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After completion of this work, another crystalline form of Li(Equation presented)IrO(Equation presented) realizing a slightly different three-dimensional, tricoordinated, honeycomb-like lattice structure of the Ir ions was reported in Ref. [42].
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After completion of this work, another crystalline form of Li(Equation presented)IrO(Equation presented) realizing a slightly different three-dimensional, tricoordinated, honeycomb-like lattice structure of the Ir ions was reported in Ref. [42].
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For the convenience of the inclined reader we provide a vesta [43] visualization file of the hyperoctagon lattice in the Supplemental Material [44].
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Further generalizations of the Kitaev models also to lattices with higher vertex coordination number, in particular the kagome, triangular, hyperkagome, and pyrochlore lattices have been studied [45-47]. It should, however, be noted that it is precisely the higher coordination number of the vertices in these lattices which prohibits following the same analytical route that can be used for the trivalent ones.
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The relative sign for the two alternative definitions of the loop operator, and subsequent “freedom†to define magnetic flux, appears also for other trivalent lattices; for instance, the two-dimensional square-octagon lattice. Using the operator (Equation presented) [Eq. (7)] to define the flux ensures that the notion of 0/(Equation presented)-flux per plaquette is consistent with the one used by Lieb [31]. In particular, the ground state of the square-octagon lattice resides in the full-flux sector according to Lieb's theorem, because the loops have lengths four and eight, respectively.
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The relative sign for the two alternative definitions of the loop operator, and subsequent “freedom†to define magnetic flux, appears also for other trivalent lattices; for instance, the two-dimensional square-octagon lattice. Using the operator (Equation presented) [Eq. (7)] to define the flux ensures that the notion of 0/(Equation presented)-flux per plaquette is consistent with the one used by Lieb [31]. In particular, the ground state of the square-octagon lattice resides in the full-flux sector according to Lieb's theorem, because the loops have lengths four and eight, respectively.
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The zeros on the diagonal are, in fact, not important for the argument.
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