Extensions of the Haldane gap proof: higher dimensions, near degeneracy, fermions and tensor-network states
Abstract
A recent remarkable paper by OpenAI showed that the spin-one Heisenberg chain with periodic boundary conditions has a spectral gap in the thermodynamic limit. The proof elegantly deduces this gap from the Hamiltonian's spectrum on a finite system. This basic idea is an inequality involving the purity of the original Gibbs state and a dual Gibbs state that is defined via space-time rotation. Here we consider a few generalizations of this purity bootstrap argument. We first extend it to arbitrary spatial dimensions. Our main result uses the space-time duality between symmetry generators and symmetry twists to obtain upper bounds on the splitting of nearly degenerate low-energy states, together with a nonzero gap above them, from finite-system information. As applications, we show that finite-system calculations for the 2+1-D toric code in a field and the 2+1-D transverse-field Ising model establish parameter regimes with four and two low-lying states whose splittings are bounded from above by e−cL and e−cL2 respectively for arbitrarily large system size L, while the gap above these states remains nonzero in the thermodynamic limit. Next, we extend it to fermionic systems and use a finite calculation to establish a symmetric mass generation phase in a 2+1-D fermionic model. Finally, we apply the bootstrap to tensor-network states and establish exponential decay of connected correlations of all local operators in the symmetric square-lattice spin-one state introduced by Jian and Zaletel.
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BibTeX
@misc{grover2026extensions,
title = {{Extensions of the Haldane gap proof: higher dimensions, near degeneracy, fermions and tensor-network states}},
author = {Tarun Grover},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07400},
eprint = {2611.07400},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07400},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extensions of the Haldane gap proof: higher dimensions, near degeneracy, fermions and tensor-network states
AU - Tarun Grover
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07400
UR - https://arxiv.org/abs/2611.07400
AB - A recent remarkable paper by OpenAI showed that the spin-one Heisenberg chain with periodic boundary conditions has a spectral gap in the thermodynamic limit. The proof elegantly deduces this gap from the Hamiltonian's spectrum on a finite system. This basic idea is an inequality involving the purity of the original Gibbs state and a dual Gibbs state that is defined via space-time rotation. Here we consider a few generalizations of this purity bootstrap argument. We first extend it to arbitrary spatial dimensions. Our main result uses the space-time duality between symmetry generators and symmetry twists to obtain upper bounds on the splitting of nearly degenerate low-energy states, together with a nonzero gap above them, from finite-system information. As applications, we show that finite-system calculations for the 2+1-D toric code in a field and the 2+1-D transverse-field Ising model establish parameter regimes with four and two low-lying states whose splittings are bounded from above by e^−cL and e^−cL^2 respectively for arbitrarily large system size L, while the gap above these states remains nonzero in the thermodynamic limit. Next, we extend it to fermionic systems and use a finite calculation to establish a symmetric mass generation phase in a 2+1-D fermionic model. Finally, we apply the bootstrap to tensor-network states and establish exponential decay of connected correlations of all local operators in the symmetric square-lattice spin-one state introduced by Jian and Zaletel.
ER -