\(XXZ\) Spin-Peierls Chain

Part of the Springer Theses book series (Springer Theses)


The phase transition and the phase diagram between the adiabatic and antiadiabatic limits in the spin-Peierls model will be investigated in this chapter. As we explained in Chap.  3, the conventional quantum Monte Carlo methods cannot efficiently calculate the model. Then, for overcoming the difficulty, we will apply the new Monte Carlo update techniques and the analysis that we developed in Chap.  2 4. We will show the precise transition point and the universality class of the spin-Peierls chain without any approximation, which has been discussed for several decades. For the XY anisotropic case, the Kosterlitz-Thouless type phase transition occurs between the Tomonaga-Luttinger liquid phase and the dimer (spin-Peierls) phase. For the Ising anisotropic case, on the other hand, the phase transition between the Néel phase and the dimer phase is the Gaussian universality. The whole phase diagram will be shown, compared to the previous result from the renormalization group method. We will discuss the correspondence that the phase diagram is consistent with the J1-J2 model that is an effective spin model in the antiadiabatic limit; that is the quantum nature of the lattice degree of freedom is relevant to the phase diagram.


Kosterlitz-Thouless transition \(J_1-J_2\) spin chain model Sine-Gordon model Adiabatic-antiadiabatic crossover 


  1. 1.
    Affleck, I., Gepner, D., Schulz, H. J., & Ziman, T. (1989). Critical behaviour of spin-s Heisenberg antiferromagnetic chains: Analytic and numerical results. Journal of Physics A: Mathematical and General, 22, 511.MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Arai, M., Fujita, M., Motokawa, M., Akimitsu, J., & Bennington, S. M. (1996). Quantum spin excitations in the spin-Peierls system CuGeO\(_3\). Physical Review Letters, 77, 3649.ADSCrossRefGoogle Scholar
  3. 3.
    Augier, D., & Poilblanc, D. (1998). Dynamical properties of low-dimensional CuGeO\(_3\) and NaV\(_2\)O\(_5\) spin-Peierls systems. European Physical Journal B: Condensed Matter Physics, 1, 19–28.ADSCrossRefGoogle Scholar
  4. 4.
    Augier, D., Poilblanc, D., Sørensen, E., & Affleck, I. (1998). Dynamical effects of phonons on soliton binding in spin-Peierls systems. Physical Review B, 58, 9110–9113.ADSCrossRefGoogle Scholar
  5. 5.
    Bakrim, H., & Bourbonnais, C. (2007). Quantum vs classical aspects of one dimensional electron-phonon systems revisited by the renormalization group method. Physical Review B, 76, 195115.Google Scholar
  6. 6.
    Binder, K. (1981). Critical properties from Monte Carlo coarse graining and renormalization. Physical Review Letters, 47, 693–696.ADSCrossRefGoogle Scholar
  7. 7.
    Binder, K., & Landau, D. P. (1984). Finite-size scaling at first-order phase transitions. Physical Review B, 30, 1477–1485.ADSCrossRefGoogle Scholar
  8. 8.
    Büchner, B., Fehske, H., Kampf, A. P., & Wellein, G. (1999). Lattice dimerization in the spinPeierls compound CuGeO\(_3\). Physica B: Condensed Matter, 259–261, 956.CrossRefGoogle Scholar
  9. 9.
    Bursill, R. J., McKenzie, R. H., & Hamer, C. J. (1999). Phase diagram of a Heisenberg spin-Peierls model with quantum phonons. Physical Review Letters, 83, 408.ADSCrossRefGoogle Scholar
  10. 10.
    Cardy, J. L. (1984). Conformal invariance and universality in finite-size scaling. Journal of Physics A: Mathematical and General, 17, L385.MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Cardy, J. L. (1986). Logarithmic corrections to finite-size scaling in strips. Journal of Physics A: Mathematical and General, 19, L1093.ADSCrossRefGoogle Scholar
  12. 12.
    Cardy, J. L. (1986). Operator content of two-dimensional conformally invariant theories. Nuclear Physics B, 270, 186.MathSciNetADSCrossRefMATHGoogle Scholar
  13. 13.
    Caron, L. G., & Bourbonnais, C. (1984). Two-cutoff renormalization and quantum versus classical aspects for the one-dimensional electron-phonon system. Physical Review B, 29, 4230–4241.ADSCrossRefGoogle Scholar
  14. 14.
