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ANNNI Model in Transverse Field

  • Sei Suzuki
  • Jun-ichi Inoue
  • Bikas K. Chakrabarti
Part of the Lecture Notes in Physics book series (LNP, volume 862)

Abstract

Competing interactions between spins or frustration give rise to intriguing many-body states with spatially modulated spin structures. The simplest model with the regular frustration is the classical axial next nearest neighbour Ising (ANNNI) model. The classical ANNNI model is described by a system of Ising spins with nearest neighbour interactions along all the lattice directions (x, y and z) as well as a competing next nearest neighbour interaction in one axial direction (z for instance). Chapter 4 discusses detailed results on quantum ANNNI models in a transverse field with a brief introduction to the classical model. Analytic studies on the basis of an interacting fermion representation of the model, real-space and field theoretic renormalisation group techniques, the numerical exact diagonalisation method, and Monte Carlo simulations have revealed a variety of ground-state phases of the quantum ANNNI chain. These are mentioned in Chap. 4. Studies of higher dimensional quantum ANNNI models are also presented there. Appendices of this chapter include details of approximate methods used in the study of the quantum ANNNI chain.

Keywords

Paramagnetic Phase Quantum Fluctuation Neighbour Interaction Transverse Field Ising System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 9.
    Allen, D., Azaria, P., Lecheminant, P.: A two-leg quantum Ising ladder: a bosonization study of the ANNNI model. J. Phys. A, Math. Gen. 34(21), L305 (2001). [1.1, 1.3, 4.3.7] MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 18.
    Arizmendi, C.M., Rizzo, A.H., Epele, L.N., García Canal, C.A.: Phase diagram of the ANNNI model in the Hamiltonian limit. Z. Phys. B, Condens. Matter 83, 273–276 (1991). [1.3, 4.3, 4.3.7] ADSCrossRefGoogle Scholar
  3. 20.
    Auerbach, A.: Interacting Fermions and Quantum Magnetism. Springer, New York (1994). [1.3, 4.1, 4.3] CrossRefGoogle Scholar
  4. 24.
    Barber, M.N., Duxbury, P.M.: A quantum Hamiltonian approach to the two-dimensional axial next-nearest-neighbour Ising model. J. Phys. A, Math. Gen. 14(7), L251 (1981). [1.1, 4.3] MathSciNetADSCrossRefGoogle Scholar
  5. 25.
    Barber, M.N., Duxbury, P.M.: Hamiltonian studies of the two-dimensional axial next-nearest-neighbor Ising (ANNNI) model. J. Stat. Phys. 29, 427–432 (1982). [3.A.2, 4.3] MathSciNetADSCrossRefGoogle Scholar
  6. 31.
    Beccaria, M., Campostrini, M., Feo, A.: Density-matrix renormalization-group study of the disorder line in the quantum axial next-nearest-neighbor Ising model. Phys. Rev. B 73, 052402 (2006). [1.3, 4.3.7] ADSCrossRefGoogle Scholar
  7. 32.
    Beccaria, M., Campostrini, M., Feo, A.: Evidence for a floating phase of the transverse ANNNI model at high frustration. Phys. Rev. B 76, 094410 (2007). [1.3, 4.3.7] ADSCrossRefGoogle Scholar
  8. 48.
    Brout, R., Müller, K., Thomas, H.: Tunnelling and collective excitations in a microscopic model of ferroelectricity. Solid State Commun. 4(10), 507–510 (1966). [1.1, 1.2, 4.5, 6.7.2, 7.1.1] ADSCrossRefGoogle Scholar
  9. 66.
    Chandra, A.K., Dasgupta, S.: Floating phase in a 2D ANNNI model. J. Phys. A, Math. Theor. 40(24), 6251 (2007). [1.1, 4.3.7] MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 67.
    Chandra, A.K., Dasgupta, S.: Floating phase in the one-dimensional transverse axial next-nearest-neighbor Ising model. Phys. Rev. E 75, 021105 (2007). [1.1, 1.3, 4.3.7] ADSCrossRefGoogle Scholar
  11. 68.
