A many-body Fredholm index for ground state spaces and Abelian anyons

Abstract

We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered $p$-dimensional ground state sector. The index is fractional with the denominator given by $p$. In particular, this yields a new short proof of the quantization of the Hall conductance and of Lieb-Schulz-Mattis theorem. In the case that the index is non-integer, the argument provides an explicit construction of Wilson loop operators exhibiting a non-trivial braiding and that can be used to create fractionally charged Abelian anyons.

8 pages, 2 figures, published in Phys. Rev. B

References in corpus (5)

• Non-Abelian Anyons and Topological Quantum Computation
• Classification of topological insulators and superconductors in three spatial dimensions
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• A Multi-Dimensional Lieb-Schultz-Mattis Theorem
• Locality in Quantum Systems

Cited by in corpus (5)

• Hall conductance and the statistics of flux insertions in gapped interacting lattice systems
• Rational indices for quantum ground state sectors
• Properties of 2D anyon gas
• On the absence of stationary currents
• Dynamical abelian anyons with bound states and scattering states