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arXiv:1305.2255 · Physical Review B

Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands

Wei Kang Zhu, D. N. Sheng, F. D. M. Haldane

Published Jul 17, 2013• 44 citations
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Abstract

We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low-energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the ground-state manifold. We extract the modular transformation matrices $\mathcal{S}$ ($\mathcal{U}$) which contain the information of mutual (self) statistics, quantum dimensions, and the fusion rule of quasiparticles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.

DOI
10.1103/physrevb.88.035122

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We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low-energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the ground-state manifold. We extract the modular transformation matrices S\mathcal{S}S (U\mathcal{U}U) which contain the information of mutual (self) statistics, quantum dimensions, and the fusion rule of quasiparticles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.
The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the ground state manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Ising anyon or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wave functions.
We show how to numerically calculate several quantities that characterize topological order starting from a microscopic fractional quantum Hall Hamiltonian. To find the set of degenerate ground states, we employ the infinite density matrix renormalization group method based on the matrix-product state representation of fractional quantum Hall states on an infinite cylinder. To study localized quasiparticles of a chosen topological charge, we use pairs of degenerate ground states as boundary conditions for the infinite density matrix renormalization group. We then show that the wave function obtained on the infinite cylinder geometry can be adapted to a torus of arbitrary modular parameter, which allows us to explicitly calculate the non-Abelian Berry connection associated with the modular T transformation. As a result, the quantum dimensions, topological spins, quasiparticle charges, chiral central charge, and Hall viscosity of the phase can be obtained using data contained entirely in the entanglement spectrum of an infinite cylinder.
We study the phase transitions of bosonic $\ensuremath{\nu}=1/2$ fractional quantum Hall (FQH) effect in different topological lattice models under the interplay of onsite periodic potential and Hubbard repulsion. Through exact diagonalization and density matrix renormalization group methods, we demonstrate that the many-body ground state undergoes a continuous phase transition between bosonic FQH liquid and a trivial (Mott) insulator induced by the periodic potential, characterized by the smooth crossover of energy and entanglement entropy. When the Hubbard repulsion decreases, we claim that this bosonic FQH liquid would turn into a superfluid state with a direct energy level crossing and a discontinuous leap of off-diagonal long-range order.
By means of finite-size exact diagonalization we theoretically study the electronic many-body effects on the nearly flat-band structure with time-reversal symmetry in a checkerboard lattice model and identify the topological nature of two quantum phases, with ninefold and threefold degeneracy, that appear, respectively, at small and large values $\ensuremath{\lambda}$ of a nearest-neighbor spin-dependent interaction. Numerical evidence from the evolution of low-lying energy spectra and Berry phases with both spin-independent and spin-dependent twisted boundary conditions reveal that these two different ground states share the same topological spin Chern number. Quantum phase transition between these two states by tuning $\ensuremath{\lambda}$ is confirmed by evaluating the closing of energy and quasispin excitation spectra. Finally, the counting rules of spin excitation spectra are demonstrated as the fingerprints of the fractionalized quantum spin Hall states.
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 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 $det|\mathbf{K}|=({m}^{2}+2m\ensuremath{-}1)$-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.
We investigate the response to modular transformations and the fractional statistics of Abelian multicomponent fractional quantum Hall (FQH) states. In particular, we analytically derive the modular matrices encoding the statistics of anyonic excitations for general Halperin states using the conformal field theories (CFTs). We validate our theory by several microscopic examples, including the spin-singlet state using the anyon condensation picture and the Halperin (221) state in a topological flat-band lattice model using numerical calculations. Our results, uncovering that the modular matrices and associated fractional statistics are solely determined by the KKK matrix, further strengthens the correspondence between the two-dimensional (2D) CFTs and (2+1)D(2+1)\mathrm{D}(2+1)D topological orders for multicomponent FQH states.