quantum phenomenon in fm & afm anisotropic xxz heisenberg chains
DESCRIPTION
QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS. Global Renormalization-Group Analysis Ferromagnetic Excitation Spectrum Gap Antiferromagnetic Spin-Wave Stiffness Ozan S. SARIYER [ Istanbul Tecnical University ] - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/1.jpg)
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)
![Page 2: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/2.jpg)
RG RG ININ 1-D CLASSICAL 1-D CLASSICAL SYSTEMSSYSTEMSIsing Model and RG
![Page 3: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/3.jpg)
RG RG ININ 1-D CLASSICAL 1-D CLASSICAL SYSTEMSSYSTEMSIsing Model and RG
![Page 4: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/4.jpg)
RG RG ININ 1-D CLASSICAL 1-D CLASSICAL SYSTEMSSYSTEMSIsing Model and RG
![Page 5: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/5.jpg)
SUZUKI – TAKANO SUZUKI – TAKANO METHODMETHOD
![Page 6: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/6.jpg)
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).
![Page 7: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/7.jpg)
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)
![Page 8: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/8.jpg)
RENORMALIZATION-GROUP
![Page 9: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/9.jpg)
RENORMALIZATION-GROUP
![Page 10: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/10.jpg)
RENORMALIZATION-GROUP
FM
AFM
SpinLiquid
Isinglike
Isinglike
![Page 11: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/11.jpg)
RENORMALIZATION-GROUP
![Page 12: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/12.jpg)
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).
![Page 13: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/13.jpg)
ANTIFERROMAGNETIC ANTIFERROMAGNETIC MODELMODELT-dependence of Correlations
M. Bortz ve F. Göhman, Eur. Phys. J. B 46, 399 (2005).
![Page 14: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/14.jpg)
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)
![Page 15: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/15.jpg)
ANTIFERROMAGNETIC ANTIFERROMAGNETIC MODELMODELSpin-wave stiffness constant
C. Kittel, Introduction to Solid State Physics, s. 441, John Wiley & Sons Inc., New York (1996).
![Page 16: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/16.jpg)
ANTIFERROMAGNETIC ANTIFERROMAGNETIC MODELMODELSpin-wave stiffness
R. Kubo, Phys. Rev. 87, 568 (1952)
![Page 17: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/17.jpg)
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)
![Page 18: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/18.jpg)
AFM AFM ANDAND FM FM CORRELATIONSCORRELATIONS
AFM FM
![Page 19: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/19.jpg)
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).
![Page 20: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/20.jpg)
FM FM ANDAND AFM SPECIFIC AFM SPECIFIC HEATHEAT
FM AFM
![Page 21: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/21.jpg)
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)
![Page 22: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/22.jpg)
LOW-TEMPERATURE LOW-TEMPERATURE ANALYSISANALYSIS
M. Takahashi, Thermodynamics of One-Dimensional Solvable Models, s. 152-158, Cambridge University Press, Cambridge (1999)
![Page 23: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/23.jpg)
HIGH-TEMPERATURE HIGH-TEMPERATURE ANALYSISANALYSIS
O. Rojas, S. M. de Souza and M. T. Thomaz, J. Math. Phys. 43, 1390 (2002).
![Page 24: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/24.jpg)
FUTURE PROJECTSFUTURE PROJECTS
Higher dimensional XXZ model
![Page 25: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/25.jpg)
FUTURE PROJECTSFUTURE PROJECTS
Falicov-Kimball model
![Page 26: QUANTUM PHENOMENON IN FM & AFM ANISOTROPIC XXZ HEISENBERG CHAINS](https://reader036.vdocuments.us/reader036/viewer/2022062423/56814326550346895daf917e/html5/thumbnails/26.jpg)
FUTURE PROJECTSFUTURE PROJECTS
Periodic Kondo lattice model
H.Tsunetsugu, M. Sigrist and K. Ueda, Rev. Mod. Phys. 69, 809 (1997).