Wave functions for fractional Chern insulators in disk geometry

Abstract

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...

Non-Abelian fractional Chern insulator in disk geometry

“…However, the quasi-degeneracy sequences are different. Edge excitations.-The bulk-edge correspondence is a key feature of the FQH states [10][11][12][13] and FCIs [25,26,48,49]. The quasi-degeneracies of FQH states are coincident well with the prediction from the chiral Luttinger liquid theory [10,11].…”

“…Similar to the Laughlin WFs, the Moore-Read state can be decomposed into anti-symmetric Slater determinants (for fermions) or symmetric monomials (for bosons) with the help of the Jack symmetric polynomials (Jacks) [17][18][19][20], which naturally reflects the generalized Pauli principle (GPP) [21][22][23][24] in the Fock space. The GPP and the Jacks have been extended [25,26] to the lattice analogs of FQH states in the absence of external magnetic field, which are named fractional Chern insulaors (FCIs) .…”

“…Here, we choose a harmonic trap with the form V = V trap r |r| 2 n r where V trap is the trap potential strength and |r| is the site position distance from the center of the lattice disk [25,26,48]. Root configurations of the stable ν = 1 NA-FCI are |20202...202 (for even boson numbers) and |20202...201 (for odd boson numbers).…”

“…Heuristically, we conjecture that these FCIs are the bosonic ν = 1 NA-FCI. Following the previous works [25,26,58], optimal trial WFs can be used to identify the FCIs. To verify our conjecture, we construct the trial WF for the NA-FCI.…”

Focus on topological physics: from condensed matter to cold atoms and optics

“…A natural generalization of the FQH liquid is fractional Chern insulator (FCI), which can be viewed as FQH liquid stabilized by a crystalline potential that breaks time-reversal symmetry. This is the subject of [9], in which He et al studied how to map the wave functions between an FCI and a FQH liquid. They tested their mapping numerically using the disc geometry, and obtained excellent results.…”