Isotropic modulating Hamiltonian approach to quantum many-body invertible phases

Understanding phase diagrams of quantum many-body systems is important in condensed matter physics, but quite often difficult due to complicated interactions. Pumping procedures, e.g., Thouless pumpings¹ and its generalizations^2,3,4,5, have been used to show the existence of noncontractible loops in the phase diagram of invertible systems --- with a unique gapped ground state on general closed lattices. In this talk, we will discuss how to find and construct isotropic noncontractible (hyper-)spheres in such a phase diagram by modulating Hamiltonians. Our result shows that those noncontractibility or homotopy groups can be related to the classification of invertible phases or the family of Hamiltonians in a one-to-one manner.

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2. A. Kapustin, & L. Spodyneiko, arxiv-preprint 2003.09519.

3. K. Shiozaki, Phys. Rev. B 106, 125108 (2022).

4. X. Wen et. al., arxiv-preprint 2112.07748.

5. Y. Yao, M. Oshikawa, & A. Furusaki, Phys. Rev. Lett. 129,017204 (2022).