Kitaev, semi-Dirac, and spinon quantum spin Hall liquids on the square-hexagon-dodecagon lattice: A parent projective symmetry group classification
Abstract
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.
Related
- Projective symmetry group classification of fermionic Z2 spin liquids on the dipolar-octupolar pyrochlore magnets
- Projective symmetry groups beyond spin-1/2: Classifying integer-spin quantum spin liquids
- Melting noncoplanar order on the square-kagome lattice: Chiral, flat-band, and staggered spin liquids from projective symmetry
- Classification and correlation signatures of chiral spin liquids on the pyrochlore lattice
- Categorical classification of non-Fermi liquids in 2+1 dimensions
BibTeX
@misc{raj2026kitaev,
title = {{Kitaev, semi-Dirac, and spinon quantum spin Hall liquids on the square-hexagon-dodecagon lattice: A parent projective symmetry group classification}},
author = {Harsh Raj and Yasir Iqbal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22965},
eprint = {2610.22965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22965},
}
RIS
TY - UNPB
TI - Kitaev, semi-Dirac, and spinon quantum spin Hall liquids on the square-hexagon-dodecagon lattice: A parent projective symmetry group classification
AU - Harsh Raj
AU - Yasir Iqbal
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22965
UR - https://arxiv.org/abs/2610.22965
AB - 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 Z_2 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 Z_2 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 N_f=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.
ER -