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## "On topology of the moduli space of gapped Hamiltonians for topological phases", Po-Shen Hsin, Zhenghan Wang, 2023

*Reviewed December 29, 2023*

*Citation:* Hsin, Po-Shen, and Zhenghan Wang. "On topology of the moduli space of gapped Hamiltonians for topological phases." Journal of Mathematical Physics 64.4 (2023).

*Web:* https://arxiv.org/abs/2211.16535

*Tags:* Ising-computer, SPT/SETs

This paper gives a general study of one of the most striking features of topological quantum computation.
Namely, the fact that symmetries of MTCs can be implemented by adiabatically deforming Hamiltonians
along homotopically non-trivial paths in the moduli space of gapped Hamiltonians realizing a given phase at low energies.
To study this phenominon, the authors make their best attempt to formally define the moduli space of
all gapped Hamiltonians realizing a given phase, and then do their best to make concrete claims and prove theorems.
The authors discover a number of interesting results, and give a few very striking conjectures
about the exact topological space that the moduli space should be homotopic to. However, it seems that
the proper definitions have not been found yet and there is more work to be done in this area going forwards.

Earlier papers have already studied these ideas:

> Hsin, Po-Shen, Anton Kapustin, and Ryan Thorngren. "Berry phase in quantum field theory: Diabolical points and boundary phenomena." Physical Review B 102.24 (2020): 245113.

> Aasen, David, Zhenghan Wang, and Matthew B. Hastings. "Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes." Physical Review B 106.8 (2022): 085122.

This paper is just a more polished and unified approach.