## Chern numbers, quaternions, and Berry’s phases in Fermi systems.(English)Zbl 0830.57020

A common property of most systems studied in the context of the Berry phase is their oddness under time reversal. The standard example is the quantum Hamiltonian $$H_B= {\mathbf B} \cdot {\mathbf J}$$ modelling a particle with $$\text{spin } J$$ in an external magnetic field $${\mathbf B}$$. If $$\Theta$$ denotes the anti-unitary operator effecting the time reversal, then $$\Theta H_B= -H_B \Theta$$. In this paper, Hamiltonians of the form $$H_Q= \sum_{i,j} Q_{ij} J_i J_j$$ that are even under time reversal are analyzed. Physically, if $$Q$$ is a real $$3\times 3$$ symmetric matrix with $$\text{trace} (Q) =0$$, such Hamiltonians describe systems with $$\text{spin } J$$ in a quadrupole electric field $$Q$$. Since for systems with integer spin, i.e. bosonic systems, the Berry phase resulting from $$H_Q$$ is trivial, only Fermi systems with half-odd-integer spin show interesting topological phenomena.
Removing the point $$Q=0$$ of level crossings from the parameter set, the eigenspaces of $$H_Q$$ give rise to a canonical bundle structure, and in the adiabatic limit the evolution induced by the (time-dependent) Schrödinger equation leads to an “adiabatic” connection for these bundles. The detailed study of the resulting adiabatic curvature and holonomy, i.e. the generalization of the Berry phase to bundles that are not necessarily line bundles, constitutes the central theme of the paper. In addition, the Chern numbers labelling the spectrum of the operator family $$H_Q$$ are computed. Quaternionic vector spaces enter into the analysis via the fact that the time reversal can be formulated as a quaternionic structure map, i.e. an antilinear map $$\Theta$$ on a complex vector space satisfying $$\Theta^2 =\pm 1$$.
As it turns out, systems with $$\text{spin } J= {3\over 2}$$ show a behaviour somewhat different from those with $$J> {3\over 2}$$. For $$J= {3\over 2}$$, the set of $$H_Q$$ is identical to the set of all Hermitian quaternionic matrices with vanishing trace. Furthermore, for unit $$Q$$ all $$H_Q$$ are unitarily equivalent, whereas for $$J> {3\over 2}$$ there are one-dimensional families not unitarily related. Also, For $$J= {3\over 2}$$ the adiabatic curvature is shown to be self-dual (with respect to the Hodge duality); on the other hand, the authors prove that for $$J> {3\over 2}$$ no bundles with self-dual or anti-self-dual curvatures exist. The second Chern numbers $$C_2$$ are calculated as $$C_2= \pm 1$$ for $$J= {3\over 2}$$, and $$C_2= (J+ {1\over 2}) (2m_T- J- {1\over 2})/2$$ for the $$m_T$$th level of $$H_Q$$ if $$J> {3\over 2}$$ (the latter result was also published separately [cf. the second and third author, Chern numbers for fermionic quadruple systems, J. Phys. A 22, No. 4, L 111–L 115 (1989)]). Finally, explicit expressions for the time evolution operator and spectral projections are derived for $$J= {3\over 2}$$ and $$J= {5\over 2}$$, respectively, implying expressions for the associated holonomies. The resulting formulas for $$J= {3\over 2}$$ are identical to the corresponding ones found previously by the third author [Non-abelian Berry’s phase, accidental degeneracy, and regular momentum, J. Math. Phys. 28, No. 9, 2102-2114 (1987)].

### MSC:

 57R99 Differential topology 58J90 Applications of PDEs on manifolds 57R57 Applications of global analysis to structures on manifolds 57R20 Characteristic classes and numbers in differential topology
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### References:

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