梅尔曼–瓦格纳定理

在量子场论和统计力学中，梅尔曼–瓦格纳定理（Mermin–Wagner定理，或称梅尔铭-瓦格纳-霍亨贝格定理、梅尔铭-瓦格纳-別列津斯基定理、科勒曼定理）阐述了维度d ≤ 2的场论没有自发对称破缺（要不然无质量的南部玻色子会有无限的相关函数）。 若 φ 是高斯自由场（一种纯量场），m是质量，维度d=2；传播子是： G ( x ) = ⟨ φ ( x ) φ ( 0 ) ⟩ = ∫ d 2 k ( 2 π ) 2 e i k ⋅ x k 2 + m 2 . {\displaystyle G(x)=\left\langle \varphi (x)\varphi (0)\right\rangle =\int {\frac {d^{2}k}{(2\pi )^{2}}}{\frac {e^{ik\cdot x}}{k^{2}+m^{2}}}.} 若m=0， ∇ 2 G = δ ( x ) . {\displaystyle \nabla ^{2}G=\delta (x).} 因为高斯定律， E = 1 2 π r . {\displaystyle E={1 \over 2\pi r}.} G ( r ) = 1 2 π log ⁡ ( r ) {\displaystyle G(r)={1 \over 2\pi }\log(r)} 若 r → 0 , ∞ {\displaystyle r\to 0,\infty } ， G ( r ) → ± ∞ {\displaystyle G(r)\to \pm \infty } ，所以一维或二维的纯量场没有明确定义的平均值。 参见墨西哥帽模型。 d=2的O(2)模型没有自发对称破缺，但是有别列津斯基-科斯特利茨-索利斯相变。 （量子相變不受影响。） 两相是： 1、 G ( r ) ∼ exp ⁡ ( − r / ξ ) {\displaystyle G(r)\sim \exp(-r/\xi )} r / ξ ≫ 1 {\displaystyle r/\xi \gg 1} 2、冪定律 (a ≪ r ≪ ξ a 是晶格常數 H = − J ∑ ⟨ i , j ⟩ S i ⋅ S j . {\displaystyle H=-J\sum _{\left\langle {i,j}\right\rangle }\mathbf {S} _{i}\cdot \mathbf {S} _{j}.} see Cardy (2002) harvtxt error: no target: CITEREFCardy2002 (help) See Gelfert & Nolting (2001) harvtxt error: no target: CITEREFGelfertNolting2001 (help). Bloch, F. Zur Theorie des Ferromagnetismus. Zeitschrift für Physik. 1930-02-01, 61 (3–4): 206–219. Bibcode:1930ZPhy...61..206B. doi:10.1007/bf01339661.  Peierls, R.E. Bemerkungen über Umwandlungstemperaturen. Helv. Phys. Acta. 1934, 7: 81. doi:10.5169/seals-110415.  Landau, L.D. Theory of phase transformations II. Phys. Z. Sowjetunion: 545.  Shiba, H.; Yamada, Y.; Kawasaki, T.; Kim, K. Unveiling Dimensionality Dependence of Glassy Dynamics: 2D Infinite Fluctuation Eclipses Inherent Structural Relaxation. Physical Review Letters. 2016, 117 (24): 245701. Bibcode:2016PhRvL.117x5701S. PMID 28009193. arXiv:1510.02546 . doi:10.1103/PhysRevLett.117.245701.  Vivek, S.; Kelleher, C.P.; Chaikin, P.M.; Weeks, E.R. Long-wavelength fluctuations and the glass transition in two dimensions and three dimensions. Proceedings of the National Academy of Sciences. 2017, 114 (8): 1850–1855. Bibcode:2017PNAS..114.1850V. PMC 5338427 . PMID 28137847. arXiv:1604.07338 . doi:10.1073/pnas.1607226113.  Illing, B.; Fritschi, S.; Kaiser, H.; Klix, C.L.; Maret, G.; Keim, P. Mermin–Wagner fluctuations in 2D amorphous solids. Proceedings of the National Academy of Sciences. 2017, 114 (8): 1856–1861. Bibcode:2017PNAS..114.1856I. PMC 5338416 . PMID 28137872. doi:10.1073/pnas.1612964114.  Cassi, D. Phase transitions and random walks on graphs: A generalization of the Mermin-Wagner theorem to disordered lattices, fractals, and other discrete structures. Physical Review Letters. 1992, 68 (24): 3631–3634. Bibcode:1992PhRvL..68.3631C. PMID 10045753. doi:10.1103/PhysRevLett.68.3631.  Merkl, F.; Wagner, H. Recurrent random walks and the absence of continuous symmetry breaking on graphs. Journal of Statistical Physics. 1994, 75 (1): 153–165. Bibcode:1994JSP....75..153M. doi:10.1007/bf02186284.  Thompson-Flagg, R.C.; Moura, M.J.B; Marder, M. Rippling of graphene. EPL. 2009, 85 (4): 46002. Bibcode:2009EL.....8546002T. arXiv:0807.2938 . doi:10.1209/0295-5075/85/46002.  Halperin, B.I. On the Hohenberg–Mermin–Wagner Theorem and Its Limitations. Journal of Statistical Physics. 2019, 175 (3–4): 521–529. Bibcode:2019JSP...175..521H. arXiv:1812.00220 . doi:10.1007/s10955-018-2202-y.