Projective Symmetry Group Research Articles

Deconfined criticalities and dualities between chiral spin liquid, topological superconductor and charge density wave Chern insulator

We propose bi-critical and tri-critical theories between chiral spin liquid (CSL), topological superconductor (SC) and charge density wave (CDW) ordered Chern insulator with Chern number C=2C=2 on square, triangular and Kagome lattices. The three CDW order parameters form a manifold of S^2S2 or S^1S1 depending on whether there is easy-plane anisotropy. The skyrmion defect of the CDW order carries physical charge 2e2e and its condensation leads to a topological superconductor. The CDW-SC transitions are in the same universality classes as the celebrated deconfined quantum critical points (DQCP) between Neel order and valence bond solid order on square lattice. Both SC and CDW order can be accessed from the CSL phase through a continuous phase transition. At the CSL-SC transition, there is still CDW order fluctuations although CDW is absent in both sides. We propose three different theories for the CSL-SC transition (and CSL to easy-plane CDW transition): a U(1)U(1) theory with two bosons, a U(1)U(1) theory with two Dirac fermions, and an SU(2)SU(2) theory with two bosons. Our construction offers a derivation of the duality between these three theories as well as a promising physical realization. The SU(2)SU(2) theory offers a unified framework for a series of fixed points with explicit SO(5), O(4)SO(5),O(4) or SO(3)× O(2)SO(3)×O(2) symmetry. There is also a transparent duality transformation mapping SC order to easy-plane CDW order. The CSL-SC-CDW tri-critical points are invariant under this duality mapping and have an enlarged SO(5)SO(5) or O(4)O(4) symmetry. The DQCPs between CDW and SC inherit the enlarged symmetry, emergent anomaly, and self-duality from the tri-critical point. Our analysis unifies the well-studied DQCP between symmetry breaking phases into a larger framework where they are proximate to a topologically ordered phase. Experimentally the theory demonstrates the possibility of a rich phase diagram and criticality through closing the Mott gap of a quantum spin liquid with projective symmetry group.

Projective symmetry group classification of Abrikosov fermion mean-field ansätze on the square-octagon lattice

We perform a projective symmetry group (PSG) classification of symmetric quantum spin liquids with different gauge groups on the square-octagon lattice. Employing the Abrikosov fermion representation for spin-$1/2$, we obtain $32$ $SU(2)$, $1808$ $U(1)$ and $384$ $\mathbb{Z}_{2}$ algebraic PSGs. Constraining ourselves to mean-field parton ans\"atze with short-range amplitudes, the classification reduces to a limited number, with 4 $SU(2)$, 24 $U(1)$ and 36 $\mathbb{Z}_{2}$, distinct phases. We discuss their ground state properties and spinon dispersions within a self-consistent treatment of the Heisenberg Hamiltonian with frustrating couplings.

Competing emergent Potts orders and possible nematic spin liquids in the kagome J1−J3 Heisenberg model

The extreme frustration in the kagome antiferromagnet makes it possible to realize a large number of closely competing magnetic phases either at the classical or the quantum level. Motivated by recent neutron scattering study on the kagome antiferromagnet vesignieite ${\mathrm{BaCu}}_{3}{\mathrm{V}}_{2}{\mathrm{O}}_{8}{(\mathrm{OH})}_{2}$, we have made systematic investigation of the phase diagram of the classical and quantum ${J}_{1}\ensuremath{-}{J}_{3}$ antiferromagnetic Heisenberg model defined on the kagome lattice (KAFHM). While it is shown previously that a large antiferromagnetic exchange between the third-neighboring spins can drive an emergent $q=4$ Potts order through the order-by-disorder mechanism, it is elusive what will happen in the region where the Luttinger-Tisza criteria fails to predict the classical ordering pattern. Through extensive Monte Carlo simulation, we find that such a region is characterized by competing emergent $q=3$ and $q=4$ Potts order, whose emergence are beyond the description of the conventional order-by-disorder mechanism. Our Schwinger boson mean-field calculation suggests the existence of the Potts-3 order in the ground state of quantum ${J}_{1}\ensuremath{-}{J}_{3}$ KAFHM in such an extremely frustrated region. The predicted Potts-3 state is found to take two different forms distinguished by their projective symmetry group character (PSG or quantum order), although they have exactly the same symmetry. The transition between these two quantum orders is found to occur at a ${J}_{3}$ value intriguingly close to the tricritical point in the Potts-3 phase of the classical phase diagram of the model. A comparison with the exact diagonalization result on a 36-site cluster shows that the intermediate region may indeed host a nematic spin liquid ground state featuring an anisotropic ring structure around the $\mathbf{q}=0$ point in its spinon dispersion relation. We find that such an extremely frustrated region should better be taken as the playground to study the rich competition between exotic emergent phases, rather than a burden on their theoretical analysis.

