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arXiv:2304.08948 · Physical review. B./Physical review. B

Halperin ( m′, m, n ) fractional quantum Hall effect in topological flat bands

Tian-Sheng Zeng

Published Aug 10, 2023• 3 citations
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Abstract

The Halperin $({m}^{\ensuremath{'}},m,n)$ fractional quantum Hall effects of two-component quantum particles are studied in topological checkerboard lattice models. Here for $m\ensuremath{\ne}{m}^{\ensuremath{'}}$, we demonstrate the emergence of fractional quantum hall effects with the associated $\mathbf{K}=\left(\begin{array}{cc}m+2& 1\\ 1& m\end{array}\right)$ 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.

DOI
10.1103/physrevb.108.085111

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The Halperin (m\ensuremath′,m,n)({m}^{\ensuremath{'}},m,n)(m\ensuremath′,m,n) fractional quantum Hall effects of two-component quantum particles are studied in topological checkerboard lattice models. Here for m\ensuremath≠m\ensuremath′m\ensuremath{\ne}{m}^{\ensuremath{'}}m\ensuremath=m\ensuremath′, we demonstrate the emergence of fractional quantum hall effects with the associated K=(m+211m)\mathbf{K}=\left(\begin{array}{cc}m+2& 1\\ 1& m\end{array}\right)K=(m+21​1m​) matrix (even m=2m=2m=2 for boson and odd m=3m=3m=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∣K∣=(m2+2m\ensuremath−1)det|\mathbf{K}|=({m}^{2}+2m\ensuremath{-}1)det∣K∣=(m2+2m\ensuremath−1)-fold ground-state degeneracies and (ii) fractionally quantized topological Chern number matrix C=K\ensuremath−1\mathbf{C}={\mathbf{K}}^{\ensuremath{-}1}C=K\ensuremath−1. Our flat band model provides a paradigmatic example of a microscopic Hamiltonian featuring fractional quantum Hall effect with partial spinpolarization.
Quantum Hall effects with multicomponent internal degrees of freedom facilitate the playground of novel emergent topological orders. Here, we explore the correlated topological phases of two-component hardcore bosons at a total filling factor ν = 2/5 in topological lattice models under the interplay of intracomponent and intercomponent repulsions. We give the numerical demonstration of the emergence of Halperin (441) fractional quantum Hall effect based on exact diagonalization and density-matrix renormalization group methods. We elucidate its topological features including the degeneracy of the ground state, fractionally quantized topological Chern number matrix and chiral edge modes.
Multicomponent quantum Hall effect, under the interplay between intercomponent and intracomponent correlations, leads us to new emergent topological orders. Here, we report the theoretical discovery of fractional quantum hall effect of strongly correlated Bose-Fermi mixtures classified by the K=(m11n)\mathbf{K}=\left(\begin{array}{cc}m& 1\\ 1& n\end{array}\right)K=(m1​1n​) matrix (even mmm for boson and odd nnn 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 K\mathbf{K}K matrix; (ii) fractionally quantized topological Chern number matrix equivalent to the inverse of the K\mathbf{K}K matrix; and (iii) two parallel-propagating chiral edge branches with level counting 1,2,5,10 consistent with the conformal field theory description.
We study the many-body ground states of three-component quantum particles in two prototypical topological lattice models under strong intercomponent and intracomponent repulsions. At band filling $\ensuremath{\nu}=3/4$ for hard-core bosons, we demonstrate the emergence of the three-component fractional quantum Hall (FQH) effect characterized by the K\mathbf{K}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 K\mathbf{K}K matrix. Finally, we discuss the possible generalization of our approach to identify non-Abelian three-component spin-singlet FQH states.
We study NNN-component interacting particles (hardcore bosons and fermions) loaded in topological lattice models with SU(N)\mathrm{SU}(N)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 SU(N)\mathrm{SU}(N)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 K\mathbf{K}K matrix and discuss the close relationship to the fractional quantum Hall physics in topological flat bands with Chern number NNN.
We study the many-body ground states of SU(N)\mathrm{SU}(N)SU(N) symmetric hardcore bosons on the topological flat-band model by using controlled numerical calculations. By introducing strong intracomponent and intercomponent interactions, we demonstrate that a hierarchy of bosonic SU(N)\mathrm{SU}(N)SU(N) fractional quantum Hall (FQH) states emerges at fractional filling factors $\ensuremath{\nu}=N/(MN+1)$ (odd M=3M=3M=3). In order to characterize this series of FQH states, we figure the effective K\mathbf{K}K matrix from the inverse of the Chern number matrix. The topological characterization of the K\mathbf{K}K matrix also reveals quantized drag Hall responses and fractional charge pumping that could be detected in future experiments. In addition, we address the general one-to-one correspondence to the spinless FQH states in topological flat bands with Chern number C=NC=NC=N at fillings $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\nu}}=1/(MC+1)$.
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.