    Chandra, A.K., Dasgupta, S.: Spin-spin correlation in some excited states of the transverse Ising model. J. Phys. A, Math. Theor. 40(20), 5231 (2007). [1.1, 4.3.7] MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 71.
    Chandra, P., Doucot, B.: Possible spin-liquid state at large s for the frustrated square Heisenberg lattice. Phys. Rev. B 38, 9335–9338 (1988). [4.3] ADSCrossRefGoogle Scholar
  13. 102.
    Derian, R., Gendiar, A., Nishino, T.: Modulation of local magnetization in two-dimensional axial-next-nearest-neighbor Ising model. J. Phys. Soc. Jpn. 75(11), 114001 (2006). [1.1, 4.3.7] ADSCrossRefGoogle Scholar
  14. 115.
    Dutta, A., Sen, D.: Gapless line for the anisotropic Heisenberg spin-\(\frac{1}{2}\) chain in a magnetic field and the quantum axial next-nearest-neighbor Ising chain. Phys. Rev. B 67, 094435 (2003). [1.1, 1.3, 4.3.7] ADSCrossRefGoogle Scholar
  15. 117.
    Duxbury, P.M., Barber, M.N.: Hamiltonian studies of the two-dimensional axial next-nearest neighbour Ising (ANNNI) model. ii. Finite-lattice mass gap calculations. J. Phys. A, Math. Gen. 15(10), 3219 (1982). [4.3] MathSciNetADSCrossRefGoogle Scholar
  16. 122.
    Elliott, R.J.: Phenomenological discussion of magnetic ordering in the heavy rare-earth metals. Phys. Rev. 124, 346–353 (1961). [4.1] ADSCrossRefGoogle Scholar
  17. 129.
    Emery, V.J., Noguera, C.: Critical properties of a spin-(1/2) chain with competing interactions. Phys. Rev. Lett. 60, 631–634 (1988). [4.3] ADSCrossRefGoogle Scholar
  18. 145.
    Fisher, M.E., Selke, W.: Infinitely many commensurate phases in a simple Ising model. Phys. Rev. Lett. 44, 1502–1505 (1980). [4.2] MathSciNetADSCrossRefGoogle Scholar
  19. 152.
    Garel, T., Pfeuty, P.: Commensurability effects on the critical behaviour of systems with helical ordering. J. Phys. C, Solid State Phys. 9(10), L245 (1976). [4.3] ADSCrossRefGoogle Scholar
  20. 165.
    Hallberg, K., Gagliano, E., Balseiro, C.: Finite-size study of a spin-1/2 Heisenberg chain with competing interactions: phase diagram and critical behavior. Phys. Rev. B 41, 9474–9479 (1990). [4.3] ADSCrossRefGoogle Scholar
  21. 166.
    Hamer, C.J., Barber, M.N.: Finite-lattice methods in quantum Hamiltonian field theory. i. O(2) and O(3) Heisenberg models. J. Phys. A, Math. Gen. 14(1), 259 (1981). [1.3, 4.3] MathSciNetADSCrossRefGoogle Scholar
  22. 167.
    Hamer, C.J., Barber, M.N.: Finite-lattice methods in quantum Hamiltonian field theory. i. The Ising model. J. Phys. A, Math. Gen. 14(1), 241 (1981). [1.3, 2.3.1, 4.3] MathSciNetADSCrossRefGoogle Scholar
  23. 171.
    Harris, A.B., Micheletti, C., Yeomans, J.M.: Quantum fluctuations in the axial next-nearest-neighbor Ising model. Phys. Rev. Lett. 74, 3045–3048 (1995). [1.3, 4.4] ADSCrossRefGoogle Scholar
  24. 181.
    Hornreich, R.M., Liebmann, R., Schuster, H.G., Selke, W.: Lifshitz points in Ising systems. Z. Phys. B, Condens. Matter 35, 91–97 (1979). [4.2] ADSGoogle Scholar
  25. 183.
    Hu, B.: The classical Ising model: a quantum renormalization group approach. Phys. Lett. A 71(1), 83–86 (1979). [2.4.1, 4.3] ADSCrossRefGoogle Scholar
  26. 187.
    Igarashi, J.i., Tonegawa, T.: Excitation spectrum of a spin-1/2 chain with competing interactions. Phys. Rev. B 40, 756–759 (1989). [4.3] ADSCrossRefGoogle Scholar
  27. 226.
