Our understanding of the geometric structures underlying the space of gapped quantum many-body states has been steadily advancing. In particular, one-dimensional gapped systems are known to support a three-form defined on the parameter space, which generalizes the conventional two-form Berry curvature familiar from quantum mechanics. In this talk, I will discuss how group actions on both the parameter space and the Hilbert space impose constraints on the global structure of invertible states, using the matrix product state representation. As an application of this framework, we demonstrate that the phase transition between the Haldane chain phase and the trivial phase acts as a source of the three-form higher Berry curvature.
Reference: Ken Shiozaki, "Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation", arXiv:2507.19932, doi.org/10.48550/arXiv.2507.19932.
