Higher Berry Phase from Projected Entangled Pair States in (2+1) dimensions

We consider families of invertible many-body quantum states in $d$ spatial dimensions that are parameterized over some parameter space $X$. The space of such families is expected to have topologically distinct sectors classified by the cohomology group $\mathrm{H}^{d+2}(X;\mathbb{Z})$. These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work, we introduced a generalized inner product for three one-dimensional many-body quantum states, (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in $\mathrm{H}^{3}(X;\mathbb{Z})$, can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in $\mathrm{H}^{4}(X;\mathbb{Z})$. Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of non-trivial parameterized families of 2d invertible states parameterized over $\mathbb{R}P^4$ and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.