Physics – Condensed Matter – Mesoscale and Nanoscale Physics
Scientific paper
2006-11-24
Phys.Rev.D77:085008,2008
Physics
Condensed Matter
Mesoscale and Nanoscale Physics
Scientific paper
10.1103/PhysRevD.77.085008
We present a diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of $\hbar $. Considering $\hbar $ as a running parameter, a differential equation connecting two diagonalization processes for two very close values of $\hbar $ is derived. The integration of this differential equation allows the recursive determination of the series expansion in powers of $\hbar $ for the diagonalized Hamiltonian. This approach results in effective Hamiltonians with Berry phase corrections of higher order in $\hbar $, and deepens previous works on the semiclassical diagonalization of quantum Hamiltonians which led notably to the discovery of the intrinsic spin Hall effect. As physical applications we consider spinning massless particles in isotropic inhomogeneous media and show that both the energy and the velocity get quantum corrections of order $\hbar ^{2}$. We also derive formally to all order in $\hbar $ the energy spectrum and the equations of motion of Bloch electrons in an external electric field.
Gosselin Pierre
Hanssen Jocelyn
Mohrbach Herve
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