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Gapped and gapless phases of frustration-free spin-1/2 chains

Bravyi, Sergey and Gosset, David (2015) Gapped and gapless phases of frustration-free spin-1/2 chains. Journal of Mathematical Physics, 56 (6). Art. No. 061902. ISSN 0022-2488. doi:10.1063/1.4922508.

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We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary two-qubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian H_n (ψ) which is defined as the sum of rank-1 projectors onto ψ applied to consecutive pairs of qubits. We show that the spectral gap of H_n (ψ) is upper bounded by 1/(n − 1) if the eigenvalues of a certain 2 × 2 matrix simply related to ψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of n (depending only on ψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.

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Additional Information:© 2015 AIP Publishing LLC. Received 9 April 2015; accepted 3 June 2015; published online 18 June 2015. D.G. thanks Edward Farhi, Jeffrey Goldstone, and Sam Gutmann for many discussions about this problem when he was a graduate student. D.G. was supported in part by NSERC and by ARO. IQC is supported in part by the Government of Canada and the Province of Ontario. S.B. would like to thank Ramis Movassagh for helpful discussions. S.B. acknowledges NSF Grant No. CCF-1110941.
Group:Institute for Quantum Information and Matter
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Army Research Office (ARO)UNSPECIFIED
Government of CanadaUNSPECIFIED
Province of OntarioUNSPECIFIED
Issue or Number:6
Record Number:CaltechAUTHORS:20150803-083252306
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Official Citation:Gapped and gapless phases of frustration-free spin-1/2 chains Sergey Bravyi and David Gosset J. Math. Phys. 56, 061902 (2015);
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:59148
Deposited By: Ruth Sustaita
Deposited On:05 Aug 2015 23:43
Last Modified:10 Nov 2021 22:15

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