Improved local spectral gap thresholds for lattices of finite dimension

Abstract

Knabe's theorem lower bounds the spectral gap of a one dimensional frustration-free local hamiltonian in terms of the local spectral gaps of finite regions. It also provides a local spectral gap threshold for hamiltonians that are gapless in the thermodynamic limit, showing that the local spectral gap much scale inverse linearly with the length of the region for such systems. Recent works have further improved upon this threshold, tightening it in the one dimensional case and extending it to higher dimensions. Here, we show a local spectral gap threshold for frustration-free hamiltonians on a finite dimensional lattice, that is optimal up to a constant factor that depends on the dimension of the lattice. Our proof is based on the detectability lemma framework and uses the notion of coarse-grained hamiltonian (introduced in [Phys. Rev. B 93, 205142]) as a link connecting the (global) spectral gap and the local spectral gap.

version 1, 18 pages, 3 figures

References in corpus (7)

• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Stochastic Error Cancellation in Analog Quantum Simulation
• Spectral gaps of frustration-free spin systems with boundary
• A class of two-dimensional AKLT models with a gap
• Gaplessness is not generic for translation-invariant spin chains
• Finite-size criteria for spectral gaps in $D$-dimensional quantum spin systems
• The AKLT model on a hexagonal chain is gapped

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• Spontaneous breaking of U(1) symmetry at zero temperature in one dimension
• Quantitatively improved finite-size criteria for spectral gaps
• Thermalization in Kitaev's quantum double models via Tensor Network techniques
• Random translation-invariant Hamiltonians and their spectral gaps
• Quadratic dispersion relations in gapless frustration-free systems
• On the critical finite-size gap scaling for frustration-free Hamiltonians
• Rigorous lower bound on dynamical exponents in gapless frustration-free systems
• Rigorous lower bound of the dynamical critical exponent of the Ising model
• A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
• On a bulk gap strategy for quantum lattice models
• The Complexity of Approximating Critical Points of Quantum Phase Transitions
• Frustration-free free fermions
• Frustration-free free fermions and beyond
• Quasi-local Frustration-Free Free Fermions