Hierarchy of Fractional Chern Insulators and Competing Compressible States

Läuchli, Andreas M.; Liu, Zhao; Bergholtz, Emil J.; Moessner, Roderich

doi:10.1103/physrevlett.111.126802

Cited by 77 publications

(129 citation statements)

References 44 publications

(47 reference statements)

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“…In the second case, the degeneracy is consistent with a ν = 13/15 FCI, however, levels return to their initial configuration already after insertion of only five flux quanta, which is unexpected. Moreover, additional interaction-generated dispersion tends to destabilize FCIs at such high fillings [15,30]. Closer inspection shows that the 15 lowenergy states can be separated into three groups of five states each, where each group shows the spectral flow expected for a denominator-five state, see Figs.…”

Section: (B)mentioning

confidence: 99%

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Combined Topological and Landau Order from Strong Correlations in Chern Bands

Kourtis

Daghofer

2014

Phys. Rev. Lett.

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We present a class of states with both topological and conventional Landau order that arise out of strongly interacting spinless fermions in fractionally filled and topologically non-trivial bands with Chern number C = ±1. These quantum states show the features of fractional Chern insulators, such as fractional Hall conductivity and interchange of ground-state levels upon insertion of a magnetic flux. In addition, they exhibit charge order and a related additional trivial ground-state degeneracy. Band mixing and geometric frustration of the charge pattern place these lattice states markedly beyond a single-band description.PACS numbers: 71.27.+a, 03.65.Vf, 71.45.Lr, The fractional quantum Hall (FQH) effect is a paradigmatic example for a correlation-driven state with topological order [1], and the proposal [2][3][4] that the concept might be extended from Landau levels arising due to a magnetic field to topologically nontrivial bands on a lattice has thus raised considerable interest. Apart from the intrinsic interest in such an effect, one motivation for the search of FCIs is their energy and thus temperature scale: as it is given by the scale of the interaction, it is expected to be considerably higher than the sub-Kelvin range of the FQH effect, especially for oxide-based proposals [7,8,18]. As one would moreover not need strong magnetic fields, both realization of such states and their potential application to qubits [22] then appear more feasible. It has been established that FCI states can persist the influence of several aspects that make bands on lattices different from perfectly flat Landau levels with uniform Berry curvature, e.g., finite dispersion [8,23], a moderate staggered chemical potential [2], disorder [24,25], or competition with a chargedensity wave (CDW) [25].Since the FQH effect can be discussed in tight-binding models instead of Landau levels [26,27], particularly intriguing features of FCI states are those that go beyond their FQH counterpart. FCIs with higher Chern numbers were discussed [23,28,29], which may be non-Abelian and thus suitable for quantum computation. It has also been noted that FCIs do not share the particle-hole symmetry of partially filled Landau levels [25,30]. All these extensions can, however, be understood by focusing exclusively on the fractionally filled Chern band. FCI states considered so far are weakly interacting, in the sense that the interactions stabilizing them are too weak to mix in the other band(s) with different Chern numbers. Those can be even projected out of the Hamiltonian, which keeps the band topology intact but obscures the impact of other aspects like lattice geometry, again reflecting the similarity to Landau levels with their weak lattice potential.In this Letter, we go beyond the limit of 'weak' interactions, into a regime where Chern bands with C = +1 and C = −1 mix. We find states that show the features of both a CDW (revealed by the charge structure factor) and of an FCI (fractional Hall conductivity and spectral flow). The states are rel...

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Section: (B)mentioning

confidence: 99%

“…FCIs with higher Chern numbers were discussed [23,28,29], which may be non-Abelian and thus suitable for quantum computation. It has also been noted that FCIs do not share the particle-hole symmetry of partially filled Landau levels [25,30]. All these extensions can, however, be understood by focusing exclusively on the fractionally filled Chern band.…”

mentioning

confidence: 99%

Combined Topological and Landau Order from Strong Correlations in Chern Bands

Kourtis

Daghofer

2014

Phys. Rev. Lett.

