In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the n...In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the number of(protected)lonely nodes.Explicit formulae as well as generating functions are obtained.We also find that,as n goes to infinity,the proportion of protected branch nodes and protected lonely nodes among all vertices of Motzkin trees with n edges approaches 4/27 and 2/9.展开更多
Node line semimetals(NLSMs) were characterized by one-dimensional band crossings in their bulk electronic structures.The nontrivial band topology of NLSM gives rise to "drumhead" surface electronic excitatio...Node line semimetals(NLSMs) were characterized by one-dimensional band crossings in their bulk electronic structures.The nontrivial band topology of NLSM gives rise to "drumhead" surface electronic excitations that exhibits exotic physical properties.The symmetries of crystalline provide the needed protection of node line from being gapped out by the perturbations that preserve the symmetry.The progress of NLSM in tungsten-based materials is reviewed with an emphasis on their symmetry-based protection,characteristic electronic band structures and their response to the spin-orbit coupling(SOC)and breaking of time-reversal symmetry.The potential exploration directions of tungsten-based NLSM in the future are also discussed.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11861045)Gansu Province Science Foundation for Youths(Grant No.20JR10RA187)the Hongliu Foundation of First-Class Disciplines of Lanzhou University of Technology,China。
文摘In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the number of(protected)lonely nodes.Explicit formulae as well as generating functions are obtained.We also find that,as n goes to infinity,the proportion of protected branch nodes and protected lonely nodes among all vertices of Motzkin trees with n edges approaches 4/27 and 2/9.
基金supported by the National Natural Foundation of China (NFSC)(Grants No.11574215)。
文摘Node line semimetals(NLSMs) were characterized by one-dimensional band crossings in their bulk electronic structures.The nontrivial band topology of NLSM gives rise to "drumhead" surface electronic excitations that exhibits exotic physical properties.The symmetries of crystalline provide the needed protection of node line from being gapped out by the perturbations that preserve the symmetry.The progress of NLSM in tungsten-based materials is reviewed with an emphasis on their symmetry-based protection,characteristic electronic band structures and their response to the spin-orbit coupling(SOC)and breaking of time-reversal symmetry.The potential exploration directions of tungsten-based NLSM in the future are also discussed.
