arXiv:2110.07245 · Physical review. B./Physical review. B

Chern-number matrix of the non-Abelian spin-singlet fractional quantum Hall effect

Tian-Sheng Zeng, Wei Kang Zhu

Published Mar 21, 2022• 7 citations

Abstract

While the internal structure of Abelian topological order is well understood, how to characterize the non-Abelian topological order is an outstanding issue. We propose a distinctive scheme based on the many-body Chern number matrix to characterize non-Abelian multicomponent fractional quantum Hall states. As a concrete example, we study the many-body ground state of two-component bosons at the filling faction $\ensuremath{\nu}=4/3$ in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fractional quantum Hall effect under three-body interaction, whose topological nature is classified by sixfold degenerate ground states and a fractionally quantized Chern-number matrix.

DOI: 10.1103/physrevb.105.125128

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arXiv:2110.07245 · Physical review. B./Physical review. B

Chern-number matrix of the non-Abelian spin-singlet fractional quantum Hall effect

Tian-Sheng Zeng, Wei Kang Zhu

Published Mar 21, 2022• 7 citations

arXiv HTML PDF

Abstract

\ensuremath{\nu}=4/3

in topological flat band models. Utilizing density-matrix renormalization group and exact diagonalization calculations, we demonstrate the emergence of non-Abelian spin-singlet fractional quantum Hall effect under three-body interaction, whose topological nature is classified by sixfold degenerate ground states and a fractionally quantized Chern-number matrix.

DOI: 10.1103/physrevb.105.125128

Related papers

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\ensuremath{\nu}=4/3

\mathbf{K}

matrix, through an exact diagonalization study of fourfold quasidegenerate ground states with a robust spectrum gap and the combined density-matrix renormalization group calculation of fractional drag charge pumping. Further, we formulate the topological characterization of FQH states of three-component Bose-Fermi mixtures at various fillings by the

\mathbf{K}

matrix. Finally, we discuss the possible generalization of our approach to identify non-Abelian three-component spin-singlet FQH states.

\mathbf{K}=\left(\begin{array}{cc}m& 1\\ 1& n\end{array}\right)

matrix (even

m

for boson and odd

n

for fermion), using topological flat band models. Utilizing the state-of-the-art exact diagonalization and density-matrix renormalization group methods, we build up the topological characterization based on three inherent aspects: (i) topological $(mn\ensuremath{-}1)$-fold ground-state degeneracy equivalent to the determinant of the

\mathbf{K}

matrix; (ii) fractionally quantized topological Chern number matrix equivalent to the inverse of the

\mathbf{K}

matrix; and (iii) two parallel-propagating chiral edge branches with level counting 1,2,5,10 consistent with the conformal field theory description.

Recent theoretical works have demonstrated various robust Abelian and non-Abelian fractional topological phases in lattice models with topological flat bands carrying Chern number

C=1

. Here we study hard-core bosons and interacting fermions in a three-band triangular-lattice model with the lowest topological flat band of Chern number

C=2

. We find convincing numerical evidence of bosonic fractional quantum Hall effect at the $\ensuremath{\nu}=1/3$ filling characterized by threefold quasidegeneracy of ground states on a torus, a fractional Chern number for each ground state, a robust spectrum gap, and a gap in quasihole excitation spectrum. We also observe numerical evidence of a robust fermionic fractional quantum Hall effect for spinless fermions at the $\ensuremath{\nu}=1/5$ filling with short-range interactions.

We study

N

-component interacting particles (hardcore bosons and fermions) loaded in topological lattice models with

\mathrm{SU}(N)

-invariant interactions based on exact diagonalization and density matrix renormalization group method. By tuning the interplay of interspecies and intraspecies interactions, we demonstrate that a class of

\mathrm{SU}(N)

fractional quantum Hall states can emerge at fractional filling factors $\ensuremath{\nu}=N/(N+1)$ for bosons [$\ensuremath{\nu}=N/(2N+1)$ for fermions] in the lowest Chern band, characterized by the nontrivial fractional Hall responses and the fractional charge pumping. Moreover, we establish a topological characterization based on the

\mathbf{K}

matrix and discuss the close relationship to the fractional quantum Hall physics in topological flat bands with Chern number

N

\mathbf{K}=\left(\begin{array}{cc}m+2& 1\\ 1& m\end{array}\right)

matrix (even

m=2

for boson and odd

m=3

for fermion) in the presence of both strong intercomponent and intracomponent repulsions. Through exact diagonalization and density-matrix renormalization group calculations, we elucidate their topological fractionalizations, including (i) the $det|\mathbf{K}|=({m}^{2}+2m\ensuremath{-}1)$-fold ground-state degeneracies and (ii) fractionally quantized topological Chern number matrix $\mathbf{C}={\mathbf{K}}^{\ensuremath{-}1}$. Our flat band model provides a paradigmatic example of a microscopic Hamiltonian featuring fractional quantum Hall effect with partial spinpolarization.