Advertisement

Related Models

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

Abstract

Chapter 10 mentions the XY model in a transverse field and the Kitaev model as related models to the transverse Ising model. The XY model in a transverse field is another simplest model, except the transverse Ising model, that exhibits a zero-temperature quantum phase transition. It also gives the pseudo-spin representation of the BCS Hamiltonian of the superconductivity. The spin-1/2 transverse XY chain can be diagonalised by using the Jordan-Wigner transformation, as the transverse Ising chain is. The Kitaev model, on the other hand, can be transformed into a free fermion model coupled with a field, which commutes with the Hamiltonian, by the Jordan-Wigner transformation, though it is defined in two dimension. Several properties of the ground state and the dynamical behaviour following a slow quench of a parameter are mentioned.

Keywords

Quantum Phase Transition Transverse Field Fermion Operator Excitation Probability Flux Phase 
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. 16.
    Anderson, P.W.: Random-phase approximation in the theory of superconductivity. Phys. Rev. 112, 1900–1916 (1958). [1.1, 1.3, 10.1, 10.1.1] MathSciNetADSCrossRefGoogle Scholar
  2. 26.
    Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957). [10.1.1] MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 27.
    Barouch, E., McCoy, B.M.: Statistical mechanics of the xy model. ii. Spin-correlation functions. Phys. Rev. A 3, 786–804 (1971). [10.1.2] ADSCrossRefGoogle Scholar
  4. 29.
    Baskaran, G., Mandal, S., Shankar, R.: Exact results for spin dynamics and fractionalization in the Kitaev model. Phys. Rev. Lett. 98, 247201 (2007). [10.2.2] ADSCrossRefGoogle Scholar
  5. 51.
    Büttner, G., Usadel, K.D.: The exact phase diagram of the quantum XY spin glass model in a transverse field. Z. Phys. B, Condens. Matter 83, 131–134 (1991). [1.3, 10.1.4] ADSCrossRefGoogle Scholar
  6. 52.
    Büttner, G., Kopeć, T., Usadel, K.: Phase diagrams of the quantum XY spin glass model in a transverse field. Phys. Lett. A 149(5–6), 248–252 (1990). [10.1.4] ADSCrossRefGoogle Scholar
  7. 73.
    Chen, H.D., Nussinov, Z.: Exact results of the Kitaev model on a hexagonal lattice: spin states, string and brane correlators, and anyonic excitations. J. Phys. A, Math. Theor. 41(7), 075001 (2008). [10.2.1] MathSciNetADSCrossRefGoogle Scholar
  8. 74.
    Cherng, R.W., Levitov, L.S.: Entropy and correlation functions of a driven quantum spin chain. Phys. Rev. A 73, 043614 (2006). [10.1.2] ADSCrossRefGoogle Scholar
  9. 99.
    Deng, S., Ortiz, G., Viola, L.: Dynamical non-ergodic scaling in continuous finite-order quantum phase transitions. Europhys. Lett. 84(6), 67008 (2008). [10.1.2] ADSCrossRefGoogle Scholar
  10. 100.
    Deng, S., Ortiz, G., Viola, L.: Anomalous nonergodic scaling in adiabatic multicritical quantum quenches. Phys. Rev. B 80, 241109 (2009). [10.1.2] ADSCrossRefGoogle Scholar
  11. 101.
    Deng, S., Ortiz, G., Viola, L.: Dynamical critical scaling and effective thermalization in quantum quenches: role of the initial state. Phys. Rev. B 83, 094304 (2011). [10.1.2] ADSCrossRefGoogle Scholar
  12. 106.
    Divakaran, U., Dutta, A., Sen, D.: Quenching along a gapless line: a different exponent for defect density. Phys. Rev. B 78, 144301 (2008). [10.1.2] ADSCrossRefGoogle Scholar
  13. 107.
