Fractional Chern Insulator

Abstract

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.

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Plain Language 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.