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arXiv:1511.03295 · Physical review. B./Physical review. B

Band flatness optimization through complex analysis

Ching Hua Lee, Daniel P. Arovas, Ronny Thomale

Published Apr 26, 2016• 48 citations
arXivHTMLPDF

Abstract

Narrow-band electron systems are particularly likely to exhibit correlated many-body phases driven by interaction effects. Examples include magnetic materials, heavy-fermion systems, and topological phases such as fractional quantum Hall states and their lattice-based cousins, the fractional Chern insulators (FCIs). Here we discuss the problem of designing models with optimal band flatness, subject to constraints on the range of electron hopping. In particular, we show how the imaginary gap, which serves as a proxy for band flatness, can be optimized by appealing to Rouch\'e's theorem, a familiar result from complex analysis. This leads to an explicit construction which we illustrate through its application to two-band FCI models with nontrivial topology (i.e., nonzero Chern numbers). We show how the imaginary-gap perspective leads to an elegant geometric picture of how topological properties can obstruct band flatness in systems with finite-range hopping.

DOI
10.1103/physrevb.93.155155

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Narrow-band electron systems are particularly likely to exhibit correlated many-body phases driven by interaction effects. Examples include magnetic materials, heavy-fermion systems, and topological phases such as fractional quantum Hall states and their lattice-based cousins, the fractional Chern insulators (FCIs). Here we discuss the problem of designing models with optimal band flatness, subject to constraints on the range of electron hopping. In particular, we show how the imaginary gap, which serves as a proxy for band flatness, can be optimized by appealing to Rouch'e's theorem, a familiar result from complex analysis. This leads to an explicit construction which we illustrate through its application to two-band FCI models with nontrivial topology (i.e., nonzero Chern numbers). We show how the imaginary-gap perspective leads to an elegant geometric picture of how topological properties can obstruct band flatness in systems with finite-range hopping.
Models with topological flat bands (TFBs) have been proposed on which the fractional Chern insulators (FCIs) have been studied. To flatten the lowest energy band, some long-range hopping terms are considered in Chern insulator (CI) models. In this paper, we study a tight-binding model on a maple leaf lattice with staggered magnetic fluxes. When the nearest-neighbor hopping term with a suitable phase factor is considered, a TFB with a large flatness ratio emerges, suggesting that staggered magnetic fluxes can give rise to TFBs. By adding the next-nearest-neighbor (NNN) hopping term, there are two types of topological states at one-third filling, i.e., the CI and the higher-order topological insulator (HOTI) states. Simultaneously, we find the CI with a high Chern number can transition to the HOTI state by tuning the NNN hopping parameter. When we consider more distant-neighbor hopping terms, a TFB with Chern number two is found. We further investigate FCI states when hard-core bosons are taken into account in these TFB models.
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.
The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest-neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.
We construct a two-band lattice model whose bands can carry the Chern numbers $C=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2$. By means of numerical exact diagonalization, we show that the most favorable situation that selects fractional Chern insulators (FCIs) is not necessarily the one that mimics Landau levels, namely a flat band with Chern number 1. First, we find that the gap, measured in units of the on-site electron-electron repulsion, can increase by almost two orders of magnitude when the bands are flat and carry a Chern number C=2C=2C=2 instead of C=1C=1C=1. Second, we show that giving a width to the bands can help to stabilize a FCI. Finally, we put forward a tool to characterize the real-space density profile of the ground state that is useful to distinguish FCI from other competing phases of matter supporting charge density waves or phase separation.
We investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from C=1C=1C=1 to C=−1C=-1C=−1. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with 1/31/31/3 filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy α(nA)α(n_A)α(nA​) term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.
Fractional Chern insulators arise in topologically nontrivial flat bands, characterized by an integer Chern number C that corresponds to the number of dissipationless edge states in the non-interacting regime. Higher Chern numbers can replicate the physics of higher Landau levels and often confer enhanced topological robustness. However, realizing correlated fractional phases with higher Chern numbers in such flat band systems remains challenging. Here, we propose an interlayer coupling scheme to generate higher Chern numbers in a flat-band system, where the interlayer coupling transforms two C = 1 bands in a bilayer checkerboard lattice into a single flat band with C = 2 by lifting their degeneracy and merging their topological indices. Exact diagonalization calculation reveals that this engineered band hosts two fractional Chern insulator states with C = 2/3 and 2/5, respectively. An experimental setup is proposed to simulate these states using cold alkaline-earth-like atoms in an effective bilayer optical lattice. Our work provides a general and widely applicable strategy for constructing higher Chern number flat bands, opening a pathway to explore exotic fractional quantum phases.