Classification of gaplessZ2spin liquids in three-dimensional Kitaev models

Abstract

Frustrated quantum magnets can harbor unconventional spin-liquid ground states in which the elementary magnetic moments fractionalize into new emergent degrees of freedom. While the fractionalization of quantum numbers is one of the recurring themes in modern condensed matter physics, it often remains a challenge to devise a controlled analytical framework tracking this phenomenon. A notable exception is the exactly solvable Kitaev model, in which spin degrees of freedom fractionalize into Majorana fermions and a Z2 gauge field. Here we discuss the physics of fractionalization in three-dimensional Kitaev models and demonstrate that the itinerant Majorana fermions generically form a (semi)metal which, depending on the underlying lattice structure, exhibits Majorana Fermi surfaces, nodal lines, or topologically protected Weyl nodes. We show that the nature of these Majorana metals can be deduced from an elementary symmetry analysis of the projective time-reversal and inversion symmetries for a given lattice. This allows us to comprehensively classify the gapless spin liquids of Kitaev models for the most elementary tricoordinated lattices in three dimensions. We further expand this classification by addressing the effects of time-reversal symmetry breaking and additional interactions.

Materials design of Kitaev spin liquids beyond the Jackeli–Khaliullin mechanism

“…In the high-spin d 7 case, the exchange processes are similar to those in the low-spin d 5 case [67,68]. The resultant Hamiltonian is given by the same form as (17). An important difference from the d 5 case is in an additional contribution from the exchange processes between the e g electrons.…”

“…This is absent in the d 5 case as the e g orbitals are empty. It was pointed out that this contribution leads to a FM Heisenberg interaction, which competes with the AFM one in (17) and possibly makes the system being closer to the pure Kitaev case when the Γ term can be neglected [67].…”

“…Therefore, the effective Hamiltonian for the j eff = 1/2 moments in the 4 f 1 case is approximately given by the dominant AFM Kitaev interaction K < 0 and the subdominant AFM Heisenberg interaction J > 0 [69,70]. There is no contribution to the Γ term in (17) in the ideal case. When considering deviations from the ideal octahedra, such as trigonal distortions, the ratio between |K| and J changes systematically, and in addition, other exchange interactions also come into play.…”

“…A recent systematic study on a series of honeycomb compounds A 2 PrO 3 (A: alkali metals) revealed that the trigonal distortions become larger for larger A-site ionic radii, and accordingly, |K| becomes smaller while J does not change significantly [70]. At the same time, a symmetric off-diagonal interaction, which is different from the Γ term in (17), arises as H Γ = Γ γ γ γ (S γ i S¯γ j + S¯γ i S γ j ) with Γ > 0.…”

“…In the quantum case, however, the ground state of the model is exactly obtained as a QSL and that the elementary excitations are described by fractional quasiparticles, itinerant Majorana fermions and localized Z 2 fluxes [10,12]. As the model and the exact solution can be straightforwardly extended to any tricoordinate structures in any spatial dimensions, they provide rare examples of the well-established QSLs in more than one dimension [13,14,15,16,17].…”