Lieb–Schultz–Mattis Type Theorems for Quantum Spin Chains Without Continuous Symmetry

  • Yoshiko OgataEmail author
  • Hal Tasaki


We prove that a quantum spin chain with half-odd-integral spin cannot have a unique ground state with a gap, provided that the interaction is short ranged, translation invariant, and possesses time-reversal symmetry or \({\mathbb{Z}_{2} \times \mathbb{Z}_{2}}\) symmetry (i.e., the symmetry with respect to the \({\pi}\) rotations of spins about the three orthogonal axes). The proof is based on the deep analogy between the matrix product state formulation and the representation of the Cuntz algebra in the von Neumann algebra \({\pi(\mathcal{A}_{R})''}\) constructed from the ground state restricted to the right half-infinite chain.


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It is a pleasure to thank HarukiWatanabe for valuable discussion which was essential for the present work, and Tohru Koma for useful discussion and comments. We also thank TakuMatsui and Bruno Nachtergaele for useful comments. The present work was supported by JSPS Grants-in-Aid for Scientific Research nos. 16K05171 (Y.O.) and 16H02211 (H.T.).


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Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoKomabaJapan
  2. 2.Department of PhysicsGakushuin UniversityMejiro, Toshima-kuJapan

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