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arXiv:1912.06167 · Physical review. B./Physical review. B

Symmetry-protected gapless spin liquids on the strained honeycomb lattice

Jiucai Wang, Zheng-Xin Liu

Published Sep 10, 2020• 13 citations
arXivHTMLPDF

Abstract

By including a material-relevant off-diagonal interaction called the $\mathrm{\ensuremath{\Gamma}}$ 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 the 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 $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ 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, i.e., their total number, their chiralities, and the way in which they are symmetry related.

DOI
10.1103/physrevb.102.094416

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The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.

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By including a material-relevant off-diagonal interaction called the \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ term into the Kitaev model and introducing spatial anisotropy in the interaction strength on the honeycomb lattice, we obtain a series of nodal Z2{Z}_{2}Z2​ quantum spin liquids (QSLs) from the 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 \mathbitk\ifmmode⋅\else\textperiodcentered\fi\mathbitp\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}\mathbitk\ifmmode⋅\else\textperiodcentered\fi\mathbitp 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, i.e., their total number, their chiralities, and the way in which they are symmetry related.
Exerting high pressure is a practical way to tune the spin-spin interactions in Mott insulators and can be applied to prepare for quantum spin liquid (QSL) ground states. We illustrate this possibility by studying a realistic spin model which contains the Kitaev and Γ\GammaΓ-interactions through extensive analytic and numerical calculations. As a consequence of the external pressure, the interaction intensities in our model are spatially anisotropic. While the anisotropic pure Kitaev model contains only two phases, the presence of non-Kitaev interactions results in a much richer phase diagram which contains several new gapless Z2Z_2Z2​ QSL phases. These QSLs are characterized by a certain number of cones in the low-energy excitation spectrum, which can be reflected in the dynamic structure factors. We provide a complete classification of nodal Z2Z_2Z2​ QSLs. Furthermore, a small out-of-plane magnetic field gaps out the cones in these gapless QSLs and results in Abelian or non-Abelian chiral spin liquids (CSLs), all belonging to the Kitaev's 16-fold classification. From the dynamic structure factor and the thermal Hall conductance of the field-induced CSLs, all the QSLs can be experimentally distinguished. Our results are instructive to search for QSL phases in related materials.
Variational Monte Carlo (VMC) studies of extended Kitaev honeycomb models reveal a series of multinode quantum spin liquids (QSLs). They have an emergent Z 2 gauge structure, a discrete number of symmetry-protected Majorana cones, and form Abelian or non-Abelian chiral spin liquids in applied magnetic fields. The large number of Z 2 projective symmetry groups for spin–orbit-coupled states on the honeycomb lattice suggests that multinode QSLs can be found experimentally in future work on proximate Kitaev materials.
Recently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ terms. To fully understand the effect of the ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ interactions, we investigate the quantum $K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z2{Z}_{2}Z2​ quantum spin liquids (QSLs) are found at ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}>0$, 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 $\ensuremath{\pi}$-flux state with a distinct PSG appears at larger ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$. The $\ensuremath{\pi}$-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 $\ensuremath{\pi}$-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 $6n+2,n\ensuremath{\in}\mathbb{Z}$ Majorana cones and a family of $\ensuremath{\pi}$-flux QSLs containing 4(6n+2)4(6n+2)4(6n+2) cones in the original BZ. It provides guidelines for experimental realization of non-Kitaev QSLs in relevant materials.
In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Γ) couplings. We investigate the quantum (S=1/2) K-J-Γ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z_{2} state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.
Abstract The S=3/2 Kitaev honeycomb model (KHM) is a quantum spin liquid (QSL) state coupled to a static Z2 gauge field. Employing an SO(6) Majorana representation of spin3/2’s, we find an exact representation of the conserved plaquette fluxes in terms of static Z2 gauge fields akin to the S=1/2 KHM which enables us to treat the remaining interacting matter fermion sector in a parton mean-field theory. We uncover a ground-state phase diagram consisting of gapped and gapless QSLs. Our parton description is in quantitative agreement with numerical simulations, and is furthermore corroborated by the addition of a [001] single ion anisotropy (SIA) which continuously connects the gapless Dirac QSL of our model with that of the S=1/2 KHM. In the presence of a weak [111] SIA, we discuss an emergent chiral QSL within a perturbation theory.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.