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.