[Paper Review] Fractional Chern Insulator

Nicolas Regnault, B. Andrei Bernevig|arXiv (Cornell University)|May 24, 2011

Topological Materials and PhenomenaPhysics and Astronomy21 references182 citations

TL;DR

This paper provides conclusive numerical evidence that a fractional quantum Hall effect (FQHE) can emerge in a Chern insulator at 1/3 filling without an external magnetic field, by diagonalizing the Hubbard interaction in a flat-band limit. The system exhibits a 3-fold degenerate ground state with a finite energy gap, incompressibility, constant momentum-space density of 1/3, and an entanglement spectrum matching Laughlin quasihole counting—strongly indicating a fractional Chern insulator phase.

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.

Motivation & Objective

• To establish the existence of a fractional quantum Hall state in a topological insulator without Landau levels.
• To distinguish the fractional Chern insulator from competing charge-density wave states.
• To demonstrate that the ground state exhibits topological order via incompressibility, degeneracy, and fractional statistics.
• To confirm the emergence of Laughlin-like quasihole excitations and a robust entanglement gap.

Proposed method

• Exact diagonalization of the Hubbard interaction in the flat-band limit of a Chern insulator model on a checkerboard lattice.
• Use of a generalized Pauli exclusion principle forbidding more than one particle in three consecutive orbitals.
• Computation of momentum-space density to rule out charge-density wave order.
• Analysis of spectral flow under flux insertion to confirm topological degeneracy.
• Calculation of the entanglement spectrum and identification of a finite entanglement gap matching Laughlin quasihole counting.
• Study of the system's response to phase transitions in the single-particle Hamiltonian, comparing topological and trivial insulator regimes.

Experimental results

Research questions

• RQ1 Can a fractional quantum Hall state emerge in a Chern insulator at 1/3 filling without an external magnetic field?
• RQ2 How can one distinguish a fractional Chern insulator from a competing charge-density wave state?
• RQ3 Do the many-body ground states exhibit topological order, including fractional statistics and incompressibility?
• RQ4 Is the entanglement spectrum of the ground state consistent with that of a Laughlin state?
• RQ5 Does the entanglement gap remain finite in the thermodynamic limit as the system undergoes a topological phase transition?

Key findings

• The system exhibits a 3-fold degenerate ground state at 1/3 filling, with a finite energy gap that persists in the thermodynamic limit.
• The momentum-space density is uniform and equal to 1/3, ruling out charge-density wave order.
• Spectral flow under flux insertion confirms topological degeneracy without level repulsion.
• The entanglement spectrum displays a clear gap, with counting below the gap matching that of Laughlin quasihole states.
• The entanglement gap remains finite and tracks the many-body energy gap across the topological phase transition.
• Excitations match the counting of Laughlin quasiholes, confirming fractional statistics and topological order.