[Paper Review] Topological Current in Fractional Chern Insulators

Tohru Koma|arXiv (Cornell University)|Apr 6, 2015

Topological Materials and Phenomena24 references3 citations

TL;DR

This paper establishes fractional quantization of the Hall conductance in interacting fermion systems on a 2D lattice without averaging over fluxes, by introducing a local current operator at the edge of a compactly supported electric potential. Using a topological argument based on twisted boundary conditions and spectral gap stability, it proves that the Hall conductance is quantized to a rational fraction for degenerate ground states in gapped systems, extending noninteracting topological quantization to interacting fractional Chern insulators.

ABSTRACT

We consider interacting fermions in a magnetic field on a two-dimensional lattice with the periodic boundary conditions. In order to measure the Hall current, we apply an electric potential with a compact support. Then, due to the Lorentz force, the Hall current appears along the equipotential line. Introducing a local current operator at the edge of the potential, we derive the Hall conductance as a linear response coefficient. For a wide class of the models, we prove that if there exists a spectral gap above the degenerate ground state, then the Hall conductance of the ground state is fractionally quantized without averaging over the fluxes. This is an extension of the topological argument for the integrally quantized Hall conductance in noninteracting fermion systems on lattices.

Motivation & Objective

• To establish fractional quantization of the Hall conductance in interacting lattice fermion systems with a spectral gap above a degenerate ground state.
• To overcome the limitation of constant electric fields in lattice systems by using a compactly supported potential to induce measurable Hall current.
• To extend the topological argument for integral quantization in noninteracting systems to interacting, fractional Chern insulator models.
• To demonstrate that fractional quantization holds without averaging over twisted boundary conditions, ensuring robustness for fixed fluxes.
• To prove the stability of the spectral gap under twisted boundary conditions, a key step in the topological argument.

Proposed method

• Introduce a local current operator at the edge of a compactly supported electric potential to measure Hall current, avoiding issues with constant electric fields on lattices.
• Define the Hall conductance as a linear response coefficient using the local current operator and the electric potential perturbation.
• Apply twisted boundary conditions to the hopping amplitudes, parameterized by angles φ₁ and φ₂, to probe topological response.
• Use the spectral gap above the degenerate ground state to ensure the system's response is well-defined and topologically protected.
• Employ a topological argument based on the Chern number of the ground state bundle over the torus of twisted boundary conditions.
• Prove that the Hall conductance is quantized to a rational fraction by integrating the current response over the torus and using periodicity of the wavefunctions.

Experimental results

Research questions

• RQ1 Can fractional quantization of the Hall conductance be rigorously established in interacting fermion systems on a 2D lattice without averaging over fluxes?
• RQ2 How can a measurable Hall current be defined in lattice systems where constant electric fields fail due to spectral changes?
• RQ3 Is the Hall conductance robust against deformations of the current operator, preserving fractional quantization?
• RQ4 Does the spectral gap above the degenerate ground state remain stable under twisted boundary conditions, enabling topological classification?
• RQ5 Can the topological argument for integral quantization in noninteracting systems be extended to interacting, fractionally quantized Hall states?

Key findings

• The Hall conductance of the ground state is fractionally quantized as a rational number when there is a spectral gap above a degenerate ground state, without averaging over fluxes.
• The fractional quantization is robust under deformation of the current operator, indicating topological stability of the response.
• The spectral gap above the degenerate ground state remains stable under twisted boundary conditions, a crucial assumption for the topological argument.
• The Hall conductance is derived as a linear response coefficient using a local current operator defined at the edge of a compactly supported potential.
• Integration of the current response over the torus of twisted boundary conditions yields a rational number, confirming fractional quantization via the Chern number.
• The proof relies on the periodicity of the ground state wavefunctions under flux threading and the unitarity of the phase evolution, ensuring the boundary terms vanish.