Abelian topological order of ν=2/5 and 3/7 fractional quantum Hall states in lattice models

Abstract

Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component =2/5 and 3/7 states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the =2/5 and 3/7 states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.

Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model

“…The charge pumping for the first three hierarchy states (r = 1, 2, 3) in Figs. 2(a), 2(c), and 2(e) have been demonstrated previously (with certain values of n φ ) [56,57,75,76], and so these results may be used to benchmark our numerics. In contrast, the charge pumping for the fourth-order hierarchy state (r = 4) in Fig.…”

“…Future work in this area may involve, for example, an analysis of the role of interaction range. There is currently a wealth of numerical evidence to suggest that Abelian Jain states favor short-range interactions [41,50,51,57], whereas exotic fractional quantum Hall states may be stabilized exclusively via long-range interactions [99,100]. It is important to establish where |C| > 1FCIs fall on this spectrum to facilitate reliable device engineering.…”

“…The frustration between the magnetic unit-cell area and the irreducible area occupied by one flux quantum results in the famous Hofstadter butterfly-a fractal spectrum of eigenenergies E as a function of n φ [49]. The interactions between the particles are modeled using a nearestneighbor density-density term, which has been shown to be favorable for stabilizing composite fermion states [41,50,51], experimentally relevant due to screening lengths in common electronic devices [52][53][54], and also computationally tractable [1,8,19,20,25,[55][56][57].…”

“…In order to find and analyze the ground state of this 2D many-body problem, we employ the iDMRG algorithm on a thin cylinder [70][71][72]. This is a well-established method that can be formulated in the tensor-network framework and has been used to successfully model fractional quantum Hall states in recent years [56,57,[73][74][75][76]. The method is defined by transcribing the Hamiltonian to a matrix product operator (MPO) and serves to optimize an ansatz wave function in the form of a matrix product state (MPS).…”

“…To complement our charge pumping computations, we additionally diagnose our quantum Hall states using a correlation-function-based approach [20,30,57,105]. To this end, we study the connected two-point density correlation functions g(x) = :ρ 0,0 ρ x,0 : − ρ 0,0 ρ ∞,0 for each of the FCIs presented in Sec.…”