The Berry curvature is a geometric quantity that underlies parameter-dependent quantum mechanics. Recently, the concept of a higher dimensional generalization of the Berry curvature, known as the higher Berry curvature, has been discussed [1]. In d-spatial dimensions, the higher Berry curvature is proposed as a (d+2)-form that is defined for short-range entangled states. Providing a computationally feasible method for calculating the higher Berry curvature for a given pure state is challenging. In our recent work [2], we propose an explicit method to compute the higher Berry curvature for a given translationally invariant matrix product state in 1-dimensional systems. This method is computationally applicable for any gapped Hamiltonian with unique ground state. We showed that summing up all the higher Berry fluxes across tetrahedra results in a nontrivial integer associated with the 3rd cohomology H^3(X,Z).
 

[1] A. Kapustin and L. Spodyneiko, Higher-dimensional generalizations of berry curvature, Phys. Rev. B 101, 235130 (2020).
[2] Ken Shiozaki, Niclas Heinsdorf, Shuhei Ohyama, Higher Berry curvature from matrix product states, arXiv:2305.08109.

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