  The higher Berry curvature was introduced by Kapustin and Spodyneiko as an
extension of the Berry curvature in quantum mechanical systems with finite
degrees of freedom to quantum many-body systems in finite spatial dimensions.
In this paper, we propose an alternative formulation of the higher Berry
curvature using translationally invariant matrix product states. They are the
ground states of a set of gapped Hamiltonians which are evolved adiabatically
through a discretized parameter space. Because matrix product states transform
under a projective representation, evaluating the Berry curvature on a closed
loop through parameter space is not sufficient to fix all the gauge degrees of
freedom. To obtain a gauge-invariant real quantity, the higher-dimensional
Berry curvature is evaluated on small tetrahedra in parameter space. Our
numerical calculations confirm that the higher Berry curvature varies
continuously throughout an adiabatic evolution and becomes quantized over a
closed 3-dimensional parameter space.
