Classification of gapped phases in d≥2
Organize classification of uniformly gapped local Hamiltonian families in under gapped-path equivalence, fixing bosonic or fermionic degrees of freedom, symmetries, ground-space structure and stabilization conventions. Use explicit subproblems for intr
Problem statement
Organize classification of uniformly gapped local Hamiltonian families in under gapped-path equivalence, fixing bosonic or fermionic degrees of freedom, symmetries, ground-space structure and stabilization conventions. Use explicit subproblems for intrinsic order, symmetry enrichment and boundaries.
Lean 4
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Background
A uniformly gapped local path induces quasi-local spectral flow under the applicable hypotheses. This is not an unrestricted assertion of exact finite-depth-circuit equality: approximation accuracy, circuit range, ancillary product states and thermodynamic conventions matter. These choices form part of the equivalence relation, not an afterthought.
What is known: The 1D bosonic unique-ground-state setting has a controlled classification: without symmetry it is trivial under the stated phase conventions, and on-site symmetry yields the familiar projective-representation/SPT classification. Symmetry breaking, degenerate ground spaces and fermionic parity introduce additional cases. The phrase 'all 1D gapped phases are trivial' would erase them. In 2D, anyon categories, symmetry-fractionalization data and invertible phases provide substantial but not universally complete classifications.
But 2D and 3D are a major, still-incomplete programme mixing physics with higher category theory and cobordism. Chiral phases (nonzero chiral central charge, no gapped boundary), the precise role of invertible/anomalous phases, fermionic and crystalline cases, and a fully rigorous equivalence relation are not settled. Related: B9.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Chen, Gu, Liu & Wen, 'Symmetry protected topological orders and the group cohomology of their symmetry group', PRB 87, 155114 (2013) link
- [2]Kitaev, 'Anyons in an exactly solved model and beyond', Ann. Phys. 321 (2006) link
- [3]Senthil, 'Symmetry-protected topological phases of quantum matter', Annu. Rev. Condens. Matter Phys. 6 (2015) link
- [4]Schuch, Pérez-García & Cirac, classification with symmetry and ground-space distinctions link
Recorded progress
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