Abstract
We introduce and analyze a class of quantum spin models defined on \(d\)-dimensional lattices \(\Lambda \subseteq {\mathbb Z}^d\), which we call Product Vacua with Boundary States (PVBS). We characterize their ground state spaces on arbitrary finite volumes and study the thermodynamic limit. Using the martingale method, we prove that the models have a gapped excitation spectrum on \({\mathbb Z}^d\) except for critical values of the parameters. For special values of the parameters we show that the excitation spectrum is gapless. We demonstrate the sensitivity of the spectrum to the existence and orientation of boundaries. This sensitivity can be explained by the presence or absence of edge excitations. In particular, we study a PVBS models on a slanted half-plane and show that it has gapless edge states but a gapped excitation spectrum in the bulk.
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Acknowledgments
This research was supported in part by the National Science Foundation: S.B. under Grant #DMS-0757581 and B.N. and A.Y. under Grant #DMS-1009502. A.Y. also acknowledges support from the National Science Foundation under Grant #DMS-0636297. A.Y. would also like to thank the Mathematisches Institut at Ludwig-Maximilians-Universität München and the Erwin Schrödinger International Institute for Mathematical Physics (ESI) in Vienna, Austria, where part of the work reported here was carried out. E. H gratefully acknowledges the support of a Fulbright scholarship spent at University of California, Davis.
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Bachmann, S., Hamza, E., Nachtergaele, B. et al. Product Vacua and Boundary State Models in \(d\)-Dimensions. J Stat Phys 160, 636–658 (2015). https://doi.org/10.1007/s10955-015-1260-7
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DOI: https://doi.org/10.1007/s10955-015-1260-7