Narrow Results

We review the formulation of gauge fields in terms of the frame of reference as well as the space in which the frame is defined. We highlighted some recent applications of gauge physics in the momentum space — in the modern fields of the spin Hall effect, the magnon Hall, the optical Magnus and the graphene valley Hall. General procedures of gauge transformation which lead to the construction of the gauge curvature and the equations of motion (EOM) are outlined. Central to this review is our intention to illustrate the impact of gauge physics on the past and future development of many new research fields emerging out of condensed matter physics, particularly in quantum nanosciences and nanoelectronics.

The molecular approach of a spin model is constructed on the Bethe lattice (BL), and then it is examined in terms of exact recursion relations. Rather than assuming that each BL site is inhabited by a single spin, each site is occupied by two spin-1/2 atoms A and B, forming a molecule. Each molecule is considered to contain two spin-1/2 atoms, as well as $q=3,4,$ or 6 nearest-neighbor molecules. In addition to the internal interactions between the atoms of each molecule, the molecules interact via their atoms in terms of bilinear interaction parameters J. Atoms of a molecule interact with $J_{A^i{B}^i}$, while the molecules interact via their atoms in terms of $J_{A^i{B}^{i+1}}=J_{B^i{A}^{i+1}}$ and $J_{A^i{A}^{i+1}}=J_{B^i{B}^{i+1}}$. After obtaining the magnetizations of each atom in the central molecule of the BL, the average magnetization of the molecule is determined. It is found that the model presents first-and second-order and random phase transitions. The model also displays tricritical, bicritical and end points, in addition to reentrant behavior for appropriate J values.