Cosmic string influence on a 2D hydrogen atom and its relationship with the Rytova-Keldysh logarithmic approximation in semiconductors

TL;DR

This work analyzes how a straight cosmic string, encoded by the conical metric with $0<\theta<2\pi\alpha$, modifies a two-dimensional hydrogen atom with the logarithmic interaction $V(r)=\frac{e^2}{2\pi\epsilon_0}\ln\left(\frac{r}{r_0}\right)$. The Schrödinger equation is solved in this geometry, yielding a radial problem with $l_{\alpha}=l/\alpha$ and an effective potential $V_{eff}(r)=\frac{l_{\alpha}^2-1/4}{r^2}+\ln r$, whose bound-state spectrum $E_{n, l_{\alpha}}$ depends on the defect parameter $\alpha$. Numerical solutions are obtained by the Finite Difference Method and validated against a linear variational method, with eigenfunctions, probability densities, and expectation values $\langle r\rangle$, $\langle r^2\rangle$ illustrating defect-induced shifts. An analogy with two-dimensional excitons in semiconductors is developed via the Rytova–Keldysh framework, showing that the RK short-distance logarithmic form leads to a dimensionless equation matching the 2D logarithmic problem, and providing exciton energies in SI units and substrate effects that align with existing literature.

Abstract

A two-dimensional hydrogen atom offers a promising alternative for describing the quantum interaction between an electron and a proton in the presence of a straight cosmic string. Reducing the hydrogen atom to two dimensions enhances its suited to capture the cylindrical/conical symmetry associated with the cosmic string, providing a more appropriate description of the physical system. After solving Schrdinger's equation, we calculate the eigenenergies, probability distribution function, and expected values for the hydrogen atom with logarithmic potential under the influence of the topological defect. The calculations for the 2D hydrogen atom are performed for the first time using the Finite Difference Method. The results are presented through graphics, tables, and diagrams to elucidate the system's physical properties. We have verified that our calculations agree with a linear variational method result. Our model leads to an interesting analogy with excitons in a two-dimensional monolayer semiconductor located within a specific semiconductor region. To elucidate this analogy, we present and discuss some interaction potentials and their exciton eigenstates by comparing them with the results from the literature.

Figures (6)

• Figure 1: The effective potential for a fixed $l = 3$ modified by the cosmic string parameter $\alpha$. According to Table \ref{['table1']}, the blue and red curves represent, respectively, both $l_{\alpha=3/4}=4$ and $l_{\alpha=3/5}=5$. The black line is the case of $l_{\alpha=1}=l=3$.
• Figure 2: The energy level diagram for a fixed $l = 3$ influenced by the cosmic string factor $\alpha$, which causes upper shifts in the energies values. The energy levels sate in blue and red serve for $l_{\alpha=3/4}=4$ and $l_{\alpha=3/5}=5$ (the black lines depict the ones for $\alpha=1$), respectively.
• Figure 3: The probability distribution function for a fixed $l = 3$ is influenced by the cosmic string factor $\alpha$, which causes shifts in probability values. In (a) for radial quantum number $n=1$ and in (b) for radial quantum number $n=2$. The black curve is for $\alpha=1$, the blue and red curves represent, respectively, both $l_{\alpha=3/4}=4$ and $l_{\alpha=3/5}=5$.
• Figure 4: The disks of probability indicating regions with higher probabilities (higher intensities). We use the same $l=3$ and $n$ as done in Fig. \ref{['prob1']}. The disks are described as follows: the first line is for $n=1$, and the second line is for $n=2$. First column for $l_{\alpha=1.0}=3$, second for $l_{\alpha=3/4}=4$ and third for $l_{\alpha=3/5}=5$.
• Figure 5: Comparison between the logarithmic approximation given by Eq. \ref{['pot-r0']}, the RK potential described in Eq. \ref{['RKpot']}, and the 3D Coulomb potential from Eq. \ref{['coulomb3d']}.