    Kimball, J.C.: The kinetic Ising model: exact susceptibilities of two simple examples. J. Stat. Phys. 21, 289–300 (1979). [4.3] ADSCrossRefGoogle Scholar
  28. 247.
    Liebmann, R.: Statistical Mechanics of Periodic Frustrated Ising Systems. Lecture Notes in Physics, vol. 251. Springer, Berlin (1986). [4.2, 4.A.3] Google Scholar
  29. 255.
    Majumdar, C.K., Ghosh, D.K.: On next-nearest-neighbor interaction in linear chain. i. J. Math. Phys. 10(8), 1388–1398 (1969). [1.3, 4.3] MathSciNetADSCrossRefGoogle Scholar
  30. 256.
    Majumdar, C.K., Ghosh, D.K.: On next-nearest-neighbor interaction in linear chain. ii. J. Math. Phys. 10(8), 1399–1402 (1969). [1.3, 4.3] MathSciNetADSCrossRefGoogle Scholar
  31. 263.
    Mattis, D.C.: Encyclopedia of Magnetism in One Dimension. World Scientific, Singapore (1994). [1.3, 4.1] Google Scholar
  32. 288.
    Müller-Hartmann, E., Zittartz, J.: Interface free energy and transition temperature of the square-lattice Ising antiferromagnet at finite magnetic field. Z. Phys. B, Condens. Matter 27, 261–266 (1977). [4.2] ADSGoogle Scholar
  33. 291.
    Nagy, A.: Exploring phase transitions by finite-entanglement scaling of MPS in the 1D ANNNI model. New J. Phys. 13(2), 023015 (2011). [1.1, 1.3, 4.3.7] ADSCrossRefGoogle Scholar
  34. 311.
    Peschel, I., Emery, V.J.: Calculation of spin correlations in two-dimensional Ising systems from one-dimensional kinetic models. Z. Phys. B, Condens. Matter 43, 241–249 (1981). [1.3, 4.3, 4.3.7] MathSciNetADSCrossRefGoogle Scholar
  35. 312.
    Pfeuty, P.: The one-dimensional Ising model with a transverse field. Ann. Phys. 57(1), 79–90 (1970). [1.1, 1.3, 2.2, 2.2.1, 2.A.3, 4.3, 5.2, 10.1.2] ADSCrossRefGoogle Scholar
  36. 314.
    Pfeuty, P., Jullien, R., Penson, K.A.: In: Real Space Renormalisation. Topics in Current Physics, vol. 30, p. 119. Springer, Heidelberg (1982) [1.3, 4.3] CrossRefGoogle Scholar
  37. 330.
    Rieger, H., Uimin, G.: The one-dimensional ANNNI model in a transverse field: analytic and numerical study of effective Hamiltonians. Z. Phys. B, Condens. Matter 101, 597–611 (1996). [4.3.7] ADSCrossRefGoogle Scholar
  38. 336.
    Ruján, P.: Critical behavior of two-dimensional models with spatially modulated phases: analytic results. Phys. Rev. B 24, 6620–6631 (1981). [1.1, 4.3] ADSCrossRefGoogle Scholar
  39. 347.
    Schiff, L.I.: Quantum Mechanics. McGraw-Hill, London (1968). [4.A.3] Google Scholar
  40. 352.
    Selke, W.: The ANNNI model—theoretical analysis and experimental application. Phys. Rep. 170(4), 213–264 (1988). [4.1, 4.2] MathSciNetADSCrossRefGoogle Scholar
  41. 353.
    Selke, W.: In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena, vol. 15. Academic Press, New York (1992). [4.1, 4.2] Google Scholar
  42. 354.
    Selke, W., Duxbury, P.M.: The mean field theory of the three-dimensional ANNNI model. Z. Phys. B, Condens. Matter 57, 49–58 (1984). [4.2] ADSCrossRefGoogle Scholar
  43. 355.
    Sen, D.: Large-S analysis of a quantum axial next-nearest-neighbor Ising model in one dimension. Phys. Rev. B 43, 5939–5943 (1991). [1.3, 4.4] ADSCrossRefGoogle Scholar
  44. 356.
