楊李定理

在统计力学和統計場論中，楊李定理（或杨李单位圆定理）說：有些铁磁配分函數的零點都是虚数。該定理是以楊振寧和李政道命名的。【T. D. Lee and C. N. Yang （1952）、（Lee & Yang 1952）】

應用

• 伊辛型
• 海森堡模型，Asano contractions^[1][2]
• Simon & Griffiths (1973)
• Newman (1974)
• Lieb & Sokal (1981)
• 黎曼猜想、黎曼ζ函數、动力系统^[3]

數學定理

哈密爾頓是

${\displaystyle H=-\sum J_{jk}S_{j}S_{k}-\sum z_{j}S_{j}}$ 

若${\displaystyle S_{j}S_{k}\geq 0}$ ，則H是鐵磁性的。

配分函数是

${\displaystyle Z=\int e^{-H}d\mu _{1}(S_{1})\cdots d\mu _{N}(S_{N})}$ 

${\displaystyle \int e^{bS^{2}}d|\mu _{j}(S)|<\infty ,\,\forall b\in \mathbb {R} .}$ 

${\displaystyle \int e^{hS}d\mu _{j}(S)\neq 0,\,\forall h\in \mathbb {H} _{+}:=\{z\in \mathbb {C} |\Re {z}>0\}}$ 

楊李定理說

${\displaystyle Z(\{z_{j}\})\neq 0,\,\forall z_{j}\in \mathbb {H} _{+}}$ 

参考文献

1. Asano, Taro. Lee-Yang Theorem and the Griffiths Inequality for the Anisotropic Heisenberg Ferromagnet. Physical Review Letters. 1970-06-22, 24 (25): 1409–1411. doi:10.1103/PhysRevLett.24.1409. 
2. Asano, Taro. Generalization of the Lee-Yang Theorem. Progress of Theoretical Physics. 1968-12-01, 40 (6): 1328–1336 [2020-03-07]. ISSN 0033-068X. doi:10.1143/PTP.40.1328. （原始内容于2020-06-23） （英语）. 
3. Knauf, Andreas. NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS. Reviews in Mathematical Physics. 1999-09, 11 (08): 1027–1060 [2020-03-07]. ISSN 0129-055X. doi:10.1142/S0129055X99000325. （原始内容于2021-04-19） （英语）. 

阅读

• Itzykson, Claude; Drouffe, Jean-Michel, Statistical field theory. Vol. 1, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1989, ISBN 978-0-521-34058-8, MR 1175176 
• Knauf, Andreas, Number theory, dynamical systems and statistical mechanics, Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics, 1999, 11 (8): 1027–1060, Bibcode:10.1.1.184.8685 请检查|bibcode=值 (帮助), ISSN 0129-055X, MR 1714352, doi:10.1142/S0129055X99000325 
• Lee, T. D.; Yang, C. N., Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model, Physical Review, 1952, 87 (3): 410–419, ISSN 0031-9007, doi:10.1103/PhysRev.87.410 
• Lieb, Elliott H.; Sokal, Alan D., A general Lee-Yang theorem for one-component and multicomponent ferromagnets, Communications in Mathematical Physics, 1981, 80 (2): 153–179 [2020-03-07], ISSN 0010-3616, MR 0623156, doi:10.1007/BF01213009, （原始内容于2020-07-17） 
• Newman, Charles M., Zeros of the partition function for generalized Ising systems, Communications on Pure and Applied Mathematics, 1974, 27 (2): 143–159, ISSN 0010-3640, MR 0484184, doi:10.1002/cpa.3160270203 
• Simon, Barry; Griffiths, Robert B., The (φ⁴)₂ field theory as a classical Ising model, Communications in Mathematical Physics, 1973, 33 (2): 145–164 [2020-03-07], Bibcode:10.1.1.210.9639 请检查|bibcode=值 (帮助), ISSN 0010-3616, MR 0428998, doi:10.1007/BF01645626, （原始内容于2020-06-10） 
• Yang, C. N.; Lee, T. D., Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation, Physical Review, 1952, 87 (3): 404–409, ISSN 0031-9007, doi:10.1103/PhysRev.87.404