野村 清英（のむら きよひで） | データ更新日：2021.06.16 |

准教授 ／
理学研究院
物理学部門
物性基礎論 統計物理学

**原著論文**

1. | T.Mashiko,
S.Moriya,
K.Nomura, Universality Class around the SU(3) Symmetric Point of the Dimer–Trimer Spin-1 Chain, J. Phys. Soc. Jpn., https://doi.org/10.7566/JPSJ.90.024005, 90, 2, 024005, 2021.02, We study critical phenomena of an SU(3) symmetric spin-1 chain when adding an SU(3) asymmetric term. To investigate such phenomena, we numerically diagonalize the dimer–trimer (DT) model Hamiltonian around the SU(3) symmetric point, named the pure trimer (PT) point. We analyze our numerical results on the basis of the conformal field theory (CFT). First of all, we discover that soft modes appear at the wave number q = 0 and ±2π/3 for the PT point, and then the system is critical. Secondly, we find that the system at the PT point can be described by the CFT with the central charge c = 2 and the scaling dimension x = 2/3. Finally, by investigating the eigenvalues of the Hamiltonian in the vicinity of the PT point, we find that there is a phase transition at the PT point from a massive phase to a massless phase. From these numerical results, the phase transition at the PT point belongs to the Berezinskii–Kosterlitz–Thouless (BKT)-like universality class that is explained by the level-1 SU(3) Wess–Zumino–Witten [SU(3)1 WZW] model.. |

2. | N. Aiba,
K. Nomura, Method to observe the anomaly of magnetic susceptibility in quantum spin systems, Phys. Rev. B, https://doi.org/10.1103/PhysRevB.102.134435, 102, 13, 134435, 2020.10, In quantum spin systems, a phase transition is studied from the perspective of magnetization curve and a magnetic susceptibility. We propose a new method for studying the anomaly of magnetic susceptibility χ that indicates a phase transition. In addition, we introduce the fourth derivative A of the lowest-energy eigenvalue per site with respect to magnetization, i.e., the second derivative of χ−1. To verify the validity of this method, we apply it to an S=1/2XXZ antiferromagnetic chain. The lowest energy of the chain is calculated by numerical diagonalization. As a result, the anomalies of χ and A exist at zero magnetization. The anomaly of A is easier to observe than that of χ, indicating that the observation of A is a more efficient method of evaluating an anomaly than that of χ. The observation of A reveals an anomaly that is different from the Kosterlitz-Thouless (KT) transition. Our method is useful in analyzing critical phenomena.. |

3. | S. Moriya,
K. Nomura, A New Method to Calculate a 2D Ising Universality Transition Point: Application near the Ashkin–Teller Multicritical Point, J. Phys. Soc. Jpn., https://doi.org/10.7566/JPSJ.89.093001, 89, 9, 093001, 2020.09, We propose a new method to numerically calculate transition points that belongs to 2D Ising universality class for quantum spin models. Generally, near the multicritical point, in conventional methods, a finite size correction becomes very large. To suppress the effect of the multicritical point, we use a z-axis twisted boundary condition and a y-axis twisted boundary condition. We apply our method to an S=1/2 bond-alternating XXZ model. The multicritical point of this model has a BKT transition, where the correlation length diverges singularly. However, with our method, the convergence of calculation is highly improved, thus we can calculate the transition point even near the multicritical point.. |

4. | 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, [URL], 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.. |

5. | 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, [URL], 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.. |

6. | Kiyomi Okamoto, Takashi Tonegawa, Hiroki Nakano, Toru Sakai, Kiyohide Nomura, Makoto Kaburagi, How to distinguish the Haldane/Large-D state and the intermediate-D state in an S = 2 quantum spin chain with the XXZ and on-site anisotropies, Journal of Physics: Conference Series, 10.1088/1742-6596/320/1/012018, 320, 012018, 2011.10, We numerically investigate the ground-state phase diagram of an S = 2 quantumspin chain with the XXZ and on-site anisotropies described by H =Σj (Sxj Sxj+1 +Syj Syj+1 +ΔSzj Szj+1) + DΣj (Szj )2, where Δ denotes the XXZ anisotropy parameter of the nearestneighbor interactions and D the on-site anisotropy parameter. We restrict ourselves to the Δ > 0 and D > 0 case for simplicity. Our main purpose is to obtain the definite conclusion whether there exists or not the intermediate-D (ID) phase, which was proposed by Oshikawa in 1992 and has been believed to be absent since the DMRG studies in the latter half of 1990’s. In the phase diagram with Δ > 0 and D > 0 there appear the XY state, the Haldane state, the ID state, the large-D (LD) state and the N´eel state. In the analysis of the numerical data it is important to distinguish three gapped states; the Haldane state, the ID state and the LD state. We give a physical and intuitive explanation for our level spectroscopy method how to distinguish these three phases.. |

