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arXiv:2610.22965

Kitaev, semi-Dirac, and spinon quantum spin Hall liquids on the square-hexagon-dodecagon lattice: A parent projective symmetry group classification

Harsh Raj, Yasir Iqbal

Published Oct 16, 2026• 0 citations
arXivHTMLPDF

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 $Z_2$ 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 $\propto r^{-3}$ 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$\times$semion, and three-fermion orders, rotation-breaking ones show $q\to -q$ 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.

DOI
10.48550/arxiv.2610.22965

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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 Z2Z_2Z2​ PSGs are the projective classes of p6mmp6mmp6mm with time reversal, from which those of every single-orbit p6mmp6mmp6mm lattice follow. In the idealized Kitaev model, where lattice operations do not rotate spins, these liquids carry a single Z2Z_2Z2​ PSG descending from the SU(2)SU(2)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\propto r^{-3}∝r−3 along three directions and r−6r^{-6}r−6 otherwise, and extended Dirac phases with up to 484848 fermion flavors, among them an Nf=12N_f=12Nf​=12 state whose PSG forbids every singlet mass. Out-of-plane Dzyaloshinskii-Moriya couplings gap the Dirac states of its U(1)U(1)U(1) class into a mean-field spinon counterpart of the Kane-Mele quantum spin Hall insulator, with spin Chern number 111 or 333 and spin-carrying monopoles. Chiral states imply Kalmeyer-Laughlin, semion×\times×semion, and three-fermion orders, rotation-breaking ones show q→−qq\to -qq→−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.
The star lattice hosts the Yao-Kivelson chiral spin liquid, an exactly solvable Kitaev spin liquid that breaks time reversal spontaneously and is non-Abelian or Abelian depending on the ratio of its two couplings. A density-matrix renormalization group study found that an added Heisenberg exchange leaves it a wide stability window for ferromagnetic but only a narrow one for antiferromagnetic Kitaev coupling. Using projective symmetry groups (PSGs) of fermionic partons, we show that the chirality is not a peculiarity of the solvable point: whenever the πππ spin rotations act on the partons as in the exact solution (spin-gauge locking), a time-reversal-symmetric mean-field state has no amplitude on the triangles, at any range, so a Kitaev-type spin liquid on this lattice must break time reversal. The Abelian and non-Abelian liquids of both signs share a single chiral PSG, within which their transition is a band inversion. The Klein duality, which reverses the Heisenberg coupling, permutes the PSGs and leaves exactly the locked ones unchanged; a Z2Z_2Z2​ spin-liquid phase mapped onto itself is therefore locked and, if it spans a connected range of couplings, contains a Kitaev point. Unprojected mean-field theory in Kitaev's Majorana representation treats both signs alike. Exact second-order perturbation theory about the Kitaev points locates the asymmetry: the triangles respond identically for both signs, whereas on the bonds between triangles the two off-diagonal Heisenberg terms create the same four visons and interfere constructively for antiferromagnetic and almost destructively for ferromagnetic coupling, admixing a low-energy four-vison state 828282 times more strongly on the antiferromagnetic side. The PSG results rest on a classification of 352352352 Z2Z_2Z2​ PSGs with spin-orbit-coupled amplitudes, which also identifies the fermionic partners of the known Schwinger-boson spin liquids.
Projective symmetry groups beyond spin-1/21/21/2: Classifying integer-spin quantum spin liquids
Wen's projective symmetry group (PSG), introduced a quarter of a century ago, reduced the classification of symmetric quantum spin liquids to an algebraic problem and has guided the study of spin-1/21/21/2 models and materials ever since. Many current candidate materials and quantum simulators, however, have spin-111 or larger moments, and no comparable classification exists for them. We formulate the PSG for fermionic partons of arbitrary spin SSS. Integer and half-odd-integer spins already differ in their gauge structure. Half-odd-integer spins give the familiar SU(2)\mathrm{SU}(2)SU(2), U(1)\mathrm{U}(1)U(1), and Z2\mathbb{Z}_2Z2​ invariant gauge groups, while integer spins give a different hierarchy that includes U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​ and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​; the latter has no spin-1/21/21/2 counterpart. Three lattice-independent results follow for integer spin. Singlet pairing of the partons is odd under site exchange. A lattice symmetry can exchange the particle and hole representations of the local constraint, which forces symmetry-related sites into opposite constraint sectors. And a time-reversal-symmetric U(1)\mathrm{U}(1)U(1) state with the same constraint on every site can be gapped only if the number of sites per unit cell is a multiple of 2S+12S+12S+1. We apply the formalism to the spin-111 diamond lattice, realized in frustrated A-site spinels, and find ten fully symmetric Ansätze up to second neighbors. In the J1J_1J1​-J2J_2J2​ Heisenberg model at the mean-field level, the ground state depends on how the constraint is imposed. With the particle and hole representations, the lowest states are fully gapped U(1)\mathrm{U}(1)U(1) states whose two sublattices sit in opposite constraint sectors, and one of them is gauge equivalent to a paired Ansatz. The classification provides a starting point for integer-spin quantum spin liquids on other lattices and in other materials.
