Advertisement

Transverse Ising System in Higher Dimensions (Pure Systems)

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

Abstract

One of the remarkable feature of a transverse Ising model is that it can be transformed into an effective classical Ising model. Chapter 3 starts from mentioning the mapping of a d-dimensional transverse Ising model to a (d+1)-dimensional classical Ising model, called the Suzuki-Trotter formalism. On the basis of this formalism, this chapter mentions numerical and analytical arguments on the pure transverse Ising models in two and three dimensions, which is followed by infinite-range models where the mean field theory provides exact results. The last part of this chapter is devoted to the scaling theory of a quantum phase transition as well as a note on real-space and field-theoretic renormalisation group techniques. The appendices include derivation of an effective classical Hamiltonian which represents a transverse Ising model and that of a quantum Hamiltonian equivalent to a classical Ising spin system.

Keywords

Partition Function Domain Wall Ising Model Quantum Phase Transition Transverse Field 
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. 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
  2. 72.
    Chayes, L., Crawford, N., Ioffe, D., Levit, A.: The phase diagram of the quantum Curie-Weiss model. J. Stat. Phys. 133, 131–149 (2008). [3.4.1] MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 82.
    Continentino, M.A.: Quantum scaling in many-body systems. Phys. Rep. 239(3), 179–213 (1994). [1.1, 1.3, 3.5] ADSCrossRefGoogle Scholar
  4. 103.
    Derrida, B.: Random-energy model: limit of a family of disordered models. Phys. Rev. Lett. 45, 79–82 (1980). [3.4.2, 6.6, 8.5.3.2] MathSciNetADSCrossRefGoogle Scholar
  5. 111.
    dos Santos, R.R., Sneddon, L., Stinchcombe, R.B.: The 2D transverse Ising model at t=0: a finite-size rescaling transformation approach. J. Phys. A, Math. Gen. 14(12), 3329 (1981). [3.6.1] ADSCrossRefGoogle Scholar
  6. 125.
    Elliott, R.J., Wood, C.: The Ising model with a transverse field. i. High temperature expansion. J. Phys. C, Solid State Phys. 4(15), 2359 (1971). [3.3] ADSCrossRefGoogle Scholar
  7. 126.
    Elliott, R.J., Pfeuty, P., Wood, C.: Ising model with a transverse field. Phys. Rev. Lett. 25, 443–446 (1970). [1.1, 3.1, 3.2] ADSCrossRefGoogle Scholar
  8. 150.
    Friedman, Z.: Critical exponents for the three-dimensional Ising model from the real-space renormalization group in two dimensions. Phys. Rev. Lett. 36, 1326–1328 (1976). [3.6.1] ADSCrossRefGoogle Scholar
  9. 151.
    Friedman, Z.: Ising model with a transverse field in two dimensions: phase diagram and critical properties from a real-space renormalization group. Phys. Rev. B 17, 1429–1432 (1978). [3.6.1] ADSCrossRefGoogle Scholar
  10. 173.
    Hertz, J.A.: Quantum critical phenomena. Phys. Rev. B 14, 1165–1184 (1976). [1.1, 3.5] ADSCrossRefGoogle Scholar
  11. 186.
    Husimi, K.: Proc. Int. Conf. Theor. Phys., 531 (1953). [3.4.1] Google Scholar
  12. 197.
    Ishizuka, H., Motome, Y., Furukawa, N., Suzuki, S.: Quantum Monte Carlo study of molecular polarization and antiferroelectric ordering in squaric acid crystals. Phys. Rev. B 84, 064120 (2011). [3.2] ADSCrossRefGoogle Scholar
  13. 198.
    Ishizuka, H., Motome, Y., Furukawa, N., Suzuki, S.: Quantum Monte Carlo study of the transverse-field Ising model on a frustrated checkerboard lattice. J. Phys. Conf. Ser. 320(1), 012054 (2011). [3.2] ADSCrossRefGoogle Scholar
  14. 205.
    Jörg, T., Krzakala, F., Kurchan, J., Maggs, A.C., Pujos, J.: Energy gaps in quantum first-order mean-field-like transitions: the problems that quantum annealing cannot solve. Europhys. Lett. 89(4), 40004 (2010). [1.3, 3.4.2, 8.5.3.1] CrossRefGoogle Scholar
  15. 208.
    Jullien, R., Pfeuty, P., Fields, J.N., Doniach, S.: Zero-temperature renormalization method for quantum systems. i. Ising model in a transverse field in one dimension. Phys. Rev. B 18, 3568–3578 (1978). [2.4, 3.5] ADSCrossRefGoogle Scholar
  16. 222.
