在本論文中發展一種新的微擾方法對Hubbard 模型之廣義自洽場(GSCF)近似結果做修正. 此微擾方法以 GSCF 解作為零階近似, 比以無交互作用電子系統作為零階近似的傳統微擾方法能在更大的參數範圍中得到收斂結果, 而傳統微擾方法只適用於弱交互作用強度(|U|/t<<1)的情況.
本論文詳細分析一維及二維排斥性Hubbard 模型. 計算基態性質, 包括:能量, 雙重佔據數, 跳躍動能, 並在一維情況下與 Bethe-ansatz 精確解結果做比較. 研究在廣泛電子濃度值,廣泛交互作用強度值,廣泛磁場值下進行. 在一定的 n, U, h 參數範圍中新的微擾近似結果更符合精確解, 且於弱耦合及強耦合極限下與精確解完全符合.
本論文首次做了二維的GSCF 近似計算, 除了以上所提之基態物理量外,還探討了化學勢, 橫向自旋, 縱向自旋, 橫向自旋波矢等量. 在特定的情況下二維系統會呈現順磁態或發生一階相變. 此現象在一維情形中不存在.

In this thesis a new perturbation method is developed to calculate corrections to generalized self-consistent field (GSCF) approximation of the Hubbard model solution. This perturbation method regards the GSCF solution as the zero order approximation and is able to get converging results in a more wide parameter range than the tranditional perturbation method. The tranditional perturbation method regards the system of the non-interacting electrons as the zero order approximation and is suitable only for the weak interaction intensities (|U|/t<<1). 
In this thesis the one- and two-dimentional repulsive Hubbard model is analysed in detail . The calculated gound state properties, namely, the energy, the double occupancy andthe kinetic energy are compared with Bethe-ansatz exact results in the one-dimensional case. Investigation is carried out for the general electronic density value, general interaction intensity value, and general magnetic field value. In certain n, U, h ranges, the perturbation and exact results are quite close and they coincide at weak and strong coupling limits.
This thesis for the first time presents the two-dimentional GSCF results. In
addition of the above mentioned ground state physical quantities, the chemical
potential, transverse and longitudinal spin, and the wave verctor of transverse spin are
discussed as well. In the two-dimentional system a paramagnetic state and some one
order phase transitions appear in specific cases. This phenomenon does not exist in the
one-dimensional case.

1	前言…..1
2	Hubbard 模型及廣義自洽場(GSCF)近似…..3
2.1	Hubbard 模型…..3
2.2	GSCF 近似…..6
2.3	正則變換…..9
2.4	自洽方程…..12
2.5	自旋結構…..13
2.6	極限情形…..17
2.6.1	一維弱耦合極限…..17
2.6.2	一維強耦合極限…..19
3	Hubbard 模型之微擾理論…..23
3.1	一般微擾理論…..23
3.2	微擾理論 A …..25
3.2.1	一般情形…..25
3.2.2	一維情形…..29
3.3	微擾理論 B …..32
3.3.1	基態波函數近似…..32
3.3.2	一階能量的修正…..34
3.3.3	二階能量的修正…..37
4	數值計算…..43
4.1	GSCF 能量之數值計算…..43
4.2	微擾理論 B 之數值計算…..47
5	一維情形的數值結果…..49
5.1	基態能量…..49
5.2	雙重佔據數…..58
5.3	跳躍動能…..65
5.4	對 q 的變分…..70
6	二維情形的數值結果…..73
6.1	基態能量…..73
6.2	雙重佔據數…..85
6.3	跳躍動能…..90
6.4	化學勢…..95
6.5	自旋…..100
6.6	橫向自旋波矢…..104
7	結論…..109
A  Wick 定理…..111
B  無磁場下微擾理論 A 之二階能量…..113

