Indirect exchange interaction between magnetic impurities in one-dimensional gapped helical states

Abstract

We investigate theoretically indirect exchange interaction between magnetic impurities mediated by one-dimensional gapped helical states. Such states, containing massive Dirac fermions, may be realized on the edge of a two-dimensional topological insulator when time-reversal symmetry is weakly broken. We find that the indirect exchange interaction consists of Heisenberg, Dzyaloshinsky–Moriya, in-plane and out-of-plane Ising terms. These terms decay exponentially when the Fermi level lies inside the bandgap whereas the Dzyaloshinsky–Moriya term has smallest amplitude. Outside the bandgap, the massive helical states modify oscillatory behaviors of the range functions so that the period of oscillations decreases near the edge of band in terms of energy gap or Fermi energy. In addition, the out-of-plane Ising term vanishes in the case of zero-gap structure. Also, the oscillation amplitude of out-of-plane Ising term increases versus energy gap but it decreases as a function of Fermi energy. While the oscillation amplitudes of other components remain constant as functions of energy gap and Fermi energy. Analytical results are also obtained for subgap and over gap regimes. Furthermore, the effects of electron–electron interactions are analyzed.

Non-Hermitian indirect exchange interaction in a topological insulator coupled to a ferromagnetic metal

“…The horizontal dash line indicates the boundary between two regimes and , i.e., over gap and subgap regimes. As usual, for , the range functions oscillate in terms of both and R 45 , 52 , 57 . With the increase of Fermi energy, the amplitude of spatial oscillations increases, but the period of oscillations decreases, originating from the increase of the Fermi surface 45 , 52 .…”

“…In the over gap regime, with increasing M , the amplitude of spatial oscillations decreases but the period of oscillations increases. Also, interestingly, with the further increase of M, near the band edge, the amplitude of range function begins to oscillate as a function of M 57 . In the subgap regime, as already mentioned, there is no free carriers.…”

“…With the increase of Fermi energy, the amplitude of spatial oscillations increases, but the period of oscillations decreases, originating from the increase of the Fermi surface 45 , 52 . On the other hand, for , the range functions remain unchanged versus , whereas the spatial dependence of them tends to zero exponentially 45 , 53 , 57 , due to the absence of Fermi surface.

Figure 6 (Color online) Dependence of the range functions on for (a) and (b) cases in the subgap regime with , , and .

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“…Given the interest to study indirect exchange interaction in topological insulators, for the sake of completeness, we will present the results in the absence of non-Hermitian terms which might have been derived previously in literature 45 , 52 , 53 , 57 . Without loss of generality, we assume , set a lattice constant as the length unit, and take as the energy unit in the numerical evaluation of the range functions.…”

Signatures of interfacial topological chiral modes via RKKY exchange interaction in Dirac and Weyl systems

“…RKKY exchange interaction behavior along the edge modes in topological spin Hall liquid [75]. On the other hand, along the direction of QW width, the long distance limit is not desired as it is restricted by the QW width.…”