Quantum spin liquids are novel phases of matter whose physical properties cannot be captured within the conventional framework of phase transitions. These quantum states are characterized by excitations, called spinons, carrying a fractional value of the integer spin. A description based on a mean-field approximation leads to effective gauge theory. Due to this gauge freedom symmetry group operations act via projective symmetry group (PSG) representations, which can be used to classify quantum states beyond the conventional method. This introducing discussion is found in Chapters 2-6. A preceding PSG classification on the square lattice, which took spin rotational symmetry breaking terms into account, showed that the resulting mean-field states could exhibit topologically non-trivial spinon bands. In Chapter 7, only those el- ements of this Classification are studied, which lead to short-ranged mean-field models. The mutual effect of spinons and visons is scrutinized by adding static configurations of a background gauge field. Spinons and visons form composite ob- jects, which in the case of topological spinon bands, are described by Majorana zero modes. An exact mapping corroborates the relation to topological supercon- ductors and vortices. The effect of multiple visons is investigated, and a possible experimental signature is exposed. Chapter 8 explains how spinon mean-field theory can be used to derive a phe- nomenological effective model for the compound Ca 10 Cr 7 O 28 . A material that was recently proposed as a quantum spin liquid. This low-energy theory consists of spinon hopping processes. The underlying bilayer kagome structure can be simpli- fied to an effective honeycomb lattice. The dynamical spin structure factor is calcu- lated and compared to neutron scattering data. It is discussed how f -wave pairing of spinons can explain small deviations from a linear temperature dependency observed in heat capacity measurements at the low temperatures. This pairing mechanism breaks the U (1) invariance group of the pure hopping model and renders a coherent picture of Ca 10 Cr 7 O 28 being a Z 2 quantum spin liquid. Chapter 9 carries out a PSG classification for the simple-, body centered-, and face centered cubic lattices. Despite a large number of possible PSG representations, only very few describe short-ranged mean-field states. These mean-field models are constructed up to third neighbor couplings and calculated in a self-consistent manner. It is found that the energy is lower for non-trivial sign patterns induced by non-trivial representations of the symmetry generators. Higher neighbor extensions, including pairing terms, yield lower mean-field ground state energies compared to extensions with only hopping terms. For the fcc lattice, a first neighbor state could be identified, which exhibits symmetry protected zero-energy modes.
Due to strong geometric frustration and quantum fluctuation, the S=1/2S=1/2S=1/2 quantum Heisenberg antiferromagnet on the kagome\text{kagome}kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S=1/2,kagomeS=1/2,\text{kagome}S=1/2,kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2{\mathbb{Z}}_{2}Z2​ topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2{\mathbb{Z}}_{2}Z2​ SLs on the kagome\text{kagome}kagome lattice. Among them we found only one gapped Z2{\mathbb{Z}}_{2}Z2​ SL (which we call the ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117205 (2007)] to be the lowest among many other candidate U(1) SLs, including the uniform resonating-valence-bond states, we propose this ${\mathbb{Z}}_{2}[0,\ensuremath{\pi}]\ensuremath{\beta}$ state to be the numerically discovered SL ground state of the HKLM.
We revisit the fermionic parton approach to S=1/2S=1/2S=1/2 quantum spin liquids with SU(2)\mathrm{SU}(2)SU(2) spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with S=1S=1S=1 and S=0S=0S=0 Majorana fermion excitations coupled to a deconfined Z2{Z}_{2}Z2​ gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous SU(2)\mathrm{SU}(2)SU(2) spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting Z2{Z}_{2}Z2​ vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such S=1S=1S=1 Majorana spin liquids may occur in models and in real systems.
A projective symmetry group (PSG) has been regarded as a classification theory of spin liquids.However, it does not include a symmetry-protected topological order of fermionic spinon excitations, and thus the classification of gapped spin liquids is incomplete.We demonstrate the classification beyond PSG by utilizing the Kitaev model on the square-octagon lattice, where two gapped spin liquids are distinguished by a topological Z 2 invariant.This Z 2 invariant can be defined solely by the time-reversal and translation symmetries on condition that the time-reversal symmetry is implemented projectively.Thus, it is a hidden class of topological Kitaev spin liquids with helical edge states, which has been ignored for a long time.This suggests that there exists an unknown classification scheme of gapped spin liquids beyond PSG.