Higher cup products on hypercubic lattices: Application to lattice models of topological phases

In this paper, we derive the explicit formula for higher cup products on hypercubic lattices based on the recently developed geometrical interpretation on the simplicial complexes. We illustrate how this formalism can elucidate lattice constructions on hypercubic lattices for various models and derive them from spacetime actions. In particular, we demonstrate explicitly that the (3 + 1)D SPT S=(1/)(2 integral)w(2)(2)+w(1)(4 )(where w1 and w(2) are the first and second Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang model constructed on the cubic lattice. Other examples include the double-semion model and also the fermionic toric code in arbitrary dimensions on hypercubic lattices. In addition, we extend previous constructions of exact boson-fermion dualities and the Gu-Wen Grassmann integral to arbitrary dimensions. Another result that may be of independent interest is a derivation of a cochain-level action for the generalized double-semion model, reproducing a recently derived action on the cohomology level.

Automated summary

In this paper, we derive explicit formula for higher cup products on hypercubic lattices and demonstrate their applications in various models including (3+1)D SPT materials, the double-semion model, and the fermionic toric code. We also extend the constructions of exact boson-fermion dualities and the Gu-Wen Grassmann integral to arbitrary dimensions, and derive a cochain-level action for the generalized double-semion model.