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“…The filling factors (10) are analogous to the hierarchy states, which have been observed in Chern number one fractional Chern insulators [14,49,50]. Their properties, in terms of quasiparticle charges and statistics were predicted by Kol and Read [11], as summarized in [40].…”

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confidence: 96%

Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number

2015

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The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number, C. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with |C| > 1. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor, ν, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors ν = r/(r|C|+1) for bosons, or ν = r/(2r|C|+1) for fermions. This series includes a bosonic integer quantum Hall state (bIQHE) in |C| = 2 bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.Recently, there has been much progress towards experimental realizations of topological flat bands, such as by light-matter coupling in cold gases [1][2][3][4][5][6][7] or via spin-orbit coupling in condensed matter systems [8][9][10]. These systems provide novel avenues for exploring fractional quantum Hall physics in new settings where lattice effects play important roles [8][9][10][11][12][13][14][15][16][17][18][19]. Furthermore, these "fractional Chern insulators" generalize the fractional quantum Hall states of interacting particles in continuum Landau levels to lattice-based systems.In cases where the underlying topological band has unit Chern number, C = 1, the states can be continuously connected to conventional fractional quantum Hall states in the continuum Landau level [20]. However, if the band has Chern number, C, of magnitude greater than 1, no such continuity is possible. The fractional quantum Hall states have features that are particular to the lattice structure. The appearance of fractional quantum Hall states for bands with |C| > 1 has been demonstrated in various lattice models, with unit cells that contain multiple states in the form of distinct sublattices, or in terms of internal degrees of freedom such as spin or color [see, e.g. 21]. In particular, this has led to the proposal of states at filling factors ν = 1/(|C| + 1) for bosons [22][23][24][25].In this Letter we show that this physics of interacting particles in the novel Chern bands can be captured within the Harper-Hofstadter model, in which a magnetic unit cell arises naturally without additional internal degrees of freedom. This model leads to a complex energy spectrum as a function of flux n φ = Φ/Φ 0 per plaquette. The low energy bands can have Chern numbers larger than one. We show that they can realize the sequences of fractional Chern insulator states for |C| > 1 discussed for other models, providing an interpretation of these states in terms of the composite fermion construction on a lattice [11, 14]. Based on these insights, we identify another sequence of fractional Chern insulator states with fil...

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“…to recently discovered fractional Chern insulators (FCIs) [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33] because of the adiabatic continuity between Hofstadter and Chern insulator states [34][35][36]. Additionally, we study a one-dimensional lattice model for which our generalized Hofstadter model can be regarded as a two-dimensional ancestor.…”

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confidence: 99%

“…These states are expected to be similar to FCIs in topological flat bands, although the net magnetic field is nonzero in our model (a gauge transformation can be used to obtain a zero net magnetic field [34]). We adopt some commonly used criteria, such as the ground-state topological degeneracy, the spectral flow under twisted boundary conditions, and the particle-cut entanglement spectrum, to identify the ground states as FQH states [19][20][21][22][23][24][25][26][27][28][29][30][31].…”

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Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices

et al. 2013

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Two-dimensional lattice models subjected to an external effective magnetic field can form nontrivial band topologies characterized by nonzero integer band Chern numbers. In this Letter, we investigate such a lattice model originating from the Hofstadter model and demonstrate that the band topology transitions can be realized by simply introducing tunable longer-range hopping. The rich phase diagram of band Chern numbers is obtained for the simple rational flux density and a classification of phases is presented. In the presence of interactions, the existence of fractional quantum Hall states in both |C| = 1 and |C| > 1 bands is confirmed which can reflect the band topologies in different phases. In contrast, when our model reduces to a one-dimensional lattice, the ground states are crucially different from fractional quantum Hall states. Our results may provide insights into the study of new fractional quantum Hall states and experimental realizations of various topological phases in optical lattices.

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Hierarchy of Fractional Chern Insulators and Competing Compressible States

Cited by 77 publications

References 44 publications

Combined Topological and Landau Order from Strong Correlations in Chern Bands

Combined Topological and Landau Order from Strong Correlations in Chern Bands

Fractional Chern Insulators in Harper-Hofstadter Bands with Higher Chern Number

Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices

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