[1] P. W. Anderson, "A dialogue on the theory of high Tc. (cover story)", Physics Today 6, 54 (1991).
[2] V. J. Emery and S. A. Kivelson, "Importance of phase fluctuations in superconductors with small superfluid density", Nature 374, 434 (1995).
[3] "The Hubbard Model", edited by Arianna Montorsi, 1992, ISBN 981-02-0585-6.
[4] "The Hubbard Model Its Physics and Mathematical Physics", edited by Dionys Baeriswyl and David K. Campbell etc, 1995, ISBN 0-306-45003-8.
[5] "Hubbard Model and Anyon Superconductivity", edited by A.P. Balachandran and E. Ercolessi etc, 1990, ISBN 981-02-0349-7.
[6] E. H. Lieb and F. Y. Wu, "Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension", Phys. Rev. Lett. 20, 1445 (1968); ibid. 21, 192 (1968).
[7] M. Takahashi, "Magnetic susceptibility for the half-filled Hubbard model", Progr. Theor. Phys.; ibid. 42, 1098 (1969); ibid. 43, 1619 (1970); ibid. 44, 348 (1970).
[8] D. C. Mattis, "The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension", ed. by Daniel C. Mattis (World Scientific, 1994).
[9] C. Yang, A. N. Kocharian and Y. L. Chiang, "Phase transitions and exact ground-state properties of the one-dimensional Hubbard model in a magnetic field", J. Phys: Condens. Matter 12, 7433 (2000).
[10] D. R. Penn, "Stability theory of the magnetc phase for a simple model of the transition metal", Phys. Rev. 142, 350 (1966).
[11] D. I. Khomskii, "Electron correlations in narrow zones (Hubbard’s model)", Fiz. Metal. Metalloved. 29, 31 (1970).
[12] W. D. Langer, M. Plischke and D. C. Mattis, "Existence of two phase transitions in Hubbard model", Phys. Rev. Lett. 23, 1448 (1969); W. D. Langer and D. C. Mattis, "Ground state energy of Hubbard model", Phys. Lett. A36, 139 (1971).
[13] A. N. Kocharian, C. Yang and Y. L. Chiang, "Self-consistent and exact studies of pairing correlations and crossover in the one-dimensional attractive Hubbard model", Phys. Rev. B59, 7458 (1999).
[14] A. N. Kocharian, C. Yang, Y. L. Chiang and T. Y. Chou, "Exact and self-consistent results in one-dimensional repulsive Hubbbard model", Int. J. Mod. Phys. B17, 5749 (2003).
[15] G. Kotliar and A. E. Ruckenstein, "New functional integral approach to strongly correlated Fermi systems: the Gutzwiller approximation as a saddle point", Phys. Rev. Lett. 57, 1362 (1986).
[16] J. R. Schrieffer, X. G.Wen and S. C. Zhang, "Dynamic spin fluctuations and the bag mechanism of high-Tc superconductivity", Phys. Rev. B39, 11663 (1989).
[17] S. Yarlagadda and S. Kurihara, "Fermi-liquid theory in the low-density two-dimensional Hubbard model", Phys. Rev. B48, 10567 (1993).
[18] M. R. Hedayati, and G. Vignale, "Ground-state energy of the one- and two-dimensional Hubbard model calculated by the method of Singwi, Tosi, Land, and Sj¨olander", Phys. Rev. B40, 9044 (1993).
[19] P. W. Anderson, "New approach to the theory of superexchange interactions", Phys. Rev. 115, 2 (1959).
[20] D. J. Klein and W. A. Seitz, "Perturbation expansion of the linear Hubbard model", Phys. Rev. B8, 2236 (1973).
[21] M. Ogata and H. Shiba, "Bethe-ansatz wave function, momentum distribution, and spin correlation in the one-dimensional strongly correlated Hubbard model", Phys. Rev. B41, 2326 (1990); Prog. Theor. Phys. Suppl. 108, 265 (1992).
[22] S. Pairault, D. S¨en¨echal and A. M. S. Tremblay, "Strong-coupling perturbation theory of the Hubbard model", Eur. Phys. J. B16, 85 (2000).
[23] Y. Nagaoka, "Ferromagnetism in a narrow, almost half-filled s band", Phys. Rev. 147, 392 (1966).
[24] J. E. Hirsch, "Two-dimensional Hubbard model : Numerical simulation study", Phys. Rev. B31, 4403 (1985).
[25] E. Dagotto, "Correlated electrons in high-temperature superconductors", Rev. Mod. Phys. 66, 763 (1994).
[26] E. Eisenberg, R. Berkovits, D. A. Huse and B. L. Altshuler, "Breakdown of the Nagaoka phase in the two-dimensional t-J model", Phys. Rev. B65, 134437-1 (2002).
[27] M. Takahashi, "Magnetization curve for the half-filled Hubbard model", Progr. Theor. Phys. 42, 1098 (1969); ibid, "Ground state energy of the one-dimensional electron system with short-range interaction. I", 44, 348 (1970); ibid, "Magnetic susceptibility for the half-filled Hubbard model", 43, 1619 (1970).
[28] V. Ya. Krivnov and A. A. Ovchinnikov, "One-dimensional Fermi gas with attraction between the electrons". Sov. Phys. JETP 40, 781 (1975).
[29] T. B. Bahder and F. Woynarovich, "Gap in the spin excitations and magnetization curve of the one-dimensional attractive Hubbard model", Phys. Rev. B33, 2114 (1986).
[30] K.-J.-B. Lee and P. Schlottmann, "Thermodynamic Bethe-ansatz equations for the Hubbard chain with an attractive interaction", Phys. Rev. B38, 11566 (1988); ibid, "Low-temperature properties of the Hubbard chain with an attractive interaction" , B40, 9104 (1989).
[31] H. Frahm, and V. E. "Critical exponents for the one-dimensional Hubbard model", Phys. Rev. B42, 10553 (1990).
[32] N. Kawakami and A. Okiji, "Caracteristics of dressed particles in the 1D correlated electron systems", "Prog. Theor. Phys. Suppl." 101, 429 (1990).
[33] Y.-L. Chiang, "Main Properties of the One-dimensional Hubbard Model in an External Magnetic Filed", Ph. D dissertation, Department of Physics Tamkang University, Tamsui, Taiwan, R. O. C. (2000).
[34] T.-Y. Chou, "GSCF Study of the Magnetic Ground State Properties of One-dimensional Repulsive Hubbard Model", Master dissertation, Department of Physics Tamkang University, Tamsui, Taiwan, R. O. C.(2001).
[35] C.-L. Lu, "Generalized Self-consistent Field Results of One-dimensional Hubbard Model and Perturbation Corrections", Master dissertation, Department of Physics Tamkang University, Tamsui, Taiwan, R. O.C. (2002).
[36] "Quantum Mechanics", by Lesle E. Ballentine, ISBN 0-13-746843-1.
[37] A. N. Kocharian, C. Yang and Y. L. Chiang, "Perturbation theory about self-consistent field solution in one-dimensional Hubbard model", Int. J. Mod. Phys. B17, 3363 (2003).
[38] C. Yang, A. N. Kocharian, Y. L. Chiang and L. Y. Chen, "Perturbation theory about generalized self-consistent field solution ", to be public, Int. J. Mod. Phys. B .
[39] Willam H. Press, Saul A. Teukolsky, Willam T. Vetterling, Brian P. Flannery, "Numerical Recipes in Fortran", Second Edition, Cambridge University Press.