Confluent supersymmetric algorithm for bilayer graphene

TL;DR

The paper develops a second-order confluent SUSY quantum-mechanical framework applied to a shifted harmonic oscillator to tailor external magnetic-field profiles for bilayer graphene, producing equidistant or partially equidistant spectra within the tight-binding model. It derives the corresponding effective Hamiltonian, analyzes spectrum properties (ordering, spacing, degeneracy), and constructs new bilayer graphene coherent states using diagonal and non-diagonal ladder operators, namely Barut-Girardello and Gilmore-Perelomov types. The work also investigates the time evolution, cyclic behavior, geometric phases, and quadrature uncertainties of these states, revealing temporal stability in many cases and showing how root structure and factorization energies govern coherence and completeness. Collectively, these results offer a principled method to engineer graphene spectra via SUSY transformations and to realize robust, controllable coherent-state dynamics in position-dependent magnetic fields with potential applications in quantum control of graphene-based systems.

Abstract

External magnetic field profiles leading to equidistant and partially equidistant bilayer graphene spectra within the tight-binding model are obtained. This is achieved by implementing the integral and differential versions of the second-order confluent algorithm to the harmonic oscillator for arbitrary real factorization energies. Additionally, new Barut-Girardello and Gilmore-Perelomov coherent states for bilayer graphene are derived, for both diagonal and non-diagonal ladder operators. Their time evolution is analyzed, finding temporal stability and cyclic evolution in some cases. This fact is contrasted with the non-cyclic evolution of bilayer graphene coherent states obtained when using two different factorization energies. Likewise, the geometric phase and uncertainty product of the quadratures for the previously obtained coherent states are studied.

Figures (14)

• Figure 1: Energy levels of bilayer graphene as a function of the level index of $H_0$. The natural ordering of the BG levels does not correspond with the standard ordering of the auxiliary problems.
• Figure 2: Applied magnetic field for two different factorization energies: $\epsilon=\omega/2$ (left) and $\epsilon=2\omega$ (right). It is shown the critical case for $w_0=0$ (dotted blue lines) and the regular case for $w_0=1$ (green lines).
• Figure 3: Probability density of the bilayer graphene ground state eigenfunction for $\epsilon = \omega/2$ in the regular (a) and critical case (b).
• Figure 4: Probability current in $y$-direction for the ground state with $\epsilon = \omega/2$.
• Figure 5: BGCS for $\epsilon = 2\omega$ and fixed $r=1$, with $\theta=0$ (green), $\theta=\pi/3$ (blue), $\theta=2\pi/3$ (orange), and $\theta=\pi$ (purple).