Topological phase transitions and flat bands on an islamic lattice.

We construct an islamic lattice by considering the nearest-neighbor (NN) hoppings with staggered magnetic fluxes and the next-NN hoppings. This model supports abundant quantum phases for various values of filling fractions. At1/4filling, Chern insulator (CI) phases with Chern numbersC=±1, -2and a zero-Chern-number topological insulator (ZCNTI) phase exist. At3/8filling, several CI phases with Chern numbersC=±1, 3and the ZCNTI phase are obtained. For the filling fraction 3/4, CI phases with Chern numbersC=±1, 2and two ZCNTI phase areas appear. Interestingly, these ZCNTI phases host both robust corner states and gapless edge states which can be characterized by the quantized polarization and quadrupole moment. We further find that staggered magnetic fluxes can give rise to the ZCNTI state at1/4and3/4fillings. Phase diagrams for filling fractions1/8,1/2,5/8and7/8are presented as well. In addition, flat bands are obtained for various filling fractions by tuning the hopping parameters. At 1/8 filling, a best topological flat band (TFB) with flatness ratio about 12 appears. Several trivial flat bands but with total Chern number|C|=1emerge in this model and exactly flat band is found at 3/8 filling. We further investigateν=1/2fractional Chern insulate state when hard-core bosons fill into this TFB model.

Topological quantum phase transitions on the breathing kagomé lattice.

We explore extensively topological quantum phase transitions (TQPTs) of the breathing kagomé lattice model in the presence of staggered fluxes. We obtain rich topological phases, including the Chern insulator (CI) and the second-order topological insulator (SOTI) phases, by tuning the dimerized hopping parametert1' and the staggered-flux parameterϕ. The CI phases can be identified on the basis of the chiral edge states and the non-zero Chern numbers. However, in sharp contrast to the CI phases, the SOTI phases are characterized by the robust corner states and the quantized polarizations. In addition, we explore the TQPTs considering the next-nearest-neighbor hopping parametert2. We demonstrate the existence of two-dimensional SOTIs with broken time-reversal symmetry and reveal the TQPTs between the CIs and the SOTIs.

Cn-symmetric Chern insulators.

Chern insulators (CIs) have attracted great interests for the realization of quantum Hall states without external magnetic field. Recently, CIs have been studied on various curved lattices, such as the cone-like lattices and the fullerenes. However, few works were reported how to identify curved-CIs and explore their topological phase transitions (TPTs). In this paper, we systemically investigate the curved-CIs with arbitraryn-fold rotational symmetry on cone-like and saddle-like lattices (also dubbed asCn-symmetric CIs), by 'cutting and gluing' unit sectors with a disk geometry. TheseCn-symmetric CIs can be identified based on the chiral edge states, the real-space Chern number and the quantized conductance. Here, we propose two ways to calculate the real-space Chern number, the Kitaev's formula and the local Chern marker. Furthermore, the TPTs of curved CIs are explored by tuning staggered flux and on-site mass.

Quantum transport in Chern insulators on Möbius strips.

Chern insulators (CIs) or quantum anomalous Hall (QAH) states have drawn more attention, with emergence of quantized Hall conductance but in absence of Landau levels. Here, we study the Haldane-type CI/QAH states on Möbius strips and focus on the quantum transport properties. The Möbius strips can be constructed from the twisted honeycomb-lattice strips with domain walls. Topological properties of the Möbius CI/QAH states can be identified by the local density of states, the real-space Chern numbers and the transport properties. Nearly perfect quantized conductance plateaus are observed in the numerical quantum-transport simulations. In addition, we found that adding magnetic flux in domain walls can induce alternating edge states. At last, we discussed in details the different lead connection schemes, and found that we can use multiple domain walls to obtain various higher quantized conductance plateaus.

Topological quantum phase transitions of Chern insulators in disk geometry.

We investigate topological quantum phase transitions (TQPTs) of Chern insulators in two-dimensional honeycomb-lattice disk with six-fold rotational symmetry. By considering the nearest-neighbor, next-nearest-neighbor hopping parameters and the staggered-flux parameter of the Haldane model, we can obtain rich topological quantum phases. The trivial and non-trivial phases of the Haldane model in disk geometry can be distinguished based on chiral edge states, real-space particle densities and local density of states. We also explore the TQPTs of Chern insulators with an external potential which varies with the radius of the disk geometry. Some interesting topological phases with large Chern numbers can be observed when we consider long-distance hoppings. Furthermore, we use a machine learning algorithm as an effective way to automatically identify various topological phases and phase diagrams for the Haldane model in disk geometry.