to hold for several models. In the follow up paper [I. Vidanović, A. Bogojević, A. Balaz, and A. Belić, Phys. Rev. E 80, 066706 (2009)] we present and analyze substantial improvements that result from the merger of this approach with the recently introduced effective-action scheme for high-precision calculation of short-time propagation.

SPEEDUP Code for Calculation of Transition Amplitudes via the Effective Action Approach

“…However, in majority of applications the time of propagation cannot be assumed to be small. The effective actions are found to have finite radius of convergence [8], and if the typical propagation times in the considered case exceed this critical value, Path Integral Monte Carlo approach must be used in order to accurately calculate the transition amplitudes and the corresponding expectation values [4,18]. As outlined earlier, in this case the time of propagation T is divided into N time steps, such that ε = T/N is sufficiently small and that the effective action approach can be used.…”

“…The error ε p for the effective action, obtained when level p effective potential is used, translates into ε p−DM/2 for a general many-body short-time amplitude. However, when amplitudes are calculated using the Path Integral Monte Carlo SPEEDUP C code [19], which will be presented in the next section, the errors of numerically calculated amplitudes are always proportional to ε p ∼1/N p , where N is number of time-steps in the discretization of the propagation time T. Therefore, accessibility of higher-order effective actions is central to the application of this approach if it is used for direct calculation of short-time amplitudes [8][9][10], as well as in the case when PIMC code is used [4,5,18]. However, increase in the level p leads to the increase in complexity of analytic expressions for the effective potential.…”

“…This has been extensively used in Refs. [8,9], where SPEEDUP codes were applied for numerical studies of several lower-dimensional models and calculation of large number of energy eigenvalues and eigenfunctions. The similar approach is used in Ref.…”

“…It gives systematic short-time expansion of amplitudes for a general potential, thus allowing accurate calculation of short-time properties of quantum systems directly, as has been demonstrated in Refs. [8][9][10]. For numerical calculations that require long times of propagation to be considered using e.g.…”

“…through extraction of the energy spectra from the partition function [2][3][4][5], and using the diagonalization of spacediscretized matrix of the evolution operator, i.e. matrix of transition amplitudes [6][7][8][9][10]. All these applications use the imaginary-time formalism [11,12], typical for numerical simulations of such systems.…”