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

TL;DR

This paper develops algebraic methods to solve $SU(2) imes SU(2)$ Hamiltonian eigensystems, enabling analysis of entanglement and thermodynamics, with applications to mesoscopic systems like layered graphene.

Contribution

It introduces a systematic algebraic approach to solve $SU(2) imes SU(2)$ Hamiltonians and constructs eigenstates based on quartic polynomial equations, linking to entanglement and thermodynamics.

Findings

Eigenstates constructed from quartic polynomial solutions.

Thermodynamic properties like partition function and purity derived.

Application to layered graphene and mesoscopic systems demonstrated.

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.