    Caron, L. G., & Moukouri, S. (1996). Density matrix renormalization group applied to the ground state of the \(XY\) spin-Peierls system. Physical Review Letters, 76, 4050.ADSCrossRefGoogle Scholar
  15. 15.
    Citro, R., Orignac, E., & Giamarchi, T. (2005). Adiabatic-antiadiabatic crossover in a spin-Peierls chain. Physical Review B, 72, 024434.Google Scholar
  16. 16.
    Cross, M. C., & Fisher, D. S. (1979). A new theory of the spin-Peierls transition with special relevance to the experiments on TTFCuBDT. Physical Review B, 19, 402.ADSCrossRefGoogle Scholar
  17. 17.
    Dashen, R. F., Hassiacher, B., & Neveu, A. (1974). Nonperturbative methods and extended-hadron models in field theory. I. semiclassical functional methods. Physical Review D, 10, 4114–4129.Google Scholar
  18. 18.
    Fehske, H., Holicki, M., & Weiße, A. (2000). Lattice dynamical effects on the peierls transition in one-dimensional metals and spin chains. Advances in Solid State Physics, 40, 235–250.CrossRefGoogle Scholar
  19. 19.
    Fradkin, E., & Hirsch, J. E. (1983). Phase diagram of one-dimensional electron-phonon systems. I. the Su-Schrieffer-Heeger model. Physical Review B, 27, 1680–1697.ADSCrossRefGoogle Scholar
  20. 20.
    Fukui, K., & Todo, S. (2009). Order-\(N\) cluster Monte Carlo method for spin systems with long-range interactions. Journal of Computational Physics, 228, 2629.MathSciNetADSCrossRefMATHGoogle Scholar
  21. 21.
    Geertsma, W., & Khomskii, D. (1996). Influence of side groups on 90 Figa superexchange: A modification of the Goodenough-Kanamori-Anderson rules. Physical Review B, 54, 3011–3014.Google Scholar
  22. 22.
    Gros, C., & Werner, R. (1998). Dynamics of the Peierls-active phonon modes in CuGeO\(_3\). Physical Review B, 58, R14677.Google Scholar
  23. 23.
    Harada, K., & Kawashima, N. (1997). Universal jump in the helicity modulus of the two-dimensional quantum XY model. Physical Review B, 55, R11949.Google Scholar
  24. 24.
    Harada, K., & Kawashima, N. (1998). Kosterlitz-thouless transition of quantum XY model in two dimensions. Journal of the Physical Society of Japan, 67, 2768.ADSCrossRefGoogle Scholar
  25. 25.
    Hase, M., Terasaki, I., & Uchinokura, K. (1993). Observation of the spin-Peierls transition in linear Cu\(^{2+}\) (spin-\(\frac{1}{2})\) chains in an inorganic compound CuGeO\(_3\). Physical Review Letters, 70, 3651.ADSCrossRefGoogle Scholar
  26. 26.
    Huizinga, S., Kommandeur, J., Sawatzky, G. A., Thole, B. T., Kopinga, K., de Jonge, W. J. M., et al. (1979). Spin-Peierls transition in N-methyl-N-ethyl-morpholinium-ditetracyanoquinodimethanide [MEM-(TCNQ)\(_2\)]. Physical Review B, 19, 4723–4732.ADSCrossRefGoogle Scholar
  27. 27.
    Jeckelmann, E., Zhang, C., & White, S. R. (1999). Metal-insulator transition in the one-dimensional Holstein model at half filling. Physical Review B, 60, 7950–7955.ADSCrossRefGoogle Scholar
  28. 28.
    Kikuchi, J., Matsuoka, T., Motoya, K., Yamauchi, T., & Ueda, Y. (2002). Absence of edge localized moments in the doped spin-Peierls system CuGe\(_{1-x}\)Si\(_x\)O\(_3\). Physical Review Letters, 88, 037603.Google Scholar
  29. 29.
    Kosterlitz, J. M., & Thouless, D. J. (1973). Ordering, metastability and phase transitions in two-dimensional systems. Journal of Physics C, 6, 1181.CrossRefGoogle Scholar
  30. 30.
    Kuboki, K., & Fukuyama, H. (1987). Spin-Peierls transition with competing interactions. Journal of the Physical Society of Japan, 56, 3126–3134.ADSCrossRefGoogle Scholar
  31. 31.
