Citation: Kapustin, Anton, and Lukasz Fidkowski. "Local commuting projector Hamiltonians and the quantum Hall effect." Communications in Mathematical Physics 373.2 (2020): 763-769.
Web: https://arxiv.org/abs/1810.07756
Tags: Quantum-hall-effect, No-go
This paper proves that there are no models for the integer or fractional quantum Hall effect which are local commuting projector Hamiltonians with finite-dimensional on-site Hamiltonians. The proof is based on the interpretation of the Hall conductance as the Chern number of a certain vector bundle associated to the Hamiltonian on a torus, as pioneered in
> Niu, Qian, Ds J. Thouless, and Yong-Shi Wu. "Quantized Hall conductance as a topological invariant." Physical Review B 31.6 (1985): 3372.
The idea then is to show using mathematical techniques that this Chern number must vanish. One way of getting around this no-go result is to include infinite dimensional on-site Hilbert spaces. This is exactly what was done by DeMarco and Wen:
> DeMarco, Michael, and Xiao-Gang Wen. "A commuting projector model with a non-zero quantized hall conductance." arXiv preprint arXiv:2102.13057 (2021).
Using this trick of putting infinite-dimensional Hilbert spaces on site Nikita Sopenko has been able to construct all chiral topological phases:
> Sopenko, Nikita. "Chiral topologically ordered states on a lattice from vertex operator algebras." arXiv preprint arXiv:2301.08697 (2023).
As the common wisdom goes: the only thing a no-go theorem proves is a lack of imagination!