Abstract
We generalize the dynamical analog of the Berry geometric phase setup to the quaternionic model of Avron et al. In our dynamical quaternionic system, the fast half-integer spin subsystem interacts with a slow two-degrees-of-freedom subsystem. The model is invariant under the 1:1:2 weighted SO(2) symmetry and spin inversion. There is one formal control parameter in addition to four dynamical variables of the slow subsystem. We demonstrate that the most elementary qualitative phenomenon associated with the rearrangement of the energy super-bands of our model consists of the rearrangement of one energy level between two energy superbands which takes place when the formal control parameter takes the special isolated value associated with the conical degeneracy of the semi-quantum eigenvalues. This qualitative phenomenon is of topological origin, and is characterized by the second Chern class of the associated semi-quantum system. The correspondence between the number of redistributed energy levels and the second Chern number is confirmed through a series of examples.
Highlights
We generalize the dynamical analog of the Berry geometric phase setup to the quaternionic model of Avron et al In our dynamical quaternionic system, the fast half-integer spin subsystem interacts with a slow two-degrees-of-freedom subsystem
Without going into strict mathematical details associated with the definition of the spectral flow in the context of the Atiyah–Singer theorem [20,21,22] we show on concrete examples that typical qualitative modifications of theband structure can be interpreted in terms of the correspondence between semi-quantum and quantum quaternionic Hamiltonians and allows to relate topological invariant associated with formation of degeneracy point for semi-quantum model to the number of redistributed quantum energy levels
Our main result in this paper is the indication of a topological origin of the rearrangement of energy levels between the superbands of quaternionic slow-fast dynamical systems
Summary
We begin by reviewing the qualitative analysis of model Hamiltonians describing interaction of two “fast” quantum states with one “slow” degree of freedom. For S = 1/2, the model Hamiltonian (1) has two degeneracy points of semi-quantum eigenvalues occurring at the north and south poles of the classical phase space P = S2 (which we denote by (+) and (−), respectively) at two different isolated values of control parameter. The generic qualitative modification of the band structure in quaternionic dynamical systems can occur if such systems possess two slow degrees of freedom providing four dynamical variables in addition to one formal control parameter In such a case, under variation of the formal control parameter, an isolated degeneracy point of two Kramers-degenerate doublets can be formed generically and can be associated with the redistribution of energy levels between superbands in the quantum version of the same system. This requirement is based on the assumption that S describes fast quantum subsystem, whereas X and Y describe the slow classical subsystem
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