Zoology of fractional Chern insulators

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Zoology of fractional Chern insulators

Yang-Le Wu, B. Andrei Bernevig, and N. Regnault

Phys. Rev. B 85, 075116 – Published 14 February 2012

Abstract

We study four different models of Chern insulators in the presence of strong electronic repulsion at partial fillings. We observe that all cases exhibit a Laughlin-like phase at filling fraction $1 / 3$ . We provide evidence of such a strongly correlated topological phase by studying both the energy and the entanglement spectra. In order to identify the key ingredients of the emergence of Laughlin physics in these systems, we show how they are affected when tuning the band structure. We also address the question of the relevance of the Berry curvature flatness in this problem. Using three-body interactions, we show that some models can also host a topological phase reminiscent of the $ν = 1 / 2$ Pfaffian Moore-Read state. Additionally, we identify the structures indicating cluster correlations in the entanglement spectra.

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Received 7 November 2011

DOI:https://doi.org/10.1103/PhysRevB.85.075116

Authors & Affiliations

Yang-Le Wu¹, B. Andrei Bernevig¹, and N. Regnault²

¹Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
²Laboratoire Pierre Aigrain, ENS and CNRS, 24 rue Lhomond, FR-75005 Paris, France

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Vol. 85, Iss. 7 — 15 February 2012

