The square-hexagon-dodecagon (SHD) lattice hosts an exactly solvable Kitaev spin liquid, while further-neighbor couplings frustrate its Heisenberg magnet into classically degenerate lines where quantum fluctuations can melt the Néel order. To place this state among the lattice's spin liquids, we classify fermionic parton mean-field states with singlet and spin-orbit-coupled triplet amplitudes by their projective symmetry groups (PSGs). As SHD sites lack site symmetry, its
Z2 PSGs are the projective classes of
p6mm with time reversal, from which those of every single-orbit
p6mm lattice follow. In the idealized Kitaev model, where lattice operations do not rotate spins, these liquids carry a single
Z2 PSG descending from the
SU(2) Lieb-flux state, and their projections are exact eigenstates. The PSG's sixfold-rotation sign, fixed by the dodecagon flux, doubles all Majorana bands, and the mirrors exclude a chiral phase in any uniform field, so only a sublattice-staggered three-spin term drives the non-Abelian Ising phase. Beyond the Kitaev state, we find a critical semi-Dirac spin liquid, with mean-field correlations
∝r−3 along three directions and
r−6 otherwise, and extended Dirac phases with up to
48 fermion flavors, among them an
Nf=12 state whose PSG forbids every singlet mass. Out-of-plane Dzyaloshinskii-Moriya couplings gap the Dirac states of its
U(1) class into a mean-field spinon counterpart of the Kane-Mele quantum spin Hall insulator, with spin Chern number
1 or
3 and spin-carrying monopoles. Chiral states imply Kalmeyer-Laughlin, semion
×semion, and three-fermion orders, rotation-breaking ones show
q→−q asymmetric neutron spectra, and sign-alternating fluxes break time reversal alone. Masses and monopoles in these many-flavor phases and the energetics of frustrated SHD Heisenberg models are natural next steps.
We examine the star lattice Kitaev model whose ground state is a chiral spin liquid. We fermionize the model such that the fermionic vacua are toric-code states on an effective Kagome lattice. This implies that the Abelian phase of the system is inherited from the fermionic vacua and that time-reversal symmetry is spontaneously broken at the level of the vacuum. In terms of these fermions we derive the Bloch-matrix Hamiltonians for the vortex-free sector and its time-reversed counterpart and illuminate the relationships between the sectors. The phase diagram for the model is shown to be a sphere in the space of coupling parameters around the triangles of the lattices. The Abelian phase lies inside the sphere and the critical boundary between topologically distinct Abelian and non-Abelian phases lies on the surface. Outside the sphere the system is generically gapped except in the planes where the coupling parameters between the vertices on triangles are zero. These cases correspond to bipartite lattice structures and the dispersion relations are similar to that of the original Kitaev honeycomb model. In a further analysis we demonstrate the threefold non-Abelian ground-state degeneracy on a torus by explicit calculation.
Projective symmetry groups are the mathematical tools which make it possible to list and classify mean-field spin liquids (SLs) based on a parton construction. The seminal work of Wen [Phys. Rev. B 65, 165113 (2002)] and its subsequent extension to bosons by Wang and Vishwanath [Phys. Rev. B 74, 174423 (2006)] concerned the so-called symmetric SLs; i.e., states that break neither lattice symmetries nor time reversal invariance. Here we generalize this tool to chiral (time reversal symmetry breaking) SLs described in a Schwinger boson mean-field approach and illustrate it on the triangular lattice, which can harbor nine different weakly symmetric SLs (two symmetric SLs and seven chiral SLs) with nearest neighbor bond operators only. Results for other lattices (square and kagome) are given in the Appendixes. Application of this new approach has recently led to the discovery of two chiral ground states on the kagome lattice [Messio et al., Phys. Rev. Lett. 108, 207204 (2012); F\aa{}k et al., Phys. Rev. Lett. 109, 037208 (2012)]. The signature of a time reversal symmetry breaking SL is the presence in the ground state of nontrivial fluxes of loop operators that break some lattice point group symmetries. The physical significance of these gauge invariant quantities is discussed both in the classical limit and in the quantum SL and their expressions in terms of spin observables are given.
We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\mathbit{k}\ensuremath{\rightarrow}\ensuremath{-}\mathbit{k}$ symmetry as well as the sixfold rotation symmetry around the $\mathrm{\ensuremath{\Gamma}}$ point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum for the cuboc1 state. Our work is based on the projective symmetry group classification of time-reversal symmetry breaking Schwinger-boson mean-field states by Messio, Lhuillier, and Misguich.
The interplay of frustrated interactions and lattice geometry can lead to a variety of exotic quantum phases. Here we unearth a particularly rich phase diagram of the Kitaev-Heisenberg model on the star lattice, a triangle decorated honeycomb lattice breaking sublattice symmetry. In the antiferromagnetic regime, the interplay of Heisenberg coupling and geometric frustration leads to the formation of valence bond solid (VBS) phases—a singlet VBS and a bond selective triplet VBS stabilized by the Kitaev exchange. We show that the ratio of the Kitaev versus Heisenberg exchange tunes between these VBS phases and chiral quantum spin-liquid regimes. Remarkably, the VBS phases host a whole variety of chiral triplon excitations with high Chern numbers in the presence of a weak magnetic field. We discuss our results in light of a recently synthesized star lattice material and other decorated lattice systems. Published by the American Physical Society 2024
Inspired by the recent discovery of a new instability towards a chiral phase of the classical Heisenberg model on the kagome lattice, we propose a specific chiral spin liquid that reconciles different, well-established results concerning both the classical and quantum models. This proposal is analyzed in an extended mean-field Schwinger boson framework encompassing time reversal symmetry breaking phases, which allows both a classical and a quantum phase description. At low temperatures, we find that quantum fluctuations favor this chiral phase, which is stable against small perturbations of second- and third-neighbor interactions. For spin-1/2, this phase may be, beyond the mean field, a chiral gapped spin liquid. Such a phase is consistent with the density matrix renormalization group results of Yan et al. [Science 332, 1173 (2011)]. Mysterious features of the low-lying excitations of exact diagonalization spectra also find an explanation in this framework. Moreover, thermal fluctuations compete with quantum ones and induce a transition from this flux phase to a planar zero flux phase at a nonzero value of the renormalized temperature (T/S2), reconciling these results with those obtained for the classical system.