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Local gap threshold for frustration-free spin systems

Gosset, David and Mozgunov, Evgeny (2016) Local gap threshold for frustration-free spin systems. Journal of Mathematical Physics, 57 (9). Art. No. 091901. ISSN 0022-2488. doi:10.1063/1.4962337.

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We improve Knabe’s spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6/(n+1), which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n^(−2)). This contrasts with gapless frustrated systems where the gap can be Θ(n^(−1)). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is O(1/ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.

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Additional Information:© 2016 AIP Publishing. Received 18 May 2016; accepted 17 August 2016; published online 15 September 2016. We thank Fernando Brandão, Yichen Huang, Alexei Kitaev, Bruno Nachtergaele, Daniel Nagaj, and John Preskill for helpful comments. We also thank Alexei Kitaev for sharing his calculation with us. We acknowledge funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant No. PHY-1125565) with support of the Gordon and Betty Moore Foundation (Grant No. GBMF-12500028).
Group:Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Gordon and Betty Moore FoundationGBMF-12500028
Issue or Number:9
Record Number:CaltechAUTHORS:20160622-145800493
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Official Citation:Local gap threshold for frustration-free spin systems Gosset, David and Mozgunov, Evgeny, Journal of Mathematical Physics, 57, 091901 (2016), DOI:
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:68605
Deposited By: Jacquelyn O'Sullivan
Deposited On:27 Jun 2016 17:50
Last Modified:11 Nov 2021 04:02

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