7. | T. Tonegawa, K. Okamoto, H. Nakano, T. Sakai,
K. Nomura, M. Kaburagi, Haldane, Large-D and Intermediate-D States in an S=2 Quantum Spin Chain with On-Site and XXZ Anisotropies, J. Phys. Soc. Jpn, 10.1143/JPSJ.80.043001, 80, 4, 043001, 2011.04, Using mainly numerical methods, we investigate the ground-state phase diagram of the S = 2 quantum spin chain described by H = ∑ j ( S j x S j +1 x + S j y S j +1 y + Δ S j z S j +1 z ) + D ∑ j ( S j z ) 2 , where Δ denotes the X X Z anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. We restrict ourselves to the case of Δ≥0 and D ≥0 for simplicity. Each of the phase boundary lines is determined by the level spectroscopy or the phenomenological renormalization analysis of numerical results of exact-diagonalization calculations. The resulting phase diagram on the Δ– D plane consists of four phases; the X Y 1 phase, the Haldane/large- D phase, the intermediate- D phase, and the Néel phase. The remarkable natures of the phase diagram are as follows: (1) there exists an intermediate- D phase which was predicted by Oshikawa in 1992; (2) the Haldane state and the large- D state belong to the same phase; (3) the shape of the phase diagram on the Δ– D plane is different from that considered so far. We note that this is the first report on the observation of the intermediate- D phase.. |

8. | T. Tonegawa, H. Nakano, Toru Sakai, Kiyomi Okamoto, K. Okunishi, Kiyohide Nomura, Half magnetization plateau of a frustrated S = 1 antiferromagnetic chain, J. Phys.: Conf. Ser., 10.1088/1742-6596/200/2/022065, 200, 022065, 2010.02, We numerically investigate the magnetization process of a frustrated S=1 chain with antiferromagnetic nearest-neighbor (nn) and next-nearest-neighbor (nnn) exchange interactions, which compete with each other, and also with uniaxial single-ion-type anisotropy energies. Special attention is paid to the half magnetization plateau which appears at the half of the saturation magnetization in the ground-state magnetization curve. We determine the parameter region where this plateau appears by employing the level spectroscopy method, and find that the plateau appears even in the absence of the single-ion-type anisotropy, if the ratio of the nnn to the nn interaction constant is in a specified region.. |

9. | 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.. |

10. | H. Otsuka and K.Nomura, Critical intermediate phase and phase transitions in triangular-lattice three-spin interaction model: Level-spectroscopy approach, Journal of Physics A, 10.1088/1751-8113/41/37/375001, 41, 37, 375001, Vol. 41, p.375001, 2008.08, We investigate infinite-order phase transitions like the Berezinskii–Kosterlitz–Thouless transition observed in a triangular-lattice three-spin interaction model. Based on a field theoretical description and the operator-production-expansion technique, we perform the renormalization-group analysis, and then clarify properties of marginal operators near the phase transition points. The results are utilized to establish criteria to determine the transition points and some universal relations among excitation levels to characterize the transitions. We verify these predictions via the numerical analysis on eigenvalue structures of the transfer matrix. Also, we discuss an enhancement of symmetry at the end points of a critical intermediate phase in connection with a transition observed in the ground state of the bilinear-biquadratic spin-1 chain.. |

11. | T. Murashima and K. Nomura, Cancellation of oscillatory behaviors in incommensurate region, J. Phys. Condens. Matter,, 10.1088/0953-8984/19/14/145210, 19, 14, 145210, Vol. 19 (2007) pp.145210, 2007.03, In several frustrated systems, incommensurate behaviours are often observed. For the S = 1 bilinear–biquadratic model, we show that the main oscillatory behaviour, which is proportional to the free edge spins, is eliminated in the incommensurate subphase, considering the average of triplet and singlet energy spectra under open boundary conditions. In the same way, the π-mode oscillation is also removed in the commensurate subphase. Moreover, we find that higher-order corrections are exponentially decaying from an analysis of small-size data.. |

12. | H. Otsuka, Y. Okabe, and K. Nomura, Global phase diagram and six-state clock universality behavior in the triangular antiferromagnetic Ising model with anisotropic next-nearest-neighbor coupling: Level-spectroscopy approach, Phys. Rev. E, 10.1103/PhysRevE.74.011104, 74, 011104, Vol.74, 011104, 2006.07, We investigate the triangular-lattice antiferromagnetic Ising model with a spatially anisotropic next-nearest-neighbor ferromagnetic coupling, which was first discussed by Kitatani and Oguchi. By employing the effective geometric factor, we analyze the scaling dimensions of the operators around the Berezinskii-Kosterlitz-Thouless (BKT) transition lines, and determine the global phase diagram. Our numerical data exhibit that two types of BKT-transition lines separate the intermediate critical region from the ordered and disordered phases, and they do not merge into a single curve in the antiferromagnetic region. We also estimate the central charge and perform some consistency checks among scaling dimensions in order to provide the evidence of the six-state clock universality. Further, we provide an analysis of the shapes of boundaries based on the crossover argument.. |

13. | 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.. |

14. | 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.. |

15. | H. Inoue and K. Nomura, Conformal field theory in the Tomonaga–Luttinger model with the 1/rβ long-range interaction , J. Phys. A, 10.1088/0305-4470/39/9/012, 39, 9, 2161, Vol. 39,p.2161., 2006.02, We attempt to construct U(1) conformal field theory (CFT) in the Tomonaga–Luttinger (TL) liquid with the 1/rβ long-range interaction (LRI). Treating the long-range forward scattering as a perturbation and applying CFT to it, we derive finite size scalings which depend on the power of the LRI. The obtained finite size scalings give nontrivial behaviours when β is odd and is close to 2. We find consistency between the analytical arguments and numerical results in the finite size scaling of energy.. |

