Lieb–Schultz–Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems
- 51 Downloads
We formulate and prove the local twist version of the Yamanaka–Oshikawa–Affleck theorem, an extension of the Lieb–Schultz–Mattis theorem, for one-dimensional systems of quantum particles or spins. We can treat almost any translationally invariant system with global U(1) symmetry. Time-reversal or inversion symmetry is not assumed. It is proved that, when the “filling factor” is not an integer, a ground state without any long-range order must be accompanied by low-lying excitations whose number grows indefinitely as the system size is increased. The result is closely related to the absence of topological order in one-dimension. The present paper is written in a self-contained manner, and does not require any knowledge of the Lieb–Schultz–Mattis and related theorems.
I wish to thank Tohru Koma, Masaki Oshikawa, and Haruki Watanabe for valuable discussions which were essential for the present work, and Ian Affleck, Hosho Katsura, Tomonari Mizoguchi, Lee SungBin, Akinori Tanaka, Masafumi Udagawa, and Masanori Yamanaka for useful discussions and correspondences. The present work was supported by JSPS Grants-in-Aid for Scientific Research No. 16H02211.
- 12.Watanabe, H., Po, H.C., Vishwanath, A., Zaletel, M.P.: Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystal. Proc. Natl. Acad. Sci. U.S.A. 112, 14551–14556. http://www.pnas.org/content/112/47/14551.short (2015)
- 13.Tasaki, H.: Low-lying excitations in one-dimensional lattice electron systems. Preprint. arXiv:cond-mat/0407616 (2004)
- 16.Wen, X.-G: Zoo of quantum-topological phases of matter. Preprint. arxiv.org/abs/1610.03911 (2016)
- 17.Zeng, B., Chen, X., Zhou, D.-L., Wen, X.-G.: Quantum Information Meets Quantum Matter: From Quantum Entanglement to Topological Phase in Many-Body Systems, (to be published from Springer). arxiv.org/abs/1508.02595
- 22.Oshikawa, M.: Private communication (1997)Google Scholar