Elastically induced magnetization at ultrafast time scales in a chiral helimagnet

The extreme frustration in the Kagome antiferromagnet makes it possible to realize a large number of closely competing magnetic phases either at the classical or the quantum level. Motivated by recent neutron scattering study on the Kagome antiferromagnet vesignieite BaCu$_{3}$V$_{2}$O$_{8}$(OH)$_{2}$, we have made systematic investigation of the phase diagram of the classical and quantum $J_{1}-J_{3}$ antiferromagnetic Heisenberg model defined on the Kagome lattice(KAFHM). While it is shown previously that a large antiferromagnetic exchange between the third-neighboring spins can drive an emergent $q=4$ Potts order through the order-by-disorder mechanism, it is elusive what will happen in the so called "grey region" where the Luttinger-Tisza criteria fails to predict the classical ordering pattern. Through extensive Monte Carlo simulation, we find that the "grey region" is characterized by competing emergent $q=3$ and $q=4$ Potts order, whose emergence are beyond the description of the conventional order-by-disorder mechanism. Our Schwinger Boson mean field calculation in the "grey region" confirms the existence of the Potts-3 order in the ground state of quantum $J_{1}-J_{3}$ KAFHM. The predicted Potts-3 state is found to take two different forms distinguished by their projective symmetry group character(PSG or quantum order), although they have exactly the same symmetry. A comparison with the exact diagonalization result on a 36-site cluster shows that the "grey region" may indeed host a nematic spin liquid ground state featuring an anisotropic ring structure around the $\mathbf{q}=0$ point in its spinon dispersion relation. We find that such "grey region" of extremely frustrated magnets should better be taken as the playground to study the rich competition between exotic emergent phases, rather than a burden on their theoretical analysis.

Projective symmetry group classification of chiral Z2 spin liquids on the pyrochlore lattice: Application to the spin- 12 XXZ Heisenberg model

We give a complete classification of fully symmetric as well as chiral $\mathbb{Z}_2$ quantum spin liquids on the pyrochlore lattice using a projective symmetry group analysis of Schwinger boson mean-field states. We find 50 independent ans\"atze, including the 12 fully symmetric nearest-neighbor $\mathbb{Z}_2$ spin liquids that have been classified by Liu et al. [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.075125]. For each class we specify the most general symmetry-allowed mean-field Hamiltonian. Additionally, we test the properties of a subset of the spin liquid ans\"atze by solving the mean-field equations for the spin-$1/2$ XXZ model near the antiferromagnetic Heisenberg point. We find four chiral spin liquids that break the screw symmetry of the lattice modulo time reversal symmetry. These states have a different symmetry than the previously studied monopole flux state and their unique characteristic is a $\frac{\pi}{3}$ flux enclosed by every rhombus of the lattice.

Competing U (1) and Z2 dipolar-octupolar quantum spin liquids on the pyrochlore lattice: Application to Ce2Zr2O7

Recent experiments on the dipolar-octupolar pyrochlore compound ${\mathrm{Ce}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$ indicate that it may realize a three-dimensional quantum spin liquid (QSL). In particular, the analyses of available data suggest that the system is in a region of parameter space, where the so-called $\ensuremath{\pi}$-flux $U(1)$ octupolar quantum spin ice (QSI) with an emergent photon could be realized. However, because the system is far from the perturbative classical spin ice regime, it is unclear whether the quantum ground state is primarily a coherent superposition of 2-in-2-out configurations as in the canonical QSI phases. In this work, we explore other possible competing quantum spin-liquid states beyond QSI using the Schwinger boson parton construction of the dipolar-octupolar pseudospin-1/2 model. After classifying all symmetric $U(1)$ and ${\mathbb{Z}}_{2}$ QSLs using the projective symmetry group (PSG), we construct a mean-field phase diagram that possesses a number of important features observed in an earlier exact diagonalization study. In the experimentally relevant frustrated region of the phase diagram, we find two closely competing gapped ${\mathbb{Z}}_{2}$ QSLs with a narrow spinon dispersion. The equal-time and dynamical spin structure factors of these states show key features similar to what was reported in neutron scattering experiments. Hence, these QSLs are closely competing ground states in addition to the $\ensuremath{\pi}$-flux $U(1)$ QSI, and they should be taken into account on equal footing for the interpretation of experiments on ${\mathrm{Ce}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$.

