杨李定理

在统计力学和统计场论中，杨李定理（或杨李单位圆定理）说：有些铁磁配分函数的零点都是虚数。该定理是以杨振宁和李政道命名的。【T. D. Lee and C. N. Yang （1952）、(Lee & Yang 1952)】

应用

• 伊辛型
• 海森堡模型，Asano contractions
• Simon & Griffiths (1973)
• Newman (1974)
• Lieb & Sokal (1981)
• 黎曼猜想、黎曼ζ函数、动力系统

数学定理

哈密尔顿是

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

若$S_{j}S_{k}\geq 0$，则H是铁磁性的。

配分函数是

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

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

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

杨李定理说

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

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, 1999, 11 (8): 1027–1060, Bibcode:1999RvMaP..11.1027K, CiteSeerX 10.1.1.184.8685 , 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, （原始内容存档于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:1973CMaPh..33..145S, CiteSeerX 10.1.1.210.9639 , ISSN 0010-3616, MR 0428998, S2CID 123201243, doi:10.1007/BF01645626, （原始内容存档于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