Two-dimensional helium-like atom in a homogeneous magnetic field: Numerically exact solutions.

A two-dimensional helium atom (2D-helium) is a real subject for current studies, particularly regarding a hot topic of negatively charged excitons (trions) in semiconducting monolayers. The present study considers a 2D-helium-like atom in a homogeneous magnetic field. We are able to rewrite its Schrödinger equation into a polynomial form concerning dynamic variables. This form is useful for utilizing the algebraic calculation by annihilation and creation operators, enabling the successful application of the Feranchuk-Komarov (FK) operator method to obtain numerically exact solutions (energies and wave functions) for this system. The polynomialization of the equation allows obtaining analytical expressions of all matrix elements, which saves the computational resources significantly. Numerical results for the case without a magnetic field are comparable to other calculations. Moreover, the precise separation of the center-of-mass motion, as provided in this study, leads to an equation for the relative motion of the electrons in a magnetic field, incorporating all previously neglected terms. This result is useful for further study of trions where the electron effective mass is comparable with the hole effective mass. Additionally, we provide a FORTRAN program designed to solve the problems above. Program Title: CHeAMF CPC Library link to program files: https://doi.org/10.17632/mp8tf2dz67.1 Licensing provisions: BSD 3-clause Programming language: FORTRAN90 Nature of problem: The Schrödinger equation for a 2D-helium-like atom in a homogeneous magnetic field is transformed into a polynomial form using the Levi-Civita transformation twice. This transformation results in a structure more conductive to applying algebraic methods based on annihilation and creation operators. Consequently, we employ the FK operator method [1] to obtain numerically exact solutions, ensuring that the calculated energies converge to a high level of precision, up to 15 decimal places in this study. This method is developed to cover a broad range of magnetic field intensities, extending up to 0.1 a.u. (2.35 × 10 4 Tesla). Moreover, it is applicable not only for the ground state but also for highly excited states. Solution method: The modified FK operator method, as introduced in reference [2], has been developed and applied to obtain precise numerical solutions for a 2D-helium-like atom. Concurrently, algebraic techniques have been employed to compute matrix elements. Subsequently, we transformed the Schrödinger equation into a linear matrix equation, which we solved using the 'dsygvx.f' subroutine from the LAPACK library [3]. This subroutine has been optimized for improved accuracy by employing real*16 variables instead of real*8. Furthermore, we have incorporated an optimal free parameter into the FORTRAN program, enhancing convergence speed. Additional comments including restrictions and unusual features: Operating system: Linux. RAM: at least 4 GByte per core. We recommend using the gFortran compiler for this program. The runtime varies from a few minutes to hours, depending on the required precision. Particularly for strong magnetic fields (γ ≥ 0.01 a.u.) or when working with excited levels, the runtime may extend to several hours to achieve a precision of 16 decimal places. In such cases, it is advisable to have at least 60 GB of RAM per core. [1] I. Feranchuk, A. Ivanov, Van-Hoang Le, A. Ulyanenkov, Non-perturbative Description of Quantum Systems, Springer, Switzerland, 2015, https://doi.org/10.1007/978-3-319-13006-4. [2] Thanh-Xuan H. Cao, Duy-Nhat Ly, Ngoc-Tram D. Hoang, Van-Hoang Le, High-accuracy numerical calculations of the bound states of a hydrogen atom in a constant magnetic field with arbitrary strength, Comput. Phys. Commun. (2019), https://doi.org/10.1016/j.cpc.2019.02.013. [3] Netlib.org. LAPACK: Linear Algebra PACKage, Subroutine dsygvx.f, https://netlib.org/lapack/explore-3.1.1-html/dsygvx.f.html. [ABSTRACT FROM AUTHOR]