Invariants for Gapped Ground State Phases

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States with a uniform spectral gap map to each other by a quasi-local quasi-adiabatic continuation (spectral flow) that commutes with on-site symmetry, so the symmetry representation on a half-infinite chain is constant along any gapped path. For 1D frustration-free matrix product ground states the transfer map has a simple eigenvalue 1 and a transfer operator gap giving exponential convergence, which lets one construct a cutoff-weighted generator and a well-defined boundary representation (AKLT: bulk spin-1 → two-dimensional spin-1/2 edge), hence a computable bulk invariant.

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1. 0:00Introduction
2. 0:12Symmetry protected phases
3. 5:50Defining ground state phases
4. 9:11Limits of ground states
5. 11:05Role of invariants
6. 14:00Detecting phase transitions
7. 16:28Understanding level crossing
8. 17:38Local symmetry properties
9. 20:53AKLT model example
10. 24:16Adiabatic evolution
11. 26:43Equivalence of ground states
12. 30:06AKLT boundary states
13. 33:41Symmetry in AKLT
14. 35:30Detecting invariants
15. 37:25MPS ground states
16. 38:40MPS formalism review
17. 44:10Properties of the map
18. 47:20Bulk calculation
19. 50:30Defining symmetry action
20. 55:13Calculation theorem
21. 1:00:14Final results
22. 1:03:44Extensions and closing