Fractional Chern Insulator

Chern insulators are band insulators exhibiting a nonzero Hall conductance but preserving the lattice translational symmetry. We conclusively show that a partially filled Chern insulator at $1/3$ filling exhibits a fractional quantum Hall effect and rule out charge-density-wave states that have not been ruled out by previous studies. By diagonalizing the Hubbard interaction in the flat-band limit of these insulators, we show the following: The system is incompressible and has a 3-fold degenerate ground state whose momenta can be computed by postulating an generalized Pauli principle with no more than 1 particle in 3 consecutive orbitals. The ground-state density is constant, and equal to $1/3$ in momentum space. Excitations of the system are fractional-statistics particles whose total counting matches that of quasiholes in the Laughlin state based on the same generalized Pauli principle. The entanglement spectrum of the state has a clear entanglement gap which seems to remain finite in the thermodynamic limit. The levels below the gap exhibit counting identical to that of Laughlin $1/3$ quasiholes. Both the 3 ground states and excited states exhibit spectral flow upon flux insertion. All the properties above disappear in the trivial state of the insulator—both the many-body energy gap and the entanglement gap close at the phase transition when the single-particle Hamiltonian goes from topologically nontrivial to topologically trivial. These facts clearly show that fractional many-body states are possible in topological insulators.

Popular Summary

Asking a condensed matter physicist what the current hot topics are in her field, you are very likely to get this answer: the fractional quantum Hall effect and topological insulators. The former, the discovery of which won a Nobel Prize in Physics, was first demonstrated in 1982 in a 2-dimensional electron gas created in a semiconductor heterostructure. Since then, it has continued to generate new research. The most recent fascination it draws from physicists lies in the promise that some fractional-quantum-Hall states can host a new type of nonelementary particles—non-Abelian anyons. These exotic particles are quantized excitations that behave neither like fermions nor like bosons, but in manners that are in-between, and are believed to lend themselves as a physical basis for robust “topological” quantum computing. Topological insulators, insulating in their interiors, but conducting along their surfaces or edges, are a set of new and remarkable materials that has galvanized condensed matter physics in the past four years. In this paper, we explore a confluence of these two topical areas, and provide, for the first time, an unambiguous proof of principle that fractional quantum Hall states can also emerge, in fact, in the absence of a magnetic field, from a simple—arguably the simplest—class of topological insulators, the fractional Chern insulators.

To investigate the nature of fractional Chern insulators, we have used two techniques. Using the first, we excite quasiparticles in a system and work out the properties of the nucleated quasi-particles. The second method relies on the recently developed concept of entanglement spectra, which allows us to extract information about the excitations only by probing the ground state with tools borrowed from the field of quantum information. In both cases, the results were unambiguous and demonstrate the fractional nature of the excitations.

Our work should lead new theoretical and experimental explorations of the fractional quantum Hall effect. Experimental realizations of fractional Chern insulators would open another avenue to an easier and more robust access to the strange world of anyons and their potential for topological quantum computing.

Figure 1

Low-energy spectrum for $N=6$, 10, and 12, $N_x=N/2$, $N_y=6$. The energies are shifted by $E_1$, the lowest energy for each system size. We only show the lowest energy per momentum sectors in addition to the 3-fold ground state. We note the good ground-state degeneracy, even for relatively small system sizes such as 6 particles.

Figure 2

Evolution of the 3-fold degenerate ground state upon flux insertion along the $y$ direction at $N=10$, $N_x=5$, and $N_y=6$. The 3-fold degenerate ground-states spectral flow into each other (inset) separated at each point in the flux insertion from the first excited state (the only one of the excited states shown here), which does not exhibit spectral flow with any of the other states.

Figure 3

Energy gap $\Delta$ for different system sizes and aspect ratio. The gap is defined as the difference between the energy of the first excited state and the highest energy of the 3-fold ground-state manifold. In each case, $N_x=3N/N_y$.

Figure 4

Ratio between the energy spread $\delta$ of the 3-fold ground-state manifold and the energy gap $\Delta$ for various system sizes and aspect ratio. The spread is defined as the difference between the highest energy and the lowest energy of the 3-fold ground-state manifold. In each case, $N_x=3N/N_y$.

