梅尔曼–瓦格纳定理

量子场论统计力学中,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-别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)。

概览

φ高斯自由场(一种纯量场),m是质量,维度d=2;传播子是:

 

若m=0,

 

因为高斯定律

 
 

  ,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

XY模型的相变

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

量子相变不受影响。)

两相是:

1、  

 

2、幂定律

(arξ

a晶格常数

海森堡模型

[1]

 

[2]

历史

[3] [4] [5]

2D晶体

[6][7][8]

[9][10]

限制

[11][12]

参考文献

  1. ^ see Cardy (2002)
  2. ^ See Gelfert & Nolting (2001).
  3. ^ 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. 
  4. ^ Peierls, R.E. Bemerkungen über Umwandlungstemperaturen. Helv. Phys. Acta. 1934, 7: 81. doi:10.5169/seals-110415. 
  5. ^ Landau, L.D. Theory of phase transformations II. Phys. Z. Sowjetunion: 545. 
  6. ^ 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. 
  7. ^ 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. 
  8. ^ 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. 
  9. ^ 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. 
  10. ^ 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. 
  11. ^ 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. 
  12. ^ 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.