Lower Bounds on Spectral Gaps of Parent Hamiltonians via Tensor Networks

Proving the existence of a spectral gap—a uniform lower bound on the energy difference between the ground state and the first excited state—is a fundamental challenge in quantum many-body physics. A spectral gap ensures that a system exhibits exponential decay of correlations and topological stability. While the 'martingale method' developed by Fannes, Nachtergaele, and Werner (FNW) established that parent Hamiltonians of Matrix Product States (MPS) are gapped, obtaining tight quantitative lower bounds on these gaps has remained difficult.

This paper revisits the martingale method, which relates the spectral gap of a many-body Hamiltonian to the local overlap of ground spaces of blocked Hamiltonian terms. The authors provide two major improvements:

1. Refined Blocking Schemes: They expand the ways in which lattice sites are grouped into super-sites, allowing for more flexible and optimized gap bounds.
2. Efficient Overlap Computation: The central technical contribution is a new method to compute the overlap of local ground spaces exactly and efficiently. By transforming the overlap calculation into an eigenvalue problem on a space of fixed dimension (scaling with the bond dimension of the MPS), they avoid the exponential costs associated with exact diagonalization.

Beyond providing a more efficient computational framework, the authors demonstrate that their approach yields significantly tighter lower bounds on spectral gaps compared to existing techniques, including the original FNW method and finite-size criteria like those of Knabe or Gosset-Mozgunov. Furthermore, the authors provide a simplified, unified proof that parent Hamiltonians of well-behaved MPS are always gapped, showing that the gap existence is a direct consequence of the convergence of their overlap eigenvalue problem.

This work bridges the gap between abstract proofs of existence and practical, quantitative analysis of quantum phases. By providing a scalable way to bound spectral gaps, researchers can now more accurately assess the stability and correlation properties of a wide range of quantum spin models, including the AKLT model and various valence bond states, without needing to perform computationally prohibitive exact diagonalizations.