Nodeless superconductivity in cuprates with Ba$$_2$$CuO$$_3$$-type structure

In this work zero-temperature phase diagrams of cuprates with Ba$_2$CuO$_3$-type CuO chain structure is investigated. The projective symmetry group analysis is employed in the strong coupling limit, and renormalization group with bosonization analysis is employed in the weak coupling limit. We find that in both of these two limits, large areas of the phase diagrams are filled with nodeless $s$-wave superconducting phases (with weak $d$-wave components), instead of pure $d$-wave phase mostly found in cuprates. This implies that nodeless $s$-wave phase is the dominant superconducting phase in cuprates with Ba$_2$CuO$_3$-type CuO chain structure in low temperature. Other phases are also found, including $(s+d)$-wave superconducting phases and Luttinger liquid phases.

Pseudoscalar U(1) spin liquids in α−RuCl3

A recent experiment has reported oscillations of the thermal conductivity of $\alpha$-RuCl$_3$ driven by an in-plane magnetic field that are reminiscent of the quantum oscillations in metals. At first glance, these observations are consistent with the presence of the long-sought-after spinon Fermi surface state. Strikingly, however, the experiment also reported vanishing thermal Hall conductivity coexisting with the oscillations of the longitudinal one. Such absence of the thermal Hall effect must originate from crystalline symmetries of $\alpha$-RuCl$_3$. But if the system was a traditional spinon fermi surface state, these symmetries would also necessarily prohibit the emergence of a magnetic field acting on the spinons, in stark contradiction with the presence of quantum oscillations in experiments. To reconcile these observations, we introduce a new class of symmetry enriched ``pseudoscalar" U(1) spin liquids in which certain crystalline symmetries act as a particle-hole conjugation on the spinons. The associated pseudoscalar spinon Fermi surface states allow for the coexistence of an emergent Landau quantizing magnetic field while having an exactly zero thermal Hall conductivity. We develop a general theory of these states by constructing Gutzwiller-projected wave-functions and describing how they naturally appear as U(1) spin liquids with a distinctive projective symmetry group implementation of crystalline symmetries in the fermionic parton representation of spins. We propose that the field induced quantum disordered state in $\alpha$-RuCl$_3$ descends from a pseudoscalar spinon fermi surface state that features compensated spinon-particle and spinon-hole pockets possibly located around the $M$ points of its honeycomb Brillouin zone. These points are connected via a wave-vector associated with the emergence of the competing zig-zag antiferromagnetic state.

Theoretical study of quantum spin liquids in S=12 hyper-hyperkagome magnets: Classification, heat capacity, and dynamical spin structure factor

Recent experiments suggest a quantum spin liquid ground state in the material PbCuTe$_2$O$_6$, where $S=1/2$ moments are coupled by antiferromagnetic Heisenberg interactions into a three dimensional structure of corner sharing triangles dubbed the hyper-hyperkagome lattice. It exhibits a richer connectivity, and thus likely a stronger geometric frustration, than the relatively well studied hyperkagome lattice. Here, we investigate the possible quantum spin liquids in the $S=1/2$ hyper-hyperkagome magnet using the complex fermion mean field theory. Extending the results of a previous projective symmetry group analysis, we identify only two $\mathbb{Z}_2$ spin liquids and a $U(1)$ spin liquid that are compatible with the hyper-hyperkagome structure. The $U(1)$ spin liquid has a spinon Fermi surface. For the $\mathbb{Z}_2$ spin liquids, one has a small excitation gap, while the other is gapless and proximate to the $U(1)$ spin liquid. We show that the gapped and gapless spin liquids can in principle be distinguished by heat capacity measurements. Moreover, we calculate the dynamical spin structure factors of all three spin liquids and find that they highly resemble the inelastic neutron scattering spectra of PbCuTe$_2$O$_6$. Implications of our work to the experiments, as well as its relations to the existing theoretical studies, are discussed.