16. | T. Murashima, K. Hijii, K. Nomura, and T. Tonegawa, Phase Diagram of S=1 XXZ Chain with Next-Nearest-Neighbor Interaction, Journal of the Physical Society of Japan, 10.1143/JPSJ.74.1544, 74, 5, 1544-1551, Vol.74p.1544-1551, 2005.01, The one dimensional S =1 XXZ model with next-nearest-neighbor interaction α and Ising-type anisotropy Δ is studied by using a numerical diagonalization technique. We discuss the ground state phase diagram of this model numerically by the twisted-boundary-condition level spectroscopy method and the phenomenological renormalization group method, and analytically by the spin wave theory. We determine the phase boundaries among the XY phase, the Haldane phase, the ferromagnetic phase and the Néel phase, and then we confirm the universality class. Moreover, we map this model onto the non-linear σ model and analyze the phase diagram in the α-1 and Δ1 region by using the renormalization group method.. |

17. | K. Hijii, K. Nomura, and A.Kitazawa, Phase diagram of S=1/2 two-leg XXZ spin-ladder systems, Physical Review B, 10.1103/PhysRevB.72.014449, 72, 1, Vol. 72pp.014449, 2005.01, We investigate the ground-state phase diagram of the S=12 two-leg XXZ spin-ladder system with an isotropic interchain coupling. In this model, there is the Berezinskii-Kosterlitz-Thouless transition which occurs at the XY-Haldane and XY-rung singlet phase boundaries. It was difficult to determine the transition line using traditional methods. We overcome this difficulty using the level spectroscopy method combined with the twisted boundary condition method, and we check the consistency. We find out that the phase boundary between XY phase and Haldane phase lies on Δ=0 line. And we show that there exist two different XY phases, which we can distinguish investigating a XX correlation function.. |

18. | H. Otsuka, K. Mori, Y. Okabe, and K. Nomura, Level spectroscopy of the square-lattice three-state Potts model with a ferromagnetic next-nearest-neighbor coupling, Phys. Rev. E, 10.1103/PhysRevE.72.046103, 72, 4, Vol. 72046103, 2005.01. |

19. | 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. . |

20. | A. Kitazawa, K. Hijii, and K. Nomura, An SU(2) symmetry of the one-dimensional spin-1 XY model, J. Phys. A, 10.1088/0305-4470/36/23/104, 36, 23, L351-L357, Vol. 36, L351-L357, 2004.01, We show that the one-dimensional spin-1 XY model has an additional SU(2) symmetry for the open boundary condition and for an artificial one. We can explain some degeneracies of excitation states which were reported in previous numerical studies.. |

21. | K. Hijii, S. Qin, and K. Nomura, Staggered dimer order and criticality in S=1/2 quantum spin ladder system with four spin exchange, Phys. Rev. B, 10.1103/PhysRevB.68.134403, 68, 134403, Vol. 68, pp. 134403, 2003.10, We study an S=12 quantum-spin ladder system with four-spin cyclic exchange, using the density matrix renormalization group (DMRG) and exact diagonalization methods. Recently, the phase transition and its universality class in this system have been studied. However, controversies remain on whether the phase transition is second-order type or another type and on the nature of critical phenomena. In addition, there are several arguments concerning where the transition point is. Analyzing DMRG data, we use an approach to determine the ordered phase which appears after the phase transition. We find that the edge state appears under the open boundary condition by investigating excitation energies of states with higher magnetizations. We also estimate the correlation length and discuss the critical behavior.. |

22. | 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.. |

23. | 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.. |

24. | 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. . |

25. | H. Inoue and K. Nomura, Magnetization plateau in the 1D S=1/2 spin chain with alternating next-nearest-neighbor coupling", Phys. Lett. A, 10.1016/S0375-9601(99)00544-7, 262, 1, 96-102, Vol.262, pp.96-102, 1999.01. |

26. | M. Nakamura, A. Kitazawa and K. Nomura, Spin-Gap Phases in Tomonaga-Luttinger Liquids, Phys. Rev. B, 10.1103/PhysRevB.60.7850, 60, 11, 7850-7862, Vol. 60 pp.7850-7862, 1999.01. |

27. | 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.. |

28. | M. Tsukano and K. Nomura, Spin-1 XXZ chains in a staggered magnetic field, Phys. Rev. B, 10.1103/PhysRevB.57.R8087, 57, 14, R8087-R8090, Vol. 57, R8087-R8090, 1998.08. |

29. | A. Kitazawa and K. Nomura, Bifurcation at the c=3/2 Takhtajan-Babujian point to the c=1 critical lines, Phys. Rev. B, 10.1103/PhysRevB.59.11358, 59, 17, 11358-11366, Vol. 59, pp.11358-11366, 1999.01. |

30. | 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. |

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