Unconventional supersymmetric quantum mechanics in spin systems

Abstract

It is shown that the eigenproblem of any $2\times 2$ matrix Hamiltonian with discrete eigenvalues is involved with a supersymmetric quantum mechanics. The energy dependence of the superalgebra marks the disparity between the deduced supersymmetry and the standard supersymmetric quantum mechanics. The components of an eigenspinor are superpartners\textemdash up to a $SU(2)$ transformation\textemdash which allows to derive two reduced eigenproblems diagonalizing the Hamiltonian in the spin subspace. As a result, each component carries all information encoded in the eigenspinor. We also discuss the generalization of the formalism to a system of a single spin-$\frac{p}{2}$ coupled with external fields. The unconventional supersymmetry can be regarded as an extension of the Fulton-Gouterman transformation, which can be established for a two-level system coupled with multi oscillators displaying a mirror symmetry. The transformation is exploited recently to solve Rabi-type models. Correspondingly, we illustrate how the supersymmetric formalism can solve spin-boson models with no need to appeal a symmetry of the model. Furthermore, a pattern of entanglement between the components of an eigenstate of a many-spin system can be unveiled by exploiting the supersymmetric quantum mechanics associated with single spins which also recasts the eigenstate as a matrix product state. Examples of many-spin models are presented and solved by utilizing the formalism.

arXiv comment15 pages, 7 figures

References in corpus (17)

• Quantum entanglement
• Area laws for the entanglement entropy - a review
• Supersymmetry and Quantum Mechanics
• Efficient classical simulation of slightly entangled quantum computations
• Entanglement renormalization
• On the Integrability of the Rabi Model
• Algorithms for entanglement renormalization
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• The quantum Rabi model: solution and dynamics
• All Majorana Models with Translation Symmetry are Supersymmetric
• Supersymmetry and Goldstino-like Mode in Bose-Fermi Mixtures
• Bogoliubov transformations and exact isolated solutions for simple non-adiabatic Hamiltonians
• Supersymmetry in quantum optics and in spin-orbit coupled systems
• Generalized Rabi models: diagonalization in the spin subspace and differential operators of Dunkl type
• Supersymmetric waves in Bose-Fermi mixtures
• Implementing supersymmetric dynamics in ultracold atom systems
• Spin Vortices and Skyrmions of a Single Electron in Inhomogeneous Magnetic Fields