QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS

1. QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS Global Renormalization-Group AnalysisFerromagnetic Excitation Spectrum GapAntiferromagnetic Spin-Wave Stiffness Ozan S. SARIYER [ Istanbul Tecnical University ] Prof. Dr. A. Nihat BERKER [ Koç Univ. - M.I.T. - Feza Gürsey Res. Inst. ] Dr. Michael HINCZEWSKI [ Feza Gürsey Res. Inst. ] (2007) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA

2. RG IN 1-D CLASSICAL SYSTEMS Ising Model and RG

3. RG IN 1-D CLASSICAL SYSTEMS Ising Model and RG

4. RG IN 1-D CLASSICAL SYSTEMS Ising Model and RG

5. SUZUKI – TAKANO METHOD

6. SUZUKI – TAKANO METHOD Applications • 2-dimensional XY model: Suzuki and Takano (1979,1981) • 1,2,3-dimensional tJ electronic model: Falicov and Berker (1995) • AF Heisenberg model on fractal (kagomé, squagome) lattices: Tomczak and Richter (1996,2003) • 3-dimensional Hubbard electronic model: Hinczewski and Berker (2005) • M. Suzuki and H. Takano, Phys. Lett. A 69, 426 (1979). • H. Takano and M. Suzuki, J. Stat. Phys. 26, 635 (1981). • A. Falicov and A. N. Berker, Phys. Rev. B 51, 12458 (1995). • P. Tomczak, Phys. Rev. B 53, R500 (1996). • P. Tomczak and J. Richter, Phys. Rev. B 54, 9004 (1996). • P. Tomczak and J. Richter, J. Phys. A 36, 5399 (2003). • M. Hinczewski and A. N. Berker, Eur. Phys. J. B 48, 1 (2005).

7. XXZ MODEL • Has been studied since the introduction of “spin” concept (Heisenberg, Bloch, Bethe, Hulthén 1930s) • Still an actual problem in 2000s (Rojas et.al., Klümper, Bortz, Göhman... 2000s) • Theory gained richness with Haldane’s studies (Haldane 1980s) • High-Tc superconductivity ↔ Antiferromagnetism (Bednorz, Müller 1980; Hinczewski, Berker 2005) • Finite-systems extrapolation (Bonner, Fisher 1964) • Linked-cluster and dimer-cluster expansions (Inawashiro, Katsura 1965; Karbach et.al. 1993) • Quantum decimation (Xi-Yao, Tuthill 1985) • Decoupling Green’s functions (Zhang, Shen, Xu, Ting 1995) • Quantum transfer matrix (Fabricius, Klümper, McCoy 1999, Klümper 2004) • High-temperature series expansion (Rojas, de Souza, Thomaz 2002) • Numerical evaluation of multiple integrals (Bortz, Göhman 2005)

8. RENORMALIZATION-GROUP

9. RENORMALIZATION-GROUP

10. RENORMALIZATION-GROUP Isinglike FM Spin Liquid AFM Isinglike

11. RENORMALIZATION-GROUP

12. CORRELATIONS SCANNEDWITH ANISOTROPY • E. Lieb, T. Schultz and D. Mattis, Ann. of Phys. 16,407 (1961). • G. Kato, M. Shiroishi, M. Takahashi and K. Sakai, J.Phys. A 37,5097 (2004). • N. Kitanine, J.M. Maillet, N.A. Slavnovand V. Terras,J. Stat. Mech. L09002 (2005). • J. Sato, M. Shiroishi, and M. Takahashi, Nucl. Phys. B729, 441 (2005). • M. Takahashi, Thermodynamics of One-DimensionalSolvable Models, pgs. 41,56, 152-158, Cambridge University Press, Cambridge (1999).

13. ANTIFERROMAGNETIC MODEL T-dependence of Correlations • M. Bortzve F. Göhman, Eur. Phys. J. B 46, 399 (2005).

14. ANTIFERROMAGNETIC MODEL T-dependence of Specific Heat • A. Klümper, Int. of Qu.Chains: Th.and App.to the Spin-1/2 XXZ Ch., Lec.Not.in Phys.645, 349 (Springer, Berlin-Heidelberg2004) • J. C. Bonner and M. E. Fisher, Phys. Rev. 135, A640 (1964) • C. Xi-Yaoand G.F. Tuthill, Phys. Rev. B 32, 7280(1985). • R. Narayanan and R.R.P. Singh, Phys. Rev. B42, 10305(1990). • K. Fabricius, A. Klümper and B.M. McCoy,Stat. Phys. on the Eve of the 21st Cent., 351 (World Scientific, Singapur 1999). • A. Klümper, Lecture Notes in Phys. 645, 349 (Springer, Berlin-Heidelberg2004)

15. ANTIFERROMAGNETIC MODEL Spin-wave stiffness constant • C. Kittel, Introduction to Solid State Physics, s. 441, John Wiley & Sons Inc., New York (1996).

16. ANTIFERROMAGNETIC MODEL Spin-wave stiffness • R. Kubo, Phys. Rev. 87, 568 (1952)

17. FERROMAGNETIC MODEL T-dependence of correlations • W. J. Zhang, J.L. Shen, J.H. Xuand C.S. Ting, Phys.Rev. B 51, 2950 (1995). • K. Fabricius, A. Klümper and B.M. McCoy, Stat. Phys. on the Eve of the 21st Cent., s.351 (World Scientific,Singapur 1999)

18. AFM AND FM CORRELATIONS AFM FM

19. FERROMAGNETIC MODEL T-dependence of Spec. Heat • S. Katsura, Phys. Rev. 127, 1508 (1962). • J. C. Bonner and M. E. Fisher, Phys. Rev. 135, A640 (1964) • C. Xi-Yaoand G.F. Tuthill, Phys. Rev. B 32, 7280(1985). • W.J. Zhang, J.L. Shen, J.H. Xuand C.S. Ting, Phys.Rev. B 51, 2950 (1995). • K. Fabricius, A. Klümper and B.M. McCoy,Stat. Phys. on the Eve of the 21st Cent., 351 (World Scientific, Singapur 1999).

20. FM AND AFM SPECIFIC HEAT FM AFM

21. FERROMAGNETIC MODEL Excitation Spectrum Gap and Exponent • F. D. M. Haldane, Phys. Rev. Lett. 45, 1358 (1980) • F. D. M. Haldane, Phys. Rev. B 25, 4925 (1982) • M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, s. 152-158, Cambridge University Press, Cambridge (1999)

22. LOW-TEMPERATURE ANALYSIS • M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, s. 152-158, Cambridge University Press, Cambridge (1999)

23. HIGH-TEMPERATURE ANALYSIS • O. Rojas, S.M. de Souza and M.T. Thomaz, J. Math.Phys. 43, 1390 (2002).

24. FUTURE PROJECTS Higher dimensional XXZ model

25. FUTURE PROJECTS Falicov-Kimball model

26. FUTURE PROJECTS Periodic Kondo lattice model • H.Tsunetsugu, M. Sigrist and K. Ueda, Rev. Mod. Phys. 69, 809 (1997).