A class of two-dimensional AKLT models with a gap

Houssam Abdul-Rahman; Marius Lemm; Angelo Lucia; Bruno Nachtergaele; Amanda Young

doi:10.1090/conm/741/14917

A class of two-dimensional AKLT models with a gap

Houssam Abdul-Rahman, Marius Lemm, Angelo Lucia, Bruno Nachtergaele, Amanda Young

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal lattice exhibits a spectral gap. In this paper, we introduce a family of variants of the two-dimensional AKLT model depending on a positive integer n, which is defined by decorating the edges of the hexagonal lattice with one-dimensional AKLT spin chains of length n. We prove that these decorated models are gapped for all n ≥ 3.

Original language	English (US)
Title of host publication	Analytic Trends in Mathematical Physics
Editors	Houssam Abdul-Rahman, Robert Sims, Amanda Young
Publisher	American Mathematical Society
Pages	1-21
Number of pages	21
ISBN (Electronic)	9781470453886
ISBN (Print)	9781470448417
DOIs	https://doi.org/10.1090/conm/741/14917
State	Published - 2020
Externally published	Yes

Publication series

Name	Contemporary Mathematics
Publisher	American Mathematical Society
Volume	741
ISSN (Print)	0271-4132

ASJC Scopus subject areas

General Mathematics

Online availability

10.1090/conm/741/14917

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A class of two-dimensional AKLT models with a gap. / Abdul-Rahman, Houssam; Lemm, Marius; Lucia, Angelo et al.
Analytic Trends in Mathematical Physics. ed. / Houssam Abdul-Rahman; Robert Sims; Amanda Young. American Mathematical Society, 2020. p. 1-21 (Contemporary Mathematics; Vol. 741).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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author = "Houssam Abdul-Rahman and Marius Lemm and Angelo Lucia and Bruno Nachtergaele and Amanda Young",

note = "Funding Information: This work arose from discussions during the follow-up workshop on Gapped Ground State Phases of Quantum Many-Body Systems to the 2018 Arizona School of Analysis and Mathematical Physics, organized by Robert Sims and two of the authors (H. A. and A. Y.) and supported by NSF Grant DMS-1800724. A. L. acknowledges support from the Walter Burke Institute for Theoretical Physics in the form of the Sherman Fairchild Fellowship as well as support from the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NFS Grant PHY-1733907). B. N. acknowledges support by the National Science Foundation under Grant DMS-1813149 and a CRM-Simons Professorship for a stay at the Centre de Recherches Math{\'e}matiques (Montr{\'e}al) during Fall 2018, where part of this work was carried out. Publisher Copyright: {\textcopyright} 2020 The Authors.",

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AB - The AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki also conjectured that the two-dimensional version of their model on the hexagonal lattice exhibits a spectral gap. In this paper, we introduce a family of variants of the two-dimensional AKLT model depending on a positive integer n, which is defined by decorating the edges of the hexagonal lattice with one-dimensional AKLT spin chains of length n. We prove that these decorated models are gapped for all n ≥ 3.

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A class of two-dimensional AKLT models with a gap

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