Imaginary time propagation code for large-scale two-dimensional eigenvalue problems in magnetic fields

Abstract

We present a code for solving the single-particle, time-independent Schrödinger equation in two dimensions. Our program utilizes the imaginary time propagation (ITP) algorithm, and it includes the most recent developments in the ITP method: the arbitrary order operator factorization and the exact inclusion of a (possibly very strong) magnetic field. Our program is able to solve thousands of eigenstates of a two-dimensional quantum system in reasonable time with commonly available hardware. The main motivation behind our work is to allow the study of highly excited states and energy spectra of two-dimensional quantum dots and billiard systems with a single versatile code, e.g., in quantum chaos research. In our implementation we emphasize a modern and easily extensible design, simple and user-friendly interfaces, and an open-source development philosophy. Nature of problem: Numerical calculation of the lowest energy solutions (up to a few thousand, depending on available memory), of a single-particle, time-independent Schrödinger equation in two dimensions with or without a homogeneous magnetic field. Solution method: Imaginary time propagation (also known as the diffusion algorithm), with arbitrary even order factorization of the imaginary time evolution operator. Additional comments: Please see the README file distributed with the program for more information. The source code of our program is also available at https: //bitbucket.org/luukko/itp2d. Running time: Seconds to hours, depending on system size. Keywords

Information analysis in free and confined harmonic oscillator

“…This resulting equation is numerically solved through a finite-difference procedure, in amalgamation with a minimization of expectation value to hit the ground state. After the original proposal that came several decades ago, a number of successful implementations [80,81,82,83,84,85,86] have been reported in the literature since then. In this work we have adopted an implementation, which has been successfully applied to a number of physical systems, such as atoms, diatomic molecules within a quantum fluid dynamical density functional theory (DFT) [87,88], as well as some model (harmonic, anharmonic, self-interacting, double-well, spiked oscillators) potentials [89,90,91,92,93], in both 1D, 2D and 3D.…”

Controllable quantum scars in semiconductor quantum dots

“…(1) using the itp2d code. 9 This code utilizes the imaginary time propagation method, which is particularly suited for 2D problems with strong perpendicular magnetic fields because of the existence of an exact factorization of the exponential kinetic energy operator in a magnetic field. 10 The eigenstates and energies can be compared to the well-known solutions of a unperturbed system.…”

Quantum Lissajous Scars

“…The Schrödinger equation for the Hamiltonian in Eq. (1) is solved by utilizing the itp2d code [33] based on arXiv:1911.09729v1 [quant-ph] 21 Nov 2019 the imaginary time propagation method. However, before considering the quantum solutions of the perturbed HO, we briefly discuss the unperturbed system, both classical and quantum.…”