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advanced_notions:symmetry_breaking:mermin-wagner_theorem

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advanced_notions:symmetry_breaking:mermin-wagner_theorem [2017/09/29 07:54]
jakobadmin [Researcher]
advanced_notions:symmetry_breaking:mermin-wagner_theorem [2021/03/18 12:36] (current)
orbital
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 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
 +<​blockquote>​A theorem due to Mermin and Wagner states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. For a discrete symmetry this lower critical dimensionality is one. This is, in fact, well known since in quantum mechanics with finitely many degrees of freedom (corresponding to one-dimensional field theory) tunneling between degenerate classical minima allows for a unique symmetric ground state. [...] The Mermin-Wagner theorem has been restated by Coleman in the framework of field theory. <​cite>​page 525 in Quantum Field Theory by Claude Itzykson, ‎Jean-Bernard Zuber </​cite>​ </​blockquote>​
  
 <tabbox Layman> ​ <tabbox Layman> ​
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 Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
 </​note>​ </​note>​
-  ​+The Mermin-Wagner theorem explains, why the existence of phase transitions like Magnetization or Freezing depends on the dimension of the material undergoing the transition. Imagine water molecules close to the freezin point in the following configurations:​ 
 + 
 +  * a one-dimensional chain 
 +  * a two dimensional monolayer of watermolecules 
 +  * a cube 
 +The molecules are moving around because of their thermal energy but also interact with each other due to hydrogen bonding. 
 +  
 +For water to freeze, the temperature has to be low enough such that the bonding between the molecules becomes strong enough to keep the molecules fixed at their place. 
 +This means that the number of water molecules that are neighbours determines how hard it is for a water molecule to move away from its position. 
 +We can imagine that there are more neighbours, the higher the dimension of the configuration is. 
 +Mermin-Wagner predicts that for certain problems there is a lower critical dimension ​ below which there are not enough neighbors to build up a strong enough bond such that all the water molecules "​freeze"​ at their position. This behaviour is also called cooperativity. 
 + 
 +Water freezing is actually a very complex problem and the above explanation is only meant to simplify the visualization and should not be taken to be physically correct. 
 +But for simple problems like the continuous Ising model it is possible to calculate this lower critical dimension to be .  ​
 <tabbox Student> ​ <tabbox Student> ​
  
advanced_notions/symmetry_breaking/mermin-wagner_theorem.1506664441.txt.gz · Last modified: 2017/12/04 08:01 (external edit)