Biphenylene is a new topological material that has attracted much attention recently.By amplifying its size of unit cell,we construct a series of planar structures as homogeneous carbon allotropes in the form of polyp...Biphenylene is a new topological material that has attracted much attention recently.By amplifying its size of unit cell,we construct a series of planar structures as homogeneous carbon allotropes in the form of polyphenylene networks.We first use the low-energy effective model to prove the topological three periodicity for these allotropes.Then,through first-principles calculations,we show that the topological phase has the Dirac point.As the size of per unit cell increases,the influence of the quaternary rings decreases,leading to a reduction in the anisotropy of the system,and the Dirac cone undergoes a transition from type II to type I.We confirm that there are two kinds of non-trivial topological phases with gapless and gapped bulk dispersion.Furthermore,we add a built-in electric field to the gapless system by doping with B and N atoms,which opens a gap for the bulk dispersion.Finally,by manipulating the built-in electric field,the dispersion relations of the edge modes will be transformed into a linear type.These findings provide a hopeful approach for designing the topological carbon-based materials with controllable properties of edge states.展开更多
Here the notion of geometric phase acquired by an electron in a one-dimensional periodic lattice as it traverses the Bloch band is carefully studied. Such a geometric phase is useful in characterizing the topological ...Here the notion of geometric phase acquired by an electron in a one-dimensional periodic lattice as it traverses the Bloch band is carefully studied. Such a geometric phase is useful in characterizing the topological properties and the electric polarization of the periodic system. An expression for this geometric phase was first provided by Zak, in a celebrated work three decades ago. Unfortunately, Zak’s expression suffers from two shortcomings: its value depends upon the choice of origin of the unit cell, and is gauge dependent. Upon careful investigation of the time evolution of the system, here we find that the system displays cyclicity in a generalized sense wherein the physical observables return in the course of evolution, rather than the density matrix. Recognition of this generalized cyclicity paves the way for a correct and consistent expression for the geometric phase in this system, christened as Pancharatnam-Zak phase. Pancharatnam-Zak geometric phase does not suffer from the shortcomings of Zak’s expression, and correctly classifies the Bloch bands of the lattice. A naturally filled band extension of the Pancharatnam-Zak phase is also constructed and studied.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos.12074156 and 12164023)the Yunnan Local College Applied Basic Research Projects (Grant No.2021Y710)。
文摘Biphenylene is a new topological material that has attracted much attention recently.By amplifying its size of unit cell,we construct a series of planar structures as homogeneous carbon allotropes in the form of polyphenylene networks.We first use the low-energy effective model to prove the topological three periodicity for these allotropes.Then,through first-principles calculations,we show that the topological phase has the Dirac point.As the size of per unit cell increases,the influence of the quaternary rings decreases,leading to a reduction in the anisotropy of the system,and the Dirac cone undergoes a transition from type II to type I.We confirm that there are two kinds of non-trivial topological phases with gapless and gapped bulk dispersion.Furthermore,we add a built-in electric field to the gapless system by doping with B and N atoms,which opens a gap for the bulk dispersion.Finally,by manipulating the built-in electric field,the dispersion relations of the edge modes will be transformed into a linear type.These findings provide a hopeful approach for designing the topological carbon-based materials with controllable properties of edge states.
文摘Here the notion of geometric phase acquired by an electron in a one-dimensional periodic lattice as it traverses the Bloch band is carefully studied. Such a geometric phase is useful in characterizing the topological properties and the electric polarization of the periodic system. An expression for this geometric phase was first provided by Zak, in a celebrated work three decades ago. Unfortunately, Zak’s expression suffers from two shortcomings: its value depends upon the choice of origin of the unit cell, and is gauge dependent. Upon careful investigation of the time evolution of the system, here we find that the system displays cyclicity in a generalized sense wherein the physical observables return in the course of evolution, rather than the density matrix. Recognition of this generalized cyclicity paves the way for a correct and consistent expression for the geometric phase in this system, christened as Pancharatnam-Zak phase. Pancharatnam-Zak geometric phase does not suffer from the shortcomings of Zak’s expression, and correctly classifies the Bloch bands of the lattice. A naturally filled band extension of the Pancharatnam-Zak phase is also constructed and studied.