Figure 5

Occupation number of each of the single-particle momentum orbitals for the three $N=10$ ground states (which occur at different total momenta) at $N_x=5$, $N_y=6$. Notice that the occupation number is uniform and close to the filling factor $1/3$.

Figure 6

Low-energy spectrum for $N=9$, $N_x=5$, $N_y=6$. The number of states below the dashed line is 19 in sectors where $K_y\mathrm{mod}3=0,18$ in other sectors, as expected from the analytical results. The total counting matches the one expected for Laughlin quasihole states. The counting for each momentum sector is given by the generalized Pauli principle and the 2-dimensional to 1-dimensional unfolding presented in the next section.

Figure 7

Low-energy spectrum for $N=7$, $N_x=8$, $N_y=3$. The number of states below the dashed line is 12 in all sectors and matches the number of expected quasihole states given by the generalized Pauli principle.

Figure 8

Evolution of the low-energy spectrum for $N=7$, $N_x=8$, $N_y=3$ at $K_x=2$, $K_y=0$ upon flux insertion along the $x$ direction. The number of states below the dashed line is 12 for any value of $\Phi$ and matches the number of expected quasihole states given by the generalized Pauli principle

Figure 9

Momentum of the 3-fold degenerate ground state for the fractional Chern insulator at filling $\nu =1/3$. The 2-dimensional lattice of momenta $k_x,k_y$ with $k_x\in [0,N_x-1]$ and $k_y\in [0,N_y-1]$ (in units of $2\pi /N_x$, $2\pi /N_y$, respectively) where we have taken $N_x$ to be a multiple of 3 (since $N_x·N_y=3N$, this can always be done) is unfolded in a 1D lattice. The total momentum of the ground states is then the same as the momentum of the three (1,3)-admissible partitions possible for the number of orbitals $N_x·N_y$ and number of particles $N=N_x·N_y/3$.

Figure 10

Example of the total momentum counting for the 3-fold degenerate ground states of the $N=4$ (top) and $N=6$ (bottom) problem. Every time the number of particles $N$ is a multiple of both $k_x,k_y$, the 3-fold degenerate ground states occur at the same total momentum and are expected to be split by the interaction for finite-size samples. Notice that for the $N=4$ problem, the total momenta at which the ground state occurs are related by inversion symmetry, as $(1,2)=(-2,-2)\mathrm{mod}(3,4)$.

Figure 11

Example of total momentum counting for the quasiholes of the $N=2$, $N_x=N_y=3$ problem.

Figure 12

Particle entanglement spectrum for $N=10$, $N_x=5$, $N_y=6$, and $N_A=5$. The $\xi$’s are the entanglement energies. The number of states below the dashed line is 776 in all the $k_x=0$, and 775 in the other sectors. This is in agreement with the (1,3) Pauli principle for every momentum sector.

Figure 13

Particle entanglement spectrum for $N=12$, $N_x=N_y=6$, and $N_A=4$. The $\xi$’s are the entanglement energies. The number of states below the dashed line is 741 in momentum sectors where $N_x\mathrm{mod}2=N_y\mathrm{mod}2=0$ and 728 elsewhere. The total number below this line (26 325) exactly matches the one predicted by the counting rule.

Figure 14

Low-energy spectrum for different values of $M$ at $N=10$, $N_x=5$, and $N_y=6$. We show only the two lowest energies per momentum sector. All energies are shifted such that the ground-state energy is 0 for each $M$ value.

Figure 15

Energy gap $\Delta$ as a function of $M$ at filling $\nu =1/3$ for $N=8$ and $N=10$ with $N_x=N/2$, $N_y=6$. For both system sizes, the transition occurs at $M=4t_2$.

Figure 16

Entanglement spectrum gap ${\Delta }_{\mathrm{PES}}$ as a function of $M$ at filling $\nu =1/3$ for $N=8$ and $N=10$ with $N_x=N/2$, $N_y=6$.