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Non-Abelian Geometrical Phase for General Three-Dimensional Quantum Systems

Ali Mostafazadeh
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Abstract

Adiabatic $U(2)$ geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of $\xC P^2$ is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians $H=Q^{i_1,\cdots i_n}J_{i_1}\cdots J_{i_n}$ and their linear combinations for spin $j=1$ systems. In particular it is shown that the nuclear quadrupole Hamiltonian $H=Q^{ij}J_iJ_j$ does actually lead to non-Abelian geometric phases for $j=1$. This system, being bosonic, is time-reversal-invariant. Therefore it cannot support Abelian adiabatic geometrical phases.

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Ali Mostafazadeh. 1996-08-20. Non-Abelian Geometrical Phase for General Three-Dimensional Quantum Systems. https://doi.org/10.1088/0305-4470/30/21/023

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