Improved bounds on the spectral gap above frustration free ground states of quantum spin chains

TL;DR

This paper improves lower bounds on the spectral gap for frustration free quantum spin chains, providing sharper estimates and applying the method to models like the ferromagnetic XXZ chain and AKLT model.

Contribution

It introduces simple improvements to existing bounds, enabling sharper spectral gap estimates for specific quantum spin models.

Findings

Sharp lower bound for ferromagnetic XXZ chain

Lower bound estimate for AKLT model's spectral gap

Enhanced bounds using the martingale method

Abstract

In \cite{Nac2,Nac1} 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.