    Kühne, R. W., & Löw, U. (1999). Thermodynamical properties of a spin-\(\frac{1}{2}\) Heisenberg chain coupled to phonons. Physical Review B, 60, 12125.Google Scholar
  32. 32.
    Landau, D. P., & Binder, K. (2005). A guide to Monte Carlo simulations in statistical physics (2nd ed.). Cambridge: Cambridge University Press.CrossRefMATHGoogle Scholar
  33. 33.
    Ludwig, A. W. W., & Cardy, J. L. (1987). Perturbative evaluation of the conformal anomaly at new critical points with applications to random systems. Nuclear Physics B, 285, 687.MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    McKenzie, R. H., Hamer, C. J., & Murray, D. W. (1996). Quantum Monte Carlo study of the one-dimensional Holstein model of spinless fermions. Physical Review B, 53, 9676–9687.ADSCrossRefGoogle Scholar
  35. 35.
    Michel, F., & Evertz, H. G. (2007). Lattice dynamics of the Heisenberg chain coupled to finite frequency bond phonons. cond-mat p. arXiv:0705.0799v2.Google Scholar
  36. 36.
    Nakano, T., & Fukuyama, H. (1981). Dimerization and solitons in one-dimensional XY-Z antiferromagnets. Journal of the Physical Society of Japan, 50, 2489–2499.ADSCrossRefGoogle Scholar
  37. 37.
    Nelson, D. R., & Kosterlitz, J. M. (1977). Universal jump in the superfluid density of two-dimensional superfuilds. Physical Review Letters, 39, 1201–1204.ADSCrossRefGoogle Scholar
  38. 38.
    Okamoto, K., & Nomura, K. (1992). Fluid-dimer critical point in \(S=\frac{1}{2}\) antiferromagnetic Heisenberg chain with next nearest neighbor interactions. Physics Letters A, 169, 433.ADSCrossRefGoogle Scholar
  39. 39.
    Olsson, P., & Minnhagen, P. (1991). On the helicity modulus, the critical temperature and Monte Carlo simulations for the two-dimensional XY-model. Physica Scripta, 43, 203–209.ADSCrossRefGoogle Scholar
  40. 40.
    Onishi, H., & Miyashita, S. (2003). Quantum narrowing effect in a spin-Peierls system with quantum lattice fluctuation. Journal of the Physical Society of Japan, 72, 392.ADSCrossRefGoogle Scholar
  41. 41.
    Pearson, C. J., Barford, W., & Bursill, R. J. (2010). Quantized lattice dynamic effects on the spin-Peierls transition. Physical Review B, 82, 144408.Google Scholar
  42. 42.
    Pouget, J. P. (2001). Microscopic interactions in CuGeO\(_3\) and organic spin-Peierls systems deduced from their pretransitional lattice fluctuations. European Physical Journal B: Condensed Matter Physics, 20, 321–333.ADSCrossRefGoogle Scholar
  43. 43.
    Pouget, J. P., Ravy, S., Schoeffel, J., Dhalenne, G., & Revcolevshi, A. (2004). Spin-Peierls lattice fluctuations and disorders in CuGeO\(_3\) and its solid solutions. European Physical Journal B: Condensed Matter Physics, 38, 581–598.ADSCrossRefGoogle Scholar
  44. 44.
    Raas, C., Bühler, A., & Uhrig, G. S. (2001). Effective spin models for spin-phonon chains by flow equations. European Physical Journal B: Condensed Matter Physics, 21, 369–374.ADSCrossRefGoogle Scholar
  45. 45.
    Raas, C., Löw, U., Uhrig, G. S., & Kühne, R. W. (2002). Spin-phonon chains with bond coupling. Physical Review B, 65, 144438.Google Scholar
  46. 46.
    Regnault, L. P., Renard, J. P., Dhalenne, G., & Revcolevschi, A. (1995). Coexistence of dimerization and antiferromagnetism in Si-doped CuGeO\(_3\). Europhysics Letters, 32, 579–584.ADSCrossRefGoogle Scholar
  47. 47.
    Reinicke, P. (1987). Analytical and non-analytical corrections to finite-size scaling. Journal of Physics A, 20, 5325.MathSciNetCrossRefGoogle Scholar
  48. 48.