Frustrated Kondo impurity triangle: A simple model of deconfinement

The concepts of deconfinement and topological order are of great current interest for quantum information science and for our understanding of quantum materials. Here, we introduce a simple model of three antiferromagnetically coupled Kondo impurities, a Kondo triangle, which can be used to further extend the application of these concepts to electronic systems. We show that, by tuning the magnetic frustration, the Kondo triangle undergoes a quantum phase transition between two phases of unbroken symmetry, signaling a phase transition beyond the Landau paradigm. We demonstrate that the frustrated spin liquid phase is described by a three-channel Kondo (3CK) fixed point and thus displays an irrational ground state degeneracy. Using an Abrikosov pseudofermion representation, this quantum state is categorized by an emergent U(1) gauge field and its projective symmetry group. The gauge theory is deconfining in the sense that a miniature Wilson loop orders and topological defects (instantons in the gauge field) are expelled. This phase persists in the presence of moderate Kondo screening until proliferation of topological defects leads to a quantum phase transition to an unfrustrated Fermi liquid phase. Based on this evidence, we propose that the 3CK phase displays topological order in a similar sense as gapless spin liquids.

Symmetric U(1) and Z2 spin liquids on the pyrochlore lattice

The geometrically frustrated 3D pyrochlore lattice has been long predicted to host a quantum spin liquid, an intrinsic long-range entangled state with fractionalized excitations. To date, most proposals for pyrochlore materials have focused on quantum spin ice, a $\mathrm{U(1)}$ quantum spin liquid whose only low energy excitations are emergent photons of Maxwell type. In this work, we explore the possibility of finding pyrochlore quantum spin liquids whose low energy theories go beyond this standard one. We give a complete classification of symmetric $\mathrm{U(1)}$ and $\mathbb{Z}_2$ spin liquids on the pyrochlore lattice within the projective symmetry group framework for fermionic spinons. We find 18 $\mathrm{U(1)}$ spin liquids and 28 $\mathbb{Z}_2$ spin liquids that preserve pyrochlore space group symmetry while, upon further imposing time reversal symmetry, the numbers of classes become 16 and 48, respectively. For each class, the most general symmetry-allowed spinon mean-field Hamiltonian is given. Interestingly, we find that several $\mathrm{U(1)}$ spin liquid classes possess an unusual gapless multi-nodal-line structure ("nodal star") in the spinon bands, which is protected by the projective actions of the three-fold rotation and screw symmetries of the pyrochlore space group. Through a simple model, we study the effect of gauge fluctuations on such a nodal star spin liquid and propose that the leading terms in the low temperature specific heat have the scaling form $C/T\sim \sqrt{T}+\sqrt{T}/\ln T$, in contrast to the form $C/T \sim T^2$ of the standard $\mathrm{U(1)}$ pyrochlore spin liquid with gapped spinons.

Topological Z2 invariant in Kitaev spin liquids: Classification of gapped spin liquids beyond projective symmetry group

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 squareoctagon 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.

Multinode quantum spin liquids on the honeycomb lattice

Recently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the $\Gamma'$ terms. To fully understand the effect of the $\Gamma'$ interactions, we investigate the quantum $K$-$\Gamma$-$\Gamma'$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z$_2$ quantum spin liquids (QSLs) are found at $\Gamma'>0$, one of which is the previously found proximate Kitaev spin liquid called the PKSL14 state which shares the same projective symmetry group (PSG) with the Kitaev spin liquid. A remarkable result is that a $\pi$-flux state with a distinct PSG appears at larger $\Gamma'$. The $\pi$-flux state is characterized by an enhanced periodic structure in the spinon dispersion in the original Brillouin zone (BZ), which is experimentally observable. Interestingly, two PKSL8 states are competing with the $\pi$-flux state and one of them can be stabilized by six-spin ring-exchange interactions. The physical properties of these nodal QSLs are studied by applying magnetic fields and the results depend on the number of cones. Our study infers that there exist a family of zero-flux QSLs that contain $6n+2, n\in\mathbb Z$ Majorana cones and a family of $\pi$-flux QSLs containing $4(6n+2)$ cones in the original BZ. It provides guidelines for experimental realization of non-Kitaev QSLs in relevant materials.

