A hierarchy of SDPs yields lower bounds on spectral gaps of frustration-free Hamiltonians that encompass and improve upon Knabe's bound on 1D spin chains.
Nachtergaele, The spectral gap for some spin chains with discrete symmetry breaking, Communications in Mathematical Physics 175, 565 (1996), arXiv:cond- mat/9410110
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A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
A hierarchy of SDPs yields lower bounds on spectral gaps of frustration-free Hamiltonians that encompass and improve upon Knabe's bound on 1D spin chains.