    Sen, D., Chakrabarti, B.K.: Large-S analysis of one-dimensional quantum-spin models in a transverse magnetic field. Phys. Rev. B 41, 4713–4722 (1990). [1.3, 4.3, 4.4] ADSCrossRefGoogle Scholar
  45. 357.
    Sen, P.: Ground state properties of a one dimensional frustrated quantum XY model. Phys. A, Stat. Mech. Appl. 186(1–2), 306–313 (1992). [4.6] CrossRefGoogle Scholar
  46. 358.
    Sen, P.: Order disorder transitions in Ising models in transverse fields with second neighbour interactions. Z. Phys. B, Condens. Matter 98, 251–254 (1995). [3.3, 4.5] ADSCrossRefGoogle Scholar
  47. 359.
    Sen, P., Chakrabarti, B.K.: Ising models with competing axial interactions in transverse fields. Phys. Rev. B 40, 760–762 (1989). [1.1, 1.3, 4.3] ADSCrossRefGoogle Scholar
  48. 360.
    Sen, P., Chakrabarti, B.K.: Critical properties of a one-dimensional frustrated quantum magnetic model. Phys. Rev. B 43, 13559–13565 (1991). [1.1, 1.3, 4.3] ADSCrossRefGoogle Scholar
  49. 361.
    Sen, P., Chakrabarti, B.K.: Frustrated transverse Ising models: a class of frustrated quantum systems. Int. J. Mod. Phys. B 6, 2439–2469 (1992). [1.1, 1.3, 4.6, 6.2, 6.3] ADSCrossRefGoogle Scholar
  50. 362.
    Sen, P., Chakraborty, S., Dasgupta, S., Chakrabarti, B.K.: Numerical estimate of the phase diagram of finite ANNNI chains in transverse field. Z. Phys. B, Condens. Matter 88, 333–338 (1992). [1.1, 1.3, 4.3] ADSCrossRefGoogle Scholar
  51. 384.
    Stratt, R.M.: Path-integral methods for treating quantal behavior in solids: mean-field theory and the effects of fluctuations. Phys. Rev. B 33, 1921–1930 (1986). [3.3, 3.4.1, 4.5] ADSCrossRefGoogle Scholar
  52. 387.
    Suzuki, M.: In: Suzuki, M. (ed.) Quantum Monte Carlo Methods, p. 1. Springer, Heidelberg (1986). [1.1, 1.3, 3.1, 4.3, 6.5, 6.A.2] Google Scholar
  53. 399.
    Tentrup, T., Siems, R.: Structure and free energy of domain walls in ANNNI systems. J. Phys. C, Solid State Phys. 19(18), 3443 (1986). [4.5] ADSCrossRefGoogle Scholar
  54. 412.
    Villain, J., Bak, P.: Two-dimensional Ising model with competing interactions: floating phase, walls and dislocations. J. Phys. Fr. 42(5), 657–668 (1981). [1.1, 4.2, 4.A.3] CrossRefGoogle Scholar
  55. 423.
    Wolf, D., Zittartz, J.: On the one-dimensional spin-1/2-chain and its related fermion models. Z. Phys. B, Condens. Matter 43, 173–183 (1981). [4.3] MathSciNetADSCrossRefGoogle Scholar
  56. 435.
    Yeomans, J.: The theory and application of axial Ising models. Solid State Phys. 41, 151–200 (1988). [4.1, 4.2] CrossRefGoogle Scholar
  57. 436.
    Yokoi, C.S.O., Coutinho-Filho, M.D., Salinas, S.R.: Ising model with competing axial interactions in the presence of a field: a mean-field treatment. Phys. Rev. B 24, 4047–4061 (1981). [4.2] ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sei Suzuki
    • 1
  • Jun-ichi Inoue
    • 2
  • Bikas K. Chakrabarti
    • 3
  1. 1.Dept. of Physics and MathematicsAoyama Gakuin UniversitySagamiharaJapan
  2. 2.Graduate School of Information Science and TechnologyHokkaido UniversitySapporoJapan
  3. 3.Saha Institute of Nuclear PhysicsKolkataIndia

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