Spectral gaps of PEPS parent Hamiltonians
For a stated two-dimensional PEPS tensor class, give verifiable sufficient tensor or boundary-state conditions implying a system-size-independent bulk gap of its parent Hamiltonian, with an explicit lower bound.
Problem statement
For a stated two-dimensional PEPS tensor class, give verifiable sufficient tensor or boundary-state conditions implying a system-size-independent bulk gap of its parent Hamiltonian, with an explicit lower bound.
Lean 4
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Background
Projected entangled-pair states (PEPS) are the 2D generalization of matrix product states (MPS); each family comes with a 'parent Hamiltonian' having the PEPS as ground state. Deciding whether that parent Hamiltonian is gapped in the thermodynamic limit controls whether the state represents a stable gapped phase.
What is known: Injective MPS have gapped parent Hamiltonians with the appropriate boundary/interaction-length conventions. In 2D, ordinary injectivity gives parent-state uniqueness under its hypotheses, whereas G-injective PEPS deliberately accommodate topological ground-state degeneracy, including on a torus. Neither property by itself supplies a uniform 2D gap. Additional boundary-factorization or finite-size conditions yield positive results in controlled classes.
Undecidability of the unrestricted Hamiltonian gap problem (U1) is important background, not automatically a theorem for each promised PEPS class: a reduction must preserve that class's assumptions. A successful sufficient certificate may leave other instances undecided. Specify tensor conditioning, boundary locality and the meaning of any transfer-operator gap rather than treating correlation length alone as a complete criterion.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Cirac, Pérez-García, Schuch & Verstraete, 'Matrix product states and projected entangled pair states: Concepts, symmetries, and theorems', Rev. Mod. Phys. 93, 045003 (2021) link
- [2]Pérez-García, Verstraete, Wolf, Cirac, 'PEPS as unique ground states of local Hamiltonians', QIC 8 (2008) link
- [3]Schuch, Cirac & Pérez-García, 'PEPS as ground states: Degeneracy and topology' (2010) link
Recorded progress
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