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Figure 1
The Haldane model on the honeycomb lattice. $A$ and $B$ sublattices are colored in red and green, respectively. The lattice translation vectors are $b_{1} = a_{2} - a_{3}$ and $b_{2} = a_{3} - a_{1}$ . The hopping between NNs is $t_{1}$ , and the hopping between NNNs is $t_{2} e^{i φ}$ in the direction of arrows. The density-density repulsion between nearest neighbors is depicted in gray ellipses.Reuse & Permissions
Figure 2
Low-energy spectrum of the Haldane model with $N = 8$ (marked by red crosses), $N = 10$ (green plus signs), and $N = 12$ (blue triangles) particles on the $N_{x} \times N_{y} = 6 \times \frac{N}{2}$ lattice, with energies shifted by $E_{0}$ , the lowest energy for each system size. We only show the lowest excited level in each momentum sector in addition to the threefold ground state.Reuse & Permissions
Figure 3
Energy gap $Δ E$ for different system sizes and aspect ratios of the Haldane model. The gap is defined as the energy difference between the first excited state and the highest of the threefold ground states. A more precise definition of $Δ E$ is given in Sec. 2d. In each case, $N_{x} = 3 N / N_{y}$ . The negative values of $Δ E$ in the $N_{y} = 3$ group mean that the actual ground state is not in the momentum sectors predicted by the theoretical counting rule; that is, the gap above the supposed topological ground state closes.Reuse & Permissions
Figure 4
Spectral flow of the low-lying states of the Haldane model with $N = 10$ particles on the $N_{x} \times N_{y} = 6 \times 5$ lattice upon flux insertion along the $x$ direction. $γ_{x}$ counts the number of fluxes inserted. The threefold ground states flow into each other and do not mix with higher states during flux insertion. It takes three full fluxes for the threefold states to return to the original configuration (inset).Reuse & Permissions
Figure 5
Low-energy spectrum of the Haldane model with $N = 7$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The number of states below the gray line is 12 in each momentum sector, in agreement with the $(1, 3)$ -admissible counting rule.Reuse & Permissions
Figure 6
Low-energy spectrum of the Haldane model with $N = 9$ particles on the $N_{x} \times N_{y} = 6 \times 5$ lattice. The number of states below the gray line is 19 in momentum sectors with $k_{x} = 0$ , 3 and 18 elsewhere, in agreement with the $(1, 3)$ -admissible counting rule.Reuse & Permissions
Figure 7
Particle entanglement spectrum of the ground state of the Haldane model of $N = 12$ particles on the $N_{x} \times N_{y} = 6 \times 6$ lattice, with $N_{B} = 8$ particles traced out. The number of states below the gray line is 741 in momentum sectors where both $k_{x}$ and $k_{y}$ are even and 728 elsewhere, in agreement with the $(1, 3)$ -admissible counting rule. The system size is exactly the same as Fig. 13 of Ref. 15.Reuse & Permissions
Figure 8
The distribution of the energy gap $Δ E$ in the $(φ, M)$ plane for the Haldane model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The color code for $Δ E$ is shown by the bar on the right, and the region with the gap closed is marked in gray. The bold red line $M = 3 \sqrt{3} \sin φ$ separates the topologically nontrivial sector of the single-particle problem from the trivial one.Reuse & Permissions
Figure 9
The distribution of the energy spread $δ E$ in the $(φ, M)$ plane for the Haldane model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The color code for $δ E$ is shown by the bar on the right. The bold red line $M = 3 \sqrt{3} \sin φ$ separates the topologically nontrivial sector of the single-particle problem from the trivial one.Reuse & Permissions
Figure 10
The distribution of the entanglement gap $Δ ξ$ in the $(φ, M)$ plane for the ground state of the Haldane model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice, with $N_{B} = 5$ particles traced out. The color code for $Δ ξ$ is shown by the bar on the right, and the region with the gap closed is marked in gray. The average spacing of the entanglement levels below the gap is at most $0.05$ (not shown in the figure); hence, the entanglement gap should be considered widely open as long as $Δ ξ ≫ 0.05$ . The line with $Δ ξ = 0.5$ is shown in magenta. The bold red line $M = 3 \sqrt{3} \sin φ$ separates the topologically nontrivial sector of the single-particle problem from the trivial one.Reuse & Permissions
Figure 11
The distribution of the standard deviation of the Berry curvature $σ_{B}$ in the $(φ, M)$ plane for the Haldane model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The bar on the right shows the color code for the value of $2 π σ_{B}$ . The bold red line $M = 3 \sqrt{3} \sin φ$ separates the topologically nontrivial sector of the single-particle problem from the trivial one. We focus on the nontrivial sector. The strong fluctuation of the Berry curvature very close to the boundary of the nontrivial sector is not shown.Reuse & Permissions
Figure 12
The two-orbital model on square lattice. The two orbitals $A$ , $B$ are colored in red and green, respectively. The intraorbital NN hopping amplitude is $t_{2}$ between the $A$ orbitals, and $- t_{2}$ between the $B$ orbitals. The interorbital NN hopping amplitude is $\pm i t_{1}$ in the $x$ direction along the arrows and $\pm t_{1}$ in the $y$ direction. The on-site Hubbard repulsion is depicted by the ellipse in gray.Reuse & Permissions