Projective symmetry group classifications of quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices

We perform extensive classifications of $\mathbb{Z}_2$ quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices using a spin-rotation invariant fermionic projective symmetry group approach. Taking into account that all three lattices share the same point group $O_h$, we apply an efficient gauge where the classification for the simple cubic lattice can be partially carried over to the other two lattices. We identify hundreds of projective representations for each of the three lattices, however, when constructing short-range mean-field models for the fermionic partons (spinons) these phases collapse to only very few relevant cases. We self-consistently calculate the corresponding mean-field parameters for frustrated Heisenberg models on all three lattices with up to third neighbor spin interactions and discuss the spinon dispersions, ground state energies and dynamical spin structure factors. Our results indicate that phases with non-uniform spinon hopping or pairing amplitudes are energetically favored. An unusual situation is identified for the fcc lattice where the spinon dispersion minimizing the mean-field energy features a network of symmetry protected line-like zero modes in reciprocal space. We further discuss characteristic fingerprints of these phases in the dynamical spin structure factor which may help to identify and distinguish them in future numerical or experimental studies.

Symmetry-protected gapless spin liquids on the strained honeycomb lattice

By including a material-relevant off-diagonal interaction called the $\Gamma$ term into the Kitaev model and introducing spatial anisotropy in the interaction strength on the honeycomb lattice, we obtain a series of nodal Z$_2$ quantum spin liquids (QSLs) from parton approach. These QSLs share the same projective symmetry group and are characterized by certain numbers of symmetry-protected Majorana cones in their low-energy excitation spectrum. We illustrate that the physical properties of the QSLs are dependent on the information of the cones. Using the $\pmb k\cdot\pmb p$ method, we analyze the chirality of every cone with respect to mass generating perturbations. Especially, for an applied external magnetic field, we provide the maximum-mass field-orientation for every cone. Thus, for arbitrarily oriented weak magnetic fields, we can immediately read out the Chern number of the system and the properties of the resultant chiral spin liquids. The new gapless QSLs predicted in our phase diagrams are promising to be realized experimentally by exerting uniaxial pressure to tune the anisotropy of the interaction strength. We further show that all these QSLs can be distinguished by measurable quantities. Based on the study of these QSL phases, we conclude that a complete classification of nodal QSLs with certain symmetry should include not only the projective symmetry groups but also the information of the cones, {\it i.e.}, their total number, their chiralities, and the way in which they are symmetry-related.

Projective symmetry group analysis of inelastic light scattering in Kitaev spin balls

Projective symmetry groups are applied to Raman observations of the Kitaev quantum spin liquids in spherical lattice geometries realized by Platonic and Archimedean polyhedra. Parton single excitations in Kitaev spin polyhedra are characterized by double-valued irreducible representations of their belonging projective symmetry groups, whereas parton geminate excitations relevant to Raman scattering are decomposed into single-valued irreducible representations of the corresponding point symmetry groups. We combine a standard point-symmetry-group analysis of the Loudon-Fleury vertices and an elaborate projective-symmetry-group analysis of itinerant spinons against the ground gauge fields to reveal $hidden$ $selection$ $rules$ for Raman scattering in $\mathbb{Z}_2$ spin liquids.

Symmetric spin liquids on the stuffed honeycomb lattice

We use a projective symmetry group analysis to determine all symmetric spin liquids on the stuffed honeycomb lattice Heisenberg model. This lattice interpolates between honeycomb, triangular and dice lattices, always preserving hexagonal symmetry, and it already has one spin liquid candidate, TbInO$_3$, albeit with strong spin-orbit coupling not considered here. In addition to the stuffed honeycomb lattice itself, we gain valuable insight into potential spin liquids on the honeycomb and triangular lattices, as well as how they might be connected. For example, the sublattice pairing state proposed on the honeycomb lattice connects to the uniform spinon Fermi surface that may be relevant for the triangular lattice with ring exchange, while there are no spin liquids competitive on both the $J_1-J_2$ honeycomb and triangular lattice limits. In particular, we find three stuffed honeycomb descendants of the U(1) Dirac spin liquid widely believed to be found on the $J_1-J_2$ triangular lattice. We also discuss how spin liquids near the honeycomb limit can potentially explain the physics of LiZn$_2$Mo$_3$O$_8$.

