Post on 22-Dec-2015
QUANTUM QUANTUM PHENOMENON PHENOMENON ININ FM & FM & AFM ANISOTROPIC AFM ANISOTROPIC XXZ HEISENBERG XXZ HEISENBERG CHAINSCHAINSGlobal Renormalization-Group Global Renormalization-Group
AnalysisAnalysis
Ferromagnetic Excitation Spectrum GapFerromagnetic Excitation Spectrum GapAntiferromagnetic Spin-Wave StiffnessAntiferromagnetic Spin-Wave Stiffness
Ozan S. SARIYER [ Istanbul Tecnical University ]Ozan S. SARIYER [ Istanbul Tecnical University ]
Prof. Dr. A. Nihat BERKER [ Koç Univ. - M.I.T. - Feza Gürsey Res. Inst. ]Prof. Dr. A. Nihat BERKER [ Koç Univ. - M.I.T. - Feza Gürsey Res. Inst. ]Dr. Michael HINCZEWSKI [ Feza Gürsey Res. Inst. ]Dr. Michael HINCZEWSKI [ Feza Gürsey Res. Inst. ]
(2007)(2007)
RG RG ININ 1-D CLASSICAL 1-D CLASSICAL SYSTEMSSYSTEMSIsing Model and RG
SUZUKI – TAKANO SUZUKI – TAKANO METHODMETHODApplications
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).
XXZ MODELXXZ 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)
RENORMALIZATION-GROUP
FM
AFM
SpinLiquid
Isinglike
Isinglike
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. Slavnov and V. Terras, J. Stat. Mech. L09002 (2005). J. Sato, M. Shiroishi, and M. Takahashi, Nucl. Phys. B 729, 441 (2005). M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, pgs. 41,56, 152-158, Cambridge University Press, Cambridge (1999).
ANTIFERROMAGNETIC ANTIFERROMAGNETIC MODELMODELT-dependence of Correlations
M. Bortz ve F. Göhman, Eur. Phys. J. B 46, 399 (2005).
ANTIFERROMAGNETIC ANTIFERROMAGNETIC MODELMODELT-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-Heidelberg 2004) J. C. Bonner and M. E. Fisher, Phys. Rev. 135, A640 (1964) C. Xi-Yao and G.F. Tuthill, Phys. Rev. B 32, 7280 (1985). R. Narayanan and R.R.P. Singh, Phys. Rev. B 42, 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-Heidelberg 2004)
ANTIFERROMAGNETIC ANTIFERROMAGNETIC MODELMODELSpin-wave stiffness constant
C. Kittel, Introduction to Solid State Physics, s. 441, John Wiley & Sons Inc., New York (1996).
ANTIFERROMAGNETIC ANTIFERROMAGNETIC MODELMODELSpin-wave stiffness
R. Kubo, Phys. Rev. 87, 568 (1952)
FERROMAGNETIC MODELFERROMAGNETIC MODEL
T-dependence of correlations
W. J. Zhang, J. L. Shen, J. H. Xu and 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)
FERROMAGNETIC MODELFERROMAGNETIC 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-Yao and G.F. Tuthill, Phys. Rev. B 32, 7280 (1985). W. J. Zhang, J. L. Shen, J. H. Xu and 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).
FM FM ANDAND AFM SPECIFIC AFM SPECIFIC HEATHEAT
FM AFM
FERROMAGNETIC MODELFERROMAGNETIC 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)
LOW-TEMPERATURE LOW-TEMPERATURE ANALYSISANALYSIS
M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, s. 152-158, Cambridge University Press, Cambridge (1999)
HIGH-TEMPERATURE HIGH-TEMPERATURE ANALYSISANALYSIS
O. Rojas, S. M. de Souza and M. T. Thomaz, J. Math. Phys. 43, 1390 (2002).
FUTURE PROJECTSFUTURE PROJECTS
Higher dimensional XXZ model