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Acta Physica Polonica A

Analysis of the Quantum Spin Hall and Quantum Anomalous Hall Effects in a Two-Dimensional Decorated Lattice Using Entanglement Spectrum

Anna M. Piekarska, Paweł Potasz, Arkadiusz Wójs

Published Jan 1, 2016• 0 citations
Publisher

Abstract

We study topological properties of a two-dimensional decorated lattice using entanglement spectrum and trace index.Using these measures, we distinguish various phases of the system predicted recently, including the quantum spin Hall effect and quantum anomalous Hall effect.The topologically trivial and non-trivial phases are identified, with assignment of a Chern number to each of the bands.

DOI
10.12693/aphyspola.129.a-87

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We study topological properties of a two-dimensional decorated lattice using entanglement spectrum and trace index.Using these measures, we distinguish various phases of the system predicted recently, including the quantum spin Hall effect and quantum anomalous Hall effect.The topologically trivial and non-trivial phases are identified, with assignment of a Chern number to each of the bands.
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We study a tight-binding model on the two-dimensional star lattice (also called the decorated honeycomb lattice) with short-range hoppings modulated by staggered fluxes. We observe a series of topological quantum phase transitions when tuning the hopping parameters, and find nontrivial topological bands with high Chern numbers. We also find in some parameter regions, the system exhibits the interesting topological flat bands with large flatness ratios of the band gaps over the bandwidths. We further illustrate the 1/21/21/2 fractional quantum Hall effect of interacting hard-core bosons filled in a lowest topological flat band of the star lattice.
In non-interacting systems, bands from non-trivial topology emerge strictly at half-filling and exhibit either the quantum anomalous Hall or spin Hall effects. Here we show using determinantal quantum Monte Carlo and an exactly solvable strongly interacting model that these topological states now shift to quarter filling. A topological Mott insulator is the underlying cause. The peak in the spin susceptibility is consistent with a possible ferromagnetic state at T = 0. The onset of such magnetism would convert the quantum spin Hall to a quantum anomalous Hall effect. While such a symmetry-broken phase typically is accompanied by a gap, we find that the interaction strength must exceed a critical value for this to occur. Hence, we predict that topology can obtain in a gapless phase but only in the presence of interactions in dispersive bands. These results explain the recent quarter-filled quantum anomalous Hall effects seen in moiré systems.
This dissertation studies quantum entanglement in relation to the geometric response known as Hall viscosity. We begin by reviewing geometric response in the quantum Hall effect in comparison to its well-known electromagnetic response, Hall conductivity. We develop an understanding of momentum transport due to Hall viscosity in analogy to charge transport under Hall conductivity. We apply our momentum transport argument to continuum and lattice models of the quantum Hall effect. We also leverage this insight to reveal a previously-unrecognized manifestation of Hall viscosity: the acoustic Faraday effect in superfluid Helium-3 B. We suggest that the acoustic Faraday effect is a new platform for the direct observation of Hall viscosity in Helium-3 B and other systems. We then turn our focus to the entanglement spectrum. We calculate the momentum polarization of lattice models of the quantum Hall effect to determine the Hall viscosity based on the entanglement spectrum, revealing the close connection between geometric response and entanglement. Finally, we turn away from the quantum Hall effect to consider other topological insulators; we use the entanglement spectrum to develop a topological classification of composite systems comprising components of topological phases in different dimensions. Our composite topological index generalizes classification of weak and antiferromagnetic topological insulators. We predict the presence of topological bound states localized to defects in systems that are trivial under all other topological classifications.
The quantum anomalous Hall effect can, in principle, be driven entirely by interparticle interactions into a new phase of matter called the topological Mott insulator. Calculations to date have shown that this effect can arise in simple models only when the interaction strength between particles is not spatially decaying. Such interactions are typically unphysical. Here, the authors study particles in a decorated honeycomb lattice to find that spatially decaying interactions do in fact lead to a topological Mott insulator with a quantum anomalous Hall effect. The results set the stage for realizing interaction-driven topological effects in quantum matter.
We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.