Improved Bounds on the Spectral Gap Above Frustration-Free Ground States of Quantum Spin Chains

Nachtergaele obtained explicit lower bounds for the spectral gap above many frustration free quantum spin chains by using the martingale method. We present simple improvements to his main bounds which allow one to obtain a sharp lower bound for the spectral gap above the spin-1/2 ferromagnetic XXZ chain. As an illustration of the method, we also calculate a lower bound for the spectral gap of the AKLT model, which is about 1/3 the size of the expected gap.