    Kawashima, N., Harada, K.: Recent developments of world-line Monte Carlo methods. J. Phys. Soc. Jpn. 73(6), 1379–1414 (2004). [3.2] ADSzbMATHCrossRefGoogle Scholar
  17. 231.
    Kogut, J.B.: An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659–713 (1979). [1.1, 2.1.1, 3.A.2] MathSciNetADSCrossRefGoogle Scholar
  18. 238.
    Krzakala, F., Rosso, A., Semerjian, G., Zamponi, F.: Path-integral representation for quantum spin models: application to the quantum cavity method and Monte Carlo simulations. Phys. Rev. B 78, 134428 (2008). [3.4.1] ADSCrossRefGoogle Scholar
  19. 241.
    Landau, D.P., Binder, K.: A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge University Press, Cambridge (2000). [3.2, 5.3, 6.3] zbMATHGoogle Scholar
  20. 275.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987). [1.3, 3.4.2, 6.1, 6.5, 6.A.3, 8.1] zbMATHGoogle Scholar
  21. 283.
    Morita, S., Suzuki, S., Nakamura, T.: Quantum-thermal annealing with a cluster-flip algorithm. Phys. Rev. E 79, 065701 (2009). [3.2] ADSCrossRefGoogle Scholar
  22. 286.
    Mühlschlegel, B., Zittartz, H.: Gaussian average method in the statistical theory of the Ising model. Z. Phys. A, Hadrons Nucl. 175, 553–573 (1963). [3.6.2] CrossRefGoogle Scholar
  23. 290.
    Nagai, O., Yamada, Y., Miyatake, Y.: In: Suzuki, M. (ed.) Quantum Monte Carlo Methods, p. 95. Springer, Heidelberg (1986). [1.3, 3.2] Google Scholar
  24. 292.
    Nakamura, T., Ito, Y.: A quantum Monte Carlo algorithm realizing an intrinsic relaxation. J. Phys. Soc. Jpn. 72(10), 2405–2408 (2003). [3.2] ADSCrossRefGoogle Scholar
  25. 301.
    Oitmaa, J., Plischke, M.: Critical behaviour of the Ising model in a transverse field. Physica B+C 86–88(Part 2), 577–578 (1977). [3.3] CrossRefGoogle Scholar
  26. 308.
    Penson, K.A., Jullien, R., Pfeuty, P.: Zero-temperature renormalization-group method for quantum systems. iii. Ising model in a transverse field in two dimensions. Phys. Rev. B 19, 4653–4660 (1979). [3.6.1] ADSCrossRefGoogle Scholar
  27. 313.
    Pfeuty, P., Elliott, R.J.: The Ising model with a transverse field. ii. Ground state properties. J. Phys. C, Solid State Phys. 4(15), 2370 (1971). [3.6.1] ADSCrossRefGoogle Scholar
  28. 349.
    Schultz, T.D., Mattis, D.C., Lieb, E.H.: Two-dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys. 36, 856–871 (1964). [1.1, 1.3, 3.1, 3.A.2, 10.1.2] MathSciNetADSCrossRefGoogle Scholar
  29. 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
  30. 379.
    Stinchcombe, R.B.: Ising model in a transverse field. i. Basic theory. J. Phys. C, Solid State Phys. 6(15), 2459 (1973). [1.1, 1.2, 1.3, 3.6.2, 6.7.2] ADSCrossRefGoogle Scholar
  31. 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
  32. 385.
    Suzuki, M.: Relationship among exactly soluble models of critical phenomena. i. Prog. Theor. Phys. 46(5), 1337–1359 (1971). [1.1, 1.3, 3.1, 10.1.2] ADSzbMATHCrossRefGoogle Scholar
  33. 386.
    Suzuki, M.: Relationship between d-dimensional quantal spin systems and (d+1)-dimensional Ising systems. Prog. Theor. Phys. 56(5), 1454–1469 (1976). [1.1, 1.3, 3.1, 5.2, 8.7.2, 9.1.2, 9.2, 9.2.4, 9.2.5, 9.2.6] ADSzbMATHCrossRefGoogle Scholar
  34. 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
  35. 391.
    Swendsen, R.H., Wang, J.S.: Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987). [1.3, 3.2] ADSCrossRefGoogle Scholar
  36. 398.
    Temperley, H.N.V.: Proc. Phys. Soc. 67, 233 (1954). [3.4.1] ADSzbMATHCrossRefGoogle Scholar
  37. 403.
    Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959). [3.1, 9.2.4] MathSciNetzbMATHCrossRefGoogle Scholar
  38. 421.
    Wiesler, A.: A note on the Monte Carlo simulation of one dimensional quantum spin systems. Phys. Lett. A 89(7), 359–362 (1982). [1.3, 3.2, 6.3] ADSCrossRefGoogle Scholar
  39. 434.
    Yanase, A.: Correlation index of the Ising model with a transverse field. J. Phys. Soc. Jpn. 42(6), 1816–1818 (1977). [3.6.1] ADSCrossRefGoogle Scholar
  40. 438.
    Young, A.P.: Quantum effects in the renormalization group approach to phase transitions. J. Phys. C, Solid State Phys. 8(15), L309 (1975). [1.1, 3.5, 3.6.2] 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

Personalised recommendations