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T. Isoyama
K. Nomura, Discrete symmetries and the Lieb–Schultz–Mattis theorem, *Progress of Theoretical and Experimental Physics*, 10.1093/ptep/ptx139, 2017, 10, 103I01, 2017.10, In this study, we consider one-dimensional (1D) quantum spin systems with 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 (LSM) theorem [E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961); K. Nomura, J. Morishige, and T. Isoyama, J. Phys. A 48, 375001 (2015)], we investigate the relation between the ground states, energy spectra, and symmetries. For half-integer spin cases, we generalize the dimer and Néel concepts using the discrete symmetries, and we can reconcile the LSM theorem with the dimer or Néel 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 discrete symmetries.. |

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Kiyohide Nomura, Junpei Morishige, Takaichi Isoyama, Extension of the Lieb-Schultz-Mattis theorem, *Journal of Physics A: Mathematical and Theoretical*, 10.1088/1751-8113/48/37/375001, 48, 37, 2015.09, Lieb, Schultz and Mattis (LSM) (1961 Ann. Phys., NY 16 407) studied the S = 1/2 XXZ spin chain. The theorems of LSM's paper can be applied to broader models. In the original LSM theorem the nonfrustrating system was assumed. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers tried to extend the LSM theorem for excited states. In the cases , the lowest energy eigenvalues are continuous for wave number q. But we found that their proofs were insufficient, and improve upon them. In addition, we can prove the LSM theory without the assumption of the discrete symmetry, which means that LSM-type theorems are applicable for Dzyaloshinskii-Moriya type interactions or other nonsymmetric models.. |

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K. Hijii and K. Nomura, Phase transition of S=1/2 two-leg Heisenberg spin ladder systems with a four-spin interaction, *Phys. Rev. B*, 10.1103/PhysRevB.80.014426, 80, 1, 014426, 2009.07, We study a phase transition and critical properties of the quantum spin ladder system with a four-spin interaction. We determine a phase boundary between a rung singlet and a staggered dimer phases numerically. This phase transition is of a second order in the weak-coupling region. We confirm that this universality class is described by the k=2 SU(2) Wess-Zumino-Witten model, analyzing the central charge and scaling dimensions. In the strong-coupling region, phase transition becomes of a first order.. |

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T. Murashima and K. Nomura, Incommensurability and edge states in the one-dimensional S=1 bilinear-biquadratic model, *Phys. Rev. B*, 10.1103/PhysRevB.73.214431, 73, 214431, Vol.73, p.214431, 2006.06, Commensurate-incommensurate change on the one-dimensional S=1 bilinear-biquadratic model [H(α)=∑i{Si∙Si+1+α(Si∙Si+1)2}] is examined. The gapped Haldane phase has two subphases (the commensurate Haldane subphase and the incommensurate Haldane subphase) and the commensurate-incommensurate change point (the Affleck-Kennedy-Lieb-Tasaki point, α=1/3). There have been two different analytical predictions about the static structure factor in the neighborhood of this point. By using the Sørensen-Affleck prescription, these static structure factors are related to the Green functions, and also to the energy gap behaviors. Numerical calculations support one of the predictions. Accordingly, the commensurate-incommensurate change is recognized as a motion of a pair of poles in the complex plane.. |

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H. Matsuo and Nomura, Berezinskii-Kosterlitz-Thouless transitions in the six-state clock model, *J. Phys. A*, 10.1088/0305-4470/39/12/006, 39, 12, 2953, 2006.03, A classical 2D clock model is known to have a critical phase with Berezinskii–Kosterlitz–Thouless (BKT) transitions. These transitions have logarithmic corrections which make numerical analysis difficult. In order to resolve this difficulty, one of the authors has proposed a method called 'level spectroscopy', which is based on the conformal field theory. We extend this method to the multi-degenerated case. As an example, we study the classical 2D six-clock model which can be mapped to the quantum self-dual 1D six-clock model. Additionally, we confirm that the self-dual point has a precise numerical agreement with the analytical result, and we argue the degeneracy of the excitation states at the self-dual point from the effective field theoretical point of view.. |