Figure 13
Low-energy spectrum of two-orbital model for $N = 8$ (marked by red crosses) and $N = 10$ (green plus signs) particles on a $(N_{x}, N_{y}) = (6, N / 2)$ lattice, with energies shifted by $E_{0}$ , the lowest energy for each system size. We only show the lowest excited level in each momentum sector in addition to the threefold ground state.Reuse & Permissions
Figure 14
Energy gap $Δ E$ of the two-orbital model for different system sizes and aspect ratios. In each case, $N_{x} = 3 N / N_{y}$ . The negative values of $Δ E$ in the $N_{y} = 3$ group mean that the actual ground state is not in the momentum sectors predicted by the theoretical counting rule.Reuse & Permissions
Figure 15
Spectral flow of the low-lying states of the two-orbital model with $N = 10$ particles on the $N_{x} \times N_{y} = 6 \times 5$ lattice upon flux insertion along the $x$ direction. $γ_{x}$ counts the number of fluxes inserted. The threefold ground states flow into each other and do not mix with higher states during flux insertion. It takes three full fluxes for the threefold states to return to the original configuration (inset).Reuse & Permissions
Figure 16
Particle entanglement spectrum of the ground state of the two-orbital model of $N = 10$ particles on the $N_{x} \times N_{y} = 6 \times 5$ lattice, with $N_{B} = 5$ particles traced out. The number of states below the gray line is 776 in momentum sectors with $k_{y} = 0$ and 775 in all the other sectors, in agreement with the analytical counting rule.Reuse & Permissions
Figure 17
The distribution of the energy gap $Δ E$ in the $(t_{2}, M)$ plane for the two-orbital model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The color code for $Δ E$ is shown by the bar on the right, and the region with the gap closed is marked in gray. The bold red line $M = 4 t_{2}$ separates the topologically nontrivial sector of the single-particle problem from the trivial one.Reuse & Permissions
Figure 18
The distribution of the energy spread $δ E$ in the $(t_{2}, M)$ plane for the two-orbital model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The color code for $δ E$ is shown by the bar on the right. The bold red line $M = 4 t_{2}$ separates the topologically nontrivial sector of the single-particle problem from the trivial one.Reuse & Permissions
Figure 19
The distribution of the entanglement gap $Δ ξ$ in the $(t_{2}, M)$ plane for the ground state of the two-orbital model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice, with $N_{B} = 5$ particles traced out. The color code for $Δ ξ$ is shown by the bar on the right, and the region with the gap closed is marked in gray. The average spacing of the entanglement levels below the gap is at most $0.05$ (not shown in the figure); hence, the entanglement gap should be considered widely open as long as $Δ ξ ≫ 0.05$ . The line with $Δ ξ = 0.5$ is shown in magenta. The bold red line $M = 4 t_{2}$ separates the topologically nontrivial sector of the single-particle problem from the trivial one.Reuse & Permissions
Figure 20
The distribution of the standard deviation of the Berry curvature $σ_{B}$ in the $(t_{2}, M)$ plane for the two-orbital model of $N = 8$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The bar on the right shows the color code for the value of $2 π σ_{B}$ . The bold red line $M = 4 t_{2}$ separates the topologically nontrivial sector of the single-particle problem from the trivial one. We focus on the nontrivial sector. The strong fluctuation of the Berry curvature very close to the boundary of the nontrivial sector is not shown.Reuse & Permissions
Figure 21
The Kagome lattice model with NN spin-orbit coupling. The three sublattices $A$ , $B$ , and $C$ are colored in red, green, blue, respectively. The lattice translation vectors are $b_{1} = 2 a_{1}$ , $b_{2} = 2 a_{2}$ . The hopping amplitude between NNs is $t_{1} - i λ_{1}$ in the direction of arrows. The two-body and the three-body interactions between NNs are illustrated by the gray ellipses and triangles, respectively.Reuse & Permissions
Figure 22
Low-energy spectrum of the Kagome lattice model with $N = 8$ (marked by red crosses), $N = 10$ (green plus signs), and $N = 12$ (blue triangles) particles on the $N_{x} \times N_{y} = 6 \times N / 2$ lattice, with energies shifted by $E_{0}$ , the lowest energy for each system size. Also shown (magenta squares) is the spectrum of $N = 12$ particles on the $N_{x} \times N_{y} = 6 \times 6$ lattice in the alternative Kagome model with NNN hoppings at the parameters suggested in the original paper,[32] namely, $(λ_{1}, λ_{2}, t_{2}) = (0.28, 0.2, - 0.3)$ .Reuse & Permissions
Figure 23
Energy gap $Δ E$ of the Kagome lattice model for different system sizes and aspect ratios. In each case, $N_{x} = 3 N / N_{y}$ .Reuse & Permissions
Figure 24
Spectral flow of the low-lying states of the Kagome lattice model with $N = 10$ particles on the $N_{x} \times N_{y} = 6 \times 5$ lattice upon flux insertion along the $x$ direction. $γ_{x}$ counts the number of fluxes inserted. The threefold ground states flow into each other and do not mix with higher states during flux insertion. It takes three full fluxes for the threefold states to return to the original configuration (inset).Reuse & Permissions
