Abstract
In this study, we consider one-dimension (1D) quantum spin systems with the translation and discrete symmetries (spin reversal, space inversion and time reversal symmetries). By combining the continuous U(1) symmetry with the discrete symmetries and using the extended Lieb-Schultz-Mattis theorem \cite{Lieb-Schultz-Mattis-1961}\cite{Nomura-Morishige-Isoyama-2015}, we investigate the relation between the ground states, energy spectra and symmetries. For half-integer spin cases, we generalize the dimer and N\'eel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or N\'eel states, since there was a subtle dilemma. Furthermore, a part of discrete symmetries is enough to classify possible phases. Thus we can deepen our understanding of the relation between the LSM theorem and the discrete symmetries.
Highlights
In many body quantum systems, it is important to investigate energy spectra, that is, whether gapless or gapped, or the degeneracy of ground states
Affleck and Lieb [3] studied general spin S and SU(N) symmetric cases, and showed the same result as the LSM theorem for half-integer S. They considered the relation between the space inversion and the spin reversal symmetries
One limitation of the traditional LSM theorem [1, 3, 5] is the assumption of the unique ground state for the finite system, which is violated in several cases with frustrations (Majumdar-Ghosh model [6, 7] and the incommensurate region)
Summary
In many body quantum systems, it is important to investigate energy spectra, that is, whether gapless or gapped, or the degeneracy of ground states. Affleck and Lieb [3] studied general spin S and SU(N) symmetric cases, and showed the same result as the LSM theorem for half-integer S. They considered the relation between the space inversion and the spin reversal symmetries. Using the twisted boundary condition, Kolb [4] showed, for half-integer spin chains, the nontrivial periodicity of the wave number q → q + π of the lowest energy dispersion in the zero magnetization subspace. He discussed the continuity of the energy dispersion for q.
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