Locality at the boundary implies gap in the bulk for 2D PEPS

Abstract

Proving that the parent Hamiltonian of a Projected Entangled Pair State (PEPS) is gapped remains an important open problem. We take a step forward in solving this problem by showing two results: first, we identify an approximate factorization condition on the boundary state of rectangular subregions that is sufficient to prove that the parent Hamiltonian of the bulk 2D PEPS has a constant gap in the thermodynamic limit; second, we then show that Gibbs state of a local, finite-range Hamiltonian satisfy such condition. The proof applies to the case of injective and MPO-injective PEPS, employs the martingale method of nearly commuting projectors, and exploits a result of Araki on the robustness of one dimensional Gibbs states. Our result provides one of the first rigorous connections between boundary theories and dynamical properties in an interacting many body system.

31 pages, 9 figures. Revised version

References in corpus (11)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Criticality, the area law, and the computational power of PEPS
• Resonating valence bond states in the PEPS formalism
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Chiral topological spin liquids with projected entangled pair states
• Chiral projected entangled-pair state with topological order
• Divide and conquer method for proving gaps of frustration free Hamiltonians
• Implementation of rigorous renormalization group method for ground space and low-energy states of local Hamiltonians
• Entanglement Hamiltonian of the quantum Néel state
• Entanglement properties of the two-dimensional SU(3) AKLT state

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• A remark on matrix product operator algebras, anyons and subfactors
• The minimal canonical form of a tensor network
• Thermalization in Kitaev's quantum double models via Tensor Network techniques
• Projector Matrix Product Operators, Anyons and Higher Relative Commutants of Subfactors
• Local topological order and boundary algebras
• Classical restrictions of generic matrix product states are quasi-locally Gibbsian
• Deformations of the Boundary Theory of the Square Lattice AKLT Model
• On a bulk gap strategy for quantum lattice models