    Sandvik, A. W., & Campbell, D. K. (1999). Spin-Peierls transition in the Heisenberg chain with finite-frequency phonons. Physical Review Letters, 83, 195.ADSCrossRefGoogle Scholar
  49. 49.
    Shastry, B. S., & Sutherland, B. (1990). Twisted boundary conditions and effective mass in Heisenberg-Ising and Hubbard rings. Physical Review Letters, 65, 243–246.MathSciNetADSCrossRefMATHGoogle Scholar
  50. 50.
    Simonet, V., Grenier, B., Villain, F., Flank, A. M., Dhalenne, G., Revcolevschi, A., et al. (2006). Effect of structural distortions on the magnetism of doped spin-Peierls CuGeO\(_3\). European Physical Journal B: Condensed Matter Physics, 53, 155–167.ADSCrossRefGoogle Scholar
  51. 51.
    Sun, P., Schmeltzer, D., Bishop, A. R. (2000). Analytic approach to the one-dimensional spin-Peierls system in the entire frequency range. Physical Review B, 62, 11,308–11,311.Google Scholar
  52. 52.
    Trebst, S., Elstner, N., & Monien, H. (2001). Renormalization of the spin-Peierls transition due to phonon dynamics. Europhysics Letters, 56(2), 268–274.ADSCrossRefGoogle Scholar
  53. 53.
    Uchinokura, K. (2002). Spin-Peierls transition in CuGeO\(_3\) and impurity-induced ordered phases in low-dimensional spin-gap systems. Journal of Physics: Condensed Matter, 14, R195–R237.ADSCrossRefGoogle Scholar
  54. 54.
    Uhrig, G. S. (1998). Nonadiabatic approach to spin-Peierls transitions via flow equations. Physical Review B, 57, R14004.Google Scholar
  55. 55.
    van Bodegom, B., Larson, B. C., & Mook, H. A. (1981). Diffuse x-ray and inelastic neutron scattering study of the spin Peierls transition in N-methyl-N-ethyl-morpholinium bistetracyanoquinodimethane [MEM (TCNQ)\(_2\)]. Physical Review B, 24, 1520–1523.ADSCrossRefGoogle Scholar
  56. 56.
    Visser, R. J. J., Oostra, S., Vettier, C., & Voiron, J. (1983). Determination of the spin-Peierls distortion in N-methyl-N-ethyl-morpholinium ditetracyanoquinodimethanide [MEM(TCNQ)\(_2\)]: Neutron diffraction study at 6 K. Physical Review B, 28, 2074–2077.ADSCrossRefGoogle Scholar
  57. 57.
    Voit, J., & Schulz, H. J. (1988). Electron-phonon interaction and phonon dynamics in one-dimensional conductors. Physical Review B, 37, 10,068–10,085.Google Scholar
  58. 58.
    Weber, H., & Minnhagen, P. (1988). Monte Carlo determination of the critical temperature for the two-dimensional XY mode. Physical Review B, 37, 5986.ADSCrossRefGoogle Scholar
  59. 59.
    Weiße, A., Hager, G., Bishop, A. R., & Fehske, H. (2006). Phase diagram of the spin-Peierls chain with local coupling: Density-matrix renormalization-group calculations and unitary transformations. Physical Review B, 74, 214426.Google Scholar
  60. 60.
    Weiße, A., Wellein, G., & Fehske, H. (1999). Quantum lattice fluctuations in a frustrated Heisenberg spin-Peierls chain. Physical Review B, 60, 6566.ADSCrossRefGoogle Scholar
  61. 61.
    Wellein, G., Fehske, H., & Kampf, A. P. (1998). Peierls dimerization with nonadiabatic spin-phonon coupling. Physical Review Letters, 81, 3956.ADSCrossRefGoogle Scholar
  62. 62.
    Werner, R., Gros, C., & Braden, M. (1999). Microscopic spin-phonon coupling constants in CuGeO\(_3\). Physical Review B, 59, 14,356–14,366.Google Scholar
  63. 63.
    Witten, E. (1984). Non-abelian bosonization in two dimensions. Communications in Mathematical Physics, 92, 455–472.MathSciNetADSCrossRefMATHGoogle Scholar
  64. 64.
    Zheng, H. (1997). Quantum lattice fluctuations in the ground state of an XY spin-Peierls chain. Physical Review B, 56, 14,414–14,422.Google Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Department of PhysicsBoston UniversityBostonUSA

Personalised recommendations