Figure 25
Particle entanglement spectrum of the ground state of the Kagome lattice model of $N = 12$ particles on the $N_{x} \times N_{y} = 6 \times 6$ lattice, with $N_{B} = 7$ particles traced out. The number of states below the gray line is 2530 in each momentum sector, in agreement with the $(1, 3)$ -admissible counting rule. The width of the entanglement gap is $Δ ξ = 4.64$ , meaning that spurious, non-FQH correlations have a probability of $e^{- 4.64} \approx 0.0097$ relative to the universal ones.Reuse & Permissions
Figure 26
The correlation between the standard deviation $σ_{B}$ of Berry curvature and the gap $Δ E$ in the energy spectrum of eight particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice. The calculations are performed at $t_{1} = 1$ . The color of each scatter point encodes the value of $λ_{1}$ per the color bar to the right. A negative correlation between $σ_{B}$ and $Δ E$ is visible.Reuse & Permissions
Figure 27
Particle entanglement spectrum of the ground state of the Kagome lattice model with NNN hoppings of $N = 12$ particles on the $N_{x} \times N_{y} = 6 \times 6$ lattice at $(λ_{1}, λ_{2}, t_{2}) = (0.28, 0.2, - 0.3)$ , with $N_{B} = 7$ particles traced out. The number of states below the gray line is 2530 in each momentum sector, in agreement with the $(1, 3)$ counting rule. The width of the entanglement gap is $Δ ξ = 1.95$ , much smaller than the gap of the model with only NN hoppings shown in Fig. 25.Reuse & Permissions
Figure 28
Low-energy spectrum of the Kagome lattice model with three-body interactions of $N = 8$ (marked by red crosses), $N = 10$ (green plus signs), and $N = 12$ (blue triangles) particles on the $N_{x} \times N_{y} = \frac{N}{2} \times 4$ lattice, with energies shifted by $E_{0}$ , the lowest energy for each system size. We only show the lowest excited level in each momentum sector in addition to the sixfold ground state.Reuse & Permissions
Figure 29
Energy gap $Δ E$ of the Kagome lattice model with three-body interactions for different system sizes and aspect ratios. In each case, $N_{x} = 2 N / N_{y}$ .Reuse & Permissions
Figure 30
Spectral flow of the low-lying states of the Kagome lattice model with three-body interaction of $N = 10$ particles on the $N_{x} \times N_{y} = 5 \times 4$ lattice upon flux insertion along the $y$ direction. $γ_{y}$ counts the number of fluxes inserted. The sixfold ground states flow into each other and do not mix with higher states during flux insertion. After insertion of every two full fluxes, the sixfold states return to the original configuration (inset).Reuse & Permissions
Figure 31
Particle entanglement spectrum of the ground state of the Kagome lattice model with three-body interactions of $N = 12$ particles on the $N_{x} \times N_{y} = 6 \times 4$ lattice, with $N_{B} = 6$ particles traced out. The number of states below the gray line is 2910 in momentum sectors with $k_{x} = 1, 3$ and $k_{y} = 0, 2$ , 2912 in sectors $(0, 0)$ and $(0, 2)$ , 2940 in sectors with $k_{x} = 5$ and sectors with $k_{x} = 2, 4$ and $k_{y} = 2, 4$ , and 2944 in all the other sectors, in agreement with the $(2, 4)$ -admissible counting rule. The width of the entanglement gap is $Δ ξ = 1.66$ .Reuse & Permissions
Figure 32
The ruby lattice model. The six sublattices 1 to 6 are colored, respectively, in red, cyan, green, yellow, blue, and magenta. The lattice translation vectors are $b_{1}$ , $b_{2}$ . The complex hopping parameters between NNs are $\tilde{t} = t_{r} + i t_{i}$ for sites having the same parity (black arrows) and ${\tilde{t}}_{1} = t_{1 r} + i t_{1 i}$ for sites having opposite parity (purple arrows), both in the direction of the arrows. The real hopping parameter on the diagonal of the square (olive dashed lines) is given by $t_{4}$ . The density-density repulsion between NNs is depicted by gray ellipses.Reuse & Permissions
Figure 33
Particle entanglement spectrum of the ground state of the ruby lattice model of $N = 12$ particles on the $N_{x} \times N_{y} = 6 \times 6$ lattice, with $N_{B} = 8$ particles traced out. The number of states below each of the three gray lines is 741, 1629, and 1645 in momentum sectors where both $k_{x}$ and $k_{y}$ are even and 728, 1612, and 1628 elsewhere. These numbers are in perfect accordance with the $(1, 3)$ -, $(3, 7)$ -, and $(3, 5)$ -admissible counting, respectively. The width of the $(1, 3)$ entanglement gap is $Δ ξ = 5.07$ . The counting of levels below the unmarked gaps do not correspond to any $(k, r)$ -admissible rule.Reuse & Permissions
Figure 34
Particle entanglement spectrum of the ground state of the Kagome lattice model of $N = 12$ particles on the $N_{x} \times N_{y} = 6 \times 6$ lattice, with $N_{B} = 9$ particles traced out. The number of states below each of the three gray lines is 138, 195, and 198 in momentum sectors where $k_{x}, k_{y} \in {0, 3}$ and 135, 192, and 195 elsewhere. These numbers are in perfect accordance with the $(1, 3)$ -, $(2, 5)$ -, and $(2, 4)$ -admissible counting, respectively.Reuse & Permissions

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