查看“︁梅尔曼–瓦格纳定理”︁的源代码

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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{{Math|''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 ==
若 {{Mvar|φ}} 是[[高斯自由场]]（一种[[纯量场]]），{{Mvar|m}}是质量，维度d=2；[[传播子]]是：

: <math>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}.</math>

若m=0，

: <math>\nabla^2 G = \delta(x).</math>

因为[[高斯定律]]，

: <math>E = {1\over 2\pi r}.</math>

: <math>G(r) = {1\over 2\pi} \log(r)</math>

若<math>r\to 0, \infty</math>，<math>G(r) \to \pm \infty</math>，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]]。

（[[量子相變]]不受影响。）

两相是：

1、 <math>G(r)\sim\exp(-r/\xi)</math> 

<math>r/\xi\gg1</math>

2、[[冪定律]]

({{Math|''a'' ≪ ''r'' ≪ ''ξ''}}

{{Mvar|a}} 是[[晶格常數]]

== [[海森堡模型]] ==
<ref>see {{Harvard citation text|Cardy|2002}}</ref> 

: <math>H = - J\sum_{\left\langle {i,j} \right\rangle } \mathbf{S}_i \cdot \mathbf{S}_j.</math>

<ref>See {{Harvard citation text|Gelfert|Nolting|2001}}.</ref>

== 历史 ==
<ref>{{Cite journal|title=Zur Theorie des Ferromagnetismus|last=Bloch|first=F|date=1930-02-01|journal=Zeitschrift für Physik|issue=3–4|doi=10.1007/bf01339661|volume=61|pages=206–219|bibcode=1930ZPhy...61..206B}}</ref> <ref>{{Cite journal|title=Bemerkungen über Umwandlungstemperaturen|last=Peierls|first=R.E.|journal=Helv. Phys. Acta|doi=10.5169/seals-110415|year=1934|volume=7|pages=81}}</ref> <ref>{{Cite journal|title=Theory of phase transformations II|last=Landau|first=L.D.|journal=Phys. Z. Sowjetunion|volume=11|pages=545}}</ref>

== 2D[[晶体]] ==
<ref>{{Cite journal|title=Unveiling Dimensionality Dependence of Glassy Dynamics: 2D Infinite Fluctuation Eclipses Inherent Structural Relaxation|last=Shiba|first=H.|last2=Yamada|first2=Y.|date=2016|journal=Physical Review Letters|issue=24|doi=10.1103/PhysRevLett.117.245701|volume=117|pages=245701|arxiv=1510.02546|bibcode=2016PhRvL.117x5701S|pmid=28009193|last3=Kawasaki|first3=T.|last4=Kim|first4=K.}}</ref><ref>{{Cite journal|title=Long-wavelength fluctuations and the glass transition in two dimensions and three dimensions|last=Vivek|first=S.|last2=Kelleher|first2=C.P.|date=2017|journal=Proceedings of the National Academy of Sciences|issue=8|doi=10.1073/pnas.1607226113|volume=114|pages=1850–1855|arxiv=1604.07338|bibcode=2017PNAS..114.1850V|pmc=5338427|pmid=28137847|last3=Chaikin|first3=P.M.|last4=Weeks|first4=E.R.}}</ref><ref>{{Cite journal|title=Mermin–Wagner fluctuations in 2D amorphous solids|last=Illing|first=B.|last2=Fritschi|first2=S.|date=2017|journal=Proceedings of the National Academy of Sciences|issue=8|doi=10.1073/pnas.1612964114|volume=114|pages=1856–1861|bibcode=2017PNAS..114.1856I|pmc=5338416|pmid=28137872|last3=Kaiser|first3=H.|last4=Klix|first4=C.L.|last5=Maret|first5=G.|last6=Keim|first6=P.}}</ref>

<ref>{{Cite journal|title=Phase transitions and random walks on graphs: A generalization of the Mermin-Wagner theorem to disordered lattices, fractals, and other discrete structures|last=Cassi|first=D.|date=1992|journal=Physical Review Letters|issue=24|doi=10.1103/PhysRevLett.68.3631|volume=68|pages=3631–3634|bibcode=1992PhRvL..68.3631C|pmid=10045753}}</ref><ref>{{Cite journal|title=Recurrent random walks and the absence of continuous symmetry breaking on graphs|last=Merkl|first=F.|last2=Wagner|first2=H.|date=1994|journal=Journal of Statistical Physics|issue=1|doi=10.1007/bf02186284|volume=75|pages=153–165|bibcode=1994JSP....75..153M}}</ref> 

== 限制 ==
<ref>{{Cite journal|title=Rippling of graphene|last=Thompson-Flagg|first=R.C.|last2=Moura|first2=M.J.B|date=2009|journal=EPL|issue=4|doi=10.1209/0295-5075/85/46002|volume=85|pages=46002|arxiv=0807.2938|bibcode=2009EL.....8546002T|last3=Marder|first3=M.}}</ref><ref>{{Cite journal|title=On the Hohenberg–Mermin–Wagner Theorem and Its Limitations|last=Halperin|first=B.I.|date=2019|journal=Journal of Statistical Physics|issue=3–4|doi=10.1007/s10955-018-2202-y|volume=175|pages=521–529|arxiv=1812.00220|bibcode=2019JSP...175..521H}}</ref>

== 参考文献 ==
{{Reflist}}
[[Category:数学物理定理]]
[[Category:物理定理]]
[[Category:量子场论]]