Non-Abelian fractional Chern insulators from long-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.

Figure 1

Flattening of the lowest band(s) with longer-ranged hopping on a square lattice. Top: Distribution of eigenvalues for the Hofstadter problem—a nearest-neighbor hopping model on a square lattice in a uniform gauge field of strength $\varphi$ quanta per plaquette, with the magnitude of the hopping matrix element set to unity. Bottom: The same distributions for the Kapit-Mueller Hamiltonian, (1), with the nearest-neighbor hopping matrix element set to unity as well and a shift of the whole spectrum for a better comparison with the top figure. On an $L_1\times L_2$ lattice with magnetoperiodic boundary conditions, the lowest $\varphi L_1{L}_2$ states (those within the shaded triangle) collapse to an exactly degenerate lowest Landau level. As $\varphi \to 1$ the hopping becomes infinite ranged in this Hamiltonian, as seen in the diverging energy of the highest band. We only study $1/8\le \varphi \le 1/2$ in this work.

Figure 2

Berry curvature, rescaled by a multiplicative factor of $2\pi /q$, of the lowest band of the Kapit-Mueller Hamiltonian with (a) $\varphi =1/2$, (b) $\varphi =1/3$, (c) $\varphi =1/4$, and (d) $\varphi =1/8$, in $1/q$ of the Brillouin zone. The Berry curvature distribution is always inhomogeneous for finite $\varphi$, but the variations quickly decay as $\varphi \to 0$. See color bars for the rapidly changing scale.

Figure 3

Numerical evidence for bosonic Moore-Read states at $\nu =1$ for (a–d) $\varphi =1/3$ and (e–h) $\varphi =1/4$, with two-body on-site and dipolar interactions, (2). We choose $U=1.0$ for $\varphi =1/3$ and $U=-0.6$ for $\varphi =1/4$. (a, e) Energy spectra for $N_b$ bosons in a square lattice of $q{N}_1\times N_2$ sites. (b, f) Finite-size scaling of both energy gap and ground-state splitting. The scaling behavior depends not only on the system size but also on the aspect ratio $L_1/L_2$ of the lattice. (f) The gap even goes up when the system size increases because the sample becomes more isotropic. (c, g) The $x$-direction spectral flow for $N_b=14$ bosons, $N_1=2$, and $N_2=7$. (d, h) Particle-cut entanglement spectra for $N_b=12$ bosons, $N_1=2$, and $N_2=6$. The number of levels below the gap (solid line) is 3430, matching the quasihole excitation counting of MR states.

Figure 4

Numerical evidence for bosonic Read-Rezayi states at $\nu =3/2$ for (a) $\varphi =1/4$ and (b–d) $\varphi =1/8$, with two-body on-site and dipolar interactions, (2). We choose $U=-0.6$ for $\varphi =1/4$ and $U=-2.7$ for $\varphi =1/8$. (a, b) Energy spectra for $N_b$ bosons in the square lattice of $q{N}_1\times N_2$ sites. (c) Finite-size scaling of both energy gap and ground-state splitting for $\varphi =1/8$. (d) Particle-cut entanglement spectra for $N_b=12$ bosons, $N_1=1$, $N_2=8$, and $\varphi =1/8$. The number of levels below the gap (solid line) is 884, matching the quasihole excitation counting of RR states.

Figure 5

Two-particle spectrum for the on-site interaction on a $12\times 12$ lattice for (a) $\varphi =1/2$, (b) $\varphi =1/3$, and (c) $\varphi =1/4$ and on a $16\times 16$ lattice for (d) $\varphi =1/8$.

Figure 6

Two-particle spectrum on a $12\times 12$ lattice for (a) $\varphi =1/3,N_1\times N_2=4\times 12$ and (b) $\varphi =1/4,N_1\times N_2=3\times 12$, with dipolar interactions, (2). We choose the on-site interaction as $U=1.0$ for $\varphi =1/3$ and $U=-0.6$ for $\varphi =1/4$. Only the largest eight levels are shown.

Figure 7

Numerical evidence for bosonic non-Abelian states at $\nu =1$ and $\nu =3/2$, with two-body interactions, (7). (a) Energy spectra and (b) finite-size scaling of the energy gap and ground-state splitting at $\nu =1$ for $\varphi =1/2$. We choose $n_{\max }=2$ with $V_1=0.65$ and $V_2=0.4$. (c) Energy spectra at $\nu =3/2$ for $\varphi =1/3$. We choose $n_{\max }=4$ with $V_1=1$, $V_2=V_3=0.55$, and $V_4=0.35$. (d) Energy spectra at $\nu =3/2$ for $\varphi =1/4$. We choose $n_{\max }=4$ with $V_1=0.9$, $V_2=0.7$, $V_3=0.6$, and $V_4=0.2$.