    Divakaran, U., Mukherjee, V., Dutta, A., Sen, D.: Defect production due to quenching through a multicritical point. J. Stat. Mech. Theory Exp. 2009(02), P02007 (2009). [10.1.2] CrossRefGoogle Scholar
  14. 110.
    dos Santos, R.R., Stinchcombe, R.B.: Finite size scaling and crossover phenomena: the XY chain in a transverse field at zero temperature. J. Phys. A, Math. Gen. 14(10), 2741 (1981). [10.1.2] ADSCrossRefGoogle Scholar
  15. 174.
    Hikichi, T., Suzuki, S., Sengupta, K.: Slow quench dynamics of the Kitaev model: anisotropic critical point and effect of disorder. Phys. Rev. B 82, 174305 (2010). [1.3, 10.2.3] ADSCrossRefGoogle Scholar
  16. 216.
    Katsura, S.: Statistical mechanics of the anisotropic linear Heisenberg model. Phys. Rev. 127, 1508–1518 (1962). [1.1, 1.3, 2.1.2, 10.1.2] ADSzbMATHCrossRefGoogle Scholar
  17. 229.
    Kitaev, A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321(1), 2–111 (2006). [1.3, 10.2.1] MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 237.
    Kramers, H.A., Wannier, G.H.: Statistics of the two-dimensional ferromagnet. Part i. Phys. Rev. 60, 252–262 (1941). [2.1.1, 10.1.2] MathSciNetADSCrossRefGoogle Scholar
  19. 245.
    Lieb, E., Schultz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16(3), 407–466 (1961). [2.2, 2.A.2, 10.1.2] MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 246.
    Lieb, E.H.: Flux phase of the half-filled band. Phys. Rev. Lett. 73, 2158–2161 (1994). [10.2.1] ADSCrossRefGoogle Scholar
  21. 287.
    Mukherjee, V., Divakaran, U., Dutta, A., Sen, D.: Quenching dynamics of a quantum XY spin-\(\frac{1}{2}\) chain in a transverse field. Phys. Rev. B 76, 174303 (2007). [10.1.2] ADSCrossRefGoogle Scholar
  22. 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
  23. 322.
    Ray, P., Chakrabarti, B.K.: Exact ground-state excitations of the XY model in a transverse field in one dimension. Phys. Lett. A 98(8–9), 431–432 (1983). [1.3, 10.1.2] ADSCrossRefGoogle Scholar
  24. 344.
    Satija, I.I.: Symmetry breaking and stabilization of critical phase. Phys. Rev. B 48, 3511–3514 (1993). [1.3, 10.1.3] ADSCrossRefGoogle Scholar
  25. 345.
    Satija, I.I.: Spectral and magnetic interplay in quantum spin chains: stabilization of the critical phase due to long-range order. Phys. Rev. B 49, 3391–3399 (1994). [1.3, 10.1.3] ADSCrossRefGoogle Scholar
  26. 346.
    Satija, I.I., Chaves, J.C.: XY-to-Ising crossover and quadrupling of the butterfly spectrum. Phys. Rev. B 49, 13239–13242 (1994). [1.3, 10.1.3] ADSCrossRefGoogle Scholar
  27. 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
  28. 364.
    Sengupta, K., Sen, D.: Entanglement production due to quench dynamics of an anisotropic xy chain in a transverse field. Phys. Rev. A 80, 032304 (2009). [10.1.2] ADSCrossRefGoogle Scholar
  29. 366.
    Sengupta, K., Sen, D., Mondal, S.: Exact results for quench dynamics and defect production in a two-dimensional model. Phys. Rev. Lett. 100, 077204 (2008). [1.3, 10.2.3] ADSCrossRefGoogle Scholar
  30. 373.
    Sokoloff, J.: Unusual band structure, wave functions and electrical conductance in crystals with incommensurate periodic potentials. Phys. Rep. 126(4), 189–244 (1985). [10.1.3] ADSCrossRefGoogle Scholar
  31. 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
  32. 407.
    Usadel, K.: Frustrated quantum spin systems. Nucl. Phys. B, Proc. Suppl. 5(1), 91–96 (1988). [10.1.4] 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