Nodal Brillouin-zone boundary from folding a Chern insulator

Abstract

Chern insulator is a building block of many topological quantum matters, ranging from quantum spin Hall insulators to fractional Chern insulators. Here, we discuss a new type of insulator, which consists of two half filled ordinary Chern insulators. On the one hand, the bulk energy spectrum is obtained from folding that of either Chern insulator. Such folding gives rise to a nodal boundary of the Brillouin zone, at which the band crossing is protected by the symmetries of the two-dimensional lattice that is invariant under combined transformations in the spatial and the spin space. It also provides one a natural platform to explore the non-abelian Berry curvature and the resultant quantum phenomena. On the other hand, these two underlying Chern insulators are distinguished from each other by nonsymmorphic operators, which lead to intriguing properties absent in conventional Chern insulators. A new degree of freedom, the parity of the nonsymmorphic symmetry, needs to be introduced for describing the topological pumping, if the edge respects the nonsymmorphic symmetry .In the band structure of a crystal, if different bands are separated from each other by finite band gaps, a Chern number [1], which is the integral of the abelian Berry curvature in the Brillouin zone (BZ), can be assigned to each individual band. A Chern insulator[2] arises if filling electrons to these bands leads to a finite total Chern number. Such Chern insulators are fundamental elements of a wide range of topological quantum matters [3][4][5]. For instance, one may obtain quantum spin Hall insulators [6][7][8] by assembling two insulators with both opposite spins and Chern numbers. Introducing interactions, fractional Chern insulators may emerge, in analogy to fractional quantum Hall states [9][10][11][12][13].The study of topological matters using ultracold atoms has been growing fast in the past a few years [14,15]. Using highly controllable atomic samples, a number of fundamentally important theoretical models have been realised, such as the Harper-Hofstadter model with a large magnetic flux per unit cell [16,17] and the topological Haldane model [18]. Meanwhile, many topological quantum quantities or phenomena, which are difficult to trace in solids, have been directly observed. For instance, both Zak phase and Chern numbers have been directly measured in optical lattices [19,20]. Topological charge pumping has also been realised recently [21,22]. Whereas most of the current studies have been focusing on abelian topological matters, it is promising that ultracold atoms may provides physicists a platform to explore non-abelian topological matters, as well as new quantum states and phenomena that have not been studied in the literature.In this Article, we study a new type of topological matter, whose bulk energy spectrum consists of two half filled ordinary Chern insulators. The band structure can be obtained from folding of that of either Chern insulator, which has BZ doubling the one of the realistic system. As a result of the folding, en...

Gain/loss effects on spin-orbit coupled ultracold atoms in two-dimensional optical lattices

“…It is shown [39] that Hermitian cases of Ĥ support nodal loops between lower/upper two bands when ∆ = 0, which is protected by nonsymmorphic symmetries; a finite ∆ breaks the symmetries and thus the nodal loops. The proof conducted in the Hermitian context, however, is invalid for non-Hermitian cases due to the complexity of bands.…”

“…Moreover, because the protection of nodal loops by nonsymmorphic symmetries found in Ref. [39] breaks down for complex bands, the presence of a purely imaginary staggered potential, which breaks the symmetries, can induce exceptional loops from Hermitian ones between lower/upper two bands; it motivates us to develop a non-Hermitian Wilson-line method for numerically calculat-ing the non-Hermitian Chern number for a subspace consisting of multiple complex bands, and find that only with dual left/right eigenvectors is the Chern number gauge independent. This method can be regarded as a non-Hermitian generalization of the non-Abelian scheme for calculating Chern numbers in Hermitian systems [40].…”

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