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Local gap threshold for frustration-free spin systems

### Abstract

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 , 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 as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.

Published by AIP Publishing.

Received 18 May 2016
Accepted 17 August 2016
Published online 15 September 2016

Acknowledgments:
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^{17} 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).

Article outline:

I. INTRODUCTION
A. Entanglement versus gap
II. SPECTRAL GAP BOUND FOR ONE-DIMENSIONAL SYSTEMS
A. Proof strategy
B. Proof of theorem 3
III. SPECTRAL GAP BOUND FOR TWO-DIMENSIONAL SYSTEMS
A. Deformed patch operators
B. Proof of theorem 5

/content/aip/journal/jmp/57/9/10.1063/1.4962337

1.

Affleck, I. , Kennedy, T. , Lieb, E. H. , and Tasaki, H. , “Rigorous results on valence-bond ground states in antiferromagnets,” Phys. Rev. Lett. 59, 799–802 (1987).

http://dx.doi.org/10.1103/PhysRevLett.59.799
2.

Arad,

I. ,

Kitaev,

A. ,

Landau,

Z. , and

Vazirani,

U. , “

An area law and sub-exponential algorithm for 1D systems,” e-print

arXiv:1301.1162 (

2013).

6.

Farhi,

E. ,

Goldstone,

J. ,

Gutmann,

S. , and

Sipser,

M. , “

Quantum computation by adiabatic evolution,” e-print

arXiv:quant-ph/0001106 (

2000).

12.

Here the notation hides a polylogarithmic function of which is present in the entanglement entropy bound from Ref. 2.

13.

Here we consider a quantum many-body system described by a sequence of Hamiltonians {H_{n}} indexed by system size n. The system is gapped if the spectral gap of H_{n} is lower bounded by a positive constant independent of n; otherwise it is gapless.

14.

If H_{i} is not a projector, we may replace it with the projector Π_{i} orthogonal to its null space. It is not hard to see that the new Hamiltonian has the same zero energy ground space as H and that the spectral gaps ϵ and ϵ′ of these Hamiltonians satisfy aϵ′ ≤ ϵ ≤ ϵ′, where a lower bounds the smallest non zero eigenvalue of each term H_{i}.

15.

In other words, for x ∈ ℝ^{2} we have T_{e1→e2}(x) = x + e_{2} − e_{1} (here e_{2}, e_{1} are identified with the coordinates of their midpoints).

16.

Irani, S. , “Ground state entanglement in one-dimensional translationally invariant quantum systems,” J. Math. Phys. 51(2), 022101 (2010).

http://dx.doi.org/10.1063/1.3254321
17.

Kitaev, A. , private communication (2015).

19.

Movassagh,

R. and

Shor,

P. W. , “

Power law violation of the area law in quantum spin chains,” e-print

arXiv:1408.1657 (

2014).

21.

Note that G(n) is invariant under a rescaling c_{j} → ac_{j}, so we are free to choose the normalization so that . To find the optimum of G(n) we minimize the numerator of (28) subject to this constraint. The solution is equal to (29) (up to an irrelevant rescaling).

http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/9/10.1063/1.4962337

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2016-09-15

2016-10-24

### Abstract

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 , 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 as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.

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