Algebraic solutions for $SU(2)\otimes SU(2)$ Hamiltonian eigensystems: generic statistical ensembles and a mesoscopic system application

TL;DR

The paper addresses the analytic solution of generic $SU(2)\otimes SU(2)$ Hamiltonian eigensystems by reducing the problem to quartic secular equations solved via Cardano-Ferrari methods. It introduces an ansatz to construct separable and entangled eigenstate bases from quartic coefficients and derives closed-form expressions for thermodynamic quantities (partition function $\mathcal{Z}(T)$, purity $\mathcal{P}(T)$) and quantum entanglement (concurrence) for both zero-temperature and finite-temperature ensembles. A distinction is drawn between separable and entangled cases, with specific constraints (e.g., $\omega$-tensor forms or $\alpha\cdot\omega\cdot\beta=0$) that simplify the eigensystem and yield tractable solutions. The framework is then specialized to Bernal-stacked bilayer graphene, mapping tight-binding parameters to $SU(2)\otimes SU(2)$ coefficients and providing explicit band-structure and momentum-resolved concurrence calculations as a function of bias, thereby illustrating wide applicability to mesoscopic systems including photonic graphene and Dirac-like quasiparticles.

Abstract

Solutions of generic $SU(2)\otimes SU(2)$ Hamiltonian eigensystems are obtained through systematic manipulations of quartic polynomial equations. An {\em ansatz} for constructing separable and entangled eigenstate basis, depending on the quartic equation coefficients, is proposed. Besides the quantum concurrence for pure entangled states, the associated thermodynamic statistical ensembles, their partition function, quantum purity and quantum concurrence are shown to be straightforwardly obtained. Results are specialized to a $SU(2)\otimes SU(2)$ structure emulated by lattice-layer degrees of freedom of the Bernal stacked graphene, in a context that can be extended to several mesoscopic scale systems for which the onset from $SU(2)\otimes SU(2)$ Hamiltonians has been assumed.

Figures (2)

• Figure 1: Positive energy bands for the branches $n=1$ (first column) and $n=2$ (second column) for $\Lambda = 0.1$ (first row), $1$ (second row), and $10$ (third row).
• Figure 2: Quantum concurrence in the first Brillouin zone for the branches $n=1$ (first column) and $n=2$ (second column) for $\Lambda = 0.1$ (first row), $1$ (second row), and $10$ (third row), in terms of a blue-green-yellow scheme quantified according to the sectional plots.