Quantization of Chern-Simons topological invariants for H-type and L-type quantum systems

TL;DR

This paper derives quantization conditions for topological invariants like chiral central charge and filling fraction in 2+1D topological phases using a cobordism approach, linking them to ground state degeneracies and spacetime manifold types.

Contribution

It introduces a cobordism-based method to define and derive quantization conditions for Chern-Simons topological invariants in topological quantum systems.

Findings

Quantization conditions depend on ground state degeneracies on Riemann surfaces.

Quantization conditions depend on the topology of spacetime manifolds.

Derived invariants relate to thermal Hall conductance and Hall conductance.

Abstract

In 2+1-dimensions (2+1D), a gapped quantum phase with no symmetry (i.e. a topological order) can have a thermal Hall conductance $\kappa_{xy}=c \frac{\pi^2 k_B^2}{3h}T$, where the dimensionless $c$ is called chiral central charge. If there is a $U_1$ symmetry, a gapped quantum phase can also have a Hall conductance $\sigma_{xy}=\nu \frac{e^2}{h}$, where the dimensionless $\nu$ is called filling fraction. In this paper, we derive some quantization conditions of $c$ and $\nu$, via a cobordism approach to define Chern--Simons topological invariants which are associated with $c$ and $\nu$. In particular, we obtain quantization conditions that depend on the ground state degeneracies on Riemannian surfaces, and quantization conditions that depend on the type of spacetime manifolds where the topological partition function is non-zero.