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arXiv:1208.2055 · Physical Review B

Band geometry of fractional topological insulators

Rahul Roy

Published Oct 30, 2014• 306 citations
arXivHTMLPDF

Abstract

Recent numerical simulations of flat-band models with interactions which show clear evidence of fractionalized topological phases in the absence of a net magnetic field have generated a great deal of interest. We provide an explanation for these observations by showing that the physics of these systems is the same as that of conventional fractional quantum Hall phases in the lowest Landau level under certain ideal conditions which can be specified in terms of the Berry curvature and the Fubini-Study or quantum metric of the topological band. In particular, we show that when these ideal conditions hold, the density operators projected to the topological band obey the ${W}_{\ensuremath{\infty}}$ algebra. Our approach leads us to propose a way of testing the suitability of topological bands for hosting fractionalized phases.

DOI
10.1103/physrevb.90.165139

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Recent numerical simulations of flat-band models with interactions which show clear evidence of fractionalized topological phases in the absence of a net magnetic field have generated a great deal of interest. We provide an explanation for these observations by showing that the physics of these systems is the same as that of conventional fractional quantum Hall phases in the lowest Landau level under certain ideal conditions which can be specified in terms of the Berry curvature and the Fubini-Study or quantum metric of the topological band. In particular, we show that when these ideal conditions hold, the density operators projected to the topological band obey the W\ensuremath∞{W}_{\ensuremath{\infty}}W\ensuremath∞​ algebra. Our approach leads us to propose a way of testing the suitability of topological bands for hosting fractionalized phases.
Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of fractional Chern insulator states. In fact, we show that an ``ideal flatband'' cannot have constant Berry curvature, equivalently, we show that the density algebra of Landau levels cannot be realised in any tight-binding lattice system.
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=1C=1C=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=2C=2C=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 consider a topologically nontrivial flat-band structure in one spatial dimension in the presence of nearest- and next-nearest-neighbor Hubbard interaction. The noninteracting band structure is characterized by a symmetry-protected topologically quantized Berry phase. At certain fractional fillings, a gapped phase with a filling-dependent ground-state degeneracy and fractionally charged quasiparticles emerges. At filling 13\frac{1}{3}31​, the ground states carry a fractional Berry phase in the momentum basis. These features at first glance suggest a certain analogy to the fractional quantum Hall scenario in two dimensions. We solve the interacting model analytically in the physically relevant limit of a large band gap in the underlying band structure, the analog of a lowest Landau level projection. Our solution affords a simple physical understanding of the properties of the gapped interacting phase. We pinpoint crucial differences to the fractional quantum Hall case by studying the Berry phase and the entanglement entropy associated with the degenerate ground states. In particular, we conclude that the ``fractional topological phase in one-dimensional flat bands'' is not a one-dimensional analog of the two-dimensional fractional quantum Hall states, but rather a charge density wave with a nontrivial Berry phase. Finally, the symmetry-protected nature of the Berry phase of the interacting phase is demonstrated by explicitly constructing a gapped interpolation to a state with a trivial Berry phase.
Recent proposals of topological flat band models have provided a new route to realize the fractional quantum Hall effect without Landau levels. We study hard-core bosons with short-range interactions in two representative topological flat band models, one of which is the well-known Haldane model (but with different parameters). We demonstrate that fractional quantum Hall states emerge with signatures of an even number of quasidegenerate ground states on a torus and a robust spectrum gap separating these states from the higher energy spectrum. We also establish quantum phase diagrams for the filling factor 1/2 and illustrate quantum phase transitions to other competing symmetry-breaking phases.
We study how the stability of the fractional quantum Hall effect is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria play a dominant role in predicting the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.
Recent theoretical papers have demonstrated the realization of fractional quantum anomalous Hall states (also called fractional Chern insulators) in topological flat band lattice models without an external magnetic field. Such newly proposed lattice systems play a vital role in obtaining a large class of fractional topological phases. Here we report the exact numerical studies of edge excitations for such systems in a disk geometry loaded with hard-core bosons, which will serve as a more viable experimental probe for such topologically ordered states. We find convincing numerical evidence of a series of edge excitations characterized by the chiral Luttinger liquid theory for the bosonic fractional Chern insulators in both the honeycomb disk Haldane model and the kagome-lattice disk model. We further verify these current-carrying chiral edge states by inserting a central flux to test their compressibility.