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

RG RG ININ 1-D CLASSICAL 1-D CLASSICAL SYSTEMSSYSTEMSIsing Model and RG

RG RG ININ 1-D CLASSICAL 1-D CLASSICAL SYSTEMSSYSTEMSIsing Model and RG

SUZUKI – TAKANO SUZUKI – TAKANO METHODMETHOD

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

RENORMALIZATION-GROUP

RENORMALIZATION-GROUP

FM

AFM

SpinLiquid

Isinglike

Isinglike

RENORMALIZATION-GROUP

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)

AFM AFM ANDAND FM FM CORRELATIONSCORRELATIONS

AFM FM

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

FUTURE PROJECTSFUTURE PROJECTS

Falicov-Kimball model

FUTURE PROJECTSFUTURE PROJECTS

Periodic Kondo lattice model

H.Tsunetsugu, M. Sigrist and K. Ueda, Rev. Mod. Phys. 69, 809 (1997).