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K. Nomura and T. Murashima, Incommensurability and Edge State in Quantum Spin Chain, *J. Phys. Soc. Jpn (Suppl.)*, 74, 42, Vol. 74 (Suppl.) pp.42, 2005.01, In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS) type points. It was difficult to study the commensurate–incommensurate (C–IC) change. On the one hand field theoretical approaches are not justified because of the short correlation length. On the other hand numerical calculations are not suitable to study the incommensurability since it is needed to treat the large size data. We discuss the relation between the edge state and the incommensurability, partially using the previous our study on the C–IC change. . |

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K. Nomura, Onset of Incommensurability in Qunatum spin chain, *J. Phys. Soc. Jpn*, 10.1143/JPSJ.72.476, 72, 3, 476-478, Vol.72, pp.476-478, 2003.03, In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS)-type solvable points, and the correlation length becomes shortest at VBS-type points. In addition, the correlation function decays purely exponentially at VBS-type points, in contrast with the two-dimensional (2D) Ornstein-Zernicke type behavior in other regions with an excitation gap. We propose a mechanism to explain the onset of the incommensurability and the shortest correlation length at VBS-like points. This theory can be applied to more general cases.. |

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K. Hijii and K. Nomura, Universality class of an S=1 quantum spin ladder system with four-spin exchange, *Phys. Rev. B*, 10.1103/PhysRevB.65.104413, 65, 10, 104413, Vol. 65, pp. 104413, 2002.01, We study a s=12 Heisenberg spin ladder with four-spin exchange. Combining numerical results with conformal field theory, we find a phase transition with central charge c=32. Since this system has an SU(2) symmetry, we can conclude that this critical theory is described by k=2 SU(2) Wess-Zumino-Witten model with Z2 symmetry breaking.. |

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S. Hirata and K. Nomura, Phase diagram of S=1/2 XXZ chain with NNN interaction, *Phys. Rev. B*, 10.1103/PhysRevB.61.9453, 61, 14, 9453-9456, Vol. 61, pp.9453-9456., 2000.01, We study the ground state properties of one-dimensional XXZ model with next-nearest neighbor coupling alpha and anisotropy Delta. We find the direct transition between the ferromagnetic phase and the spontaneously dimerized phase. This is surprising, because the ferromagnetic phase is classical, whereas the dimer phase is a purely quantum and nonmagnetic phase. We also discuss the effect of bond alternation which arises in realistic systems due to lattice distortion. Our results mean that the direct transition between the ferromagnetic and spin-Peierls phase occur. . |

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K. Nomura and A. Kitazawa, SU(2)/Z_2 symmetry of the BKT transition and twisted boundary condition, *J. Phys. A*, 10.1088/0305-4470/31/36/008, 31, 36, 7341-7362, Vol. 31, pp.7341-7362, 1998.01, The Berezinskii-Kosterlitz-Thouless (BKT) transition, the transition of the two-dimensional sine-Gordon model, plays an important role in low-dimensional physics. We relate the operator content of the BKT transition to that of the SU(2) Wess-Zumino-Witten model, using twisted boundary conditions. With this method, in order k - 1 to determine the BKT critical point, we can use the level crossing of the lower excitations instead of those for the periodic boundary case, thus the convergence to the transition point is highly improved. We verify the efficiency of this method by applying it to the S = 1, 2 spin chains.. |

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M. Tsukano and K. Nomura, Berezinski-Kosterlitz-Thouless Transition of Spin-1 XXZ Chains in a staggered Magnetic Field, *J. Phys. Soc. Jpn.*, 10.1143/JPSJ.67.302, 67, 1, 302-306, Vol. 67, pp.302-306, 1998.01. |