Fractional Hall Conductivity and Spin-c Structure in Solvable Lattice Hamiltonians

TL;DR

This paper demonstrates that certain (2+1)d $U(1)$ symmetric abelian topological orders with Hall conductivity can be approximated by controllably solvable lattice Hamiltonians, overcoming a known no-go theorem, and introduces a lattice spin-c structure for fermionic cases.

Contribution

It constructs low-energy controllably solvable lattice Hamiltonians for $U(1)$ symmetric abelian topological orders, and introduces the lattice spin-c structure for fermionic topological orders.

Findings

Constructed lattice Hamiltonians that are controllably solvable at low energies.

Showed the no-go theorem does not prevent lattice study of these topological orders.

Introduced the lattice notion of spin-c structure for fermionic topological orders.

Abstract

The Kapustin-Fidkowski no-go theorem forbids $U(1)$ symmetric topological orders with non-trivial Hall conductivity in (2+1)d from admitting commuting projector Hamiltonians, where the latter is the paradigmatic method to construct exactly solvable lattice models for topological orders. Even if a topological order would intrinsically have admitted commuting projector Hamiltonians, the theorem forbids so once its interplay with $U(1)$ global symmetry which generates Hall conductivity is taken into consideration. Nonetheless, in this work, we show that for all (2+1)d $U(1)$ symmetric abelian topological orders of such kind, we can construct a lattice Hamiltonian that is controllably solvable at low energies, even though not "exactly" solvable; hence, this no-go theorem does not lead to significant difficulty in the lattice study of these topological orders. Moreover, for the fermionic topological orders in our construction, we introduce the lattice notion of spin-c structure -- a concept important in the continuum that has previously not been adequately introduced in the lattice context.