Classical and quantum order in hyperkagome antiferromagnets

Motivated by recent experiments and density functional theory calculations on choloalite PbCuTe$_2$O$_6$, which possesses a Cu-based three-dimensional hyperkagome lattice, we propose and study a $J_1$-$J_2$-$J_3$ antiferromagnetic Heisenberg model on a hyperkagome lattice. In the classical limit, possible ground states are analyzed by two triangle rules, i.e., the "hyperkagome triangle rule" and the "isolated triangle rule," and classical Monte Carlo simulations are exploited to identify possible classical magnetic ordering and explore the phase diagram. In the quantum regime, Schwinger boson theory is applied to study possible quantum spin liquid states and long-range magnetically ordered states on an equal footing. These quantum states with bosonic partons are classified and analyzed by using projective symmetry groups (PSGs). It is found that there are only four types of algebraic PSGs allowed by the space group $P4_{1}32$ on a hyperkagome lattice. Moreover, there are only two types of PSGs that are compatible with the $J_1$-$J_2$-$J_3$ Heisenberg model. These two types of $Z_2$ bosonic states are distinguished by the gauge-invariant flux on the elementary ten-site loops on the hyperkagome network, called zero-flux state and $\pi$-flux state respectively. Both the zero-flux state and the $\pi$-flux state are able to give rise to quantum spin liquid states as well as magnetically ordered states, and the zero-flux states and the $\pi$-flux states can be distinguished by the lower and upper edges of the spectral function $S(\bm{q},\omega)$, which can be measured by inelastic neutron scattering experiments.

Competing orders in pyrochlore magnets from a Z2 spin liquid perspective

The pyrochlore materials have long been predicted to harbor a quantum spin liquid, an intrinsic long-range-entangled state supporting fractionalized excitations. Existing pyrochlore experiments, on the other hand, have discovered several weakly ordered states and a tendency of close competition amongst them. Motivated by these facts, we give a complete classification of spin-orbit-coupled ${\mathbb{Z}}_{2}$ spin-liquid states on the pyrochlore lattice by using the projective symmetry group (PSG) approach for bosonic spinons. For each spin liquid, we construct a mean-field Hamiltonian that can be used to describe phase transitions out of the spin liquid via spinon condensation. Studying these phase transitions, we establish phase diagrams for our mean-field Hamiltonians that link magnetic orders to specific spin liquids. In general, we find that seemingly unrelated magnetic orders are intertwined with each other and that the conventional spin orders seen in the experiments are accompanied by more exotic hidden orders. Our critical theories are categorized into $z=1$ and $z=2$ types, based on their spinon dispersion and Hamiltonian diagonalizability, and are shown to give distinct signatures in the heat capacity and the spin structure factor. This study provides a clear map of pyrochlore phases for future experiments and variational Monte Carlo studies in pyrochlore materials.

Non-Kitaev spin liquids in Kitaev materials

We point out that the Kitaev materials may not necessarily support Kitaev spin liquid. It is well-known that having a Kitaev term in the spin interaction is not the sufficient condition for the Kitaev spin liquid ground state. Many other spin liquids may be stabilized by the competing spin interactions of the systems. We thus explore the possibilities of non-Kitaev spin liquids in the honeycomb Kitaev materials. We carry out a systematic classification of gapped $\mathbb{Z}_2$ spin liquids using the Schwinger boson representation for the spin variables. The presence of strong spin-orbit coupling in the Kitaev materials brings new ingredients into the projective symmetry group classification of the non-Kitaev spin liquid. We predict the spectroscopic properties of these gapped non-Kitaev spin liquids. Moreover, among the gapped spin liquids that we discover, we identify the spin liquid whose spinon condensation leads to the magnetic Bragg peak structure of the zig-zag magnetic order that was observed in Na$_2$IrO$_3$ and $\alpha$-RuCl$_3$. We further discuss the possibility of gapped $\mathbb{Z}_2$ spin liquid and the deconfined quantum criticality from the zig-zag magnetic order to spin dimerization in pressurized $\alpha$-RuCl$_3$.