Double semions in arbitrary dimension

TL;DR

This paper generalizes the double semion topological quantum field theory to higher dimensions, constructing a local Hamiltonian and analyzing excitations, ground state degeneracy, and the role of higher Betti numbers in the wavefunction.

Contribution

It introduces a higher-dimensional extension of the double semion theory with a local Hamiltonian and explores its topological properties and relation to other models.

Findings

For odd dimensions, related to the toric code by a local unitary transformation.

For even dimensions, ground state degeneracy differs from the toric code.

Higher Betti numbers influence the ground state wavefunction structure.

Abstract

We present a generalization of the double semion topological quantum field theory to higher dimensions, as a theory of $d-1$ dimensional surfaces in a $d$ dimensional ambient space. We construct a local Hamiltonian which is a sum of commuting projectors and analyze the excitations and the ground state degeneracy. Defining a consistent set of local rules requires the sign structure of the ground state wavefunction to depend not just on the number of disconnected surfaces, but also upon their higher Betti numbers through the semicharacteristic. For odd $d$ the theory is related to the toric code by a local unitary transformation, but for even $d$ the dimension of the space of zero energy ground states is in general different from the toric code and for even $d>2$ it is also in general different from that of the twisted $Z_2$ Dijkgraaf-Witten model.