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Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractionalization phenomena to the conventional fractional quantum Hall (FQH) systems. Using the mapping relationship between the FQH states and the FCI/FQAH states, we construct the many-body wave functions of the fermionic FCI/FQAH states in disk geometry with the aid of the generalized Pauli principle (GPP) and Jack polynomials. Compared with the ground state by the exact diagonalization method, the wave-function overlap is higher than 0.97, even when the Hilbert space dimension is as large as 3×10 6 . We also use the GPP and the Jack polynomials to construct edge excitations for the fermionic FCI/FQAH states. The quasi-degeneracy sequences of fermionic FCI/FQAH systems reproduce the prediction of the chiral Luttinger liquid theory, complementing the exact diagonalization results with larger lattice sizes and more particles. OPEN ACCESS RECEIVED exclusion principle corresponds to the Fermi statistics, the GPP corresponds to the fractional statistics of anyons.Recently, finding the suitable trial wave functions of FCI/FQAH states has become an outstanding problem. It is conjectured that there is a one-to-one mapping between IQH and CI/QAH states [10,[34][35][36][37]. In other words, the single-particle orbitals of an LL can be mapped to the ones in a TFB. Based on one-dimensional (1D) maximally localized Wannier functions, Qi et al have used a mapping between the FQH and the FCI/FQAH states to construct generic wave functions of FCI/FQAH states in cylinder geometry with these 1D Wannier functions [10,[34][35][36][37]. In terms of this mapping relationship, the Haldane pseudo-potential [26] for these FCI/ FQAH states can be constructed [38,39] through a suitable gauge choice for the Wannier functions. An improved prescription has been adopted to construct variational wave functions of FCI/FQAH states in torus geometry by utilizing gauge-fixed (non-maximally) localized Wannier states [40][41][42]. From another aspect, conventional FQH states can also be obtained for 2D lattices analytically by using conformal field theory [43][44][45]. In contrast to the above analytical or semi-analytical approaches, we here pursue a very direct yet effective purely numerical prescription to construct FCI/FQAH states in disk geometry, without any variational parameter or adjustable gauge freedom, but just utilizing the powerful GPP and the Jacks structure of FQH states, and the information from exact numerical single-particle orbitals of TFB models.In this paper, we exploit the single-particle wave functions of CI/QAH states (in the Kagomé model [5, 46] and the Haldane [3, 9] model) with TFB parameters in disk geometry, and explore the polynomial structure of the continuum Laughlin wave functions [25] to establish the many-body wave functions of the FCI/FQAH states. We firstly construct finite-size lattices in...

Topological quantum states have been proposed and investigated on two-dimensional flat surfaces or lattices with different geometries like the plane, cylinder and torus. Here, we study quantum anomalous Hall (QAH) or Chern insulator (CI) states on two-dimensional singular surfaces (such as conical and helicoid-like surfaces). Such singular geometries can be constructed based on the disk geometry and a defined unit sector with n-fold rotational symmetry. The singular geometry induces novel and intriguing features of CI/QAH states, such as in-gap and in-band core states, charge fractionalization, and multiple branches of edge excitations.

The fractional quantum anomalous Hall (FQAH) states or fractional Chern insulator (FCI) states have been studied on two-dimensional (2D) flat lattices with different boundary conditions. Here, we propose the geometry-dependent FCI/FQAH states that interacting particles are bounded on 2D singular lattices with arbitrary n-fold rotational symmetry. Based on the generalized Pauli principle, we construct trial wave functions for the singular-lattice FCI/FQAH states with the aid of an effective projection approach, and compare them with the exact diagonalization results. High wave-function overlaps show that the singular-lattice FCI/FQAH states are certainly related to the geometric factor β. More interestingly, we observe some exotic degeneracy sequences of edge excitations in these singular-lattice FCI/FQAH states, and provide an explanation that two branches of edge excitations mix together. PACS numbers:Introduction.-Laughlin states [1] with a simple math expression but profound implications can be used to describe the ground state (GS) of fractional quantum Hall (FQH) effect [2] which shows the electron fractionalization in two-dimensional (2D) systems. Over the following decades, the quantum anomalous Hall (QAH) or Chern insulator (CI) states were proposed [3] and realized in ultracold fermion systems [4]. Meanwhile, the fractional quantum anomalous Hall (FQAH) or fractional Chern insulator (FCI) states have been studied based on the topological flat band (TFB) models [5][6][7] with various boundary conditions, like the torus, cylinder and disk geometries [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]25]. One of the most palpable features for the FCI/FQAH states is the edge excitations which are characterized by the chiral Luttinger liquid theory [26]. The edge excitation spectra have been directly or indirectly observed via different numerical techniques [23,24]. These edge excitation spectra have been predicted and approximately obtained [23,25] base on the generalized Pauli principle (GPP) [27][28][29][30]. In terms of the one-to-one mapping relationship between the FQH and FCI/FQAH states, there are various ways to construct the optimal trial wave functions (WFs) for the FCI/FQAH states with analytical, semi-analytical [15,16,20,21] and purely numerical approaches [25]. Recently, inspired by the analytic expression of the Laughlin WF, we have proposed a direct yet effective prescription to construct the FCI/FQAH states on flat disk geometry [25] with the aid of the GPP and the Jack polynomials (Jacks) [31][32][33].

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.

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 arbitrary n-fold rotational symmetry on cone-like and saddle-like lattices (also dubbed as C n -symmetric CIs), by ‘cutting and gluing’ unit sectors with a disk geometry. These C n -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.

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