Lower Bounds on Spectral Gaps of Parent Hamiltonians via Tensor Networks

• URL: http://arxiv.org/abs/2607.19078v1
• Date: Tue, 21 Jul 2026 13:10:20 GMT
• Title: Lower Bounds on Spectral Gaps of Parent Hamiltonians via Tensor Networks
• Abstract summary: Proving spectral gaps above the ground space is a central problem in quantum many-body physics.<n>We revisit the martingale method to prove the existence of spectral gaps of parent Hamiltonians of Matrix Product States (MPS)<n>We devise a new technique which allows to compute the key quantity in the martingale method exactly and efficiently.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Proving spectral gaps above the ground space is a central problem in quantum many-body physics. Yet, finding lower bounds on the gap is notoriously difficult. We revisit the martingale method, originally developed by Fannes, Nachtergaele, and Werner [Fannes '92, Nachtergaele '96] to prove the existence of spectral gaps of parent Hamiltonians of Matrix Product States (MPS), and provide improvements to the different steps of the method. Most importantly, we devise a new technique which allows to compute the key quantity in the martingale method -- the overlap of local ground spaces -- exactly and efficiently. This enables a clear improvement of the method, allowing it to outperform other existing techniques to lower bound gaps, which we demonstrate by benchmarking on several models. Remarkably, our -- numerically motivated -- approach at the same time also yields a significantly simplified proof of the fact that parent Hamiltonians of any (well-behaved) MPS are always gapped.

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